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Approximating Relativistic Quantum Field Theories with Continuous Tensor Networks

Approximating Relativistic Quantum Field Theories with Continuous Tensor Networks Approximating Relativistic Quantum Field Theories with Continuous Tensor Networks 1 1 Tom Shachar and Erez Zohar Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel. (Dated: October 5, 2021) We present a continuous tensor-network construction for the states of quantum fields called cPEPS (continuous projected entangled pair state), which enjoys the same spatial and global symmetries of ground-states of relativistic field theories. We explicitly show how such a state can approximate and eventually converge to the free field theory vacuum and suggest a regularization-independent way of estimating the convergence via a universal term in the fidelity per-site. We also present a detailed bottom-up construction of the cPEPS as the continuum limit of the conventional lattice Projected Entangled Pair State (PEPS). I. INTRODUCTION ativistic problems. Recently, aiming at extending the cMPS to a relativistic setting, the one dimensional con- struction of RcMPS (relativistic cMPS) was introduced Tensor network states have proven to be a prolific theo- [25]. It uses a different operator basis, with respect to retical and numerical framework for advancement in the the particle creation and annihilation operators, which understanding of many-body quantum systems. These is more adequate for dealing with relativistic problems. states, which are constructed by contractions of local ten- In other tensor network techniques involving continuous sors, capture very well the physics of ground-states, low fields, one may use continuous fields to contract lattice lying excited states and thermal states of local Hamiltoni- tensor networks for spin models [26]. Finally, another ans [1–5]. By construction, they have the relevant physi- type of tensor network states, MERA (multiscale entan- cal properties, including the right entanglement structure glement renormalization ansatz) [27] has its continuum (area law [6]), possibility to encode symmetries - both version - cMERA - too [28, 29]. But in general, the field global [7, 8] and local [9–11], topological properties [12], of continuous tensor network states is still far from be- and, of course, a large set of numerical methods appli- ing fully studied and understood, in spite of its obvious cable to different scenarios and purposes - e.g. [13–17] importance and relevance, and hence provides an inter- and more. These approaches, including the 1 + 1 dimen- esting and challenging scientific playground. sional Matrix Product States (MPSs) and their higher dimensional extension PEPS (Projected Entangled Pair In this work, we present a continuous, field-theoretic version of PEPS which is built from quantum fields. We States) [5], are almost entirely restricted to lattice mod- therefore refer to them as cPEPS - continuous PEPS. els, with very few extensions, so far, to quantum field The reason this construction sheds new light on contin- theories defined in the continuum. uous tensor-networks is twofold. To begin with, these Originally, tensor network state methods were mostly states are well defined in any dimension and they are applied to condensed matter problems. In the recent well suited to deal with relativistic field theories simply years, however, there has been a growing effort in ap- from the way they are formalized. Moreover, they inherit plying them also to high energy physics, in particular many of the appealing attributes of the PEPS such their for the study of lattice gauge theories. That includes ability to account for symmetries. Second, the cPEPS the benchmarking of known results, as well as computa- are formalized in a field-theoretic way. This simple fact tions that go beyond the state of the art of conventional, is crucial as it brings this particular manifestation of a Monte-Carlo methods (see, e.g., [18] for a contemporary continuous tensor-network closer in spirit to relativistic review of this field). Tensor network contractions of path quantum field theory, which has its own wide array of integrals, rather than states, have also been introduced computational techniques [30, 31] - specifically also of and successfully applied to these models, using the TRG quantum informational calculations (see [32] and refer- (tensor renormalization group) toolbox [19]. However, ences therein). Thus, it may hopefully serve as a bridge as the fundamental field theories of particle physics are for implementing tensor-network techniques in quantum continuous, it is also interesting to develop tensor net- field theories and vice versa. work formulations enabling their study directly, without applying any lattice discretization schemes. In section II, we present the cPEPS construction in Continuous tensor network states (cTNs) were first in- detail and it is rigorously demonstrated how the ground- troduced in [20], in the form of cMPS (continuous Matrix state of a free scalar field (Klein-Gordon) theory can be Product States). This formalism was further developed obtained from it, both exactly (with unlimited compu- in later works [21], including higher dimensional exten- tational resources) or approximately. We study the ac- sions [22, 23] and even used as a numerical ansatz [24]. curacy of the approximation, show that it depends on However, as these constructions rely on Fock space con- the choice of a regularization scheme and propose a uni- cepts, they are seemingly less equipped to deal with rel- versal (regularization-independent) approach for tackling arXiv:2110.01603v1 [quant-ph] 4 Oct 2021 2 this problem. In section III, we present a bottom-up con- positive definite symmetric matrix and V is a general an- struction of the cPEPS as the continuum limit of a con- alytic functional of the fields and their first derivatives. ventional, lattice PEPS construction, and explain how a Following previous papers [20]-[25], we shall call D the cPEPS with continuous translational and rotational sym- (generalized) bond dimension. metry can be built. Finally, in section IV we explain in To demonstrate what could be done with such states, detail how states that are invariant under a general global we will restrict ourselves now to a case which can be symmetry can be directly constructed. handled analytically: Gaussian states, in which A is In what follows we adhere to the common choice of quadratic in both the physical and virtual fields and their dimensions ~ = c = 1. The Einstein summation con- first derivatives (other than being analytically tractable, vention where repeated indices are being summed over is Gaussian states are interesting in their own right as they assumed. serve as the ground-states of non-interacting theories). The most general A of such a state may be written as follows: II. CPEPS 1 1 A = − Z ∇v (x)∇v (x) − A v (x) v (x) αβ α β αβ α β We will begin our discussion of cPEPS by showing its 2 2 construction for, perhaps, the simplest possible physical + z ∇v (x)∇φ (x) + a v (x) φ (x) − φ (x) α α α α case, of a real scalar field in a d + 1 dimensional flat (Minkowski) spacetime. The Hilbert space of the field (5) theory is constructed by choosing a foliation of space, and Although one could, in principle, choose by defining the field and its respective conjugate momen- {Z , A , z , a , c} to be position dependent, we αβ αβ α α tum field over each slice (equal time surface), where the did not do this, and thus the above expression is two satisfy the canonical commutation algebra. Dictated translationally invariant. Thus, it is natural from this by the dynamics of the theory, there is then a unitary point onward to work in Fourier (momentum) space. evolution from one slice to the other (time evolution). using the Fourier transform We choose the standard foliation in which spacetime is Z d −ik·x divided into constant d dimensional (spatial) time slices. φ (k) = d k e φ (x) . (6) On each slice of constant time t we introduce the field operator φ (x, t ), and its conjugate momentum operator (and similarly for the virtual fields) we obtain: π (x, t ), satisfying the canonical commutation relation d k A[v (k),φ(k)] (2π) (d) |ψi = Dφ (k)Dv (k) e |{φ (k)}i [φ (x, t ) , π (y, t )] = iδ (x − y) . (1) 0 0 2 ∗ Choosing an arbitrary time-slice (that is, fixing the A = − A + Z k v (k) v (k) + αβ αβ β time) we denote by |{φ (x)}i a field configuration state: 2 ∗ a + z k v (k) φ (k) − φ (k) φ (k) . α β |{φ (x)}i = ⊗|φ (x)i (2) (7) Since these are real fields, they additionally satisfy - that is, a product of eigenstates of the field operator φ (k) = φ (−k), even though it shall not be of impor- everywhere. The field configuration states form a basis, tance for this paper. After integrating out the virtual in which one can span any field state. In particular, we fields we will end with a Gaussian state |ψ i, quadratic choose to express our cPEPS in this basis (and not using in φ: Fock terms as in [23]), and define it as Z Z d 1 d k ∗ − φ (k)ω (k)φ(k) d D 2 d d xA[v ,∇v ,φ,∇φ] (2π) α α |ψ i = Dφe |{φ (k)}i (8) |ψi = Dφ (x) Dv (x) e |{φ (x)}i D (3) where where −1 2 2 2 ω (k) = c − a + z k A + Zk a + z k (9) D α α β β αβ A = − Z ∇v ∇v + V [v ,∇v , φ,∇φ] . (4) αβ α β α α Here φ (x) is a real scalar field - the physical field of A. Approximation of the Free Vacuum interest. Additionally, {v (x)} is a set of D fields α=1 which are called henceforth virtual fields and are being A more explicit expression for ω (k) can be put forth integrated over (their role is to allow for a locally con- by using the matrix inversion formula: tracted state of the physical field, in full analogy with adj A + Zk lattice PEPS contractions, as we shall later show). Like −1 αβ A + Zk = (10) αβ 2 φ, the virtual fields are also real scalars. Z is a D × D αβ det (A + Zk ) 3 where adj (M) is the adjugate matrix of M. From that we which is satisfied by the following choice of parameters can tell that ω (k) is a rational function in the argument P : k , of order D over D. The parameters which appear in 2 2 the Gaussian cPEPS (3) will be collectively denoted by A + Z k = (δ + δ ) ik αβ αβ α,β−1 α,β+1 P = {Z , A , z , a , c}. We thus write αβ αβ α α ( m α = 2n n ∈ N 2 2D + 2δ × 2 p (P ) + p (P ) k + ... + p (P ) k αβ 0 1 D ω (k) = (11) α = 2n + 1 , n ∈ N 2 2D 1 + q (P ) k + ... + q (P ) k 1 D a = 0 z = δ c = m α α α1 We have used the freedom to set one of the param- eters of the rational function to 1. p (P ) , q (P ) are α α (16) non-linear maps which are non-injective; note that while We see that when D is infinite, the ground-state can be 2D − 1 parameters uniquely determine ω , the number constructed as a cPEPS, exactly: the continued fraction of parameters in the set P is of order D . This redun- is known to converge to ω (k) as D → ∞. However, dancy can be eliminated by ”gauge-fixing” conditions, what if the continued fraction is truncated after D steps? common for tensor networks which are usually redun- dant constructions [5], as addressed in previous contin- uum works [23–25]. Yet this will not be necessary for our B. Quantum Fidelity and Universality purposes. With (8) being a Gaussian state, it is natural to inspect We would like to inquire how well the Gaussian how well it approximates the ground-state (vacuum) of a cPEPS (5) with a finite D approximates the free vac- free field theory, which is given by the Hamiltonian (see uum |Ω i (which is also a Gaussian cPEPS, but with appendix B for more details): an infinite D). A natural probe that we explore for that task is the quantum fidelity, which is defined for two pure states as: 1 d k ∗ 2 ∗ H = Π (k) Π (k) + ω (k) φ (k) φ (k) , (2π) F (Ω , ψ ; Λ) = |hψ |Ω i| (17) f D D f (12) with the relativistic dispersion relation: To compute this quantity, we have chosen a Wilsonian 2 2 2 regularization where the momentum modes are truncated ω (k) = m + k . (13) at a cutoff k = Λ. The fidelity depends also on the The free vacuum |Ω i is given by: f cutoff Λ. The two states are assumed to be normal- ized, which bounds the fidelity from above and below: 1 ∗ − ω (k)φ (k)φ(k) |Ω i = Dφ (k) e |{φ (k)}i (14) 0 ≤ F (Ω , ψ ; Λ) ≤ 1. f D The fidelity takes maximal value, 1, only when the two states |ψ i and |Ω i are exactly identical (and hence Given the functional form of ω (k), the approxi- D D f mation to some desired dispersion relation such as ω the two functions ω (k) and ω (k)). Since the fidelity D f is bounded, any difference between the two functions is an approximation by a rational function, known as the Pad´e approximant of a function [33]. There are must result in a smaller value for the fidelity. Assum- ing the fidelity is also a smooth functional of ω (k) and infinitely many different approximations which converge to ω (k) as D → ∞. This fact can be seen, for instance, ω (k) (which holds true in this particular case, as to be shown below), it will increase as the two functions ω (k) by noting that the Pad´e approximant, like the Taylor series it is based on, is localized around a particular and ω (k) get closer to each other within the interval 0 ≤ k ≤ Λ. Hence we can choose ω (k) to be the Pad´e base point of momentum k , and the choice of such 0 D approximant of ω (k) around some chosen base point k . base point is of course arbitrary. Nevertheless, there is f 0 both a theoretical and effective difference between these The approximant diverges from the original function for large values of k ≫ k in this case. This can be seen distinct approximations, which will be addressed in the following subsection. as ω (k) asymptotically behaves as k, a behavior which a rational function in k can ever achieve. On the in- It is of importance in rigorously demonstrating how the terval 0 ≤ k ≤ Λ the convergence to ω (k) is uniform. Although since the approximation is local, the rate of dispersion relation of a free relativistic field theory can be obtained from the Gaussian cPEPS (3). In appendix A convergence does not depend on Λ. In conclusion, the fi- delity increases when D is increased while Λ is kept fixed, we concretely construct such a state with mass m using the continued fraction representation of the square root: and decreases when D is kept fixed while Λ is increased. This may be alarming since it seems that as the cutoff 2 2 grows, a larger and larger bond dimension will be re- m + k = m + , (15) quired to sustain a reasonable overlap between the two 2m + states. Furthermore, it implies that the bond dimension 2m + . has no physical significance as it depends on the way . 4 the theory is regularized. However, in case we allow the Given the Gaussianity of the particular case of study, parameters P of the approximation to depend upon the the fidelity can be explicitly calculated (and even later ex- cutoff themselves, it allows for rate of convergence to de- panded around the large cutoff limit). The cPEPS |ψ i pend on Λ as well. In which case, it enables, as the cutoff and the free ground-state |Ω i both factorize into tensor is sent to infinity, to tune the parameters in a way that products over momentum modes of harmonic oscillator makes the fidelity decay much less quickly, or perhaps ground-states: even saturate to a non-zero value. Λ Λ 1/4 Y Y ω (k) 1 ∗ − ω (k)φ (k)φ(k) |Ω i = |Ω (k)i = dφ (k) e |φ (k)i f f k k (18) Λ Λ 1/4 Y Y 1 ∗ ω (k) − ω (k)φ (k)φ(k) |ψ i = |ψ (k)i = dφ (k) e |φ (k)i D D k k The fidelity therefore factorizes as well: 1   1   1 4 4 2 ω (k) ω (k) 1 ω (k) ω (k) 2π f D f D − (ω (k)+ω (k))φ (k)φ(k) f D hψ (k)|Ω (k)i = dφ (k) e = D f 2 2 π π ω (k) + ω (k) f D (19) We finally obtain Λ Λ 2 Y Y 2 |ω (k)| ω (k) D f F (Ω , ψ ; Λ) = |hψ (k)|Ω (k)i| = (20) f D D f |ω (k) + ω (k)| D f k k In order to deal with the product inside (20) in the in many examples (e.g. [34, 35]) and in particular has continuum limit Λ → ∞, we need to be more precise been shown to also be generic for Matrix Product States with the definition of the Wilsonian regularization. We (MPS) [36] (other possibilities may also occur [37],[35]). begin from regularizing the field theory by placing it on a In principle, the fidelity would go to zero as Λ → ∞ or 1 N lattice with N sites and spacing . The volume V = V → ∞ even if the two dispersion relations are infinites- Λ Λ of the d-dimensional spatial space is taken to be finite yet imally close to each other but not exactly equal (this arbitrarily large and likewise the cutoff Λ is kept fixed. phenomena is also known as Anderson’s orthogonality Consequently, upon taking the logarithm the sum can be catastrophe [38]). In which case, we see that a relatable turned into an integral: quantity which bears its own significance exists, this is the fidelity per-site, defined by: 2 |ω (k)|ω (k) V d k D f log F = log (21) 2 |ω (k) + ω (k)| (2π) D f ΩΛ N F = lim F (23) N→∞ Where is an integral in the domain 0 ≤ k ≤ Λ. This may be defined alternatively in terms of the loga- The scaling of the problem becomes more transparent by ¯ rithm: log F (Ω , ψ ; Λ) = lim log F (Ω , ψ ; Λ). using the dimensionless variable: k = : f D f D N→∞   The fidelity per-site F is well defined in the contin- ¯ ¯ d 2 ω k ω k ¯ uum limit Λ → ∞, V → ∞ as it has no dependence D f N d k   log F = log (22) on the volume or the cutoff - unless inserted explicitly ¯ ¯ 2 ω k + ω k Ω (2π) D f through ω through the parameters P . We now explore this option, and for that we shall rescale the dispersion where Ω is the unit sphere. It is then seen that the relations ω (k), ω (k) in the dimensions of the cutoff, D f fidelity decays exponentially with the system’s size (as similarly to the definition of k. Hence we define: what multiplies N must be a negative number since each harmonic ground-state of a single mode k was taken to be ¯ ¯ ω k = Λω˜ k f f normalized). Such behavior between two distinguished (24) ¯ ¯ states of a quantum many-body system is known to hold ω k = Λω˜ k D D 5 We see that we may expand ω˜ (k) around Λ → ∞: In summary, we deduce that while in this particular example the fidelity itself is doomed to vanish in the con- 2 2 4 m 1 m m tinuum limit, there exist a quantity with a universal term ¯ ¯2 ¯ ω˜ k = k + = k + + O ¯2 that can in principle be variationaly optimized. Λ 2k Λ Λ (25) and III. CPEPS FROM PEPS 2 2D ¯ ¯ p˜ (P ) + p˜ (P ) k + ..p˜ (P ) k 0 1 D ω˜ k = (26) 2 2D ¯ ¯ 1 + q˜ (P ) k + ..q˜ (P ) k 1 D A. PEPS With Fields The rescaled parameters can be related back to the Starting from the lattice, we use the PEPS formalism original ones by: to construct a state whose continuum limit is the cPEPS (3) or a generalization thereof. We will first show how to 1−2α p (P ) = Λ p˜ (P ) α α do it for the simple case of a single scalar field, and then (27) −2α generalize to more possibilities, including scenarios with q (P ) = Λ q˜ (P ) α α global symmetries. We note that this is an arbitrary choice, since the extra On a constant time slice where the Hilbert space is power of Λ could have been otherwise extracted from defined, we consider a spatial lattice Z , whose sites are the denominator. Similarly to (25), an expansion around labelled by vectors of the form x ∈ ǫZ ≡ L, where ǫ > 0 Λ → ∞ for ω˜ k can be made as well. The form of is the lattice spacing. We use {eˆ } to denote the unit this expansion will depend of course on the way we have vectors in all the positive directions. Thus, for example, chosen the parameters P to depend on the cutoff Λ. We the sites x and x + ǫeˆ are nearest neighbours in the therefore chose p˜ , q˜ to be separated very generally into α α 1 ≤ i ≤ d direction, separated by a single unit of lattice a cutoff independent term and an irrelevant term: spacing, ǫ. We would like to construct a PEPS for the state of a (0) (irr) p˜ (P ) = p˜ (P ) + p˜ (P ; Λ) α α lattice scalar field φ (x): on each lattice site x, we in- (28) (0) (irr) troduce a physical Hilbert space, H (x), spanned by phys q˜ (P ) = q˜ (P ) + q˜ (P ; Λ) α α eigenstates of the on-site field operator φ (x). This can (irr) (irr) be either a real scalar field (as before) or a complex one p˜ , q˜ are defined to vanish as Λ → ∞. This α α (corresponding, as usual, to a pair of real scalar fields). definition is necessary since the fidelity is bounded for Thus, a good basis for the local physical Hilbert space is normalized states, therefore the parameters P cannot be |φ (x)i in the real case and φ (x) , φ (x) in the complex chosen in such a way that the fidelity diverges. one (where z represents complex conjugation). Below, Since the decay of ω˜ k in Λ is polynomial, it is an we will adapt the notation of the complex case, and the evident choice to let ω˜ k decay polynomially as well, real case will follow from it in a straightforward manner. thus an expansion in powers of for ω˜ k is possible The physical Hilbert space of the whole system is as well, similarly to (25). However, the decay can be in principle of a different class instead, such as an exponen- H = H (x) (32) phys phys tial one. x∈L Derived from this assumption, ω˜ k may be ex- panded in the same way around Λ → ∞: However, as usual in the construction of PEPS, the phys- ical degrees of freedom are not sufficient: to contract the local ingredients together, we have to introduce auxiliary (0) (irr) ¯ ¯ ω˜ k = ω˜ k + ω˜ k; Λ (29) D D or virtual fields to our system. That is, on each lattice site x ∈ L, we introduce 2d extra Hilbert spaces, asso- where ciated with the legs going out of and into it. On each (0) (0) (0) 2 2D ¯ ¯ outgoing leg, in direction i, we define the virtual fields p˜ (P ) + p˜ (P ) k + ..p˜ (P ) k (0) 0 1 D ω˜ k = (30) (0) (0) {χ (x)} , while on each ingoing leg we introduce sim- ¯2 ¯2D i α=1 1 + q˜ (P ) k + ..q˜ (P ) k ilar fields {η (x)} . All virtual fields are either real α=1 To conclude, putting together (25) and (26) yields: or complex scalars, depending on the nature of the phys- r ical field φ (x). As in the physical case, we will generally   (0) treat them as complex below. The number of fields we ¯ ¯ Z 2 ω˜ k k d D   1 d k (irr) introduce on each leg, D, is nothing but the bond dimen-   log F = log + F (Λ) d   (0) 2 sion mentioned above; we are free to choose any D ≥ 1 ¯ ¯ Ω (2π) ω˜ k + k that we want to build our PEPS - increasing D will allow (31) our PEPS to depend on more parameters - which implies The first term has no dependence in Λ at all, and more freedom for the optimization problem whose solu- (irr) F (Λ) is a function that vanishes as Λ → ∞. tion is sought with the PEPS. Different physical scenarios 6 will impose different constraints on the required minimal In a sense, χ (x) = (χ (x) , ..., χ (x)) and η (x) = 1 d bond dimension, and we have seen an example for that in (η (x) , ..., η (x)) can be thought of as spatial vector 1 d our previous construction. However, whatever D is, we fields. unite the Hilbert spaces of all the virtual fields of x ∈ L On each site x ∈ L we can therefore define the local to H (x). We denote its configuration states by virt Hilbert space H (x) = H (x) ⊗ H (x) . (34) physical virt |χ (x) , η (x)i := |χ (x)i|η (x)i (33) i i i=1 involving both the physical and virtual degrees of free- (where the α indices were omitted for simplicity). dom. Any state |A (x)i ∈ H (x) may be expanded as |A (x)i = (dχ (x) dη (x)) dφ (x) A (φ (x) , χ (x) , η (x))|φ (x)i ⊗ |χ (x) , η (x)i (35) i i (where integration on complex conjugates and/or different α components of the virtual fields is implicitely assumed). |A (x)i is not the state we need: it is both a product state, with no correlations among the lattice sites, and it x∈L also involves some virtual degrees of freedom which have nothing to do with the physical Hilbert space H . Both phys issues are addressed, as usual, by projecting the virtual degrees of freedom onto maximally entangled pair states, connecting the virtual fields of both sides of each link; in our case, we choose |L (x)i = dχ (x) dη (x + ǫeˆ ) δ (χ (x) − η (x + ǫeˆ )) δ (χ¯ (x) − η¯ (x + ǫeˆ ))|χ (x)i|η (x + ǫeˆ )i (36) i i i i i i i i i i i i i defined on each link. The extension to D > 1 is straight- forward, and in the real field case the second delta func- tion(s) are removed. The PEPS is obtained by projecting |A (x)i onto x∈L the maximally entangled states on all the links: O O |ψi = hL (x)| |A (x)i ∈ H (37) i phys x∈L,i=1,...,d x∈L A two-dimensional example for the construction detailed above is depicted in Fig. (1). We have not only eliminated the virtual degrees of freedom, we actually used them for inducing correlations among the lattice sites. Furthermore, this state will sat- isfy the area law, since for any bipartition we wish to make, we will get that the entanglement entropy comes from the maximally entangled states on the links which cross the boundary, all contributing the same. FIG. 1. An example of a two-dimensional PEPS with fields. B. Spatial Symmetries The lattice sites are marked with the black dots and the vir- tual fields with the ingoing and outgoing blue lines to each site. Two outgoing and ingoing virtual fields of two neigh- We would like to require the PEPS |ψi to be invariant bouring sites are contracted by the links L , which is illus- under translations and under the transformations which trated by the oval connecting the two lines. preserve the lattice (its point group). This is a reasonable condition if we wish the state to become invariant under continuous translations and rotations SO (d) in the con- tinuum limit (the rest of the Poincar´e group is explicitly discrete lattice translations and also under rotations pre- broken by the choice of the quantization scheme). To serving the lattice. As for the latter, where it is for the that end we require the PEPS |ψi to be invariant under simple choice of a square lattice we will suffice to require 7 invariance under rotations of in all possible planes (we product of all link states remains the same). This is do not make any prior demands regarding reflections and satisfied by the states we picked for our demonstration parity). above. The discrete translations T (a) for some displacement In order to obtain the desired form of the cPEPS (3) vector a on the lattice are defined to act in a very simple in the continuum limit, we write A (φ, χ, η) as an expo- way on both the physical and virtual fields, nential: A(φ,χ,η) T (a)|φ (x)i = |φ (x + a)i (38) A (φ, χ, η) = e (43) It acts exactly the same on the virtual fields χ , η . If not i i Here, A respects the spatial symmetries (38) and (42). broken to begin with, the discrete translational symme- We will now focus on a more specific ansatz, try on the lattice will be straightforwardly enhanced to a continuous translational symmetry in the continuum A = R (χ, η) + K (χ, η) + V (χ, η, φ) (44) limit. Imposing translation symmetry on the PEPS is where V is a general analytic function of the fields that is very simple, and can be done by requiring all the func- required to be invariant under (42). R and K are defined tions A (φ (x) , χ (x) , η (x)) from (35) to be the same all by around the lattice (independent of the coordinates). Sim- ilarly, we use the same |L (x)i states on all the links in the same direction. R (χ, η) = − N |χ (x) − χ (x)| 0 i j Next we consider lattice rotations. Denote by Λ ro- ij π i<j tations of in the plane spanned by eˆ and eˆ (with no i j 2 (45) 2 2 + |χ (x) − η (x)| + |η (x) − η (x)| loss of generality, we assume that i < j). The coordinate j i i j rotation is given by + |η (x) − χ (x)| j i Λ x = Λ (x , .., x , .., x , ..x ) = (x , ..,−x , .., x , ..x ) ij ij 1 i j d 1 j i d (39) and The physical field is scalar and thus its eigenstates transform as: 0 2 K (χ, η) = − |χ (x) − η (x)| (46) i i |φ (x)i → |φ (Λ x)i (40) ij The function K will eventually play the role of the Nevertheless, the virtual fields must transform in a dif- kinetric term, whereas R will be shown to be necessary ferent manner: for continuous rotational invariance of SO (d) to emerge |χ (x)i → |χ (Λ x)i in the continuum limit. i j ij |χ (x)i → |−η (Λ x)i j i ij (41) |η (x)i → |η (Λ x)i i j ij C. The Continuum Limit |η (x)i → |−χ (Λ x)i j i ij The projection onto the maximally entangled link To explain why, we recall that these fields are associated states |L (x)i allows us to use the delta functions, elimi- with directions - they are components of spatial vector nate the η Hilbert spaces and exchange any appearance fields. Therefore, they must follow the rotation of links, of η (x) by χ (x − ǫeˆ ). Collecting all of the above, the i i i giving rise to the above rotation rules. physical state may then be written as (here D = 1 but Requiring the fiducial state |A (x)i to be invariant un- the generalization to higher bond dimensions is straight der (41) is equivalent to requiring the wave-function in forward): (35), A (φ (x) , χ (x) , η (x)) to be symmetric under the Z P permutations of the virtual fields dictated by (41) for A(φ(x),χ(x)) |ψi = Πdφ (x) Πdχ (x) e Π|φ (x)i|χ (x)i all i < j: i x x x,i (47) A (φ, .., χ , .., χ , ..η , .., η , ..) i j i j In the above expression appears the same function A as = A (φ, .., χ , ..,−η , ..η , ..,−χ , ..) j i j i it was presented previously, only now with the insertions (42) = A (φ, ..,−η , ..,−η , .. − χ , ..,−χ , ..) of η (x) replaced with χ (x − ǫeˆ ) as explained. i j i j i i i From this point the continuum limit is taken in the = A (φ, ..,−η , .., χ , .. − χ , .., η , ..) j i j i conventional way. That is, the dimensionless fields and One also has to make sure that the maximally entangled parameters are exchanged by their continuum counter- states on the links, |L (x)i are rotated in an invariant parts which obtain their mass dimension by an appropri- way (in this case, we need to demand that in an i − ate power of ǫ. j rotation, the states belonging to these two directions Namely, the fields φ, χ which appear in (47) are de- are exchange, and the others are left intact; overall, the fined on the lattice and are dimensionless. For taking 8 the continuum limit, we need to exchange summation by integration in the exponent of (47). Since the integration X X measure is dimensionful, this requires the introduction (R) (R) d −d+2[χ] R (χ) → − d xN ǫ χ (x) − χ (x) i j of dimensionful fields to compensate for the originally i<j dimensionless expression. [φ], the dimension of the phys- (49) ical field φ is chosen to have the standard dimension for −d+2[χ] Defining N = N ǫ , we may send ǫ → 0 and d−1 0 the free relativistic field [39]: [φ] = . This is a nec- tune N accordingly to obtain a finite N (the sub-leading essary choice as this degree of freedom also serves in a term will then scale as Nǫ and will hence vanish in the physical state such as the free vacuum (14). continuum limit). We define the ”renormalized” physical field Next, we do the same with the function K (χ). Due (R) −[φ] φ (x) = ǫ φ (x). Since the virtual fields are to its derivative nature, its leading order term is from a eventually integrated out there exists a freedom in higher order in ǫ: choosing their dimension (see appendix A for more details). We hence do not commit to a specific choice and denote their dimension by [χ]. We define X X 2 (R) d 2−d+2[χ] (R) K (χ) → − d xZ ǫ ∇ χ (x) −[χ] 0 i χ (x) = ǫ χ (x) accordingly. i 2 x i (50) 2−d+2[χ] The rest of the procedure is done by expanding (recall We then define Z = ǫ Z , and tune Z as we 0 0 that we have replaced η (x) with χ (x − ǫeˆ )): i i i send ǫ → 0 to obtain a finite Z. (R) (R) (R) While the continuum limit of K (χ) is symmetric under χ (x − ǫeˆ ) = χ (x) − ǫ∇ χ (x) + O ǫ i i i i i (48) discrete rotations of , it is not under the larger group (R) (R) (R) 2 φ (x − ǫeˆ ) = φ (x) − ǫ∇ φ (x) + O ǫ i i of continuous rotations SO (d). (This issue has already been pointed out in [22]). and keeping the leading order term in ǫ inside A. Sub- Therefore, we introduce the additional function R (χ): leading terms will vanish in the continuum limit ǫ → 0 it enhances (47) to have the full SO (d) symmetry in since the parameters will be tuned with respect to the its continuum limit. We note that this function has leading term, as we will show in more detail. Starting a lot of resemblance with the ansatz first proposed in [23]. with the function R (χ), we observe that by its cyclic nature there are no cancellations and the leading term is To clearly see how, we first make the separation (R) (R) ′ (R) (R) (R) therefore a non-derivative. All of the four terms in R (χ) A χ , φ = A χ , φ +R χ . At this point collapse into a single one proportional to: we write (47) as n oE R R d (R) d ′ (R) (R) (R) (R) d xR χ (x) d xA χ (x),φ (x) (R) [ ] [ ] |ψi = Dφ (x)Dχ (x) e e φ (x) (51) The function R helps as we formally send N → ∞. In the limit, the integral (51) becomes exclusively dominated by (R) (R) field configurations of the (complex) fields χ which satisfies the constraints χ (x) = χ (x) for all i 6= j. The set i j (R) of the virtual fields {χ } collapses into a single virtual field which we call v (x). Hence, now we can write (51) i=1 as: n oE d ′ (R) (R) (R) (R) d xA v (x),φ (x) (R) [ ] |ψi = Dφ (x)Dv (x) e φ (x) (52) where A = K + V . lowing all the steps described above, the functional (R) (R) (R) K becomes the standard kinetic term for the field v: V v ,∇v , φ ((3) also depends on ∇φ, we will later show how this is achieved in the end of this sec- tion). K (v) = − ∇ v¯∇ v (53) i i The mapping between the two can be defined in a for- After the collapse to a single virtual field, there is no mal way through the series expansion of V in powers of association to a spatial direction and this term is now the fields in their first derivatives and then identifying invariant under the full rotational group SO (d). (R) (R) v (x) with χ (x) and ∇ v (x) with χ (x) − η (x). Finally, as was done before in detail for R and i i i i K, the function V (χ , η , φ) similarly becomes, by fol- For example, V = a (∇v) v can be construtced from 0 i i 9 V = a |χ (x) − η (x)| χ + (permutations). The per- have to lay on the same lattice - the virtual fields after all 0 0 i i i mutations under (47) ensure the rotational symmetry. connect between the physical Hilbert spaces which may Similarly to before, the connection between the parame- contain more than one lattice site. 1 2−d+3[χ] 1 ters is given by a = ǫ a ; the factor of comes It is interesting to note that this simple operation can- 4 4 to account for the 4 terms conncted by the rotational not be applied to the virtual fields since they are re- transformation which collapse into a single one in the stricted to lay on the boundary of the block rather then continuum limit. inside its bulk. This fact seems to hinder the ability to Now that we are fully done, we can conveniently drop construct higher derivatives for the virtual fields. the R superscript as well as the prime from A . Addi- tional virtual fields can easily be added by following the IV. SYMMETRIES same steps above, so we may consider a general bond di- mension D > 1. We then arrive to the continuum limit of (47) which is the desired cPEPS: PEPS are well-known for the ease of describing sym- metries with them: one can parametrize families of PEPS d xA[v ,∇v ,φ] α α which will be invariant under some symmetry group |ψi = Dφ (x)Dv (x) e |{φ (x)}i [5, 7, 8]. (54) Consider, at first, the case of a complex scalar field. A = − Z ∇ v ∇ v + V [v ,∇v , φ] αβ i α i β α α We can define a global U(1) transformation on it, such that iθ U (θ)|{φ (x)}i = e φ (x) (56) For any θ ∈ [0, 2π). If we wish our PEPS to be invariant under the symmetry operation, we will need to define some transformation rules for the virtual fields too, e.g. iθ v (x) → e v (x) (57) α α Changing the integration variables (the integration mea- sure Dφ (x) is invariant under a unitary change of vari- ables), we will get that U (θ)|ψi = |ψi if iθ iθ iθ V [v ,∇v , φ] = V e v , e ∇v , e φ (58) α α α α If, for example, we construct a Gaussian cPEPS, this will be satisfied if we pick 1 1 A = − Z ∇v (x)∇v (x) − A v (x) v (x) αβ α β αβ α β 4 4 + z ∇v (x)∇φ (x) + a v (x) φ (x) − |φ (x)| + c.c α α α α (59) This can also be achieved through the PEPS construc- FIG. 2. An example for a two-dimensional square lattice with tion, by properly parametrizing the states |A (x)i and a H of larger support. The block is of size 2×2 and marked phys |L (x)i (see, e.g. [10]); but can also be done directly in in a dashed blue line. The black dotes stand for the lattice the continuum. sites which hosts the physical fields whereas the solid blue We can think of more general settings. Suppose that lines for the four virtual fields of the ingoing and outgoing direction to the block. instead of the field φ (x) we introduce a vector, or a mul- tiplet of fields, either real or complex, {φ (x)} , trans- a=1 In fact, (3) has dependence on ∇φ too. We close this forming as some representation j of dimension r of some section be showing how this and generelization thereof is group G; that is, for each group element g ∈ G, we define achieved. The local physical Hilbert space H may be the unitary transformation phys defined on a block of lattice sites rather on a single point, n oE U (g)|{φ (x)}i = D (g) φ (x) (60) as illustrated in Fig. 2. a b ab H (x) = H (55) where D (g) is the j representation matrix of g (of size phys φ(x+a ˆ) ab a ˆ∈block r × r). If we wish to construct a cPEPS |ψi with the symmetry The larger the block, the higher derivative terms for property the physical field φ that can be written. We now make the observation that the virtual and physical fields do not U (g)|ψi = |ψi ∀g ∈ G (61) 10 we will need to use virtual fields that are also charged fields can be constructed by taking tensor products and under this group, that is, the v fields will have to carry symmetrizing/anti-symmetrizing. an a index, forming multiplets of the group G. One can Furthermore, an important benefit of this bottom-up use several multiplets and copies thereof, as long as the construction is that tools and ideas that apply for PEPSs multiplets are fully included. The virtual and physical are transferred to cPEPSs - section IV is an important ex- representations do not even have to be the same, as long ample for that. The PEPS framework has been elevated as they are coupled properly. Thus, in general, the virtual to include physical gauge fields as well [10, 40], with a fields transform as very clear prescription on how to do so. We expect that following through this procedure to the continuum will v (x) → D (g) v (x) (62) α,a α,b ab result in ”minimal-coupling” of (54) (in the same sense that an action is minimally coupled). Nevertheless we with some (irreducible or reducible) representation J. leave this to a more thorough investigation in the future. Our cPEPS will be defined with The fidelity was used for estimating the accuracy of the 1 cPEPS approximation to the real, physical ground-state. A = − Z ∇ v ∇ v + V [v ,∇v , φ ] (63) αβ i α,a i β,a α,a α,a a In most cases however, the exact form of the ground-state is unknown, and hence it is the tensor-network state that and the symmetry condition will be serves as an ansatz that is designed to capture most of the ground-states important properties. If so, one im- h i mediate implementation of the cPEPS is to serve as a J J V [v ,∇v , φ ] = V D (g) v , D (g)∇v , D (g)φ α,a α,a a α,b α,b b ab ab variational ansatz which is numerically adjusted to give ab a minimal expectation value for some Hamiltonian of in- (64) terest. To deal with the usual divergences that appear which can also be simplified, in a straightforward man- in the ground-state energy of quantum field theories, it ner, in the Gaussian case (it can also be seen as a con- was demonstrated to be efficient to minimize the normal- tinuum limit of the case studied in [40], when changing ordered/renormalized Hamiltonian instead of the bare to scalar fields). one [25],[41]. Here again the question about the univer- Finally, if we keep A real, we will also have a charge- sality of this method is raised. For instance, if a differ- conjugation symmetry in the case of complex fields. ent renormalization scheme would be used, will a smaller bond dimension be able to yield a similar result? In our V. DISCUSSION non-interacting example, we have found that indeed there is some universal content to the problem, yet the larger picture remains open for future study. We have developed a continuous Projected Entangled Along with this, there is another context to which the Pair State (cPEPS) for quantum fields and explicitly cPEPS can be closely related. A tensor-network state can shown how such a state can approach the free field theory be thought of as a representative of a class of states with vacuum as the bond dimension is increased. In addition a specific structure of symmetry and entanglement. Since we have tackled the question on whether this approxi- the critical properties of a phase transitions is normally mation is meaningful in the real continuum sense, or in dependent upon the robust features of the two phases, other words if it has a universal significance. To that end one can build tensor-network states which lay in the uni- we have used the quantum fidelity as a measure of dis- versality class of each phase and in that way ”engineer” a tinguishability between the cPEPS and the real physical quantum phase transition [42]. There is then great merit state that it approximates, and found that it encapsulates in calculating the fidelity of two nonidentical cPEPSs be- a universal term that is independent on the short-scale longing to different symmetry classes - which can be eas- behavior of the problem. ily created with the tools portrayed in this paper. Subsequently we have built the cPEPS in a bottom-up approach. Starting from the lattice, a PEPS paired with Fidelity of many-body quantum states as a probe for a unique ansatz was used to produce the wanted result quantum phase transitions has been a popular source of in the continuum. A central feature of the cPEPS is that research (see [35] and references therein). The quantum it enjoys a global symmetry under any desired symmetry fidelity has proven to be a useful analytical tool for gain- group of interest, as well as translational and continuous ing insight about the characteristics of the phase tran- rotational symmetries. sition and it has a unique critical behavior around the While we have focused on scalars, fields with higher transition point on its own. Exact form of the fidelity spin may also be taken in consideration. A similar con- is in general not easily calculable, and known examples struction as the one done in section III is expected to exist in large for integrable models in where the ground- be possible for physical and virtual fields with higher state has a known form. We therefore hope that the spin, albeit left beyond the scope of this paper. In cPEPS together with standard perturbative techniques fact, only two generalizations are required for spin 1 can open a way to a new class of phase transitions that vector fields and spin fermions. Once they are laid can be studied via the fidelity approach and we leave that down, higher integer or respectively half-integer spin to future work. 11 ACKNOWLEDGEMENTS virtual fields (as we show below) results in: 2 2 The authors would like to thank Michael Smolkin and B k ω (k) = A + D 0 2 2 Patrick Emonts for helpful and inspiring discussions. B k Z m + This research was supported by the Israel Science Foun- 1 2 2 B k dation (grant No. 523/20). A m + 2 2 B k D−1 Z m + ... 2 2 B k Z m + D−1 A m Appendix A: A Conctinued Fraction Approximation (68) In this form it is a rather simple task to make the cPEPS (66) approach the vacuum state of a free field theory |Ω i: Hereby we shall introduce a specific and simpler choice of the more general Gaussian cPEPS (3) which enables the presentation of ω (k) in the form of a continued fraction. It allows for a rather elegant representation of 1 ∗ − ω (k)φ (k)φ(k) |Ω i = Dφ (k) e |{φ (k)}i (69) the ground-state of a free field theory. Furthermore, con- tinued fractions in general have many appealing prop- erties, allowing one to gain more knowledge about the convergence. Given that, we narrow down to the follow- 2 2 ω (k) = k + m has a continued fraction represen- ing choice of (3): tation: ω (k) = m + (70) iB k α k A + Z k = (δ + δ ) αβ αβ α,β−1 α,β+1 2m + 2m + mA α = 2n n ∈ N . + δ × 2 αβ Z k α = 2n + 1 , n ∈ N (65) It is then clear that as we set A = m, B = B = 1 0 0 α and Z = A = 2, ω (k) approaches ω (k) as D grows. α α D f The convergence however is not uniform as the remain-   der diverges as k increases. This can be seen by the fact 2 2 Z k /m iB k /m 1 1 that ω (k) asymptotically behaves for large values of k 2 2 iB k /m mA iB k /m  1 2 2   2 2 2 as k or as a constant depending if the truncation of the iB k /m Z k /m iB k /m  2 3 3    continued fraction is even or odd, whereas ω (k) asymp-   2 2 iB k /m iB k /m 3 D−1 totically behaves as k. iB k /m mA D−1 D In fact, the continued fraction representation is not (66) unique. This is not surprising given the fact that so is iB And a = 0, z = δ , c = A . The bond dimen- α α α1 0 m the more general Pad´e approximant. sion D is assumed to be even. Eq. (68) is obtained inductively. To that end we inte- We first need to take care of the fact that there is grate out the virtual fields one by one, and it is useful to more than one way of fixing the mass dimension [χ] of present the following notation: the virtual fields χ . We find it useful to choose the di- mension to be the same as that of the physical field φ: D R d−1 d Y d k [χ] = [φ] = . Recall that the dimension of the physi- A [v (k),φ(k)] D α 2 d (2π) |ψi = Dφ (k) Dv (k) e |{φ (k)}i cal field [φ] is completely set by the Klein-Gordon action. α=1 Under this convention the dimensions of the parameters D−1 d k defined in (3) are as follows: A [v (k),φ(k)] D−1 α (2π) = Dφ (k) Dv (k) e |{φ (k)}i α=1 [c] = 1 , [A ] = [a ] = 1 , [Z ] = [z ] = −1. (67) αβ α αβ α = ... (71) We note that ω (k) = A (k) + A . D 0 0 Notice that we have set the physical mass m which ap- pears in ω as a single energy scale and used it to fix For obtaining (68) we start by integrating χ (k). f α=D the dimensions of all of the rest of the parameters. Thus To clarify that, we first write (71) in a form where B , Z and A are dimensionless. Integrating out the χ (k) is separated from the rest of the virtual fields: α α α α=D 12 Appendix B: A Note on Parent Hamiltonians m iB k 2 D−1 A = − A |v (k)| + v (k) v (k) D D D D−1 2 m A natural question put in the context of tensor- D−2 networks is about the properties and uniqueness (or Z k 1 D−1 2 2 ∗ − |v (k)| − Z + Ak v (k) v (k) D−1 β α non-uniqueness) of the parent Hamiltonian H , whose αβ D 2m 2 α,β ground states is the tensor network state - in our case, (3). Since (3) is a Gaussian state there must exist a quadratic iB k A 0 0 ∗ 2 + v (k) φ (k) − |φ (k)| parent Hamiltonian, and in fact there is a family of these m 2 (72) with the following form: Integrating out χ then leads to:   1 d k D−1 4 H (φ, π) = a (k) Π (k) Π (k) 1 Z k k D D−1 2 d   2 A = − + |v (k)| (2π) D−1 D−1 (76) 2 m mA + b (k) φ (k) φ (k) iB k m D−2 2 + v (k) v (k) − A |v (k)| D−2 D−2 D−2 D−1 m 2 D−3 2 ∗ b(k) − Z + Ak v (k) v (k) where ω (k) = . In this aspect, ω (k) may be in- β D α D αβ a(k) α,β terpreted as a dispersion relation, obviously dependent on the bond dimension D and the parameters which ap- iB k A 0 0 + v (k) φ (k) − |φ (k)| pear in the Gaussian cPEPS (8). All of the Hamiltoni- m 2 ans (76) have the same spectrum as they are canonically (73) equivalent to each other. We note that the canonical Next integrating out χ leads to: D−1 transformation which connects the two is in general k dependent.   k B D−2   A = − mA + |v (k)| This in fact raises an issue if one reverses this argu- D−2 D−2 2 D−2 4 2 k B m k Z 2 D−1 D−1 + ment, since in principle the fields φ, π which appear in m mA (8) do not have to be the same as the ones in H (12). D−3 X 2 1 iB k 2 ∗ ∗ Then for any ω (k) one can choose canonically trans- − Z + Ak v (k) v (k) + v (k) φ (k) α 1 αβ 2 m formed fields and momenta in such a way that (8) is the α,β exact ground-state of H . In that sense it may seem that there is no sense at all in seeking a variational technique. − |φ (k)| (74) We can relief this obstruction by at least two argu- In case that k 6= 0, the fraction can be reduced by 2 ments. First, the free Hamiltonian and its spectrum can and give the last iteration of the continued fraction: be decomposed to a tensor product of individual modes of momenta k. This property no longer holds once inter- actions are added, reducing the possibility to transform B k D−2 each mode separately. Furthermore, canonical transfor- mA + (75) D−2 2 B k D−1 mZ − mations are more subtle in quantum field theories given D−1 mA their divergent nature (a (k) , b (k) are in principle regu- Since D is even we can now repeat this process by larization dependent). A simple example is spontaneous induction, treating the physical field φ as χ with symmetry breaking; two ground-states on the vacuum α=0 A = B and B = B . This then results in (68). manifold are connected by a unitary transformation but α=0 0 α=0 0 In the case where k = 0 the physical field is completely are nevertheless physically distinct in the infinite volume decoupled from from the virtual fields and we immedi- limit. Then to conclude, to avoid these issues we choose ately have that ω (0) = m. Which also equals of course the fields φ, π which appear in (8) and in H (12) or in D f ω (k = 0). Therefore the equality holds also for k = 0. (76) to be the same. [1] J. I. Cirac and F. Verstraete, Journal of Physics A: Math- [3] U. Schollw¨ock, Annals of Physics 326, 96 (2011). ematical and Theoretical 42, 504004 (2009). [4] R. Oru´s, Annals of Physics 349, 117 (2014). [2] F. Verstraete, V. Murg, and J. Cirac, [5] I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Advances in Physics 57, 143 (2008). “Matrix product states and projected entangled pair 13 states: Concepts, symmetries, and theorems,” (2021), Phys. Rev. Research 3, 023059 (2021). arXiv:2011.12127 [quant-ph]. [25] A. Tilloy, “Relativistic continuous matrix product [6] J. Eisert, M. Cramer, and M. B. Plenio, states for quantum fields without cutoff,” (2021), Rev. Mod. Phys. 82, 277 (2010). arXiv:2102.07741 [quant-ph]. [7] M. Sanz, M. M. Wolf, D. P´erez-Garc´ıa, and J. I. Cirac, [26] A. E. B. Nielsen, B. Herwerth, J. I. Cirac, and G. Sierra, Phys. Rev. A 79, 042308 (2009). Phys. Rev. B 103, 155130 (2021). [8] A. Molnar, J. Garre-Rubio, D. P´erez-Garc´ıa, N. Schuch, [27] G. Vidal, Phys. Rev. Lett. 101, 110501 (2008). and J. Cirac, New Journal of Physics 20, 113017 (2018). [28] J. Haegeman, T. J. Osborne, H. Verschelde, and F. Ver- [9] J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, straete, Phys. Rev. Lett. 110, 100402 (2013). and F. Verstraete, Phys. Rev. X 5, 011024 (2015). [29] M. Nozaki, S. Ryu, and T. Takayanagi, [10] E. Zohar and M. Burrello, JHEP 10, 193 (2012), arXiv:1208.3469 [hep-th]. New Journal of Physics 18, 043008 (2016). [30] M. Srednicki, Quantum Field Theory (Cambridge Uni- [11] I. Kull, A. Molnar, E. Zohar, and J. Cirac, versity Press, 2007). Ann. Phys. 386, 199 (2017). [31] S. Weinberg, The Quantum theory of fields. Vol. 1 (Cam- [12] N. Schuch, J. Cirac, and D. P´erez-Garc´ıa, bridge University Press, 2005). Annals of Physics 325, 2153 (2010). [32] T. Nishioka, Rev. Mod. Phys. 90, 035007 (2018). [13] S. White, Physical Review Letters 69, 2863 (1992). [33] G. A. Baker and P. Graves-Morris, Pad´e Approximants, [14] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003). 2nd ed., Encyclopedia of Mathematics and its Applica- [15] F. Verstraete, J. J. Garc´ıa-Ripoll, and J. I. Cirac, tions (Cambridge University Press, 1996). Phys. Rev. Lett. 93, 207204 (2004). [34] M. M. Rams and B. Damski, [16] M. Zwolak and G. Vidal, Phys. Rev. Lett. 106, 055701 (2011). Phys. Rev. Lett. 93, 207205 (2004). [35] S.-J. GU, International Journal of Modern Physics B 24, 4371 (2010), [17] J. Jordan, R. Oru´s, G. Vidal, F. Verstraete, and J. I. https://doi.org/10.1142/S0217979210056335. Cirac, Phys. Rev. Lett. 101, 250602 (2008). [36] M. Cozzini, R. Ionicioiu, and P. Zanardi, [18] M. C. Ban˜uls and K. Cichy, Phys. Rev. B 76, 104420 (2007). Reports on Progress in Physics 83, 024401 (2020). [37] L. Campos Venuti, H. Saleur, and P. Zanardi, [19] Y. Meurice, R. Sakai, and J. Unmuth-Yockey, Phys. Rev. B 79, 092405 (2009). arXiv:2010.06539 [hep-lat] (2020). [38] P. W. Anderson, Phys. Rev. Lett. 18, 1049 (1967). [20] F. Verstraete and J. I. Cirac, [39] This is set for instance from the (dimensionless) Klein- d+1 1 μ 1 2 2 Phys. Rev. Lett. 104, 190405 (2010). Gordon action: S = d x − ∂ φ∂ − m φ (the 2 2 [21] J. Haegeman, J. Cirac, T. Osborne, and F. Verstraete, the kinetic term’s coefficient is normalized to one by con- Phys. Rev. B 88, 085118 (2013). vention). [22] D. Jennings, C. Brockt, J. Haegeman, [40] E. Zohar and J. Cirac, T. J. Osborne, and F. Verstraete, Physical Review D 97 (2018), 10.1103/PhysRevD.97.034510. New Journal of Physics 17, 063039 (2015). [41] S. Rychkov and L. G. Vitale, [23] A. Tilloy and J. I. Cirac, Phys. Rev. X 9, 021040 (2019). Phys. Rev. D 91, 085011 (2015). [24] T. D. Karanikolaou, P. Emonts, and A. Tilloy, [42] M. M. Wolf, G. Ortiz, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 97, 110403 (2006). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png High Energy Physics - Theory arXiv (Cornell University)

Approximating Relativistic Quantum Field Theories with Continuous Tensor Networks

Shachar, Tom; Zohar, Erez
High Energy Physics - Theory , Volume 2021 (2110) – Oct 4, 2021
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Abstract

Approximating Relativistic Quantum Field Theories with Continuous Tensor Networks 1 1 Tom Shachar and Erez Zohar Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel. (Dated: October 5, 2021) We present a continuous tensor-network construction for the states of quantum fields called cPEPS (continuous projected entangled pair state), which enjoys the same spatial and global symmetries of ground-states of relativistic field theories. We explicitly show how such a state can approximate and eventually converge to the free field theory vacuum and suggest a regularization-independent way of estimating the convergence via a universal term in the fidelity per-site. We also present a detailed bottom-up construction of the cPEPS as the continuum limit of the conventional lattice Projected Entangled Pair State (PEPS). I. INTRODUCTION ativistic problems. Recently, aiming at extending the cMPS to a relativistic setting, the one dimensional con- struction of RcMPS (relativistic cMPS) was introduced Tensor network states have proven to be a prolific theo- [25]. It uses a different operator basis, with respect to retical and numerical framework for advancement in the the particle creation and annihilation operators, which understanding of many-body quantum systems. These is more adequate for dealing with relativistic problems. states, which are constructed by contractions of local ten- In other tensor network techniques involving continuous sors, capture very well the physics of ground-states, low fields, one may use continuous fields to contract lattice lying excited states and thermal states of local Hamiltoni- tensor networks for spin models [26]. Finally, another ans [1–5]. By construction, they have the relevant physi- type of tensor network states, MERA (multiscale entan- cal properties, including the right entanglement structure glement renormalization ansatz) [27] has its continuum (area law [6]), possibility to encode symmetries - both version - cMERA - too [28, 29]. But in general, the field global [7, 8] and local [9–11], topological properties [12], of continuous tensor network states is still far from be- and, of course, a large set of numerical methods appli- ing fully studied and understood, in spite of its obvious cable to different scenarios and purposes - e.g. [13–17] importance and relevance, and hence provides an inter- and more. These approaches, including the 1 + 1 dimen- esting and challenging scientific playground. sional Matrix Product States (MPSs) and their higher dimensional extension PEPS (Projected Entangled Pair In this work, we present a continuous, field-theoretic version of PEPS which is built from quantum fields. We States) [5], are almost entirely restricted to lattice mod- therefore refer to them as cPEPS - continuous PEPS. els, with very few extensions, so far, to quantum field The reason this construction sheds new light on contin- theories defined in the continuum. uous tensor-networks is twofold. To begin with, these Originally, tensor network state methods were mostly states are well defined in any dimension and they are applied to condensed matter problems. In the recent well suited to deal with relativistic field theories simply years, however, there has been a growing effort in ap- from the way they are formalized. Moreover, they inherit plying them also to high energy physics, in particular many of the appealing attributes of the PEPS such their for the study of lattice gauge theories. That includes ability to account for symmetries. Second, the cPEPS the benchmarking of known results, as well as computa- are formalized in a field-theoretic way. This simple fact tions that go beyond the state of the art of conventional, is crucial as it brings this particular manifestation of a Monte-Carlo methods (see, e.g., [18] for a contemporary continuous tensor-network closer in spirit to relativistic review of this field). Tensor network contractions of path quantum field theory, which has its own wide array of integrals, rather than states, have also been introduced computational techniques [30, 31] - specifically also of and successfully applied to these models, using the TRG quantum informational calculations (see [32] and refer- (tensor renormalization group) toolbox [19]. However, ences therein). Thus, it may hopefully serve as a bridge as the fundamental field theories of particle physics are for implementing tensor-network techniques in quantum continuous, it is also interesting to develop tensor net- field theories and vice versa. work formulations enabling their study directly, without applying any lattice discretization schemes. In section II, we present the cPEPS construction in Continuous tensor network states (cTNs) were first in- detail and it is rigorously demonstrated how the ground- troduced in [20], in the form of cMPS (continuous Matrix state of a free scalar field (Klein-Gordon) theory can be Product States). This formalism was further developed obtained from it, both exactly (with unlimited compu- in later works [21], including higher dimensional exten- tational resources) or approximately. We study the ac- sions [22, 23] and even used as a numerical ansatz [24]. curacy of the approximation, show that it depends on However, as these constructions rely on Fock space con- the choice of a regularization scheme and propose a uni- cepts, they are seemingly less equipped to deal with rel- versal (regularization-independent) approach for tackling arXiv:2110.01603v1 [quant-ph] 4 Oct 2021 2 this problem. In section III, we present a bottom-up con- positive definite symmetric matrix and V is a general an- struction of the cPEPS as the continuum limit of a con- alytic functional of the fields and their first derivatives. ventional, lattice PEPS construction, and explain how a Following previous papers [20]-[25], we shall call D the cPEPS with continuous translational and rotational sym- (generalized) bond dimension. metry can be built. Finally, in section IV we explain in To demonstrate what could be done with such states, detail how states that are invariant under a general global we will restrict ourselves now to a case which can be symmetry can be directly constructed. handled analytically: Gaussian states, in which A is In what follows we adhere to the common choice of quadratic in both the physical and virtual fields and their dimensions ~ = c = 1. The Einstein summation con- first derivatives (other than being analytically tractable, vention where repeated indices are being summed over is Gaussian states are interesting in their own right as they assumed. serve as the ground-states of non-interacting theories). The most general A of such a state may be written as follows: II. CPEPS 1 1 A = − Z ∇v (x)∇v (x) − A v (x) v (x) αβ α β αβ α β We will begin our discussion of cPEPS by showing its 2 2 construction for, perhaps, the simplest possible physical + z ∇v (x)∇φ (x) + a v (x) φ (x) − φ (x) α α α α case, of a real scalar field in a d + 1 dimensional flat (Minkowski) spacetime. The Hilbert space of the field (5) theory is constructed by choosing a foliation of space, and Although one could, in principle, choose by defining the field and its respective conjugate momen- {Z , A , z , a , c} to be position dependent, we αβ αβ α α tum field over each slice (equal time surface), where the did not do this, and thus the above expression is two satisfy the canonical commutation algebra. Dictated translationally invariant. Thus, it is natural from this by the dynamics of the theory, there is then a unitary point onward to work in Fourier (momentum) space. evolution from one slice to the other (time evolution). using the Fourier transform We choose the standard foliation in which spacetime is Z d −ik·x divided into constant d dimensional (spatial) time slices. φ (k) = d k e φ (x) . (6) On each slice of constant time t we introduce the field operator φ (x, t ), and its conjugate momentum operator (and similarly for the virtual fields) we obtain: π (x, t ), satisfying the canonical commutation relation d k A[v (k),φ(k)] (2π) (d) |ψi = Dφ (k)Dv (k) e |{φ (k)}i [φ (x, t ) , π (y, t )] = iδ (x − y) . (1) 0 0 2 ∗ Choosing an arbitrary time-slice (that is, fixing the A = − A + Z k v (k) v (k) + αβ αβ β time) we denote by |{φ (x)}i a field configuration state: 2 ∗ a + z k v (k) φ (k) − φ (k) φ (k) . α β |{φ (x)}i = ⊗|φ (x)i (2) (7) Since these are real fields, they additionally satisfy - that is, a product of eigenstates of the field operator φ (k) = φ (−k), even though it shall not be of impor- everywhere. The field configuration states form a basis, tance for this paper. After integrating out the virtual in which one can span any field state. In particular, we fields we will end with a Gaussian state |ψ i, quadratic choose to express our cPEPS in this basis (and not using in φ: Fock terms as in [23]), and define it as Z Z d 1 d k ∗ − φ (k)ω (k)φ(k) d D 2 d d xA[v ,∇v ,φ,∇φ] (2π) α α |ψ i = Dφe |{φ (k)}i (8) |ψi = Dφ (x) Dv (x) e |{φ (x)}i D (3) where where −1 2 2 2 ω (k) = c − a + z k A + Zk a + z k (9) D α α β β αβ A = − Z ∇v ∇v + V [v ,∇v , φ,∇φ] . (4) αβ α β α α Here φ (x) is a real scalar field - the physical field of A. Approximation of the Free Vacuum interest. Additionally, {v (x)} is a set of D fields α=1 which are called henceforth virtual fields and are being A more explicit expression for ω (k) can be put forth integrated over (their role is to allow for a locally con- by using the matrix inversion formula: tracted state of the physical field, in full analogy with adj A + Zk lattice PEPS contractions, as we shall later show). Like −1 αβ A + Zk = (10) αβ 2 φ, the virtual fields are also real scalars. Z is a D × D αβ det (A + Zk ) 3 where adj (M) is the adjugate matrix of M. From that we which is satisfied by the following choice of parameters can tell that ω (k) is a rational function in the argument P : k , of order D over D. The parameters which appear in 2 2 the Gaussian cPEPS (3) will be collectively denoted by A + Z k = (δ + δ ) ik αβ αβ α,β−1 α,β+1 P = {Z , A , z , a , c}. We thus write αβ αβ α α ( m α = 2n n ∈ N 2 2D + 2δ × 2 p (P ) + p (P ) k + ... + p (P ) k αβ 0 1 D ω (k) = (11) α = 2n + 1 , n ∈ N 2 2D 1 + q (P ) k + ... + q (P ) k 1 D a = 0 z = δ c = m α α α1 We have used the freedom to set one of the param- eters of the rational function to 1. p (P ) , q (P ) are α α (16) non-linear maps which are non-injective; note that while We see that when D is infinite, the ground-state can be 2D − 1 parameters uniquely determine ω , the number constructed as a cPEPS, exactly: the continued fraction of parameters in the set P is of order D . This redun- is known to converge to ω (k) as D → ∞. However, dancy can be eliminated by ”gauge-fixing” conditions, what if the continued fraction is truncated after D steps? common for tensor networks which are usually redun- dant constructions [5], as addressed in previous contin- uum works [23–25]. Yet this will not be necessary for our B. Quantum Fidelity and Universality purposes. With (8) being a Gaussian state, it is natural to inspect We would like to inquire how well the Gaussian how well it approximates the ground-state (vacuum) of a cPEPS (5) with a finite D approximates the free vac- free field theory, which is given by the Hamiltonian (see uum |Ω i (which is also a Gaussian cPEPS, but with appendix B for more details): an infinite D). A natural probe that we explore for that task is the quantum fidelity, which is defined for two pure states as: 1 d k ∗ 2 ∗ H = Π (k) Π (k) + ω (k) φ (k) φ (k) , (2π) F (Ω , ψ ; Λ) = |hψ |Ω i| (17) f D D f (12) with the relativistic dispersion relation: To compute this quantity, we have chosen a Wilsonian 2 2 2 regularization where the momentum modes are truncated ω (k) = m + k . (13) at a cutoff k = Λ. The fidelity depends also on the The free vacuum |Ω i is given by: f cutoff Λ. The two states are assumed to be normal- ized, which bounds the fidelity from above and below: 1 ∗ − ω (k)φ (k)φ(k) |Ω i = Dφ (k) e |{φ (k)}i (14) 0 ≤ F (Ω , ψ ; Λ) ≤ 1. f D The fidelity takes maximal value, 1, only when the two states |ψ i and |Ω i are exactly identical (and hence Given the functional form of ω (k), the approxi- D D f mation to some desired dispersion relation such as ω the two functions ω (k) and ω (k)). Since the fidelity D f is bounded, any difference between the two functions is an approximation by a rational function, known as the Pad´e approximant of a function [33]. There are must result in a smaller value for the fidelity. Assum- ing the fidelity is also a smooth functional of ω (k) and infinitely many different approximations which converge to ω (k) as D → ∞. This fact can be seen, for instance, ω (k) (which holds true in this particular case, as to be shown below), it will increase as the two functions ω (k) by noting that the Pad´e approximant, like the Taylor series it is based on, is localized around a particular and ω (k) get closer to each other within the interval 0 ≤ k ≤ Λ. Hence we can choose ω (k) to be the Pad´e base point of momentum k , and the choice of such 0 D approximant of ω (k) around some chosen base point k . base point is of course arbitrary. Nevertheless, there is f 0 both a theoretical and effective difference between these The approximant diverges from the original function for large values of k ≫ k in this case. This can be seen distinct approximations, which will be addressed in the following subsection. as ω (k) asymptotically behaves as k, a behavior which a rational function in k can ever achieve. On the in- It is of importance in rigorously demonstrating how the terval 0 ≤ k ≤ Λ the convergence to ω (k) is uniform. Although since the approximation is local, the rate of dispersion relation of a free relativistic field theory can be obtained from the Gaussian cPEPS (3). In appendix A convergence does not depend on Λ. In conclusion, the fi- delity increases when D is increased while Λ is kept fixed, we concretely construct such a state with mass m using the continued fraction representation of the square root: and decreases when D is kept fixed while Λ is increased. This may be alarming since it seems that as the cutoff 2 2 grows, a larger and larger bond dimension will be re- m + k = m + , (15) quired to sustain a reasonable overlap between the two 2m + states. Furthermore, it implies that the bond dimension 2m + . has no physical significance as it depends on the way . 4 the theory is regularized. However, in case we allow the Given the Gaussianity of the particular case of study, parameters P of the approximation to depend upon the the fidelity can be explicitly calculated (and even later ex- cutoff themselves, it allows for rate of convergence to de- panded around the large cutoff limit). The cPEPS |ψ i pend on Λ as well. In which case, it enables, as the cutoff and the free ground-state |Ω i both factorize into tensor is sent to infinity, to tune the parameters in a way that products over momentum modes of harmonic oscillator makes the fidelity decay much less quickly, or perhaps ground-states: even saturate to a non-zero value. Λ Λ 1/4 Y Y ω (k) 1 ∗ − ω (k)φ (k)φ(k) |Ω i = |Ω (k)i = dφ (k) e |φ (k)i f f k k (18) Λ Λ 1/4 Y Y 1 ∗ ω (k) − ω (k)φ (k)φ(k) |ψ i = |ψ (k)i = dφ (k) e |φ (k)i D D k k The fidelity therefore factorizes as well: 1   1   1 4 4 2 ω (k) ω (k) 1 ω (k) ω (k) 2π f D f D − (ω (k)+ω (k))φ (k)φ(k) f D hψ (k)|Ω (k)i = dφ (k) e = D f 2 2 π π ω (k) + ω (k) f D (19) We finally obtain Λ Λ 2 Y Y 2 |ω (k)| ω (k) D f F (Ω , ψ ; Λ) = |hψ (k)|Ω (k)i| = (20) f D D f |ω (k) + ω (k)| D f k k In order to deal with the product inside (20) in the in many examples (e.g. [34, 35]) and in particular has continuum limit Λ → ∞, we need to be more precise been shown to also be generic for Matrix Product States with the definition of the Wilsonian regularization. We (MPS) [36] (other possibilities may also occur [37],[35]). begin from regularizing the field theory by placing it on a In principle, the fidelity would go to zero as Λ → ∞ or 1 N lattice with N sites and spacing . The volume V = V → ∞ even if the two dispersion relations are infinites- Λ Λ of the d-dimensional spatial space is taken to be finite yet imally close to each other but not exactly equal (this arbitrarily large and likewise the cutoff Λ is kept fixed. phenomena is also known as Anderson’s orthogonality Consequently, upon taking the logarithm the sum can be catastrophe [38]). In which case, we see that a relatable turned into an integral: quantity which bears its own significance exists, this is the fidelity per-site, defined by: 2 |ω (k)|ω (k) V d k D f log F = log (21) 2 |ω (k) + ω (k)| (2π) D f ΩΛ N F = lim F (23) N→∞ Where is an integral in the domain 0 ≤ k ≤ Λ. This may be defined alternatively in terms of the loga- The scaling of the problem becomes more transparent by ¯ rithm: log F (Ω , ψ ; Λ) = lim log F (Ω , ψ ; Λ). using the dimensionless variable: k = : f D f D N→∞   The fidelity per-site F is well defined in the contin- ¯ ¯ d 2 ω k ω k ¯ uum limit Λ → ∞, V → ∞ as it has no dependence D f N d k   log F = log (22) on the volume or the cutoff - unless inserted explicitly ¯ ¯ 2 ω k + ω k Ω (2π) D f through ω through the parameters P . We now explore this option, and for that we shall rescale the dispersion where Ω is the unit sphere. It is then seen that the relations ω (k), ω (k) in the dimensions of the cutoff, D f fidelity decays exponentially with the system’s size (as similarly to the definition of k. Hence we define: what multiplies N must be a negative number since each harmonic ground-state of a single mode k was taken to be ¯ ¯ ω k = Λω˜ k f f normalized). Such behavior between two distinguished (24) ¯ ¯ states of a quantum many-body system is known to hold ω k = Λω˜ k D D 5 We see that we may expand ω˜ (k) around Λ → ∞: In summary, we deduce that while in this particular example the fidelity itself is doomed to vanish in the con- 2 2 4 m 1 m m tinuum limit, there exist a quantity with a universal term ¯ ¯2 ¯ ω˜ k = k + = k + + O ¯2 that can in principle be variationaly optimized. Λ 2k Λ Λ (25) and III. CPEPS FROM PEPS 2 2D ¯ ¯ p˜ (P ) + p˜ (P ) k + ..p˜ (P ) k 0 1 D ω˜ k = (26) 2 2D ¯ ¯ 1 + q˜ (P ) k + ..q˜ (P ) k 1 D A. PEPS With Fields The rescaled parameters can be related back to the Starting from the lattice, we use the PEPS formalism original ones by: to construct a state whose continuum limit is the cPEPS (3) or a generalization thereof. We will first show how to 1−2α p (P ) = Λ p˜ (P ) α α do it for the simple case of a single scalar field, and then (27) −2α generalize to more possibilities, including scenarios with q (P ) = Λ q˜ (P ) α α global symmetries. We note that this is an arbitrary choice, since the extra On a constant time slice where the Hilbert space is power of Λ could have been otherwise extracted from defined, we consider a spatial lattice Z , whose sites are the denominator. Similarly to (25), an expansion around labelled by vectors of the form x ∈ ǫZ ≡ L, where ǫ > 0 Λ → ∞ for ω˜ k can be made as well. The form of is the lattice spacing. We use {eˆ } to denote the unit this expansion will depend of course on the way we have vectors in all the positive directions. Thus, for example, chosen the parameters P to depend on the cutoff Λ. We the sites x and x + ǫeˆ are nearest neighbours in the therefore chose p˜ , q˜ to be separated very generally into α α 1 ≤ i ≤ d direction, separated by a single unit of lattice a cutoff independent term and an irrelevant term: spacing, ǫ. We would like to construct a PEPS for the state of a (0) (irr) p˜ (P ) = p˜ (P ) + p˜ (P ; Λ) α α lattice scalar field φ (x): on each lattice site x, we in- (28) (0) (irr) troduce a physical Hilbert space, H (x), spanned by phys q˜ (P ) = q˜ (P ) + q˜ (P ; Λ) α α eigenstates of the on-site field operator φ (x). This can (irr) (irr) be either a real scalar field (as before) or a complex one p˜ , q˜ are defined to vanish as Λ → ∞. This α α (corresponding, as usual, to a pair of real scalar fields). definition is necessary since the fidelity is bounded for Thus, a good basis for the local physical Hilbert space is normalized states, therefore the parameters P cannot be |φ (x)i in the real case and φ (x) , φ (x) in the complex chosen in such a way that the fidelity diverges. one (where z represents complex conjugation). Below, Since the decay of ω˜ k in Λ is polynomial, it is an we will adapt the notation of the complex case, and the evident choice to let ω˜ k decay polynomially as well, real case will follow from it in a straightforward manner. thus an expansion in powers of for ω˜ k is possible The physical Hilbert space of the whole system is as well, similarly to (25). However, the decay can be in principle of a different class instead, such as an exponen- H = H (x) (32) phys phys tial one. x∈L Derived from this assumption, ω˜ k may be ex- panded in the same way around Λ → ∞: However, as usual in the construction of PEPS, the phys- ical degrees of freedom are not sufficient: to contract the local ingredients together, we have to introduce auxiliary (0) (irr) ¯ ¯ ω˜ k = ω˜ k + ω˜ k; Λ (29) D D or virtual fields to our system. That is, on each lattice site x ∈ L, we introduce 2d extra Hilbert spaces, asso- where ciated with the legs going out of and into it. On each (0) (0) (0) 2 2D ¯ ¯ outgoing leg, in direction i, we define the virtual fields p˜ (P ) + p˜ (P ) k + ..p˜ (P ) k (0) 0 1 D ω˜ k = (30) (0) (0) {χ (x)} , while on each ingoing leg we introduce sim- ¯2 ¯2D i α=1 1 + q˜ (P ) k + ..q˜ (P ) k ilar fields {η (x)} . All virtual fields are either real α=1 To conclude, putting together (25) and (26) yields: or complex scalars, depending on the nature of the phys- r ical field φ (x). As in the physical case, we will generally   (0) treat them as complex below. The number of fields we ¯ ¯ Z 2 ω˜ k k d D   1 d k (irr) introduce on each leg, D, is nothing but the bond dimen-   log F = log + F (Λ) d   (0) 2 sion mentioned above; we are free to choose any D ≥ 1 ¯ ¯ Ω (2π) ω˜ k + k that we want to build our PEPS - increasing D will allow (31) our PEPS to depend on more parameters - which implies The first term has no dependence in Λ at all, and more freedom for the optimization problem whose solu- (irr) F (Λ) is a function that vanishes as Λ → ∞. tion is sought with the PEPS. Different physical scenarios 6 will impose different constraints on the required minimal In a sense, χ (x) = (χ (x) , ..., χ (x)) and η (x) = 1 d bond dimension, and we have seen an example for that in (η (x) , ..., η (x)) can be thought of as spatial vector 1 d our previous construction. However, whatever D is, we fields. unite the Hilbert spaces of all the virtual fields of x ∈ L On each site x ∈ L we can therefore define the local to H (x). We denote its configuration states by virt Hilbert space H (x) = H (x) ⊗ H (x) . (34) physical virt |χ (x) , η (x)i := |χ (x)i|η (x)i (33) i i i=1 involving both the physical and virtual degrees of free- (where the α indices were omitted for simplicity). dom. Any state |A (x)i ∈ H (x) may be expanded as |A (x)i = (dχ (x) dη (x)) dφ (x) A (φ (x) , χ (x) , η (x))|φ (x)i ⊗ |χ (x) , η (x)i (35) i i (where integration on complex conjugates and/or different α components of the virtual fields is implicitely assumed). |A (x)i is not the state we need: it is both a product state, with no correlations among the lattice sites, and it x∈L also involves some virtual degrees of freedom which have nothing to do with the physical Hilbert space H . Both phys issues are addressed, as usual, by projecting the virtual degrees of freedom onto maximally entangled pair states, connecting the virtual fields of both sides of each link; in our case, we choose |L (x)i = dχ (x) dη (x + ǫeˆ ) δ (χ (x) − η (x + ǫeˆ )) δ (χ¯ (x) − η¯ (x + ǫeˆ ))|χ (x)i|η (x + ǫeˆ )i (36) i i i i i i i i i i i i i defined on each link. The extension to D > 1 is straight- forward, and in the real field case the second delta func- tion(s) are removed. The PEPS is obtained by projecting |A (x)i onto x∈L the maximally entangled states on all the links: O O |ψi = hL (x)| |A (x)i ∈ H (37) i phys x∈L,i=1,...,d x∈L A two-dimensional example for the construction detailed above is depicted in Fig. (1). We have not only eliminated the virtual degrees of freedom, we actually used them for inducing correlations among the lattice sites. Furthermore, this state will sat- isfy the area law, since for any bipartition we wish to make, we will get that the entanglement entropy comes from the maximally entangled states on the links which cross the boundary, all contributing the same. FIG. 1. An example of a two-dimensional PEPS with fields. B. Spatial Symmetries The lattice sites are marked with the black dots and the vir- tual fields with the ingoing and outgoing blue lines to each site. Two outgoing and ingoing virtual fields of two neigh- We would like to require the PEPS |ψi to be invariant bouring sites are contracted by the links L , which is illus- under translations and under the transformations which trated by the oval connecting the two lines. preserve the lattice (its point group). This is a reasonable condition if we wish the state to become invariant under continuous translations and rotations SO (d) in the con- tinuum limit (the rest of the Poincar´e group is explicitly discrete lattice translations and also under rotations pre- broken by the choice of the quantization scheme). To serving the lattice. As for the latter, where it is for the that end we require the PEPS |ψi to be invariant under simple choice of a square lattice we will suffice to require 7 invariance under rotations of in all possible planes (we product of all link states remains the same). This is do not make any prior demands regarding reflections and satisfied by the states we picked for our demonstration parity). above. The discrete translations T (a) for some displacement In order to obtain the desired form of the cPEPS (3) vector a on the lattice are defined to act in a very simple in the continuum limit, we write A (φ, χ, η) as an expo- way on both the physical and virtual fields, nential: A(φ,χ,η) T (a)|φ (x)i = |φ (x + a)i (38) A (φ, χ, η) = e (43) It acts exactly the same on the virtual fields χ , η . If not i i Here, A respects the spatial symmetries (38) and (42). broken to begin with, the discrete translational symme- We will now focus on a more specific ansatz, try on the lattice will be straightforwardly enhanced to a continuous translational symmetry in the continuum A = R (χ, η) + K (χ, η) + V (χ, η, φ) (44) limit. Imposing translation symmetry on the PEPS is where V is a general analytic function of the fields that is very simple, and can be done by requiring all the func- required to be invariant under (42). R and K are defined tions A (φ (x) , χ (x) , η (x)) from (35) to be the same all by around the lattice (independent of the coordinates). Sim- ilarly, we use the same |L (x)i states on all the links in the same direction. R (χ, η) = − N |χ (x) − χ (x)| 0 i j Next we consider lattice rotations. Denote by Λ ro- ij π i<j tations of in the plane spanned by eˆ and eˆ (with no i j 2 (45) 2 2 + |χ (x) − η (x)| + |η (x) − η (x)| loss of generality, we assume that i < j). The coordinate j i i j rotation is given by + |η (x) − χ (x)| j i Λ x = Λ (x , .., x , .., x , ..x ) = (x , ..,−x , .., x , ..x ) ij ij 1 i j d 1 j i d (39) and The physical field is scalar and thus its eigenstates transform as: 0 2 K (χ, η) = − |χ (x) − η (x)| (46) i i |φ (x)i → |φ (Λ x)i (40) ij The function K will eventually play the role of the Nevertheless, the virtual fields must transform in a dif- kinetric term, whereas R will be shown to be necessary ferent manner: for continuous rotational invariance of SO (d) to emerge |χ (x)i → |χ (Λ x)i in the continuum limit. i j ij |χ (x)i → |−η (Λ x)i j i ij (41) |η (x)i → |η (Λ x)i i j ij C. The Continuum Limit |η (x)i → |−χ (Λ x)i j i ij The projection onto the maximally entangled link To explain why, we recall that these fields are associated states |L (x)i allows us to use the delta functions, elimi- with directions - they are components of spatial vector nate the η Hilbert spaces and exchange any appearance fields. Therefore, they must follow the rotation of links, of η (x) by χ (x − ǫeˆ ). Collecting all of the above, the i i i giving rise to the above rotation rules. physical state may then be written as (here D = 1 but Requiring the fiducial state |A (x)i to be invariant un- the generalization to higher bond dimensions is straight der (41) is equivalent to requiring the wave-function in forward): (35), A (φ (x) , χ (x) , η (x)) to be symmetric under the Z P permutations of the virtual fields dictated by (41) for A(φ(x),χ(x)) |ψi = Πdφ (x) Πdχ (x) e Π|φ (x)i|χ (x)i all i < j: i x x x,i (47) A (φ, .., χ , .., χ , ..η , .., η , ..) i j i j In the above expression appears the same function A as = A (φ, .., χ , ..,−η , ..η , ..,−χ , ..) j i j i it was presented previously, only now with the insertions (42) = A (φ, ..,−η , ..,−η , .. − χ , ..,−χ , ..) of η (x) replaced with χ (x − ǫeˆ ) as explained. i j i j i i i From this point the continuum limit is taken in the = A (φ, ..,−η , .., χ , .. − χ , .., η , ..) j i j i conventional way. That is, the dimensionless fields and One also has to make sure that the maximally entangled parameters are exchanged by their continuum counter- states on the links, |L (x)i are rotated in an invariant parts which obtain their mass dimension by an appropri- way (in this case, we need to demand that in an i − ate power of ǫ. j rotation, the states belonging to these two directions Namely, the fields φ, χ which appear in (47) are de- are exchange, and the others are left intact; overall, the fined on the lattice and are dimensionless. For taking 8 the continuum limit, we need to exchange summation by integration in the exponent of (47). Since the integration X X measure is dimensionful, this requires the introduction (R) (R) d −d+2[χ] R (χ) → − d xN ǫ χ (x) − χ (x) i j of dimensionful fields to compensate for the originally i<j dimensionless expression. [φ], the dimension of the phys- (49) ical field φ is chosen to have the standard dimension for −d+2[χ] Defining N = N ǫ , we may send ǫ → 0 and d−1 0 the free relativistic field [39]: [φ] = . This is a nec- tune N accordingly to obtain a finite N (the sub-leading essary choice as this degree of freedom also serves in a term will then scale as Nǫ and will hence vanish in the physical state such as the free vacuum (14). continuum limit). We define the ”renormalized” physical field Next, we do the same with the function K (χ). Due (R) −[φ] φ (x) = ǫ φ (x). Since the virtual fields are to its derivative nature, its leading order term is from a eventually integrated out there exists a freedom in higher order in ǫ: choosing their dimension (see appendix A for more details). We hence do not commit to a specific choice and denote their dimension by [χ]. We define X X 2 (R) d 2−d+2[χ] (R) K (χ) → − d xZ ǫ ∇ χ (x) −[χ] 0 i χ (x) = ǫ χ (x) accordingly. i 2 x i (50) 2−d+2[χ] The rest of the procedure is done by expanding (recall We then define Z = ǫ Z , and tune Z as we 0 0 that we have replaced η (x) with χ (x − ǫeˆ )): i i i send ǫ → 0 to obtain a finite Z. (R) (R) (R) While the continuum limit of K (χ) is symmetric under χ (x − ǫeˆ ) = χ (x) − ǫ∇ χ (x) + O ǫ i i i i i (48) discrete rotations of , it is not under the larger group (R) (R) (R) 2 φ (x − ǫeˆ ) = φ (x) − ǫ∇ φ (x) + O ǫ i i of continuous rotations SO (d). (This issue has already been pointed out in [22]). and keeping the leading order term in ǫ inside A. Sub- Therefore, we introduce the additional function R (χ): leading terms will vanish in the continuum limit ǫ → 0 it enhances (47) to have the full SO (d) symmetry in since the parameters will be tuned with respect to the its continuum limit. We note that this function has leading term, as we will show in more detail. Starting a lot of resemblance with the ansatz first proposed in [23]. with the function R (χ), we observe that by its cyclic nature there are no cancellations and the leading term is To clearly see how, we first make the separation (R) (R) ′ (R) (R) (R) therefore a non-derivative. All of the four terms in R (χ) A χ , φ = A χ , φ +R χ . At this point collapse into a single one proportional to: we write (47) as n oE R R d (R) d ′ (R) (R) (R) (R) d xR χ (x) d xA χ (x),φ (x) (R) [ ] [ ] |ψi = Dφ (x)Dχ (x) e e φ (x) (51) The function R helps as we formally send N → ∞. In the limit, the integral (51) becomes exclusively dominated by (R) (R) field configurations of the (complex) fields χ which satisfies the constraints χ (x) = χ (x) for all i 6= j. The set i j (R) of the virtual fields {χ } collapses into a single virtual field which we call v (x). Hence, now we can write (51) i=1 as: n oE d ′ (R) (R) (R) (R) d xA v (x),φ (x) (R) [ ] |ψi = Dφ (x)Dv (x) e φ (x) (52) where A = K + V . lowing all the steps described above, the functional (R) (R) (R) K becomes the standard kinetic term for the field v: V v ,∇v , φ ((3) also depends on ∇φ, we will later show how this is achieved in the end of this sec- tion). K (v) = − ∇ v¯∇ v (53) i i The mapping between the two can be defined in a for- After the collapse to a single virtual field, there is no mal way through the series expansion of V in powers of association to a spatial direction and this term is now the fields in their first derivatives and then identifying invariant under the full rotational group SO (d). (R) (R) v (x) with χ (x) and ∇ v (x) with χ (x) − η (x). Finally, as was done before in detail for R and i i i i K, the function V (χ , η , φ) similarly becomes, by fol- For example, V = a (∇v) v can be construtced from 0 i i 9 V = a |χ (x) − η (x)| χ + (permutations). The per- have to lay on the same lattice - the virtual fields after all 0 0 i i i mutations under (47) ensure the rotational symmetry. connect between the physical Hilbert spaces which may Similarly to before, the connection between the parame- contain more than one lattice site. 1 2−d+3[χ] 1 ters is given by a = ǫ a ; the factor of comes It is interesting to note that this simple operation can- 4 4 to account for the 4 terms conncted by the rotational not be applied to the virtual fields since they are re- transformation which collapse into a single one in the stricted to lay on the boundary of the block rather then continuum limit. inside its bulk. This fact seems to hinder the ability to Now that we are fully done, we can conveniently drop construct higher derivatives for the virtual fields. the R superscript as well as the prime from A . Addi- tional virtual fields can easily be added by following the IV. SYMMETRIES same steps above, so we may consider a general bond di- mension D > 1. We then arrive to the continuum limit of (47) which is the desired cPEPS: PEPS are well-known for the ease of describing sym- metries with them: one can parametrize families of PEPS d xA[v ,∇v ,φ] α α which will be invariant under some symmetry group |ψi = Dφ (x)Dv (x) e |{φ (x)}i [5, 7, 8]. (54) Consider, at first, the case of a complex scalar field. A = − Z ∇ v ∇ v + V [v ,∇v , φ] αβ i α i β α α We can define a global U(1) transformation on it, such that iθ U (θ)|{φ (x)}i = e φ (x) (56) For any θ ∈ [0, 2π). If we wish our PEPS to be invariant under the symmetry operation, we will need to define some transformation rules for the virtual fields too, e.g. iθ v (x) → e v (x) (57) α α Changing the integration variables (the integration mea- sure Dφ (x) is invariant under a unitary change of vari- ables), we will get that U (θ)|ψi = |ψi if iθ iθ iθ V [v ,∇v , φ] = V e v , e ∇v , e φ (58) α α α α If, for example, we construct a Gaussian cPEPS, this will be satisfied if we pick 1 1 A = − Z ∇v (x)∇v (x) − A v (x) v (x) αβ α β αβ α β 4 4 + z ∇v (x)∇φ (x) + a v (x) φ (x) − |φ (x)| + c.c α α α α (59) This can also be achieved through the PEPS construc- FIG. 2. An example for a two-dimensional square lattice with tion, by properly parametrizing the states |A (x)i and a H of larger support. The block is of size 2×2 and marked phys |L (x)i (see, e.g. [10]); but can also be done directly in in a dashed blue line. The black dotes stand for the lattice the continuum. sites which hosts the physical fields whereas the solid blue We can think of more general settings. Suppose that lines for the four virtual fields of the ingoing and outgoing direction to the block. instead of the field φ (x) we introduce a vector, or a mul- tiplet of fields, either real or complex, {φ (x)} , trans- a=1 In fact, (3) has dependence on ∇φ too. We close this forming as some representation j of dimension r of some section be showing how this and generelization thereof is group G; that is, for each group element g ∈ G, we define achieved. The local physical Hilbert space H may be the unitary transformation phys defined on a block of lattice sites rather on a single point, n oE U (g)|{φ (x)}i = D (g) φ (x) (60) as illustrated in Fig. 2. a b ab H (x) = H (55) where D (g) is the j representation matrix of g (of size phys φ(x+a ˆ) ab a ˆ∈block r × r). If we wish to construct a cPEPS |ψi with the symmetry The larger the block, the higher derivative terms for property the physical field φ that can be written. We now make the observation that the virtual and physical fields do not U (g)|ψi = |ψi ∀g ∈ G (61) 10 we will need to use virtual fields that are also charged fields can be constructed by taking tensor products and under this group, that is, the v fields will have to carry symmetrizing/anti-symmetrizing. an a index, forming multiplets of the group G. One can Furthermore, an important benefit of this bottom-up use several multiplets and copies thereof, as long as the construction is that tools and ideas that apply for PEPSs multiplets are fully included. The virtual and physical are transferred to cPEPSs - section IV is an important ex- representations do not even have to be the same, as long ample for that. The PEPS framework has been elevated as they are coupled properly. Thus, in general, the virtual to include physical gauge fields as well [10, 40], with a fields transform as very clear prescription on how to do so. We expect that following through this procedure to the continuum will v (x) → D (g) v (x) (62) α,a α,b ab result in ”minimal-coupling” of (54) (in the same sense that an action is minimally coupled). Nevertheless we with some (irreducible or reducible) representation J. leave this to a more thorough investigation in the future. Our cPEPS will be defined with The fidelity was used for estimating the accuracy of the 1 cPEPS approximation to the real, physical ground-state. A = − Z ∇ v ∇ v + V [v ,∇v , φ ] (63) αβ i α,a i β,a α,a α,a a In most cases however, the exact form of the ground-state is unknown, and hence it is the tensor-network state that and the symmetry condition will be serves as an ansatz that is designed to capture most of the ground-states important properties. If so, one im- h i mediate implementation of the cPEPS is to serve as a J J V [v ,∇v , φ ] = V D (g) v , D (g)∇v , D (g)φ α,a α,a a α,b α,b b ab ab variational ansatz which is numerically adjusted to give ab a minimal expectation value for some Hamiltonian of in- (64) terest. To deal with the usual divergences that appear which can also be simplified, in a straightforward man- in the ground-state energy of quantum field theories, it ner, in the Gaussian case (it can also be seen as a con- was demonstrated to be efficient to minimize the normal- tinuum limit of the case studied in [40], when changing ordered/renormalized Hamiltonian instead of the bare to scalar fields). one [25],[41]. Here again the question about the univer- Finally, if we keep A real, we will also have a charge- sality of this method is raised. For instance, if a differ- conjugation symmetry in the case of complex fields. ent renormalization scheme would be used, will a smaller bond dimension be able to yield a similar result? In our V. DISCUSSION non-interacting example, we have found that indeed there is some universal content to the problem, yet the larger picture remains open for future study. We have developed a continuous Projected Entangled Along with this, there is another context to which the Pair State (cPEPS) for quantum fields and explicitly cPEPS can be closely related. A tensor-network state can shown how such a state can approach the free field theory be thought of as a representative of a class of states with vacuum as the bond dimension is increased. In addition a specific structure of symmetry and entanglement. Since we have tackled the question on whether this approxi- the critical properties of a phase transitions is normally mation is meaningful in the real continuum sense, or in dependent upon the robust features of the two phases, other words if it has a universal significance. To that end one can build tensor-network states which lay in the uni- we have used the quantum fidelity as a measure of dis- versality class of each phase and in that way ”engineer” a tinguishability between the cPEPS and the real physical quantum phase transition [42]. There is then great merit state that it approximates, and found that it encapsulates in calculating the fidelity of two nonidentical cPEPSs be- a universal term that is independent on the short-scale longing to different symmetry classes - which can be eas- behavior of the problem. ily created with the tools portrayed in this paper. Subsequently we have built the cPEPS in a bottom-up approach. Starting from the lattice, a PEPS paired with Fidelity of many-body quantum states as a probe for a unique ansatz was used to produce the wanted result quantum phase transitions has been a popular source of in the continuum. A central feature of the cPEPS is that research (see [35] and references therein). The quantum it enjoys a global symmetry under any desired symmetry fidelity has proven to be a useful analytical tool for gain- group of interest, as well as translational and continuous ing insight about the characteristics of the phase tran- rotational symmetries. sition and it has a unique critical behavior around the While we have focused on scalars, fields with higher transition point on its own. Exact form of the fidelity spin may also be taken in consideration. A similar con- is in general not easily calculable, and known examples struction as the one done in section III is expected to exist in large for integrable models in where the ground- be possible for physical and virtual fields with higher state has a known form. We therefore hope that the spin, albeit left beyond the scope of this paper. In cPEPS together with standard perturbative techniques fact, only two generalizations are required for spin 1 can open a way to a new class of phase transitions that vector fields and spin fermions. Once they are laid can be studied via the fidelity approach and we leave that down, higher integer or respectively half-integer spin to future work. 11 ACKNOWLEDGEMENTS virtual fields (as we show below) results in: 2 2 The authors would like to thank Michael Smolkin and B k ω (k) = A + D 0 2 2 Patrick Emonts for helpful and inspiring discussions. B k Z m + This research was supported by the Israel Science Foun- 1 2 2 B k dation (grant No. 523/20). A m + 2 2 B k D−1 Z m + ... 2 2 B k Z m + D−1 A m Appendix A: A Conctinued Fraction Approximation (68) In this form it is a rather simple task to make the cPEPS (66) approach the vacuum state of a free field theory |Ω i: Hereby we shall introduce a specific and simpler choice of the more general Gaussian cPEPS (3) which enables the presentation of ω (k) in the form of a continued fraction. It allows for a rather elegant representation of 1 ∗ − ω (k)φ (k)φ(k) |Ω i = Dφ (k) e |{φ (k)}i (69) the ground-state of a free field theory. Furthermore, con- tinued fractions in general have many appealing prop- erties, allowing one to gain more knowledge about the convergence. Given that, we narrow down to the follow- 2 2 ω (k) = k + m has a continued fraction represen- ing choice of (3): tation: ω (k) = m + (70) iB k α k A + Z k = (δ + δ ) αβ αβ α,β−1 α,β+1 2m + 2m + mA α = 2n n ∈ N . + δ × 2 αβ Z k α = 2n + 1 , n ∈ N (65) It is then clear that as we set A = m, B = B = 1 0 0 α and Z = A = 2, ω (k) approaches ω (k) as D grows. α α D f The convergence however is not uniform as the remain-   der diverges as k increases. This can be seen by the fact 2 2 Z k /m iB k /m 1 1 that ω (k) asymptotically behaves for large values of k 2 2 iB k /m mA iB k /m  1 2 2   2 2 2 as k or as a constant depending if the truncation of the iB k /m Z k /m iB k /m  2 3 3    continued fraction is even or odd, whereas ω (k) asymp-   2 2 iB k /m iB k /m 3 D−1 totically behaves as k. iB k /m mA D−1 D In fact, the continued fraction representation is not (66) unique. This is not surprising given the fact that so is iB And a = 0, z = δ , c = A . The bond dimen- α α α1 0 m the more general Pad´e approximant. sion D is assumed to be even. Eq. (68) is obtained inductively. To that end we inte- We first need to take care of the fact that there is grate out the virtual fields one by one, and it is useful to more than one way of fixing the mass dimension [χ] of present the following notation: the virtual fields χ . We find it useful to choose the di- mension to be the same as that of the physical field φ: D R d−1 d Y d k [χ] = [φ] = . Recall that the dimension of the physi- A [v (k),φ(k)] D α 2 d (2π) |ψi = Dφ (k) Dv (k) e |{φ (k)}i cal field [φ] is completely set by the Klein-Gordon action. α=1 Under this convention the dimensions of the parameters D−1 d k defined in (3) are as follows: A [v (k),φ(k)] D−1 α (2π) = Dφ (k) Dv (k) e |{φ (k)}i α=1 [c] = 1 , [A ] = [a ] = 1 , [Z ] = [z ] = −1. (67) αβ α αβ α = ... (71) We note that ω (k) = A (k) + A . D 0 0 Notice that we have set the physical mass m which ap- pears in ω as a single energy scale and used it to fix For obtaining (68) we start by integrating χ (k). f α=D the dimensions of all of the rest of the parameters. Thus To clarify that, we first write (71) in a form where B , Z and A are dimensionless. Integrating out the χ (k) is separated from the rest of the virtual fields: α α α α=D 12 Appendix B: A Note on Parent Hamiltonians m iB k 2 D−1 A = − A |v (k)| + v (k) v (k) D D D D−1 2 m A natural question put in the context of tensor- D−2 networks is about the properties and uniqueness (or Z k 1 D−1 2 2 ∗ − |v (k)| − Z + Ak v (k) v (k) D−1 β α non-uniqueness) of the parent Hamiltonian H , whose αβ D 2m 2 α,β ground states is the tensor network state - in our case, (3). Since (3) is a Gaussian state there must exist a quadratic iB k A 0 0 ∗ 2 + v (k) φ (k) − |φ (k)| parent Hamiltonian, and in fact there is a family of these m 2 (72) with the following form: Integrating out χ then leads to:   1 d k D−1 4 H (φ, π) = a (k) Π (k) Π (k) 1 Z k k D D−1 2 d   2 A = − + |v (k)| (2π) D−1 D−1 (76) 2 m mA + b (k) φ (k) φ (k) iB k m D−2 2 + v (k) v (k) − A |v (k)| D−2 D−2 D−2 D−1 m 2 D−3 2 ∗ b(k) − Z + Ak v (k) v (k) where ω (k) = . In this aspect, ω (k) may be in- β D α D αβ a(k) α,β terpreted as a dispersion relation, obviously dependent on the bond dimension D and the parameters which ap- iB k A 0 0 + v (k) φ (k) − |φ (k)| pear in the Gaussian cPEPS (8). All of the Hamiltoni- m 2 ans (76) have the same spectrum as they are canonically (73) equivalent to each other. We note that the canonical Next integrating out χ leads to: D−1 transformation which connects the two is in general k dependent.   k B D−2   A = − mA + |v (k)| This in fact raises an issue if one reverses this argu- D−2 D−2 2 D−2 4 2 k B m k Z 2 D−1 D−1 + ment, since in principle the fields φ, π which appear in m mA (8) do not have to be the same as the ones in H (12). D−3 X 2 1 iB k 2 ∗ ∗ Then for any ω (k) one can choose canonically trans- − Z + Ak v (k) v (k) + v (k) φ (k) α 1 αβ 2 m formed fields and momenta in such a way that (8) is the α,β exact ground-state of H . In that sense it may seem that there is no sense at all in seeking a variational technique. − |φ (k)| (74) We can relief this obstruction by at least two argu- In case that k 6= 0, the fraction can be reduced by 2 ments. First, the free Hamiltonian and its spectrum can and give the last iteration of the continued fraction: be decomposed to a tensor product of individual modes of momenta k. This property no longer holds once inter- actions are added, reducing the possibility to transform B k D−2 each mode separately. Furthermore, canonical transfor- mA + (75) D−2 2 B k D−1 mZ − mations are more subtle in quantum field theories given D−1 mA their divergent nature (a (k) , b (k) are in principle regu- Since D is even we can now repeat this process by larization dependent). A simple example is spontaneous induction, treating the physical field φ as χ with symmetry breaking; two ground-states on the vacuum α=0 A = B and B = B . This then results in (68). manifold are connected by a unitary transformation but α=0 0 α=0 0 In the case where k = 0 the physical field is completely are nevertheless physically distinct in the infinite volume decoupled from from the virtual fields and we immedi- limit. Then to conclude, to avoid these issues we choose ately have that ω (0) = m. Which also equals of course the fields φ, π which appear in (8) and in H (12) or in D f ω (k = 0). Therefore the equality holds also for k = 0. (76) to be the same. [1] J. I. Cirac and F. Verstraete, Journal of Physics A: Math- [3] U. Schollw¨ock, Annals of Physics 326, 96 (2011). ematical and Theoretical 42, 504004 (2009). [4] R. Oru´s, Annals of Physics 349, 117 (2014). [2] F. Verstraete, V. Murg, and J. Cirac, [5] I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Advances in Physics 57, 143 (2008). “Matrix product states and projected entangled pair 13 states: Concepts, symmetries, and theorems,” (2021), Phys. Rev. Research 3, 023059 (2021). arXiv:2011.12127 [quant-ph]. [25] A. Tilloy, “Relativistic continuous matrix product [6] J. Eisert, M. Cramer, and M. B. Plenio, states for quantum fields without cutoff,” (2021), Rev. Mod. Phys. 82, 277 (2010). arXiv:2102.07741 [quant-ph]. [7] M. Sanz, M. M. Wolf, D. P´erez-Garc´ıa, and J. I. Cirac, [26] A. E. B. Nielsen, B. Herwerth, J. I. Cirac, and G. Sierra, Phys. Rev. A 79, 042308 (2009). Phys. Rev. B 103, 155130 (2021). [8] A. Molnar, J. Garre-Rubio, D. P´erez-Garc´ıa, N. Schuch, [27] G. Vidal, Phys. Rev. Lett. 101, 110501 (2008). and J. Cirac, New Journal of Physics 20, 113017 (2018). [28] J. Haegeman, T. J. Osborne, H. Verschelde, and F. Ver- [9] J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, straete, Phys. Rev. Lett. 110, 100402 (2013). and F. Verstraete, Phys. Rev. X 5, 011024 (2015). [29] M. Nozaki, S. Ryu, and T. Takayanagi, [10] E. Zohar and M. Burrello, JHEP 10, 193 (2012), arXiv:1208.3469 [hep-th]. New Journal of Physics 18, 043008 (2016). [30] M. Srednicki, Quantum Field Theory (Cambridge Uni- [11] I. Kull, A. Molnar, E. Zohar, and J. Cirac, versity Press, 2007). Ann. Phys. 386, 199 (2017). [31] S. Weinberg, The Quantum theory of fields. Vol. 1 (Cam- [12] N. Schuch, J. Cirac, and D. P´erez-Garc´ıa, bridge University Press, 2005). Annals of Physics 325, 2153 (2010). [32] T. Nishioka, Rev. Mod. Phys. 90, 035007 (2018). [13] S. White, Physical Review Letters 69, 2863 (1992). [33] G. A. Baker and P. Graves-Morris, Pad´e Approximants, [14] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003). 2nd ed., Encyclopedia of Mathematics and its Applica- [15] F. Verstraete, J. J. Garc´ıa-Ripoll, and J. I. Cirac, tions (Cambridge University Press, 1996). Phys. Rev. Lett. 93, 207204 (2004). [34] M. M. Rams and B. Damski, [16] M. Zwolak and G. Vidal, Phys. Rev. Lett. 106, 055701 (2011). Phys. Rev. Lett. 93, 207205 (2004). [35] S.-J. GU, International Journal of Modern Physics B 24, 4371 (2010), [17] J. Jordan, R. Oru´s, G. Vidal, F. Verstraete, and J. I. https://doi.org/10.1142/S0217979210056335. Cirac, Phys. Rev. Lett. 101, 250602 (2008). [36] M. Cozzini, R. Ionicioiu, and P. Zanardi, [18] M. C. Ban˜uls and K. Cichy, Phys. Rev. B 76, 104420 (2007). Reports on Progress in Physics 83, 024401 (2020). [37] L. Campos Venuti, H. Saleur, and P. Zanardi, [19] Y. Meurice, R. Sakai, and J. Unmuth-Yockey, Phys. Rev. B 79, 092405 (2009). arXiv:2010.06539 [hep-lat] (2020). [38] P. W. Anderson, Phys. Rev. Lett. 18, 1049 (1967). [20] F. Verstraete and J. I. Cirac, [39] This is set for instance from the (dimensionless) Klein- d+1 1 μ 1 2 2 Phys. Rev. Lett. 104, 190405 (2010). Gordon action: S = d x − ∂ φ∂ − m φ (the 2 2 [21] J. Haegeman, J. Cirac, T. Osborne, and F. Verstraete, the kinetic term’s coefficient is normalized to one by con- Phys. Rev. B 88, 085118 (2013). vention). [22] D. Jennings, C. Brockt, J. Haegeman, [40] E. Zohar and J. Cirac, T. J. Osborne, and F. Verstraete, Physical Review D 97 (2018), 10.1103/PhysRevD.97.034510. New Journal of Physics 17, 063039 (2015). [41] S. Rychkov and L. G. Vitale, [23] A. Tilloy and J. I. Cirac, Phys. Rev. X 9, 021040 (2019). Phys. Rev. D 91, 085011 (2015). [24] T. D. Karanikolaou, P. Emonts, and A. Tilloy, [42] M. M. Wolf, G. Ortiz, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 97, 110403 (2006).

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Published: Oct 4, 2021

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