Tunneling into a finite Luttinger liquid coupled to noisy capacitive leads
Tunneling into a finite Luttinger liquid coupled to noisy capacitive leads
Štrkalj, Antonio;Ferguson, Michael S.;Wolf, Tobias M. R.;Levkivskyi, Ivan;Zilberberg, Oded
2018-09-05 00:00:00
Tunneling into a nite Luttinger liquid coupled to noisy capacitive leads Antonio Strkalj, Michael S. Ferguson, Tobias M. R. Wolf, Ivan Levkivskyi, and Oded Zilberberg Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland (Dated: September 6, 2018) Tunneling spectroscopy of one-dimensional interacting wires can be profoundly sensitive to the boundary conditions of the wire. Here, we analyze the tunneling spectroscopy of a wire coupled to capacitive metallic leads. Strikingly, with increasing many-body interactions in the wire, the impact of the boundary noise becomes more prominent. This interplay allows for a smooth crossover from standard 1D tunneling signatures into a regime where the tunneling is dominated by the
uctuations at the leads. This regime is characterized by an elevated zero-bias tunneling alongside a universal power-law decay at high energies. Furthermore, local tunneling measurements in this regime show a unique spatial-dependence that marks the formation of plasmonic standing waves in the wire. Our result oers a tunable method by which to control the boundary eects and measure the interaction strength (Luttinger parameter) within the wire. Advances in control and design of mesoscopic sys- distributions [25, 40]. Interestingly, despite the fact that tems have made it possible to realize a variety of ultra- the many-body interactions profoundly alter the emer- small electronic tunnel-junctions [1, 2]. In such junc- gent quasiparticle excitations in the wire relative to the tions, many-body interactions and coherent eects com- electronic boundaries, the wire{boundary interplay can- pete with the charge
uctuations and impedance of the not be revealed in DC-transport measurements due to environment to profoundly impact the resulting tunnel- the suppression of electron backscattering in clean wires ing characteristics; the tunneling inside the junction ex- [38, 46, 47]. In contrast, a tunnel-junction between a cites the electromagnetic modes of an external circuit superconducting or metallic scanning tunneling micro- making it extremely sensitive to the circuit's impedance scope (STM) and the wire is ideally suited to sense these [1{3]. This competition alters the tunneling density of eects, since it gives access to the wire's energy distribu- states (TDOS) of the various device constituents, with tion function [48, 49], or to the (local) TDOS [50] of the a wide variety of such eects seen in, e.g., normal-metal wire, respectively. The latter commonly displays power- tunnel-junctions [4], Josephon junctions [5] and transmis- law scaling dependent on the extent of many-body inter- sion lines [6]. Particular examples of such eects include, actions in the system [38, 51] { quanti ed by the Lut- among others, the Coulomb blockade [7], the Kondo ef- tinger parameter K { and is strongly impacted by the fect [8{10] and Andreev bound modes [11{14]. boundaries, i.e. impedance of the environment [3]. Tunnel-junctions involving one-dimensional (1D) In this work, we study the impact of noisy capacitive quantum wires are especially intriguing, since many- metallic leads on tunneling into an interacting quantum body interactions fundamentally alter the emergent wire. The capacitance in the leads imposes a nite re- many-body physics compared with conventional Fermi- sponse time in the wire, suppressing its fast high-energy liquid metals. Interacting wires are better described excitations. Surprisingly, with increasing many-body in- using Tomonaga-Luttinger liquid (TLL) theory [15{17]: teractions, the impact of the boundary noise on the wire the low-energy elementary excitations in 1D appear as is enhanced, and its TDOS displays this interplay by collective bosonic plasmon modes | in stark contrast to entering a regime where it is dominated by the classi- the constitutive fermionic electrons. Consequently, 1D cal impedance of the capacitive reservoirs: at low ener- systems show exotic phenomena, such as charge frac- gies, the nite length of the wire cuts o the expected tionalization of injected electrons [18, 19], spin-charge 1D tunneling zero-bias anomaly [44, 51], and a zero-bias separation [20, 21], and zero-bias anomalies (ZBA) [22{ tunneling peak appears instead as a function of the en- 25], all of which uniquely interplay with disorder [26, 27], vironment capacitance; at high energies, the characteris- quasi-disorder [28], and dissipation [29, 30]. Such 1D tic power-law growth is replaced by a universal ! de- eects are ubiquitous and have been observed in a wide cay [3]. Interestingly, this wire{environment competition variety of systems, including nanotubes [31, 32], GaAs introduces a unique spatial dependence to the TDOS, wires [20, 21], quantum Hall edges [33{35], as well as, thus oering an external handle by which to control the chains of spins or atoms [36, 37]. correlations in the wire, such that its Luttinger parame- More recently, signi cant progress was made in the de- ter can be tunably detected. scription of realistic nite-sized 1D wires with bound- ary conditions both in- and out-equilibrium [25, 38{41]. We consider a nite 1D wire coupled to metallic leads, These can generally be grouped into wires (i) with open depicted as an outer circuit that is characterized by an boundaries [42{44], (ii) connected to ohmic contacts [45], ohmic resistance R and the capacitance C , and probed by or (iii) coupled to inherently out-of-equilibrium charge a nearby STM, see Fig 1(a). The STM signal measures arXiv:1809.01631v1 [cond-mat.mes-hall] 5 Sep 2018 2 (a) (b) S ( ω) tem. This decay manifests as a power-law in the Green's S TM S ( ω) function [39, 41, 54] −1 R C ω θ( ω) quan tu m w e 2 −1 ( τ ω) R C L i 1 −1 C 0. 5 τ C R C lim G (x; t) = ; (2) L!1 2v (1 + it) where L is the length of the wire, is the bandwidth of −1 0 τ the electronic system, v is the Fermi velocity, and = R C F K + K =2 1 is the interaction-dependent power- (c) S TM law exponent for the Luttinger parameter K . For non- r r t B B A A interacting systems K = 1, and therefore = 1. − − − e e e In a nite wire, the eects of many-body interactions path B path A compete with the noise arising at the boundaries [25, 39, δj ( t ) 40, 45]. The latter is characterized, in our case, by a power spectral-density [55{57] lead lead interacting region ! (1 f (!)) FD S(!) j (!)j ( !) = ; (3) L=R L=R 2 2 δj ( t ) 1 + ! RC 0 L where = RC is the discharge time of the capacitor in RC the outer circuit, and f (!) = (1 + exp[ !=k T ]) is FD B FIG. 1. (a) A 1D metallic quantum wire of length L is con- the Fermi-Dirac distribution in the left and right leads { nected to metallic leads, depicted as an outer circuit that is characterized by an ohmic resistance R and the capacitance C . assumed here to be identical and uncorrelated. The main The leads act as electron reservoirs with well-de ned Fermi- dierence between (3) and the power spectral-density of Dirac distributions. The tunneling density of states [Eq. (1)] ideal ohmic leads is that the RC-circuit acts as an addi- at position x along the wire is probed by a nearby scanning tional low-pass lter [39, 52], see Fig. 1(b). tunneling microscope (STM). (b) The zero temperature power We are interested in how the boundary noise (3) and spectral-density S(!) of the RC-circuit's noise [Eq. (3)] (blue interaction-induced 1D plasmons manifest in the elec- solid line) and two asymptotic limits: (i) ! 1 (or- RC tronic correlations in the wire, e.g., in G (x; t). While ange dot-dashed line) corresponding to the behavior of ideal the noise is characterized by the discharge time , we ohmic leads [52] and (ii) ! 1 (red dashed line) where RC RC high-energy
uctuations are damped by the circuit's capaci- shall see below that the plasmonic waves are charac- tance. (c) (Top) An electron from the STM induces 1D plas- terized by their time-of-
ight through the nite wire, monic excitations, for which the nite wire acts as an eective cf. Eq. (9). We provide here rst a brief overview of our Fabry-P erot interferometer with re
ection (transmission) co- main results: the nite discharge time of the leads im- ecients r (t ). (Bottom) A schematic view of the wire A;B A;B poses two distinct regimes, (i) strong-capacitance regime as left/right-propagating modes connected to two identical (see Fig. 2), where the time-of-
ight is much shorter than leads, that impart current
uctuations j (t) onto the wire L=R [Eq. (3)]. the discharge time, , and (ii) the more com- RC monly studied complementary weak-capacitance regime with . The latter shows a standard TLL be- RC the local TDOS at position x along the wire [53] havior for short times t , whereas for long times the nite wire acts as a 0D Fabry-P erot cavity for the plas- i!t > < mons and free-electron correlations are reobtained (cf. (x; !) = i dt e G (x; t) G (x; t) ; (1) Refs. [3, 44]). Case (i) shows a richer behavior: at short times (t ; ), the boundary noise inhibits highly- RC where ! is the electron's energy, and G (x; t) and excited plasmons and consequently suppresses tunneling, G (x; t) are the lesser and greater Green's functions, re- whereas at long times (t ; ), both the interactions RC spectively. We work in natural units, where h; e = 1. In and noise correlations are averaged-out to yield a sim- < > equilibrium, G (x; t) = G (x; t) [53] and it suces ilar 0D plasmonic Fabry-P erot behavior. Interestingly, < y to analyze G (x; t) = i (x; t) (x; 0) , where we wrote at intermediate times ( < t < ), a competition be- RC its de nition using the electronic eld-operator (x; t), tween the TLL correlations and the boundary response and the average is taken with respect to the equilibrium ensues, showing both Fabry-P erot oscillations, as well ground state. as non-trivial power-laws in the electronic correlations, In 1D, interacting electrons form a TLL with collec- cf. Eq. (10) and see Fig. 2(a). Furthermore, the power- tive wave-like plasmonic excitations [15{17, 38, 54]. An laws show an unexpected dependence on the STM's po- electron injected from the STM into the wire excites plas- sition [58] that can be observed through [see Fig. 2(b)] monic modes that propagate away such that the proba- G (x; t) bility amplitude for the excitation to tunnel back into g ~(x; t) : (4) G (L=2; t) the STM decreases faster than in a non-interacting sys- ir 3 τ τ R C (a) In Fig. 3(a), we plot the TDOS in the strong- πv capacitance regime. The spatial dependence can be seen in the intermediate energy regime, see Fig. 3(b). For 2, comparison, in Fig. 3(c) we plot the TDOS for both L/ nite- and in nite-length interacting wires. The rela- tively
at peak of the TDOS at low energies is a re- -Re[ G (x, t )] sult of the nite-length of the wire that suppresses the Im[ G (x, t )] ZBA of an in nite TLL [Fig. 3(c)], and is in agreement with the free-electron behavior of the Green's function 0 50 100 150 200 t [ L/v ] at long times, cf. Fig. 2(a) and Refs. [44, 59]. At high F τ τ (b) -5 R C energies, interaction-induced Fabry-P erot oscillations ap- × 10 I m [ G < ( x , t ) ] pear but there is no interaction-dependent power-law x/L = 0 .5 growth as compared with both the nite- and in nite- x/L = 0 )]) x/L TLL, where the TDOS grows as (!)= / ! , with 0 = x, x/L = (K + K )=2 and = (!; = 0; K = 1) the 0 RC (( −5 ˜g x/L TDOS into a non-interacting metal with zero capaci- tance. This is a consequence of a linear, interaction- 0 [log −1 0 independent growth of the Green's function at short Re times, see Fig. 2(a). Hence, the noise of the capacitive leads suppresses the power-law growth and causes the 3 −1 5 TDOS to drop as = / ! , in similitude to high- -1 0 1 2 10 10 10 10 t [ L/v ] impedance tunnel-junctions [1, 3]. F To obtain our results, we closely follow the derivation FIG. 2. The Green's function of the wire in the strong- used in Refs. [44, 59]. We consider the Hamiltonian den- capacitance limit. (a) The imaginary (thin green line) sity of a single-channel wire [38, 39, 41, 44, 54] and real (thick red line) part of a lesser Green's function G (L=2; t) [Eq. (7)]. The dashed lines show the analyti- y y H(x) = i v (x) @ (x) (x) @ (x) cally obtained asymptotic limits for long (t ) and short F x R x L RC R L Z (t ) times. The shaded region (light blue) marks the RC time interval < t < where the interaction-induced cor- RC 0 0 + dy V (x y) (x) (y); (5) relations in the wire compete with the RC noise. (b) The ; =L;R real part of log(g ~(x; t)) [Eq. (4)] exhibiting the non-trivial power-law behavior of the Green's function depending on the where the left- and right-moving electrons ( = L; R) position of the STM tip (solid lines). Furthermore, our ana- are described by eld operators (x), and V (x) is lytical asymptotic result (dashed lines) [Eq. (11)] agrees with the electronic interaction between (normal-ordered) den- the numerical result (solid lines). In all plots, we use an ex- perimentally realizable interaction parameter U=v = 15, see, sity operators (x) = : (x) (x):. The rst term de- e.g., Refs. [31, 32], and large capacitance, = = 100. RC scribes the kinetic contribution for a linearized dispersion E(k) = v k around the Fermi momentum k , such F F ik x ik x F F that the electron eld (x) ' e (x) +e (x). L R We further assume that the eective electron-electron Green's function of a nite wire, we obtain interaction is point-like, i.e. V 0 (x) = U (x). Note that the linearized dispersion is associated with a band- i 1 < 2 width serving as a high-energy cut-o. Using bosoniza- G (x; t) = exp ( (x; t) (x; 0)) ; (7) 2v 2 tion [54], we introduce new bosonic eld operators (x) related to the electron density by (x) = @ =2, where we have used the fact that the charge-
uctuations with commutation relations (x); @ (y) = L=R x L=R at the boundaries are Gaussian distributed, and that 2i (x y). These elds are de ned via (x) =: 1=2 i (x) F F = 1. Note that the overall Green's function is ^ ^ F (=[2v ]) e , where the Klein factors F en- < < G (x; t) = G (x; t) + G (x; t) [53]. Using the equations L R sure fermionic anti-commutation of . In this lan- of motion for the elds [59], we nd (in similitude to guage, the Hamiltonian takes a simple quadratic form Ref. [44]) that G (x; t) i=(2v ) exp( I (x; t)) with [38, 39, 54] the integral v U U H(x) = + (@ ) + @ @ : (6) x x L x R 2 2 4 8 4 d! =L;R i!t I (x; t) = (1 e )F (x; !) S(!) ; (8) Substituting the bosonization identities into the lesser 4 where S(!) is as in Eq. (3). The structure-function (a) ω [2 π v /L ] -2 -1 0 1 2 10 10 10 10 10 1 x 1 + 2 cos(!) cos 2! ( ) 2 L F (x; !) (9) 3 1 cos(2!) x/L = 0 .1 captures both interaction eects through the param- x/L = 0 .2 1 2 2 -3 eter (1 + 8 v =U + 8 (v =U ) ) = [1 F F 10 x/L = 0 .3 2 2 4 1 8K =(1 + 6K +K )] , and the nite-length of the wire x/L = 0 .4 -6 through the time-of-
ight of the plasmonic excitations x/L = 0 .5 1 1=2 = (L=v ) (1 + U=v ) . This structure-function F F -9 is equivalent to that of a plasmonic Fabry-P erot interfer- 2π /τ 2π /τ R C (b) (c) ometer of length L. Indeed, the same expression is ob- -1 tained when describing a free-particle that is injected at a 2 L = ∞ position x and is re
ected from the two boundaries with L = ∞ re
ection and tunneling coecients r r, t t, A,B A,B 2 4 1 respectively, where = 2 r (1 + r ) [cf. Fig. 1(c) and Refs. [18, 40, 44, 60]]. This implies that the plasmonic character of excitations in the wire (due to interactions) -1 causes re
ections from the free-electron boundaries. 0 1 -2 0 2 We can now (i) evaluate G (x; t) numerically using 10 10 10 10 10 ω [2 π v /L ] ω [2 π v /L ] Eqs. (3) and (7)-(9) for dierent devices with vary- F ing = and U=v [59], as well as (ii) nd analytical RC F asymptotic results for the speci c time windows men- FIG. 3. (a) The normalized TDOS = in the strong- tioned above. In the latter, we assume that the STM is capacitance regime ( = = 100) calculated for ve dierent RC STM positions in the wire. At high energies, ! 2= , placed in proximity to the middle of the wire, such that RC the tunneling is suppressed and the TDOS exhibits a power- (1=2 x=L) 1. law decay. Interaction-induced Fabry-P erot oscillations with Strong-capacitance regime ( ) For short RC a period of 2= are present at high energies. For low ener- times, t , the real-part of the Green's func- RC gies, the TDOS is constant that depends on the value of . RC tion is linear, while its imaginary-part reaches a nite (b) A zoom-in on (a) where the TDOS is rescaled by a factor < 1 value, i.e., G (x; t ! 0) = (v ) (i =2 t= ), F RC ! such that the dierence between dierent measuring posi- see Fig. 2(a). This behavior leads to the reduced TDOS tions inside the wire can be seen more clearly. (c) The TDOS of a nite (blue solid line) and in nite (black dashed line) TLL at high energies, see Eq. (1) and Fig. 3(a). The large ca- when the capacitance in the leads is set to zero. In a nite- pacitance in the leads eectively acts as a low-pass lter length TLL, the zero-bias TDOS does not vanish but satu- for the plasmonic modes, and inhibits the conversion of rates at a nite value [44]. Note that the normalization of the high-energy STM electrons into plasmons. TDOS is with respect to the value of non-interacting TDOS At intermediate times, t , the main weight RC with vanishing capacitance, . The interaction strength used of the integral I (x; t) [Eq. (8)] lies at ! , where the RC in all plots is U=v = 15. 2 1 spectral function is approximated as S(!) 1= ! . RC We expand the cosine terms in Eq. (9) in small =t 1, to obtain gies, ! ( ) , where the spectral function is ap- RC i 1 proximated as S(!) ! (1 f (!)), see Fig. 1(b). < < FD G (x; t) G (L=2; t) ; (10) (x) Furthermore, for t, the structure function is con- 2v (1 + it) 2 2 stant, i.e. F (x; !) 1 + O( =t ). Hence, the leading with a spatially-dependent exponent term in Eq. (8) becomes I (x; t) =
+ log(t= ) +i=2 E RC 2 with
the Euler constant, resulting in a free-electron re- 2 2 2 1 x (K 1) < 1 (x) = : (11) sponse, G (t) = (v ) exp(
) =t [cf. Eq. (2)]. F E RC 3 2 2 L 2K RC The plasmons created by the STM re
ect back and forth multiple times between the boundaries such that their The rst factor in Eq. (10) does not depend on the po- interference 'washes out' the eects of 1D interactions, sition within the wire. Remarkably, however, the second and a 0D plasmonic cavity forms [3, 44]. factor has the same power-law form as that of the Green's function of an in nite TLL, see Eq. (2) { with the notable The interplay between noisy capacitive boundaries and dierence that the exponent has a spatial dependence. many-body interactions in a nite quantum wire can This exponent can be extracted from g ~(x; t) as de ned in smoothly alter its temporal and spatial correlations. Eq. (4), see Fig. 2 (b). Speci cally, we nd that the many-body interactions In the long time limit, t, the main weight drive the wire to display a TDOS with features that RC of the integral I (x; t) [Eq. (8)] stems from small ener- are dominated by the classical
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(9) were derived microscopically. Our results correspond to the sharp boundary limit of Ref. [40] where the re
ection coecient is r = (1 K )=(1 + K ). [61] M. A. Cazalilla, Reviews of Modern Physics 83, 1405 (2011). [62] B. Yang, Y.-Y. Chen, Y.-G. Zheng, H. Sun, H.-N. Dai, X.-W. Guan, Z.-S. Yuan, and J.-W. Pan, Phys. Rev. Lett. 119, 165701 (2017). [63] T. Giamarchi, Physics 10, 115 (2017). [64] M. J. Gullans, J. D. Thompson, Y. Wang, Q.-Y. Liang, V. Vuleti c, M. D. Lukin, and A. V. Gorshkov, Phys. Rev. Lett. 117, 113601 (2016). 1 Supplemental Material for Tunneling into a nite Luttinger liquid coupled to noisy capacitive leads Antonio Strkalj, Michael S. Ferguson, Tobias M. R. Wolf, Ivan Levkivskyi, and Oded Zilberberg Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland In the main text, we analyze a nite-length 1D wire subject to charge-
uctuations at its boundaries [described by a power spectral-density S(!) as in Eq. (3) in the main text]. The electronic modes in the wire are naturally described by plasmonic modes according to the Hamiltonian density v U U F 2 H(x) = + (@ ) + @ @ ; x x L x R 2 2 4 8 4 =L;R where are bosonic elds, v is the Fermi velocity and U is the (repulsive) interaction strength, see Eq. (6) in L;R F the main text. The bosonic elds satisfy commutation relations [ (x); @ (y)] = 2i (x y). In Section I, we show how to obtain the eigenmodes of H(x) for given boundary conditions imposed by the current ucations at the interface between the wire and the outer circuit. In Section II and Section III we elaborate on the calculations for the weak and strong capacitance regimes discussed in the main text. In Section IV, we show in more detail how the real part of the wire's Green's function depends on the length of the wire L and the discharge time of the outer circuit's capacitor. In Section V, we show how the TDOS behaves for dierent wire lengths L and RC values of . RC I. EQUATIONS OF MOTION The equations of motion for the modes in the wire can be obtained from the Hamiltonian and the commutation relation for the left- and the right-moving bosonic elds (x; t) [3], i.e., U U @ (x; t) = i[H; (x; t)] = v + @ (x; t) + @ (x; t); (I.1) t L L F x L x R 2 2 U U @ (x; t) = i[H; (x; t)] = v + @ (x; t) @ (x; t): (I.2) t R R F x R x L 2 2 The quadratic Hamiltonian density H(x) is diagonal in the basis of the elds (x; t) = ( (x; t) (x; t)) , L R whose equations of motion directly follow from Eqs. (I.1) and (I.2), i.e., @ (x; t) = v @ (x; t); t + F x @ (x; t) = v 1 + @ (x; t); (I.3) t F x + 1=2 where the de nition of the Luttinger parameter naturally emerge as K = 1 + . L i!t The system of dierential equations (I.3) can be solved in frequency space, i.e. (x; t) = d! e (x; !), 2v resulting in 2 2 ~ ~ @ (x; !) + (!) (x; !) = 0 (I.4) with a single parameter (!) = K !=v . The solutions are plane waves of the form i (!) x i (!) x (x; !) = c e + c e ; (I.5) 1; 2; where the four coecients c ; c are still related through Eqs. (I.3), leading to the general solution 1; 2; 1 ix ix ix ix p (x; !) = c
e + c
e L 1 + 2 (x; !) = c e + c e + 1 2 , ; (I.6) 1 1 1 ix ix ix ix ~ ~ (x; !) = c e c e (x; !) = c
e + c
e 1 2 R 1 2 + K K 2 2 where
= 1 K and c , c are two independent coecients determined by boundary conditions. The latter 1 2 are given by the continuity equation for the
uctuating currents j (t) at the left and right reservoir [see Fig. 1(c) L;R in the main text] @ (L; t) = 2 j (t); @ (0; t) = 2 j (t): (I.7) t L L t R R Transforming Eq. (I.7) to frequency space and substituting the general solution Eq. (I.6) then results in iL iL e c +
e c = j (!) + 1 2 L i ! c +
c = j (!) (I.8) 1 + 2 R i ! where j (!); j (!) are the Fourier transforms of j (t). Solving for the coecients c and substituting them L R L;R 1;2 into Eq. (I.6) then yields 2 ix 2 ix 1 1 (x; !)
e
e 2i
sin((L x)) j (!) L + L = i : (I.9) 2 i(L x) 2 i(L x) 2 2 ~ iL iL 2i
sin(x)
e
e j (!) (x; !) !
e
e + R R + II. WEAK CAPACITANCE REGIME ( ) RC In the case where is the smallest time scale, the spectral function of the boundary
uctuations [cf. Eq. (3) RC in the main text] is approximated as S(!) = ! (1 f (!)) [see Fig. 1(b) in a main text]. In this case the integral FD I (x; t) in Eq. (9) of the main text can be evaluated in both asymptotic limits t and t . In the short time limit (t ), the fast-oscillating cosines in Eq. (10) of the main text can be averaged. The spatial dependance near the middle of the wire (x L=2) vanishes so that F (x; !) = (1 + )=(1 ) and hence I (t) = log(1 + it). Combining the two approximations together, we obtain Eq. (2) in the main text for the Green's function of the wire, i.e., i 1 G (x L=2; t ) = : (II.1) 2v (1 + it) This result can be interpreted as follows: an electron injected from the STM forms plasmons, which for short times t do not have time to propagate to and re
ect from the boundaries. Hence, there is no Fabry-P erot interference and we observe the Green's function behavior of (in nite) interacting 1D wires. In the long time limit (t ), the cosines in Eq. (10) of the main text can be expanded in small =t 1 such that 2 2 F (x; !) 1 +O( =t ) and hence I (t) = log(1 + it). The Green's function then takes the form i 1 G (x L=2; t ) = : (II.2) 2v 1 + it Comparing this result with Eq. (3) in the main text, we recover the result of free electrons ( = 1) similar to the long-time limit of strong capacitance regime in the main text. This is not surprising, since the long-time limit the emergent 0D Fabry-Perot cavity should always tend to the result of non-interacting electrons. III. STRONG CAPACITANCE REGIME ( ) RC We can write the integral from Eq. (9) of the main text as I (t) = I (t) +I (t) and evaluate it analytically in cos sin the limit of strong capacitance [1, 2]: t= t= RC RC ! 1 cos(!t) t e t e t I (t) = lim d! =
+ log Ei[ ] Ei[ ]; (III.1) cos E 2 2 2 2 !0 ! + 1 + ! 2 2 RC RC RC 0 RC ! sin(!t) t= RC I (t) = lim d! = 1 e ; (III.2) sin 2 2 2 2 !0 ! + 1 + ! 2 0 RC where
= 0:577::: is an Euler's constant and Ei[x] = dy e =y is the exponential integral for real non-zero values of x. The asymptotic limits of the exponential integral Ei[t= ] are: RC 3 + log for t E RC RC Ei[ ] ' t= RC RC for t RC t= RC This leads to the asymptotic result i for t RC RC I (t) = : (III.3) + log + i for t E RC RC IV. GREEN'S FUNCTION AS FUNCTION OF DISCHARGE TIME AND LENGTH L RC In Sup. Fig. 1(a), we show the behavior of the real part of the Green's function G (x; t) for dierent values of the discharge time when the time of
ight is xed (by xing the length L and interaction strength U=v ). We RC F can see the smooth transition from the weak-capacitance regime to the strong capacitance regime. The former is characterized by a 1=t dependence at short times (t ) due to the interactions and a 1=t free electron behavior at long times (t ). The latter shows interaction-independent linear dependence at short times and a free electron behavior 1=t at long times. Furthermore, in the weak capacitance limit, Fabry-P erot oscillations with length-scale 2 can be seen. In Sup. Fig. 1(b), we show the same interpolation between weak- and strong-capacitance regime but now keeping the value of the discharge time xed while instead changing the length L of the wire (and consequently RC ). (a) −1 x, −3 Re −1 0 1 2 3 10 10 10 10 10 t [L/v ] (b) R C x, [ −1 Re −2 −1 0 1 10 10 10 10 t [L/v ] Supplementary Figure 1. Dependence of the Green's function of a nite wire embedded in a capacative circuit on (a) the discharge time and (b) the length L of the wire. In both cases, the blue lines mark the weak capacitance regime, RC = 1 and the red lines mark the the strong capacitance regime = 1. The black curves show how the Green's RC RC function interpolates between the two regimes as the respective paramter is varied. In both plots, interaction strength is xed at U=v = 15. π π 4 V. TDOS FOR DIFFERENT WIRE LENGTHS L AND VALUES OF RC In the Sup. Fig. 2 we show the behavior of the tunneling density of states = for dierent lengths of the wire, both for vanishing discharge time, i.e., = 0 [Fig. 2(a-c)] and for nite [Fig. 2(d-f )]. RC RC (a) (c) (b) τ = 0 L = 0. 01 R C L = 1 τ = 0 L = 100 τ = 0 R C R C −1 (d) (e) (f) 4 τ > 0 τ > 0 L = 0. 01 L = 1 L = 100 τ > 0 R C R C R C −2 -1 0 10 10 −2 0 2 0 2 0 2 10 10 10 10 10 10 10 ω [2 π v /L ] ω [2 π v /L ] ω [2 π v /L ] F F F Supplementary Figure 2. The normalized TDOS = for dierent lengths L [in units of v ] of the wire for (a-c) in absence 0 F of an external capacative circuit ( = 0) and (d-f ) in presence of one ( 6= 0). The normalization is w.r.t. to the TDOS RC RC 0 in the non-interacting case and in absence of the capacitance, i.e., = 0. (a) For a short wire, the constant TDOS indicates RC the free-electron behavior caused by multiple re
ections of the plasmons against the boundaries. As L is further decreased, the TDOS will approach . (b) For intermediate lengths, the TDOS shows both free-electron behavior (constant TDOS) at low energies and TLL behavior (power-law growth) at high energies, see Fig. 3(c) in the main text. Furthermore, Fabry-P erot oscillations with a period of 2= appear. (c) For a long wire, the TDOS follows a power-law behavior characteristic for the in nite size TLL-s. In presence of a capacative outer circuit, the TDOS of (d) a short wire is completely determined by the uctuations with an elevated zero-bias peak and universal ! decay at high energies. (e) At intermediate lengths, the result from the main text is recovered, see Fig. 3(a). Fabry-P erot oscillations appear at higher energies, but the overall shape is the same as in the case of (d). (f ) In case of a long wire, we see that the TDOS for low and high energies behaves as in (d), but in addition an intermediate regime appears, in which the power-law behavior of (b) is recovered [see also the inset of (f )]. This intermediate region is characteristic for the tunneling into a TLL. Throughout, we used a nite interaction strength U=v = 15. [1] E. G. Idrisov, I. P. Levkivskyi and E. V. Sukhorukov, Phys. Rev. B, 96, 155408 (2017). [2] I. S. Gradshte n and I. M. Ry zik, Table of Integrals, Series, and Products, 8th Edition, (Academic Press, 2004). [3] Y. V. Nazarov, A. A. Odintsov and D. V. Averin, EPL (Europhysics Letters) 37, 213 (1997)
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Tunneling into a finite Luttinger liquid coupled to noisy capacitive leads