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Physical Review X
, Volume 7 (3) – Jul 1, 2017

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- American Physical Society (APS)
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- 2160-3308
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- 10.1103/PhysRevX.7.031006
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PHYSICAL REVIEW X 7, 031006 (2017) Black Hole on a Chip: Proposal for a Physical Realization of the Sachdev-Ye-Kitaev model in a Solid-State System 1 2 D. I. Pikulin and M. Franz Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 and Quantum Matter Institute, University of British Columbia, Vancouver British Columbia, Canada V6T 1Z4 (Received 14 February 2017; revised manuscript received 24 May 2017; published 13 July 2017) A system of Majorana zero modes with random infinite-range interactions—the Sachdev-Ye-Kitaev (SYK) model—is thought to exhibit an intriguing relation to the horizons of extremal black holes in two-dimensional anti–de Sitter space. This connection provides a rare example of holographic duality between a solvable quantum-mechanical model and dilaton gravity. Here, we propose a physical realization of the SYK model in a solid-state system. The proposed setup employs the Fu-Kane superconductor realized at the interface between a three-dimensional topological insulator and an ordinary superconductor. The requisite N Majorana zero modes are bound to a nanoscale hole fabricated in the superconductor that is threaded by N quanta of magnetic flux. We show that when the system is tuned to the surface neutrality point (i.e., chemical potential coincident with the Dirac point of the topological insulator surface state) and the hole has sufficiently irregular shape, the Majorana zero modes are described by the SYK Hamiltonian. We perform extensive numerical simulations to demonstrate that the system indeed exhibits physical properties expected of the SYK model, including thermodynamic quantities and two-point as well as four-point correlators, and discuss ways in which these can be observed experimentally. DOI: 10.1103/PhysRevX.7.031006 Subject Areas: Condensed Matter Physics, Interdisciplinary Physics, Superconductivity, Topological Insulators I. INTRODUCTION supersymmetry [14], interesting quantum phase transitions [15,16], and higher-dimensional extensions [17,18], as well Models of particles with infinite-range interactions have as a version that does not require randomness [19].Given a long history in nuclear physics dating back to the its fascinating properties it would be of obvious interest to pioneering works of Wigner [1] and Dyson [2] and in have an experimental realization of the SYK model or its condensed matter physics in studies describing spin glass variants. Thus far a realization of the Sachdev-Ye model and spin liquid states of matter [3–5]. More recently, Kitaev (with complex fermions) has been proposed using ultracold [6] and Maldacena and Stanford [7] formulated and studied gases [20], and a protocol for digital quantum simulation of a Majorana fermion version of the model with all-to-all both the complex and Majorana fermion versions of the random interactions first proposed by Sachdev and Ye [4]. model has been discussed [21]. A natural realization of the The resulting Sachdev-Ye-Kitaev (SYK) model, defined by SYK model in a solid-state system is thus far lacking. the Hamiltonian Eq. (1.1), is solvable in the limit of large Recent years have witnessed numerous proposals for number N of fermions and exhibits a host of intriguing experimental realizations of unpaired Majorana zero modes properties. The SYK model is believed to be holographic in solid-state systems [22–26], with compelling experi- dual of extremal black hole horizons in two-dimensional mental evidence for their existence gradually mounting in anti–de Sitter (AdS ) space and has been argued to possess several distinct platforms [27–35]. The purpose of this remarkable connections to information theory, many-body paper is to propose a physical realization of the SYK model thermalization, and quantum chaos [8–13]. Various exten- in one of these platforms. The SYK Hamiltonian we sions of the SYK model have been put forth containing implement is given by H ¼ J χ χ χ χ ; ð1:1Þ SYK ijkl i j k l i<j<k<l Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. where J are random independent coupling constants and ijkl Further distribution of this work must maintain attribution to χ represent the Majorana zero-mode operators that obey the author(s) and the published article’s title, journal citation, and DOI. the canonical anticommutation relations 2160-3308=17=7(3)=031006(17) 031006-1 Published by the American Physical Society D. I. PIKULIN and M. FRANZ PHYS. REV. X 7, 031006 (2017) flux quanta can be trapped in the hole. The SC phase θ will fχ ; χ g¼ δ ; χ ¼ χ : ð1:2Þ i j ij j then wind by 2πN around the hole, forming effectively an N-fold vortex with N Majorana zero modes bound to the The proposed device, depicted in Fig. 1, employs an hole. If, furthermore, the hole is designed to have an interface between a 3D topological insulator (TI) and an irregular shape, the Majorana wave functions will have ordinary superconductor such as Nb or Pb. Fu and Kane random spatial structure and their overlaps will give rise to [36] showed theoretically that magnetic vortices in such an the required randomness in the coupling constants J . interface host unpaired Majorana zero modes, and signa- ijkl tures consistent with this prediction have been reported in This randomness is related to random classical trajectories Bi Te =NbSe heterostructures [34,35]. Under ordinary inside such a hole, or “billiards,” as it is commonly called in 2 3 2 circumstances these vortices tend to form an Abrikosov the quantum chaos literature [39,40]. We note a related lattice and the low-energy effective theory is dominated by proposal to realize the SYK model using semiconductor two-fermion terms iK χ χ , with the hopping amplitudes quantum wires coupled to a disordered quantum dot ij i j advanced in the recent work [41]. K decaying exponentially with the distance between ij In the rest of the paper we provide the necessary vortex sites jr − r j. Four-fermion interaction terms of i j background on our proposed system and support its relation the type required to implement the SYK Hamiltonian to the SYK model by physical arguments and by detailed Eq. (1.1) are generically also present but are subdominant model calculations. We first review the Fu-Kane model [36] and also decay exponentially with distance. Realizing the for the TI-SC interface and numerically calculate the SYK model in this setup therefore entails two key chal- Majorana wave functions localized in a hole threaded by lenges: (i) one must find a way to suppress the two-fermion N magnetic flux quanta in the presence of disorder. tunneling terms and (ii) render the four-fermion interactions Assuming that the constituent electrons interact via effectively infinite ranged. In addition, the four-fermion screened Coulomb potential, we then explicitly calculate coupling constants J must be sufficiently random. In the ijkl the four-fermion coupling constants J between the ijkl following we show how these challenges can be overcome Majorana zero modes. We, finally, use these as input data by judicious engineering of various aspects of the device for the many-body Majorana Hamiltonian which we depicted in Fig. 1. diagonalize numerically for N up to 32 and study its The first challenge can be met by tuning the surface state thermodynamic properties, level statistics, as well as two- of the TI into its global neutrality point such that the and four-point correlators. We show that these behave chemical potential μ lies at the Dirac point. At the neutrality precisely as expected of the SYK model with random point the interface superconductor is known to acquire an independent couplings. We also discuss the effect of small extra chiral symmetry which prohibits any two-fermion residual two-fermion terms that will inevitably be present terms [37]. In other words, the symmetry requires K ¼ 0 ij in a realistic device and propose ways to experimentally and the low-energy Hamiltonian is then dominated by the detect signatures of the SYK physics using tunneling four-fermion terms [38]. The second requirement of effec- spectroscopy. tively infinite-ranged interactions can be satisfied by localizing all Majorana zero modes in the same region II. SYK MODEL FROM INTERACTING of space. In our proposed device this is achieved by MAJORANA ZERO MODES AT THE fabricating a hole in the superconducting (SC) layer, as TI-SC INTERFACE illustrated in Fig. 1. If the sample is cooled in a weak applied magnetic field, an integer number N of magnetic A. Fu-Kane superconductor The surface of a canonical 3D TI, such as Bi Se , hosts a 2 3 single massless Dirac fermion protected by time-reversal symmetry. When placed in the proximity of an ordinary superconductor, the surface state is described by the Fu-Kane Hamiltonian [36], ˆ ˆ H ¼ d rΨ H ðrÞΨ ; ð2:1Þ FK r FK r † † where Ψ ¼ðc ;c ;c ;−c Þ is the Nambu spinor and r ↑r ↓r ↓r ↑r SC z z x y H ¼ τ v σ · p− τ A − μ þ τ Δ − τ Δ : ð2:2Þ 3D TI FK F 1 2 Here, v is the velocity of the surface state, p ¼ −iℏ∇ FIG. 1. The proposed setup for a solid-state realization of the SYK model. denotes the momentum operator, Δ ¼ Δ þ iΔ is the SC 1 2 031006-2 BLACK HOLE ON A CHIP: PROPOSAL FOR A PHYSICAL … PHYS. REV. X 7, 031006 (2017) order parameter, and σ, τ are Pauli matrices in spin and In the geometry of Fig. 1 the Majorana modes we discuss Nambu spaces, respectively. To describe the geometry above can equivalently be viewed as living at the boundary depicted in Fig. 1, we take between a magnetically gapped TI surface on the inside and a SC region on the outside of the hole. The existence of 0 r<RðφÞ such modes is well known and has been discussed in ΔðrÞ¼ ð2:3Þ iNφ several papers [45,46]. Δ e r>RðφÞ; The existence and properties of the zero modes in the Fu- Kane Hamiltonian have been extensively tested by analytic where φ is the polar angle and RðφÞ denotes the hole radius. and numerical approaches for a single vortex [36], pair of The vector potential is taken to yield total flux through the vortices [47,48], periodic Abrikosov lattices [49,50],as hole dl · A ¼ NΦ , with Φ ¼ hc=2e the SC flux 0 0 well as the “giant vortex” geometry [51] similar to our quantum and the contour C taken to encircle the hole at proposed setup. This body of work firmly establishes the a radius well beyond the effective magnetic penetration 2 existence of exact Majorana zero modes for μ ¼ 0 in depth of the SC film, λ ¼ 2λ =d. (Here, λ is the London eff L L accordance with the Jackiw-Rossi-Weinberg index theo- penetration depth of the bulk SC and d the film thickness.) rem. Away from neutrality it is found that the zero modes The Hamiltonian Eq. (2.2) respects the particle-hole y y 2 are split due to two-fermion tunneling terms K ∝ μ, where ij (p-h) symmetry generated by Ξ ¼ σ τ K (Ξ ¼þ1), where the constant of proportionality is related to the wave K denotes complex conjugation. For a purely real gap function overlap between χ and χ . In addition, it has function Δ and zero magnetic field B ¼ ∇ × A, it also i j been found that for a singly quantized vortex at neutrality obeys the physical time-reversal symmetry Θ ¼ iσ K the zero mode is separated from the rest of the spectrum by (Θ ¼ −1). In the presence of vortices Δ becomes complex a gap ∼Δ , where Δ is the SC gap magnitude far from the and the time-reversal symmetry is broken. The Fu-Kane 0 0 vortex. We shall see that for a judiciously chosen hole size model with vortices therefore falls into symmetry class D in this convenient hierarchy of energy scales remains in place the Altland-Zirnbauer classification [42], which, in accor- with N zero modes separated by a large gap from the rest of dance with Ref. [37], implies a Z classification for the the spectrum. zero modes associated with vortices. Physically, this means that a system with total vorticity N will have N ¼ ðN mod 2Þ exact zero modes, in accord with the expect- B. Low-energy effective theory ation that any even number of Majorana zero modes will Having established a convenient platform that hosts N generically hybridize and form complex fermions at non- Majorana zero modes with wave functions localized in zero energy. the same region of space, we now proceed to derive the When μ ¼ 0, the Hamiltonian Eq. (2.2) also respects a effective low-energy theory in terms of the Majorana zero- x x fictitious time-reversal symmetry generated by Σ ¼ σ τ K mode operators χ . To this end, we write the full second- (Σ ¼þ1). It is important to note that unlike the physical quantized Hamiltonian of the system as time-reversal, this symmetry remains valid even in the presence of the applied magnetic field and vortices. At the ðNÞ H ¼ H þ δH þ H : ð2:4Þ FK int FK neutrality point, the two symmetries Ξ and Σ define a BDI z z class with chiral symmetry Π ¼ ΞΣ ¼ σ τ . This, in ðNÞ Here, H stands for the part of the Fu-Kane Hamiltonian accordance with Ref. [37], implies an integer classification FK Eq. (2.2) that obeys the fictitious time-reversal symmetry Σ of zero modes associated with point defects. A system with and exhibits, therefore, N exact zero modes. δH contains FK total vorticity N will thus exhibit N ¼ N exact zero V V all the remaining fermion bilinears that break Σ such as the modes, irrespective of the precise geometric arrangement of chemical potential term. H defines the four-fermion the individual vortices and other details. This remains true int interactions that have been ignored thus far but will play in the presence of any disorder that does not break the Σ a pivotal role in the physics of the SYK model we study. We symmetry. Specifically, randomness in v and Δ will not assume that electrons are subject to screened Coulomb split the zero modes but random contributions to μ will. interactions described by Another way to establish the existence of exact zero modes in the Hamiltonian Eq. (2.2) with μ ¼ 0 is ZZ to recognize it as a version of the Jackiw-Rossi Hamiltonian 2 2 0 0 0 H ¼ d rd r ρ ˆðrÞVðr − r Þρ ˆðr Þ; ð2:5Þ int [43], well known in particle physics. An index theorem for this Hamiltonian, conjectured by Jackiw and Rossi and where VðrÞ is the interaction potential and ρ ˆðrÞ¼ c c is σr later proven by Weinberg [44], equates the number N of its σr the electron charge density operator. exact zero modes in region M to the total vorticity, Now imagine we have solved the single-electron prob- N ¼½1=ð2πÞ dl · ∇θ, contained in that region. A ∂M ðNÞ region threaded by N magnetic flux quanta will thus lem defined by the Hamiltonian H for the device V FK contain N exact zero modes. geometry sketched in Fig. 1 with N flux quanta threaded 031006-3 D. I. PIKULIN and M. FRANZ PHYS. REV. X 7, 031006 (2017) through the hole. We thus have the complete set of single- and ρ ðrÞ¼ði=2ÞΦ ðrÞτ Φ ðrÞ is the charge density ij j ðNÞ associated with the pair of zero modes χ and χ .We particle eigenfunctions Φ ðrÞ and eigenenergies ε of H . i j n n FK The corresponding second-quantized Hamiltonian can then ~ observe that at the neutrality point, when K ¼ 0, the low- ij be written in a diagonal form, energy effective Hamiltonian Eq. (2.10) coincides with the SYK model. Equations (2.11) and (2.12) allow us to ðNÞ 0 † ˆ ˆ H ¼ ε ψ ψ þ E ; ð2:6Þ calculate the relevant two- and four-fermion coupling FK n n g constants from the knowledge of the Majorana wave where functions in the noninteracting system. We carry out this program in Sec. IV for a specific physically relevant model † system. Here, we finish by discussing some general ψ ˆ ¼ d rΦ ðrÞΨ ð2:7Þ n r properties of the Hamiltonian Eq. (2.10) that follow from symmetry considerations. is the eigenmode operator belonging to the eigenvalue ε . n The reality condition Eq. (2.8) for the Majorana wave The sum over n is restricted to the positive energy function implies the following spinor structure of Φ ðrÞ in eigenvalues and E is a constant representing the the Nambu space: ground-state energy. At the neutrality point, according to our preceding discussion, N of the ψ ˆ eigenmodes j Φ ¼ ; ð2:13Þ coincide with the exact zero modes mandated by the iσ η Jackiw-Rossi-Weinberg index theorem. We denote these χ with j ¼ 1; …;N. Because ε ¼ 0, these modes do not where η ðrÞ is a two-component complex spinor. We thus j j j contribute to the Hamiltonian Eq. (2.6). The zero-mode have eigenfunctions Φ ðrÞ can be chosen as eigenstates of the † † p-h symmetry generator Ξ. They then satisfy the reality ρ ¼ ðη η − c:c:Þ¼ −Imðη η Þ: ð2:14Þ ij j j i i condition The charge density is thus purely real and antisymmetric y y σ τ Φ ðrÞ¼ Φ ðrÞ; ð2:8Þ j j under i ↔ j. In the simplest case, the Σ-breaking part of the Fu-Kane Hamiltonian will simply be δH ðrÞ¼ −μτ .In FK which implies that χ ¼ χ ; the zero modes are Majorana ~ j this situation, Eq. (2.11) implies that K ¼ 4μ d rρ ðrÞ. ij ij operators. Thus, K is purely real and antisymmetric, as required for ij As noted before, the N zero modes are separated by a gap H to be Hermitian. eff from the rest of the spectrum. As long as δH and H FK int Because of the anticommutation property Eq. (1.2) remain small compared to this gap we may construct the of the Majorana operators, it is clear that only the fully effective low-energy theory of the system by simply antisymmetric part of J contributes to the Hamiltonian ijkl projecting onto the part of the Hilbert space generated Eq. (2.10). As we define in Eq. (2.12), J is already ijkl by N Majorana zero modes. In practical terms this is antisymmetric under i ↔ j and k ↔ l due to the anti- accomplished by inverting Eq. (2.7) to obtain symmetry ρ ¼ −ρ . With this in mind we can rewrite the ij ji ˆ Hamiltonian Eq. (2.10) in a more convenient form: Ψ ¼ Φ ðrÞψ ˆ ; ð2:9Þ r n n X X H ¼ i K χ χ þ J χ χ χ χ ; ð2:15Þ eff ij i j ijkl i j k l ˆ i<j i<j<k<l then substituting Ψ into δH and H and retaining only r FK int those terms that contain zero-mode operators χ but no with finite-energy eigenmodes. We thus obtain 1 1 ~ ~ ~ ~ ~ X X K ¼ ðK − K Þ;J ¼ ðJ − J þ J Þ i 1 ij ij ji ijkl ijkl ikjl lijk ~ ~ 2 3 H ¼ K χ χ þ J χ χ χ χ ; ð2:10Þ eff ij i j ijkl i j k l 2! 4! i;j i;j;k;l ð2:16Þ now fully antisymmetric. In the following, we are interested where in situations where coupling constants are random and we characterize the coupling strengths by two parameters K iK ¼ 2! d rΦ ðrÞδH ðrÞΦ ðrÞ; ð2:11Þ ij FK j and J defined by ZZ 4! 2 2 0 0 0 2 2 2 2 J ¼ d rd r ρ ðrÞVðr − r Þρ ðr Þ; ð2:12Þ K ¼ NK;J ¼ J ; ð2:17Þ ijkl ij lk ij ijkl 3! 031006-4 BLACK HOLE ON A CHIP: PROPOSAL FOR A PHYSICAL … PHYS. REV. X 7, 031006 (2017) where the bar represents an ensemble average over 3V 1 J ¼ 0; J J ¼ δ ; ð2:21Þ I I J IJ 2 3 randomness. 8ζ M C. Structure and statistics of the coupling constants J where the uppercase label represents a group of four ijkl indices I ¼fijklg. The coupling constants given In order to approximate the SYK Hamiltonian, the by Eq. (2.20) are asymptotically independent with the coupling constants J we give in the previous section ijkl higher-order correlators vanishing as higher powers of M , must behave as independent random variables. To assess −5 e.g., J J J ∼ M . ijkl klmn mnij s this condition, we now discuss their structure and statistics. The above analysis suggests that under reasonable We make two reasonable assumptions: (i) the interaction assumptions coupling constants defining the many-body potential in Eq. (2.12) is short ranged and well approxi- Hamiltonian Eq. (2.15) can be considered independent mated by VðrÞ ≃ V δðrÞ and (ii) there exists a length scale random variables. When additionally K can be taken as ij ζ beyond which Majorana wave functions Φ ðrÞ can be negligible, we expect the Hamiltonian to approximate the treated as random independent variables. SYK model. Building on the experience gained from We coarse grain the Majorana wave functions on the grid Refs. [14,16], we furthermore expect our Hamiltonian to with sites r and spacing ∼ζ. This amounts to replacing 2 2 describe an interesting non-Fermi-liquid phase even away η ðrÞ → η¯ ðr Þ=ζ and d r → ζ in Eqs. (2.13) and j j n n from the limit when J’s are independent variables. For (2.12). The discretized spinor wave functions then have the instance, certain specific correlations present in J’s are following structure on each site: known to lead to a very interesting supersymmetric version 1 2 of the SYK model [14] and a whole family of SYK-like ϕ ðnÞþ iϕ ðnÞ j j models discussed in Ref. [16] . η¯ ðr Þ¼ ; ð2:18Þ j n 3 4 ϕ ðnÞþ iϕ ðnÞ Recent work [41] performed a mathematical analysis of j j deviations in J’s from ideal random independent variables where ϕ ðnÞ are real independent random variables with in a model qualitatively similar to ours. Here, we adopt a different approach and proceed by evaluating the effect of β such deviations on the observable physical properties of the α α αβ ϕ ðnÞ¼ 0; ϕ ðnÞϕ ðnÞ¼ δ δ : ð2:19Þ i i j ij many-body model defined by the Hamiltonian Eq. (2.15). 8M We find that coupling constants that follow from the giant 2 2 Here, M ¼ πR =ζ is the total number of grid sites in the vortex geometry indeed give rise to a phenomenology that hole and the second equality follows from the normaliza- is consistent with the SYK model. tion of Φ ðrÞ. Combining Eqs. (2.12), (2.14), (2.16), and (2.18) it III. LARGE-N SOLUTION AND THE is possible to express the antisymmetrized coupling con- CONFORMAL LIMIT stants as When the number of Majorana fermions N is large, M the SYK model becomes analytically solvable in the 0 β μ α ν low-energy limit. Specifically, the Euclidean-space time- J ¼ − ϵ ϕ ðnÞϕ ðnÞϕ ðnÞϕ ðnÞ; ð2:20Þ ijkl αβμν 2 i j k l n¼1 ordered propagator defined as GðτÞ¼hT χðτÞχð0Þi ð3:1Þ where ϵ is the totally antisymmetric tensor and sum- αβμν mation over repeated indices is implied. For a general value can be expressed in the Matsubara frequency domain of M , the many-body Hamiltonian defined by coupling through the self-energy Σðω Þ as constants Eq. (2.20) represents a variant of the original SYK model similar to models studied in Refs. [14,16].As −1 Gðω Þ¼½−iω − Σðω Þ : ð3:2Þ n n n such, it might be amenable to the large-N analysis using iω τ approaches described in those works. Here, we focus on the n Here, Gðω Þ¼ dτe GðτÞ, and β ¼ 1=k T is the n B limit M ≫ N, which works when the hole radius R is large inverse temperature. At nonzero temperatures the propa- and the wave functions can be considered random on short gator and the self-energy are defined for discrete Matsubara scales ζ. In this limit each J defined in Eq. (2.20) is given ijkl frequencies ω ¼ πTð2n þ 1Þ, with n integer and taking by a sum of a large number of random terms given by k ¼ 1 here and henceforth. Using the replica trick to products of four random amplitudes ϕ ðnÞ. The central average over disorder configurations, or alternately sum- limit theorem then assures us that J’s will be random ming the leading diagrams in the 1=N expansion, one variables with a distribution approaching the Gaussian obtains (see, for example, Ref. [7]) the following expres- distribution irrespective of the detailed statistical properies sion for the self-energy appropriate for the Hamiltonian of ϕ ðnÞ. It is furthermore easy to show that Eq. (2.15): 031006-5 D. I. PIKULIN and M. FRANZ PHYS. REV. X 7, 031006 (2017) 2 2 3 ΣðτÞ¼ K GðτÞþ J G ðτÞ: ð3:3Þ (a) A , J =1 For arbitrary given parameters K, J, and β the self- consistent equations (3.2) and (3.3) can be solved by A , K =0.2 3 f numerical iteration. Analytical solutions are available in various limits and are reviewed below. In subsequent 0.4 sections we compare these with numerical results based 0.6 on the model described above. 0.6 0.4 0.2 0.0 0.2 0.4 0.6 A. Free-fermion limit When J ¼ 0 the theory becomes noninteracting and an analytic solution to Eqs. (3.2) and (3.3) can be given for all (b) conformal limit temperatures. Specifically, the self-energy in Eq. (3.3) can K=0.01 K=0.10 be written in the frequency domain as Σðω Þ¼ K Gðω Þ n n K=0.50 and substituted into Eq. (3.2). Solving for Gðω Þ then gives 2i G ðω Þ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ð3:4Þ f n 2 2 ω þ sgnðω Þ ω þ 4K n n n This implies the high-frequency limit G ðω Þ ≃ i=ω and f n n the low-frequency limit G ðω Þ ≃ i=sgnðω ÞK. f n n 0.01 0.1 1 It is useful to extract the single-particle spectral function /J from Eq. (3.4) defined as AðωÞ¼ð1=πÞImGðω →−iωþδÞ, FIG. 2. (a) Spectral functions, measurable in a tunneling by analytically continuing from Matsubara to real frequen- experiment, in the conformal (strongly interacting) limit cies to obtain the retarded propagator. We thus find (red lines) and free-fermion limit (blue lines). (b) Numerically sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ evaluated large-N Matsubara Green’s functions for J ¼ 1.0, 1 ω T ¼ 0.001, and different values of K. Red dashed line shows A ðωÞ¼ Re 1 − ; ð3:5Þ πK 2K the conformal limit behavior Eq. (3.7) while the thick green and brown lines correspond to free-fermion result Eq. (3.4) with the usual semicircle law. For this zero-dimensional system K ¼ 0.1 and 0.5, respectively. AðωÞ coincides with the local density of states DðωÞ averaged over all Majorana sites, which is experimentally It is important to note that the low-frequency behaviors measurable in a tunneling experiment. Specifically, the of A and A are quite different with the former saturating at f c tunneling conductance gðωÞ¼ðdI=dVÞ is propor- ω¼eV 1=πK and the latter divergent. Thus, it should be possible to tional to the local density of states DðωÞ. distinguish the free-fermion and the interaction-dominated behaviors, illustrated in Fig. 2(a), by performing a tunnel- B. Conformal limit ing experiment. We discuss the measurement in more detail in Sec. VI. When K ¼ 0 and T ≪ J the system is strongly interact- ing but, nevertheless, an asymptotic solution of Eqs. (3.2) and (3.3) can be found by appealing to their approximate C. Crossover region reparametrization invariance [6,7] that becomes exact in the When both K and J are nonzero, as is the case in a low-frequency limit when one can neglect the −iω term in typical experimental setup, analytical solutions are not Eq. (3.2). The conformal limit solution reads available, but one can still understand the behavior of the system from approximate analytical considerations and sgnðω Þ 1=4 G ðω Þ¼ iπ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ð3:6Þ c n numerical solutions. Let us focus on the T ¼ 0 limit and Jjω j study the effect of K and J on the self-energy Σðω Þ in Eq. (3.3). To this end, it is useful to consider the and the corresponding spectral function is propagators G and G in the imaginary time domain. f c 1 1 For long times τ, one obtains A ðωÞ¼ pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ð3:7Þ 3=4 2π Jjωj 1 sgnðτÞ 1 sgnðτÞ These expressions are valid for jωj ≪ J and must cross G ðτÞ¼ ;G ðτÞ¼ pﬃﬃﬃﬃﬃ : ð3:8Þ f c 1=2 1=4 πK jτj jτj π 2J over to the 1=ω behavior at large frequencies. 031006-6 -iJG( ) A( ) n BLACK HOLE ON A CHIP: PROPOSAL FOR A PHYSICAL … PHYS. REV. X 7, 031006 (2017) Consider the K ¼ 0 limit and then slowly turn K on. IV. NUMERICAL RESULTS: THE UNDERLYING Initially, G ðτÞ is a valid solution. However, for any NONINTERACTING SYSTEM nonzero K it is clear that the first term on the right-hand In this section, we provide support for the ideas we side of Eq. (3.3) will dominate at sufficiently long times present above by performing extensive numerical simula- τ > τ . At such long times one then expects a crossover to tion and modeling of the system we describe in Sec. II.We the behavior resembling the free-fermion propagator G ðτÞ. start by formulating a lattice model for the surface of a TI in The corresponding crossover time τ can be estimated by contact with a superconductor. We then find the wave equating the two terms on the right-hand side of Eq. (3.3), functions of the Majorana zero modes by numerically 2 2 3 K G ðτ Þ¼ J G ðτ Þ, which gives f c diagonalizing the corresponding Bogoliubov–de Gennes (BdG) Hamiltonian for the geometry depicted in Fig. 1 with pﬃﬃﬃ π J N flux quanta threading the hole. In the following section, τ ¼ ; ð3:9Þ 8 using Eqs. (2.11) and (2.12), we calculate the coupling constants K and J , which we then use to construct and ij ijkl and the corresponding crossover frequency diagonalize the many-body interacting Hamiltonian Eq. (2.15) for N up to 32. The resulting many-body spectra pﬃﬃﬃ 2π K and eigenvectors are used to calculate various physical ω ¼ ¼ 16 π : ð3:10Þ quantities (entropy, specific heat, two- and four-point τ J correlators), which are then compared to the results previously obtained for the SYK model with random We thus expect the spectral function to behave as indicated independent couplings. in Eq. (3.7) for ω < ω ≪ J with the divergence at small ω cut off below ω and saturate to ∼1=πK. To confirm the above behavior, we solved Eqs. (3.2) A. Lattice model for the TI surface and (3.3) numerically. We find it most convenient to A surface of a 3D TI is characterized by an odd number work with Matsubara Green’s functions at very low but of massless Dirac fermions protected by time-reversal nonzero temperatures. To this end, we rewrite Eq. (3.3) in symmetry Θ. The well-known Nielsen-Ninomyia theorem Matsubara frequency domain where the last term becomes [52,53] assures us that, as a matter of principle, it is a convolution and substitute the self-energy into Eq. (3.2). impossible to construct a purely 2D, Θ-invariant lattice We obtain a single equation, model with an odd number of massless Dirac fermions. This fact causes a severe problem for numerical approaches −1 2 2 2 G ¼ −iω − K G − J T G G G ; ð3:11Þ to 3D TIs because one is forced to perform an expensive n n n k l n−k−l k;l simulation of the 3D bulk to describe the anomalous 2D surface. A workaround has been proposed [54] that circum- for G ≡ Gðω Þ that must be solved self-consistently. n n vents the Nielsen-Ninomyia theorem by simulating a pair Results obtained by iterating Eq. (3.11) are displayed in of TI surfaces with a total even number of Dirac fermions. Fig. 2(b). For very small K ¼ 0.01J, we observe that This approach enables efficient numerical simulations in a numerically calculated Gðω Þ coincides with the conformal quasi-2D geometry while fully respecting Θ. limit for a range of frequencies consistent with our Here, because the physical time-reversal symmetry is discussion above. For K ¼ 0.1J, this range becomes ultimately broken by the presence of vortices and is smaller and completely disappears for K ¼ 0.5J. therefore not instrumental, we opt for an even simpler We conclude that for any nonzero K the ultimate low- model which breaks Θ from the outset but nevertheless energy behavior is controlled by the free-fermion fixed captures all the essential physics of the TI-SC interface. We point, as expected on general grounds. Nevertheless, when start from the following momentum-space normal-state K is sufficiently small in comparison to J, there can be a Hamiltonian defined on a simple 2D square lattice: significant range of energies in which the physics is x y z dominated by the SYK fixed point. At low temperatures h ðkÞ¼ λðσ sin k þ σ sin k Þþ σ M − μ; ð4:1Þ 0 x y k the corresponding range of frequencies is given by with M ¼m½ð2−cosk −cosk Þ− ð2−cos2k −cos2k Þ. k x y x y pﬃﬃﬃ Here, σ are Pauli matrices in spin space and λ, m are 16 π < ω ≪ J: ð3:12Þ model parameters. The term proportional to λ respects Θ and gives four massless Dirac fermions in accordance In this range we expect the spectral function to obey the with the Nielsen-Ninomyia theorem. The M term breaks conformal scaling form given by Eq. (3.7). A tunneling Θ and has the effect of gapping out all the Dirac fermions experiment in this regime should therefore reveal the SYK except the one located at Γ ¼ð0; 0Þ. The resulting energy behavior of the underlying strongly interacting system. spectrum, 031006-7 D. I. PIKULIN and M. FRANZ PHYS. REV. X 7, 031006 (2017) B. Solution in the giant vortex geometry To study the nonuniform system with magnetic field and vortices, we must write the Hamiltonian in the position space. The normal-state piece Eq. (4.1) is most conven- iently written in second-quantized form as X X m=0 † † α z H ¼ iλ ðψ σ ψ − H:c:Þþ ψ mσ − μ ψ 0 r rþα r r 1 r;α r † † z z − ð4ψ σ ψ − ψ σ ψ þ H:c:Þ; ð4:4Þ r rþα r rþ2α m=0.5 r;α M X M where we define on each lattice site r a two-component spinor ψ ¼ðc ;c Þ and α ¼ x, y. The magnetic field is r r↑ r↓ FIG. 3. Band structure (4.2) of the lattice model Eq. (4.1) for included through the standard Peierls substitution, which λ ¼ 1 and m ¼ 0 (blue dashed) and m ¼ 0.5 (red solid line). X replaces tunneling amplitudes on all bonds according and M denote the ð0; πÞ and ðπ; πÞ points of the Brillouin zone, † † rþα respectively. to ψ ψ → ψ ψ expf−i½e=ðℏcÞ dl · Ag. The full r rþα r rþα second-quantized BdG Hamiltonian then reads qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 † † εðkÞ¼ λ ðsin k þ sin k Þþ M − μ; ð4:2Þ H ¼ H þ ðΔ c c þ H:c:Þ; ð4:5Þ x y k BdG 0 r r↑ r↓ is depicted in Fig. 3. In the vicinity of the Γ point, we where Δ is the pair potential on site r, which takes the form observe a linearly dispersing spectrum characteristic of a TI indicated in Eq. (2.3). In accord with our discussion in the surface state. It is to be noted that for small jkj we previous section, H given in Eq. (4.5) represents a BdG 1 4 have M ≃ mk , so the amount of Θ breaking can be 8 version of the Fu-Kane Hamiltonian Eq. (2.1) regularized considered small in the physically important part of the on a square lattice. This lattice model is suitable for momentum space near the Γ point. numerical calculations and we expect it to reproduce all Proximity-induced superconducting order is imple- the low-energy features of the Fu-Kane Hamiltonian. In mented by constructing the BdG Hamiltonian: particular, we show shortly that it yields N Majorana zero modes mandated by the Jackiw-Rossi-Weinberg index h ðkÞ Δ theorem that are of central importance for the SYK model. H ðkÞ¼ : ð4:3Þ BdG y y It is most convenient to solve the problem defined by Δ −σ h ð−kÞσ Hamiltonian Eq. (4.5) on a lattice with L × L sites and Writing H in terms of σ and τ matrices it can be easily periodic boundary conditions which ensure that no spuri- BdG ous edge states exist at low energies in addition to the checked that it respects the particle-hole symmetry Ξ we expected N Majorana zero modes bound to the hole. To define in Sec. II. A. The μ and M terms both break the implement periodic boundary conditions, it is useful to fictitious time reversal Σ that protects the Majorana zero perform a singular gauge transformation, modes in our setup. As before, μ must be tuned to zero to achieve protection. On the other hand, it is crucial to iNφ=2 ψ → e ψ ; ð4:6Þ r r remember that M has been introduced only to circumvent the Nielsen-Ninomyia theorem and allow us to efficiently which has the effect of removing the phase winding simulate a single two-dimensional Dirac fermion on the from Δ and changing the Peierls phase factors to lattice. Breaking of Σ by M is therefore not a concern in rþα expði dl · ΩÞ, with the experimental setup: in a real TI tuned to the neutrality point, Σ is unbroken. Expanding H ðkÞ in the vicinity of BdG 1 2e Ω ¼ N∇φ − A : ð4:7Þ Γ to leading order in small k we recover the Fu-Kane 2 ℏc Hamiltonian H defined in Eq. (2.2). We thus conclude FK that at low energies our lattice model indeed describes the We note that N must be even because only for integer number of fundamental flux quanta hc=e ¼ 2Φ in the TI-SC interface and should exhibit the desired phenom- system can one impose periodic boundary conditions. For enology, including Majorana zero modes bound to vortices. N even, the transformation Eq. (4.6) is single valued and We show that this is indeed the case. The only repercussion that follows from the weakly broken Σ (present in the the issue of branch cuts that renders the analogous problem higher-order terms in the above expansion) is a very small with singly quantized vortices [55,56] more complicated splitting of the zero-mode energies that has no significant does not arise here. After the transformation the total effect on our results. effective flux seen by the electrons dSð∇ × ΩÞ vanishes 031006-8 BLACK HOLE ON A CHIP: PROPOSAL FOR A PHYSICAL … PHYS. REV. X 7, 031006 (2017) and numerical diagonalization of the transformed to the fact that Σ symmetry is weakly broken in our lattice Hamiltonian Eq. (4.5) with periodic boundary conditions simulation by the M term. For nonzero μ or w , the energy k μ becomes straightforward. splitting increases in proportion to these Σ-breaking per- As a practical matter, it is easiest to define a regular- turbations. In the following, we include these terms in shaped hole and introduce disorder through a replacement: δH and incorporate them in our many-body calculation FK via K terms given by Eq. (2.11). ij ðμ; λ;Δ Þ → ðμ; λ;Δ Þþðδμ ; δλ ; δΔ Þ: ð4:8Þ r r r r r Figures 4(c)–4(f) show examples of zero-mode wave function amplitudes jΦ ðrÞj . The wave functions are Here, ðδμ ; δΔ ; δλ Þ are independent random variables r r r shown to exhibit random spatial structure, which depends uniformly distributed in the interval ð−w =2;w =2Þ for μ μ sensitively on the specific disorder potential realization. δμ and similarly for δΔ and δλ . We choose a stadium- r r r Importantly, all the zero-mode wave functions are localized shaped hole sketched in Fig. 4(a), which is known to in the same region of space defined by the hole and its support classically chaotic trajectories [39,40]. In our immediate vicinity. One therefore expects Eq. (2.12) to quantum simulation we find that much smaller disorder produce strong random couplings J connecting all zero ijkl strength is required to achieve sufficiently random modes χ once the interactions are included. Majorana wave functions for a stadium-shaped hole than, e.g., with circular hole. We furthermore choose magnetic field B to be uniform inside the radius R that contains the V. NUMERICAL RESULTS: THE MANY-BODY hole and zero otherwise. We find that our results are SYK PROBLEM insensitive to the detailed distribution of B as long as Having obtained the zero-mode wave functions it is the total flux remains NΦ and is centered around the hole straightforward to calculate couplings K and J from ij ijkl (we test various radii R as well as a Gaussian profile). Eqs. (2.11) and (2.12) and construct the many-body SYK Typical results of the numerical simulations we describe Hamiltonian Eq. (2.15). In the following, we assume that above are displayed in Fig. 4. In Fig. 4(b), we observe the the system has been tuned to its global neutrality point behavior of the energy eigenvalues E of H . For zero n BdG μ ¼ 0 and include in δH only the random component of FK magnetic flux, there are several states inside the SC gap the on-site potential δμ . For the interaction term we (Andreev states bound to the hole) but no zero modes. For consider the screened Coulomb potential defined as N ¼ 24, these are converted into 24 zero modes required by the Jackiw-Rossi-Weinberg index theorem. For μ¼w ¼0 2 −r=λ TF 2πe e used in the simulation, their energies are very close to zero VðrÞ¼ ; ð5:1Þ −4 (∼10 λ), where the small residual splitting is attributable ϵ r 40 40 0.6 (c) (e) (a) 30 30 0.4 20 20 RB 0.2 10 10 10 20 30 40 10 20 30 40 0 2 40 40 (d) (f) -0.2 30 30 20 20 N=0 -0.4 N=24 (b) 10 10 -0.6 10 20 30 40 10 20 30 40 -100 -50 0 50 100 xx FIG. 4. Numerical simulations of the BdG Hamiltonian Eq. (4.5). (a) Stadium-shaped hole geometry employed in the simulations. R parametrizes the hole size whereas R denotes the radius inside which the magnetic field is nonzero. (b) Energy levels E of the BdG B n Hamiltonian Eq. (4.5) calculated for N ¼ 0 and N ¼ 24. Energies have been sorted in ascending order and plotted as a function of their integer index n. The shaded band represents the SC gap region. (c)–(f) Density plots of the typical zero-mode wave function amplitudes for N ¼ 24. The dashed circle in (c) has radius R . The following parameters are used to obtain these results: λ ¼ 1, m ¼ 0.5, Δ ¼ 0.3, B 0 μ ¼ w ¼ 0, w ¼ w ¼ 0.1, L ¼ 42, R ¼ 10, and R ¼ 15. μ λ Δ B 031006-9 n D. I. PIKULIN and M. FRANZ PHYS. REV. X 7, 031006 (2017) (a) 0.4 where ϵ is the dielectric constant and λ denotes the TF N=26 Thomas-Fermi screening length. We furthermore assume 0.3 that λ is short, so that in the lattice model the interaction TF is essentially on site. The expression for J then ijkl 0.2 random Gaussian simplifies to giant vortex, K=0 0.1 giant vortex, K=0.4 J ≃ 12V d rρ ðrÞρ ðrÞ; ð5:2Þ ijkl 0 ij lk 0.0 N=30 2 2 0.3 with V ¼ d rVðrÞ¼ 2πe λ =ϵ. Coupling constants 0 TF K and J are easy to evaluate using Eq. (2.16) and ij ijkl 0.2 the Majorana wave functions Φ ðrÞ obtained in the previous section. To facilitate comparisons with the 0.1 existing literature, we quantify the average strength of 0.0 these terms using parameters K and J defined in Eq. (2.17). 0.0 0.2 0.4 0.6 0.8 1.0 Specifically, we adjust w and V to obtain the desired T/J μ 0 values of K and J. In the following section, we connect (b) these values to the parameters expected in realistic physical systems. 0.01 A. Thermodynamic properties and many-body level statistics N =26 Once the coupling constants K and J are determined ij ijkl 0.00 as described above, one can construct a matrix representa- random Gaussian giant vortex, K =0 tion of the many-body Majorana Hamiltonian Eq. (2.15) giant vortex, K =0.4 and find its energy eigenvalues E by exact numerical 0.01 diagonalization. From the knowledge of the energy levels, it is straightforward to calculate any thermodynamic property. In Fig. 5, we display the thermal entropy SðTÞ N =30 and the heat capacity C ðTÞ. These are calculated from 0.00 0.0 0.5 1.0 1.5 2.0 T/J 2 2 hEi − F hE i − hEi S ¼ ;C ¼ ; ð5:3Þ T T FIG. 5. Thermodynamic properties of the many-body Hamil- tonian Eq. (2.15). (a) Thermal entropy per particle and (b) heat α α −E =T where hE i¼ð1=ZÞ E e , F ¼ −T ln Z is the free n capacity per particle. Dashed lines show the expected behavior −E =T energy, and Z ¼ e the partition function. for the SYK model with random independent couplings, solid lines show results for the couplings obtained from the giant The entropy per particle is seen to saturate at high vortex system. In all panels the same parameters are used as in temperature to S =N ¼ ln 2 ≃ 0.3465, as expected for a Fig. 4 with V adjusted so that J ¼ 1. system of N Majorana fermions. The behavior of SðTÞ calculated for the giant vortex system is qualitatively similar to that obtained from the SYK model with random The heat capacity C ðTÞ, displayed in Fig. 5(b), likewise independent couplings. The small deviations that exist are behaves as expected for the SYK model with random clearly becoming smaller as N grows, suggesting that they independent couplings with small deviations becoming vanish in the thermodynamic limit. Nonzero two-body negligible in the large-N limit. C ðTÞ is, in principle, coupling K is seen to modify the entropy slightly at low measurable, and we can see from Fig. 5(b) that its temperature. For large N and K ¼ 0, the entropy per high-temperature behavior could be used to gauge the particle is expected to attain a nonzero value ∼0.24 as T → relative strength of two- and four-fermion terms in the 0 due to the extensive near ground-state degeneracy of the system. SYK model. Our largest system is not large enough to show As discussed in Refs. [12,13], many-body level statistics this behavior (in agreement with previous numerical provides a sensitive diagnostic for the SYK physics results) although Fig. 5(a) correctly captures the expected encoded in the Hamiltonian Eq. (2.15). To apply this suppression of the low-T entropy in the presence of two- analysis to our results, we arrange the many-body energy body couplings, which tend to remove the extensive levels in ascending order E <E < and form ratios 1 2 ground-state degeneracy. between the successive energy spacings: 031006-10 C (T )/N C (T )/N S(T )/N S(T )/N V V J=0, K=1 J=1, K=0 BLACK HOLE ON A CHIP: PROPOSAL FOR A PHYSICAL … PHYS. REV. X 7, 031006 (2017) TABLE I. Gaussian ensembles for even N. anticipated distributions for GOE, GUE, and GSE given in Eq. (5.5). Unambiguous agreement with the pattern indi- Nðmod 8Þ 02 4 6 cated in Table I is observed, lending further support to the Level stat. GOE GUE GSE GUE notion that our proposed system realizes the SYK model. β 12 4 2 We check that the Z periodic pattern persists for all N pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ Z 4π=ð81 3Þ 4π=ð729 3Þ 4π=ð81 3Þ 27 down to 16. Additionally, the above results should be contrasted with the level statistics in the noninteracting case J ¼ 0, K ¼ 1 displayed in the bottom row of Fig. 6. In the E − E nþ1 n absence of interactions, Z periodicity is absent and the r ¼ : ð5:4Þ E − E nþ2 nþ1 distribution of the ratio r follows Poisson level statistics, According to Refs. [12,13], the SYK Hamiltonian can be PðrÞ¼ ; ð5:6Þ constructed as a symmetric matrix in the Clifford algebra ð1 þ rÞ Cl whose Bott periodicity gives rise to a Z classi- 0;N−1 8 for all N. It is to be noted that no adjustable parameters are fication with topological index ν ¼ N mod 8. As a result, employed in the level-statistics analysis we present above. statistical distributions of the ratios PðrÞ cycle through Wigner-Dyson random matrix ensembles with Z perio- B. Green’s function dicity as a function of N. Specifically, Gaussian orthogonal (GOE), Gaussian unitary (GUE), and Gaussian symplectic Computing the Green function of the model is perhaps (GSE) ensembles occur with distributions given by the the most straightforward way of comparing the behavior of “Wigner surmise,” the system at finite N to the large-N limit solutions we discuss in Sec. III. At the same time, computation of 2 β 1 ðr þ r Þ propagators is numerically more costly because in addition PðrÞ¼ ; ð5:5Þ 2 1þ3β=2 Z ð1 þ r þ r Þ to many-body energy levels, one requires the correspond- ing eigenstates. We compute the on-site retarded Green and parameters Z and β summarized in Table I for even N function defined as relevant to our system. As emphasized in Ref. [13], the level spacing analysis must be performed separately in the R 0 0 0 G ðt − t Þ¼ −iθðt − t Þhfχ ðtÞ; χ ðt Þgi: ð5:7Þ i i i two-fermion parity sectors of the Hamiltonian Eq. (2.15). Figure 6 shows statistical distributions of the ratios r Fourier transforming and using Lehmann representation in computed for N ¼ 24, 26, 28, 30, and 32 in our system. terms of the eigenstates jni of the many-body Hamiltonian For the sake of clarity Pðln rÞ is plotted along with the Eq. (2.15), one obtains, at T ¼ 0, N =24 N =26 N =28 N =30 N =32 0.8 0.8 0.8 0.8 0.8 GOE GUE GSE GUE GOE 0.6 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 0.8 0.8 0.8 0.8 0.8 Poi Poi Poi Poi Poi 0.6 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 ln r ln r ln r ln r ln r FIG. 6. Level statistics analysis. Top row: Histograms of ln r obtained from the energy levels of the SYK Hamiltonian Eq. (2.15) with coupling constants taken from the giant vortex model with V and w adjusted so that J ¼ 1 and K ¼ 0. Solid lines indicate the expected 0 μ distributions GOE (green), GUE (red), and GSE (orange) specified in Eq. (5.5). Bottom row: Results for the noninteracting case J ¼ 0, K ¼ 1. Black solid line represents the Poisson distribution Eq. (5.6). Histograms in all panels are averaged over eight independent realizations of disorder except for N ¼ 24, where 16 realizations are employed to obtain satisfactory statistics, and N ¼ 32, for which a single realization is used. 031006-11 P(ln r ) P(ln r ) D. I. PIKULIN and M. FRANZ PHYS. REV. X 7, 031006 (2017) " # X exponential growth of the Hamiltonian matrix size with N. jhnjχ j0ij G ðωÞ¼ þðE ↔ E Þ ; ð5:8Þ 0 n More sophisticated numerical techniques, such as the ω þ E − E þ iδ 0 n quantum Monte Carlo method, could possibly reach larger system sizes. Spectral functions calculated for the giant vortex setup where δ is a positive infinitesimal. From Eq. (5.8), R exhibit larger statistical fluctuations compared to those the spectral function A ðωÞ¼ð1=πÞImG ðωÞ is readily computed with random Gaussian coupling constants J ijkl extracted. but are qualitatively similar when averaged over indepen- In Fig. 7, we display the spectral function AðωÞ¼ dent disorder realizations. Therefore, we conclude that the ð1=NÞ A ðωÞ averaged over all Majorana zero modes. Green function behavior at finite N supports the notion that Physically this corresponds to a tunneling experiment with our proposed system realizes the SYK model. a large probe that allows for tunneling into all sites inside the hole. In agreement with the existing numerical results on the complex fermion version of the SYK model [57],we C. Out-of-time-order correlators find that for system sizes we can numerically access (up to and scrambling N ¼ 30), the conformal limit is approached only in a Scrambling of quantum information—a process in which narrow interval of frequencies. In the low-frequency limit, quantum information deposited into the system locally gets numerical results approach a constant value instead of the distributed among all its degrees of freedom—is central to pﬃﬃﬃﬃ the ∼1= ω divergence expected in the large-N limit. The the conjectured duality between the SYK model and AdS dependence on N is very weak, with the larger values Einstein gravity. Black holes are thought to scramble with showing reduced statistical fluctuations but otherwise the maximum possible efficiency: they exhibit quantum qualitatively similar behavior. To convincingly demonstrate chaos. For a quantum theory to be the holographic dual of a the conformal scaling of the Green function at the lowest black hole, its dynamics must exhibit similar fast scram- frequencies, numerical calculations using larger values of bling behavior. N would be necessary. Unfortunately, these are currently The out-of-time-order correlator (OTOC), defined in our out of reach for the exact diagonalization method due to the system as 0.8 F ðtÞ¼hχ ðtÞχ ð0Þχ ðtÞχ ð0Þi; ð5:9Þ ij j i j i N =28, Giant vortex large N 0.6 allows us to quantify the quantum chaotic behavior. For single realization black holes in Einstein gravity scrambling occurs expo- averaged λ t 0.4 nentially fast with 1 − FðtÞ ∼ e =N, where the decay rate is given by the Lyapunov exponent λ ¼ 2πT [9]. Similarly, for the SYK model in the large-N limit, one 0.2 expects [6,7] 0.0 λ t 1 − FðtÞ ∼ e : ð5:10Þ N=28, Random Gaussian NT 0.6 large N single realization Previous works [10,57] gave numerical evaluations of 0.4 averaged FðtÞ in the SYK model for N up to 14 but found these system sizes to be too small to clearly show the expected 0.2 J-independent Lyapunov exponent. Here, we numerically evaluate OTOC for N up to 22 and show that coupling 0.0 constants obtained from the giant vortex geometry give 0.0 1.0 2.0 3.0 4.0 5.0 /J qualitatively the same behavior as those for random independent coupling constants. Our results are summa- FIG. 7. Spectral function AðωÞ computed at zero temperature rized in Fig. 8, where we compute the on-site OTOC F ðtÞ ii for coupling constants J obtained from the giant vortex ijkl averaged over all sites. calculation (top) and taken from the Gaussian distribution For J ¼ 1, the OTOC is seen to rapidly decay to zero, (bottom). Thin gray lines represent individual disorder realiza- consistent with previous works on the SYK model [10,57]. tions corresponding to a physical measurement in a system with The rate of decay is controlled by J: as in Refs. [10,57],we quenched disorder. Thick lines reflect the average over 25 find that N ¼ 22 is not large enough to observe the independent disorder realizations. Dashed lines represent the theoretically predicted J-independent Lyapunov exponent expected low-frequency behavior in the large-N conformal limit controlled by temperature, even when J ≫ T. In addition, Eq. (3.7). All parameters are as in Fig. 4 with J ¼ 1, K ¼ 0 and broadening δ ¼ 0.04 in Eq. (5.8). we observe that adding a sizable two-body tunneling term 031006-12 A( ) A( ) BLACK HOLE ON A CHIP: PROPOSAL FOR A PHYSICAL … PHYS. REV. X 7, 031006 (2017) 1.0 we discuss in more detail the experimentally relevant constraints on the proposed device as well as possible 0.8 ways to detect manifestations of the SYK physics in a 0.6 realistic setting. N=22, Giant vortex J=1.0, K=0.0 0.4 J=1.0, K=1.0 A. Device geometry, length, and energy scales J=0.0, K=1.0 0.2 The key controllable design feature is the size of the 0.0 hole, parametrized by its radius R. For simplicity, in the estimates below we assume a circular hole, but it should 0.8 be understood that in a real experiment irregular shape is required to promote randomness of the zero-mode wave 0.6 N= 22, Random Gaussian functions. For the desired number N of Majorana zero 0.4 J=1.0, K=0.0 modes, the hole must be large enough to pin N vortices. J=1.0, K=1.0 Vortex pinning occurs because the SC order parameter Δ is 0.2 J=0.0, K=1.0 suppressed to zero in the vortex core, which costs con- densation energy. Vortices therefore prefer to occupy 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 regions where Δ has been locally suppressed by defects, tT or in our case, by an artificially fabricated hole. The optimal hole size R for N vortices in our setup can thus be FIG. 8. Out-of-time-order correlators for our giant vortex estimated from the requirement that all the electronic states system (top) and random Gaussian couplings (bottom). Nonzero inside the hole that reside within the SC gap are trans- temperature T ¼ 1 has been taken and other parameters as in Fig. 4. Correlators are displayed for a single disorder realization formed into zero modes, (unaveraged), but different realizations give very similar results for all interacting cases. Detailed shape of the oscillations πR dεDðεÞ¼ N; ð6:1Þ apparent in the J ¼ 0 curves depends sensitively on the specific −Δ disorder realization, but all realizations show qualitatively similar behavior. 2 2 where DðεÞ¼ jεj=2πv ℏ is the density of states of the TI surface. This gives K has only very modest effect on the behavior of FðtÞ when pﬃﬃﬃﬃﬃﬃﬃ the interaction strength is maintained. However, in the R ¼ πξ 2N; ð6:2Þ noninteracting case ðJ ¼ 0;K ¼ 1Þ, OTOC behavior changes qualitatively with the fast decay replaced by with ξ ¼ ℏv =πΔ the BCS coherence length. In the oscillations whose amplitude slowly increases. absence of interactions, a hole of this size will produce an energy spectrum similar to that depicted in Fig. 4(b), with N zero modes maximally separated from the rest of the VI. OUTLOOK: TOWARDS THE EXPERIMENTAL spectrum. REALIZATION AND DETECTION OF In reality, if the SC film is in the type-II regime, a THE SYK MODEL somewhat larger hole might be required to reliably pin N Our theoretical results we present above indicate that vortices in a stable configuration and not create vortices low-energy fermionic degrees of freedom in a device with nearby. The latter condition is that B<B , where B is c1 c1 geometry depicted in Fig. 1 provide a physical realization the lower critical field. Thus, the magnetic field to get the of the SYK model. Additionally, all the ingredients are necessary flux is currently in place to begin experimental explorations of the proposed system. Superconducting order has been induced 2 πðR þ λ Þ B ¼ NΦ ; ð6:3Þ N eff 0 and observed at the surface of several TI compounds by multiple groups [58–64]. Importantly, Ref. [62] already where λ is the effective penetration depth of a thin SC eff demonstrated the ability to tune the chemical potential in film defined below Eq. (2.3). This gives ðBi Sb ÞSe thin flakes through the neutrality point in the x 2−x 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ presence of superconductivity induced by Ti or Al contacts NΦ by a combination of chemical doping (tuning x) and R ≥ − λ : ð6:4Þ N eff πB backgate voltage. This is almost exactly what we require c1 for the implementation of the SYK model. Well-developed Taking the standard expression for the lower critical field, techniques (such as focused ion milling) exist to fabricate 2 −1 patterns, such as a hole with an irregular shape, in a SC film B ¼ðΦ =4πλ ÞK ðκ Þ, where κ ¼ λ =ξ, Eq. (6.4) c1 0 0 eff eff eff eff deposited on the TI surface. In the remainder of this section becomes 031006-13 F(t) F(t) D. I. PIKULIN and M. FRANZ PHYS. REV. X 7, 031006 (2017) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ quanta in the hole as can be stabilized. For the “ideal” hole −1 R ≥ λ 2N=K ðκ Þ − 1 ; ð6:5Þ N eff 0 eff size R ¼ R , given by Eq. (6.2),wehave M ¼ 2π N and N s the dependence on N drops out. The amplitude of J will In the type-II regime, λ > ξ and Eq. (6.5) will generally eff then depend only on the coherence length ξ, screening imply larger hole size than Eq. (6.2). A larger hole size length λ , and dielectric constant ϵ of the system. To get TF would reduce the spectral gap to some extent, but N zero an idea about the possible size of J we assume λ ≈ ξ TF modes will remain robustly present. If the SC film remains and ϵ ≈ 50, appropriate for the surface of a TI such as in the type-I regime, then there is no additional constraint Bi Se . Equation (6.7) then gives J ≈ ð1 Å=ξÞ17.8 meV. It 2 3 on R , but the applied field must be kept below the is clear that using a superconductor with a large gap and thermodynamic critical field B of the film. short coherence length would aid the observation of the These considerations impose some practical constraints SYK physics in this system at reasonable energy and on the material composition and thickness d of the SC film. temperature scales. Taking Pb as a concrete example, we In general, we want the film to be sufficiently thin so that have ξ ≃ 52 nm, for d> 20 nm. Equation (6.5) does not scanning tunneling spectroscopy of the hole region can be impose additional restrictions on the hole diameter, and one performed. On the other hand, we want it to be either in the obtains J in the range of tens of μeV. This energy scale is type-I or weakly type-II regime such that Eq. (6.5) does not accessible to scanning tunneling spectroscopy (STS), enlarge the hole size significantly beyond the ideal radius which, as we argue below, constitutes the most convenient given by Eq. (6.2). For Pb, we have ðξ; λ Þ¼ ð83; 37Þ nm. experimental probe. Taking d ¼ 20 nm results in λ ≃ 137 nm, and Eq. (6.5) eff imposes only a mild increase in the hole size compared to B. Experimental detection the ideal, which should not adversely affect the zero modes. For Al, we have ðξ; λ Þ¼ð1600; 16Þ nm, and one can go In our proposed setup the experimental detection of the down to very thin films and still remain in the type-I signatures of the SYK state can be achieved using tunneling regime. spectroscopy. Either a planar tunneling measurement with a The TI film must be sufficiently thick so that it exhibits fixed probe weakly coupled to the TI surface or a scanning well-developed gapless surface states. For the Bi Se tunnel probe can be used. STS has the advantage of 2 3 family of materials this means thickness larger than 5 unit simultaneously being able to image the topography of cells. TI films close to this critical thickness will also be the device with nanoscale resolution and measure the easiest to bring to the neutrality point by backgating. tunneling conductance gðωÞ, which is proportional to the Using a hole close to the ideal size given by Eq. (6.2) will spectral function of the system AðωÞ. A recently developed also promote the interaction strength. Intuitively, it is clear technique [65] combines a STS tip with a miniature Hall that screened Coulomb interaction between electrons will probe, which allows additional measurement of the local have maximum effect on the zero modes if their wave magnetic field B at the sample surface. Such a probe is functions are packed as closely together as possible. With ideally suited for the proposed SYK model setup as it can this in mind one can give a crude estimate of the expected be used to independently determine the magnetic flux and interaction strength J as follows. Starting from Eq. (2.20) thus the number N of Majorana fermions in the system. In the large-N limit of the SYK model, AðωÞ exhibits the with V ¼ 2πe λ =ϵ and using Eq. (2.19) it is easy to 0 TF pﬃﬃﬃﬃﬃﬃ show that characteristic 1= jωj singularity [illustrated in Fig. 2(a)], which should be easy to distinguish from the semicircle rﬃﬃﬃﬃﬃﬃ 3 1=2 3 2 distribution that prevails in a system dominated by random N N 2πe λ 12 TF J ¼ J ¼ ; ð6:6Þ ijkl 2 3=2 two-fermion tunneling terms. In the large-N limit and at 3! 6 ϵξ sufficiently low temperature k T ≪ J, the detection of the SYK behavior via tunneling spectroscopy should therefore where we identify the length scale ζ with the SC coherence be relatively straightforward. length ξ. We can obtain a physically more transparent 2 2 In a realistic setup the number of flux quanta N will be expression by introducing the Bohr radius a ¼ ℏ =m e ≃ 0 e finite and perhaps not too large. In this case, our results in 0.52 Å and the corresponding Rydberg energy E ¼ pﬃﬃﬃﬃﬃﬃ 2 Fig. 7 show that the characteristic 1= jωj singularity is cut e =2a ≃ 13.6 eV: off such that Að0Þ is finite and grows with N only very sﬃﬃﬃﬃﬃﬃﬃ slowly. Additional results assembled in Fig. 9 indicate that 48π N a λ 0 TF even in this situation it is possible to distinguish the pﬃﬃﬃ J ¼ E : ð6:7Þ 3 2 M ϵξ s interaction-dominated SYK behavior from the behavior characteristic of the weakly interacting system with random Several remarks are in order. Equation (6.7) implies that two-fermion couplings. For J ≳ K, we observe a relatively for a fixed hole size R the coupling strength grows as smooth spectral density peaked at ω ¼ 0 characteristic of 3=2 J ∼ N . It is therefore advantageous to put as many flux the strongly interacting regime. In the opposite limit J ≲ K, 031006-14 BLACK HOLE ON A CHIP: PROPOSAL FOR A PHYSICAL … PHYS. REV. X 7, 031006 (2017) and experimental techniques. The proposal is to use 1.0 =2 /4 the surface of a 3D TI at its global neutrality point = /4 proximitized by a conventional superconductor with an =0 0.8 irregular-shaped hole and magnetic flux threaded through the hole. We demonstrate that the conventional screened 0.6 Coulomb interaction between electrons in such a setup leads to a Majorana fermion Hamiltonian at low energies 0.4 with requisite random four-fermion couplings. Detailed analysis indicates behavior consistent with that expected 0.2 of the SYK model. We give estimates for model parameters in the realistic systems and suggest experimental tests for the SYK behavior. This work thus provides connec- 0.0 -4.0 -2.0 0.0 2.0 4.0 tions between seemingly unrelated areas of research— /J mesoscopic physics, spin liquids, general relativity, and quantum chaos—and could lead to experimental insights FIG. 9. Spectral function AðωÞ calculated for the giant vortex into phemomena that are of great current interest. geometry with N ¼ 28 and coupling constants ðJ; KÞ¼ J ðcos θ; sin θÞ taken to interpolate between the fully interacting and noninteracting limits. ACKNOWLEDGMENTS The authors are indebted to J. Alicea, O. Can, A. Kitaev, nonuniversal fluctuations that strongly depend on the E. Lantagne-Hurtubise, M. Rozali, C.-M. Jian, I. Martin, specific disorder realization become increasingly promi- and S. Sachdev for illuminating discussions. We thank nent. Eventually, when J ≪ K, the spectral function con- Microsoft, NSERC, CIfAR, and Max Planck–UBC Centre sists of a series on N sharp peaks. These peaks occur at for Quantum Materials for support. D. I. P. is grateful to the eigenvalues of the N × N random Hermitian matrix iK KITP, where part of the research was conducted with ij and represent the single-particle excitations of the non- support of the National Science Foundation Grant No. NSF interacting problem at J ¼ 0. At large N, these peaks PHY11-25915. Numerical simulations were performed in merge to form a continuous distribution described by the part with computer resources provided by WestGrid and semicircle law. Compute Canada Calcul. Full numerical diagonalization of the SYK Hamiltonian is feasible for N up to 32 on a desktop computer and 15 15 involves a matrix of size 2 × 2 in each parity sector. By going to a supercomputer one can plausibly reach N ¼ 42 [1] E. P. Wigner, On the Statistical Distribution of the [66], but larger system sizes are out of reach due to the Widths and Spacings of Nuclear Resonance Levels, Proc. Cambridge Philos. Soc. 47, 790 (1951). exponential growth of the Hamiltonian matrix with N. [2] F. J. 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Physical Review X – American Physical Society (APS)

**Published: ** Jul 1, 2017

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