Interaction Quench in Nonequilibrium Luttinger Liquids
IInteraction Quench in Nonequilibrium Luttinger Liquids
St´ephane Ngo Dinh, Dmitry A. Bagrets, and Alexander D. Mirlin , , Institut f¨ur Theorie der Kondensierten Materie and DFG Center for Functional Nanostructures,Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, 50937 K¨oln, Germany Institut f¨ur Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia (Dated: September 10, 2018)We study the relaxation dynamics of a nonequilibrium
Luttinger liquid after a sudden interactionswitch-on (“quench”), focussing on a double-step initial momentum distribution function. In theframework of the non-equilibrium bosonization, the results are obtained in terms of singular Fred-holm determinants that are evaluated numerically and whose asymptotics are found analytically.While the quasi-particle weights decay exponentially with time after the quench, this is not a relax-ation into a thermal state, in view of the integrability of the model. The steady-state distributionemerging at infinite times retains two edges which support Luttinger-liquid-like power-law singular-ities smeared by dephasing. The obtained critical exponents and the dephasing length are found todepend on the initial nonequilibrium state.
PACS numbers: 71.10.Pm, 67.85.Lm, 03.75.Ss
I. INTRODUCTION
Quantum physics of interacting one-dimensional (1D)systems represents a fascinating research area . Mostimportant experimental realizations include electrons in1D nanostructures (quantum Hall and topological insu-lator edges, carbon nanotubes, semiconductor quantumwires), quantum spin chains, and cold atoms (bosons aswell as fermions) in optical traps. One of central direc-tions of current research is the physics of nonequilibriumphenomena in these structures.In the cold atoms context (see Ref. 2 for a review, themost frequently considered nonequilibrium setting is aquantum quench : one explores a quantum evolutionof the system after a sudden change of one of the param-eters. In particular, one can modify the optical lattice po-tential confining the atoms. An alternative possibility isto suddenly change the interaction strength by employinga strong dependence of interaction on the magnetic fieldin the vicinity of Feshbach resonance. It has been recog-nized that, upon an interaction quench, a Luttinger liq-uid shows an interesting dynamics and eventually evolvesinto a non-thermal state characterized by nonequilibriumpower-law correlations . Subsequent theoretical worksexplored various generalizations of this problem .On the other hand, in the context of electronic sys-tems, typical nonequilibrium setups are of steady-statecharacter (as obtained by applying bias voltages to someof source electrodes). Recent experiments have addressednonequilibrium spectroscopy of carbon nanotubes andquantum Hall edge states as well as nonequilibriumedge state interferometry . On the theory side, oneof important recent theoretical advances was a develop-ment of the method of nonequilibrium bosonization that permits, in particular, to treat Luttinger liquids withdistribution functions of incoming electrons that havemultiple Fermi edges. It was shown that this leads to a multiple-branch zero-bias anomaly with exponents anddephasing rates controlled by the nonequilibrium state ofthe system. Related results have been obtained for theproblems of quantum Hall edge state spectroscopy and Mach-Zehnder interferometry .While two types of nonequilibrium Luttinger liquid se-tups (a temporal interaction quench and a steady-statewith multiple Fermi edges originating from applied volt-ages) are quite different, there is a remarkable similaritybetween the results. In both cases, one finds non-trivialpower-law exponents that are essentially different fromthe equilibrium ones. In this paper, we show that bothkinds of nonequilibrium settings can be treated withina unified framework of the nonequilibrium bosonization.We employ this formalism to explore the dynamics af-ter an interaction quench in an initially nonequilibriumLuttinger liquid. The Fermi-edge exponents and dephas-ing are controlled by charge fractionalization at tempo-ral (and possibly spatial) boundaries of the interactionregion and by the input nonequilibrium distributions. II. EQUILIBRIUM QUENCH IN THELUTTINGER MODEL
The Luttinger model describes the low-energy physicsof interacting 1D fermions. It turns out that low-energy excitations can be fully captured in terms ofbosonic modes. A free 1D fermionic system with right-(left-)moving modes ψ + ( ψ − ) close to the Fermi pointscan be mapped onto a free 1D bosonic system with alinear spectrum H = − iv F : (cid:90) d x (cid:16) ψ † + ∂ x ψ + − ψ †− ∂ x ψ − (cid:17) := πv F : (cid:90) d x (cid:0) (cid:37) + (cid:37) − (cid:1) : a r X i v : . [ c ond - m a t . s t r- e l ] O c t with density operators (cid:37) ± ( x ) =: ψ †± ( x ) ψ ± ( x ) : and Fermivelocity v F . The fermionic operators can be likewise ex-pressed in terms of bosonic operators ψ η ( x ) ∼ η √ πa e iφ η , η = ± , where the phase operators φ η are related to densitiesvia (cid:37) η ( x ) = ( η/ π ) ∂ x φ η ( x ). While counterpropagatingdensity modes are uncoupled for originally free fermions,they are mixed by interaction. The interaction Hamilto-nian for short-range interaction is H int ( t ) = g ( t )2 : (cid:90) d x (cid:0) (cid:37) + (cid:37) − (cid:1) :+ g ( t ) : (cid:90) d x (cid:37) + ( x ) (cid:37) − ( x ) : . To describe the interaction quench we let the couplingparameters g j ( t ) = g j θ ( t ) be time dependent. In morephysical terms we assume that the switching time is muchshorter than all characteristic time scales of the problemset for example by the inverse voltage U , see Sec. III Abelow. In the presence of interaction the new bosoniceigenmodes, the “plasmons” ˜ (cid:37) η , are obtained by the Bo-goliubov transformation (cid:18) (cid:37) + (cid:37) − (cid:19) = (cid:18) c ss c (cid:19) (cid:18) ˜ (cid:37) + ˜ (cid:37) − (cid:19) , c ≡ K √ K , s ≡ − K √ K (1)with the Luttinger parameter K = (cid:114) πv F + g − g πv F + g + g . The full Hamiltonian after the quench then reads H = H + H int ( t >
0) = πu : (cid:90) d x (cid:0) ˜ (cid:37) + ˜ (cid:37) − (cid:1) : (2)with the plasmon velocity u = v F (cid:113) (1 + g / (2 πv F )) + ( g / (2 πv F )) . In thermal equilibrium the many-body density matrixˆ (cid:37) = Z − e − H/T is a function of H , thus it is straightfor-wardly expressed as an exponential of bilinears of bosonicfields.Ref. 8,14 considered time-evolution after a sudden in-teraction switch-on. The initially noninteracting sys-tem is prepared in the thermal equilibrium state ˆ (cid:37) = Z − e − H /T which after the quench no longer representsequilibrium (with respect to the full Hamiltonian H ).However, the time-evolution of ˆ (cid:37) , (cid:37) η and ψ η can bededuced by the Bogoliubov transformation (1). Calcula-tions there were performed with a finite interaction range R ∼ v F / Λ as short-distance regularization. At long dis-tances ¯ x (cid:29) R results are insensitive to the regularizationscheme, and momentum-dependent coupling parameters(associated with finite interaction range) can be replaced by their zero momentum values, g j ( q ) ≈ g j ( q = 0). Theequal-time correlation function then is G < eq+ (¯ x, ¯ t ; 0 , ¯ t ) = G < (¯ x, (cid:12)(cid:12)(cid:12)(cid:12) R ¯ x (cid:12)(cid:12)(cid:12)(cid:12) ˜ γ (cid:12)(cid:12)(cid:12)(cid:12) (2 u ¯ t ) − ¯ x (2 u ¯ t ) (cid:12)(cid:12)(cid:12)(cid:12) ˜ γ / , (3)where the exponent is determined by ˜ γ ≡ (1 − K ) / K and G < (¯ x,
0) is the free fermionic Green’s function.For short times such that 2 u ¯ t (cid:28) | ¯ x | the correlationfunction G < + (¯ t ; ¯ x, ≈ Z (¯ t ) G < (¯ x ; 0) can be interpretedas the Green’s function of an effective time-dependentFermi liquid with “Landau quasiparticle weight” Z (¯ t ) = ( R / u ¯ t ) ˜ γ (4)which gives rise to a discontinuity in the momentum dis-tribution function n + ( p ) at Fermi momentum p = p F .According to (4) the jump decays algebraically withtime ¯ t .For large times ¯ t → ∞ the system reaches a time-independent steady state with power-law correlations G < + (¯ t (cid:29) ¯ x/u ; ¯ x,
0) = G < (0 , ¯ x ) | R / ¯ x | ˜ γ . The corresponding momentum distribution function nolonger exhibits a discontinuity at p = p F , but insteadhas a power-law singularity ∼ | p − p F | ˜ γ . This behavioris very similar to that observed in an equilibrium Lut-tinger liquid , however with an exponent ˜ γ that differsfrom the equilibrium one, 2 γ = (1 − K ) / K . Hence,while interactions drive the dynamical evolution whichdestroy the Fermi-liquid character of the spectral func-tion, the integrability prevents the system from relax-ation into thermal equilibrium. III. QUENCH IN THE OUT-OF-EQUILIBRIUMLUTTINGER MODEL
In this section we consider the quench problem in theLuttinger liquid prepared in a nonequilibrium initial statewith double-step distribution functions. First we presentthe key details of our calculations within the nonequilib-rium bosonization framework and then discuss the ob-tained results.
A. Solution via nonequilibrium bosonization
In our paper we consider initial states in whichfermionic right-/left-moving single-particle states ( ± , (cid:15) i )are independently occupied according to distributionfunctions f ± ( (cid:15) i ) = (1 − a ) θ ( − (cid:15) i ) + aθ ( U − (cid:15) i ) whichhave two sharp Fermi edges at (cid:15) = 0 and U . For suchnonequilibrium states the initial density matrix is not anexponential of an operator quadratic in the plasmonic xt (cid:142) t xt (cid:45) c cs (cid:45) s s (cid:45) c (cid:45) s c (cid:45) cs cscs (cid:45) cs v F t ∆ (cid:43) (cid:144) Π v F t ∆ (cid:45) (cid:144) Π xt (cid:142) t xt (cid:45) c c s (cid:45) s s (cid:45) c (cid:45) s c (cid:45) cs cs cs (cid:45) cs v F t ∆ (cid:43) (cid:144) Π v F t ∆ (cid:45) (cid:144) Π FIG. 1: Trajectories of density peaks and the corresponding counting phases δ ± for the Green’s function (5) in the case of shorttimes 2 u ¯ t < ¯ x ( left ) and long times 2 u ¯ t > ¯ x > right ). The x-axis corresponds to the time t = 0 when the quench takes place. modes. This makes the nonequilibrium problem consid-erably more complicated in comparison with the equilib-rium one.A general framework to deal with this class of prob-lems has been developed in Ref. 37 where the operatorbosonization method was combined with the Keldysh ac-tion technique. Within this approach the right-movers’single-particle Green’s function iG < + (¯ x, ¯ t ; 0 , ¯ t ) ≡ − (cid:68) ψ † + (0 , ¯ t ) ψ + (¯ x, ¯ t ) (cid:69) (5)= iG < eq+ (¯ x, ¯ t ; 0 , ¯ t ) (cid:89) η = ± ∆ η [ δ η ] / ∆ [ δ η ]can be expressed in terms of a functional determinant ofa Fredholm operator,∆ η [ δ η ] = Det (cid:2) − f η + e iδ η f η (cid:3) . While the distribution function f η is diagonal in energyrepresentation, the counting phase δ η ( t ) = 4 πv F lim ˜ t →−∞ (cid:90) t − t d τ (cid:37) qη ( ηv F τ, ˜ t ) (6)is diagonal in the conjugate time representation. Theequilibrium value ∆ [ δ η ] of ∆ η [ δ η ] is obtained by replac-ing f η by the equilibrium distribution function f ( (cid:15) ) = θ ( − (cid:15) ). The counting phase is sensitive to the asymp-totic behavior of the density trajectory (cid:37) qη , which is theadvanced solution of the classical equations of motion (cid:104) ∂ t + (cid:16) v F + g π (cid:17) ∂ x (cid:105) (cid:37) q + ( x, t ) + g π ∂ x (cid:37) q − ( x, t ) = J ( x, t ) , (cid:104) ∂ t − (cid:16) v F + g π (cid:17) ∂ x (cid:105) (cid:37) q − ( x, t ) − g π ∂ x (cid:37) q + ( x, t ) =0 . (7)The source term J ( x, t ) ≡ δ ( t − ¯ t ) [ δ ( x ) − δ ( x − ¯ x )] onthe right-hand side of the equation corresponds to the in-jection of a right-moving fermion at position 0 and time¯ t and its removal at (¯ x, ¯ t ). After the quench, t >
0, theinteraction couples right- and left-moving density modes.To decouple them we apply the Bogoliubov transforma-tion (1), yielding[ ∂ t + u∂ x ] ˜ (cid:37) q + ( x, t ) = cJ ( x, t ) , [ ∂ t − u∂ x ] ˜ (cid:37) q − ( x, t ) = − sJ ( x, t ) . (8) The charge configuration ˜ (cid:37) qη describes the “advancedcharge response”to the aforementioned injection and re-moval of the right-moving fermion. It is a superpositionof δ -peaks moving with velocities ± u after the quenchand ± v F prior to it.Solving the linear Eqs. (8) we find that at times t > (cid:37) q + ( x, t ) = − c θ (¯ t − t ) × { δ [ x − u ( t − ¯ t )] − δ [ x − ¯ x − u ( t − ¯ t )] } , ˜ (cid:37) q − ( x, t ) = 12 s θ (¯ t − t ) × { δ [ x + u ( t − ¯ t )] − δ [ x − ¯ x + u ( t − ¯ t )] } . Requiring continuity at t = 0 and inverting the Bogoli-ubov transformation we obtain the charge density priorto the quench, t < (cid:37) q + ( x, t ) = − (cid:110) c ( δ [ x + u ¯ t − v F t ] − δ [ x − ¯ x + u ¯ t − v F t ]) − s ( δ [ x − u ¯ t − v F t ] − δ [ x − ¯ x − u ¯ t − v F t ]) (cid:111) ,(cid:37) q − ( x, t ) = − cs (cid:110) ( δ [ x + u ¯ t + v F t ] − δ [ x − ¯ x + u ¯ t + v F t ]) − ( δ [ x − u ¯ t + v F t ] − δ [ x − ¯ x − u ¯ t + v F t ]) (cid:111) . The latter yields the counting phases δ + ( t ) = 2 π (cid:40) c (cid:18) θ (cid:20) t − − ¯ x + u ¯ tv F (cid:21) − θ (cid:20) t − u ¯ tv F (cid:21)(cid:19) (9) − s (cid:18) θ (cid:20) t − − ¯ x − u ¯ tv F (cid:21) − θ (cid:20) t − − u ¯ tv F (cid:21)(cid:19)(cid:41) ,δ − ( t ) = − πcs (cid:40)(cid:18) θ (cid:20) t − − u ¯ tv F (cid:21) − θ (cid:20) t − ¯ x − u ¯ tv F (cid:21)(cid:19) − (cid:18) θ (cid:20) t − u ¯ tv F (cid:21) − θ (cid:20) t − ¯ x + u ¯ tv F (cid:21)(cid:19)(cid:41) . (10)We visualize the above solutions in Fig. 1 which showsthe δ -peak density trajectories in the ( x, t )-plane. Thefilled circles indicate the fermion injection and removalat time ¯ t . Each fermion fractionalizes into right- and left-moving plasmonic modes with weights c and − s . At thetime of quench ( t = 0) the plasmonic peaks disintegrateinto bare particle-hole pair excitations (small circles on x -axis) which propagate with smaller velocity v F . Thecounting phases δ ± are calculated at ˜ t → −∞ .The left panel of Fig. 1 depicts the situation of 2 u ¯ t < | ¯ x | and the right panel corresponds to the case 2 u ¯ t > | ¯ x | .In the second case the phases δ η split into two indepen-dent pulses δ (1) η , δ (2) η of duration | ¯ x | /v F which are shiftedby 2 u ¯ t/v F . In the long-time limit 2 u ¯ t (cid:29) | ¯ x | , the coher-ence of right- and left-moving plasmons is negligible andthe determinant ∆ η [ δ η ] (cid:39) ∆ η [ δ (1) η ]∆ η [ δ (2) η ] factorizes intotwo single-pulse determinants which are of Toeplitz type.Following Ref. 38 we regularize the determinants∆ η [ δ η ] by introducing an ultraviolet cutoff Λ and dis-cretize times in steps ∆ t = π/ Λ. For counting phases δ η which are vanishing outside some time interval of length τ , the discretization gives rise to N × N -matrices with N = τ Λ /π . Here we are interested in δ η ( t ) which arepiecewise constant functions. Such dependence leads tothe matrices of the generalized Toeplitz form (see Ap-pendix A). Various mathematical results exist for thelong-time asymptotic behavior of their determinants.In the simplest situation that δ η are single rectangularpulses [as e.g. δ (1) η ( t )] matrices are of Toeplitz form withsymbols exhibiting Fisher-Hartwig singularities. TheFisher-Hartwig conjecture then gives the leading expo-nential and power-law contribution to ∆ η [ δ η ] for N (cid:29) δ ± ( t ) shown in Fig. 1 are not of a simple rectangular form. They stemfrom the superposition of two rectangular pulses andtherefore possess four step-like discontinuities in time.This leads to a class of matrices that are a generaliza-tion of Toeplitz matrices. The asymptotic behavior ofthe corresponding determinants can be found by a fur-ther generalization of the Fisher-Hartwig conjecture (seeAppendix A) which was put forward in Ref. 39. Thisconjecture was supported both by analytical and nu-merical arguments.In Sec. III B we present and discuss the results for theGreen’s function (5) obtained by means of the analysisof corresponding singular Fredholm determinants. Thedetails of these calculation can be found in Appendix B. B. Results
In general the determinants entering Eq. (5) can be ef-ficiently evaluated numerically . For sufficiently longtimes u ¯ t (cid:29) v F /U analytical asymptotics can be found aswe discuss below.For an arbitrary relation between ¯ x and u ¯ t we havefound that the Green’s function is a linear combination ofterms involving different powers of | ¯ x | , | u ¯ t + ¯ x | , | u ¯ t − ¯ x | ,and 2 u ¯ t . For simplicity we will focus on two limitingcases: (i) long-distance behavior at finite times, ¯ x (cid:29) u ¯ t ,and (ii) the stationary limit ¯ t → ∞ . We also assume amoderate repulsive interaction.For finite times ¯ t and long distances ¯ x (cid:29) u ¯ t (cid:29) v F /U the Green’s function is determined by two dominantterms, G < + (¯ x, ¯ t ; 0 , ¯ t ) = G < (¯ x, e − ¯ t/ (2 τ ϕ ) (cid:34) ˜Γ (cid:48) (cid:18) u ¯ t Λ πv F (cid:19) T (cid:48) (cid:18) πU Λ (cid:19) V (cid:48) + ˜Γ (cid:48) e i ¯ xU/v F (cid:18) u ¯ t Λ πv F (cid:19) T (cid:48) (cid:18) πU Λ (cid:19) V (cid:48) (cid:35) (11)with exponents T (cid:48) j ( V (cid:48) j ) = − Re (cid:34)(cid:18) − j + s + 2 β (cid:19) + (cid:16) s + 2 ˜ β (cid:17) − ± c s (cid:35) , j = 1 , , and decay rate τ − ϕ = − U uv F Im ( β + ˜ β ) , (12)where β = 12 πi ln (cid:104) ae − πis + 1 − a (cid:105) , (13)˜ β = 12 πi ln (cid:2) ae πics + 1 − a (cid:3) , (14)and ˜Γ (cid:48) j are numerical prefactors. Similar to the equilibrium quench, the entire ¯ x -dependence of the interacting Green’s function after thequench is given by the noninteracting factor, G > ∝ ¯ x − so that correlations drop off with distance in a Fermi-liquid-like manner. Correspondingly, the momentum dis-tribution function has discontinuities at p = p F and p = p F + U/v F , signaling the existence of Landau quasi-particle states (see Fig. III B for K = 0 . a = 0 . a (cid:76) (cid:72) p (cid:45) p F (cid:76)(cid:144)(cid:64) U (cid:144) v F (cid:68) n (cid:43) (cid:72) p (cid:76) b (cid:76) i n c r ea s i ng t (cid:72) p (cid:45) p F (cid:76)(cid:144)(cid:64) U (cid:144) v F (cid:68) n (cid:43) (cid:72) p (cid:76) FIG. 2: (a) The momentum distribution function for K = 0 . a = 0 . t Uu/v F =1, 10, 25, and in the stationarylimit ¯ t → ∞ , as obtained by a numerical evaluation of the determinants in Eq. (5). Clear discontinuities at the edges p = p F and p = p F + U/v F [see zoom (b)] are visible which decrease with time ¯ t according to Eq. (15) and eventually vanish. exhibits quasiparticles with (in general different) weights Z ∝ e − ¯ t/ (2 τ ϕ ) ¯ t T (cid:48) U V (cid:48) , Z ∝ e − ¯ t/ (2 τ ϕ ) ¯ t T (cid:48) U V (cid:48) . (15)In striking contrast to the equilibrium situation, thequasiparticle weights are not simply algebraically sup-pressed with time, but also exhibit exponential decaywith characteristic time τ ϕ due to nonequilibrium de-phasing. Let us now turn to the long-time limit, ¯ t → ∞ . Thequenched system then relaxes to a stationary state with-out Fermi liquid discontinuities, but with critical power-law correlations characteristic for Luttinger liquid. In thecase of moderate repulsive interaction √ − ≤ K ≤ x (cid:29) v F U − leading contributions to the Green’sfunction read G < + (¯ t (cid:29) ¯ x/u ; ¯ x,
0) = G < (0; ¯ x ) e − κ | ¯ x | (cid:32) ˜Γ (cid:12)(cid:12)(cid:12)(cid:12) Λ¯ xπv F (cid:12)(cid:12)(cid:12)(cid:12) X (cid:12)(cid:12)(cid:12)(cid:12) πU Λ (cid:12)(cid:12)(cid:12)(cid:12) V + ˜Γ e iU ¯ x/v F (cid:12)(cid:12)(cid:12)(cid:12) Λ¯ xπv F (cid:12)(cid:12)(cid:12)(cid:12) X (cid:12)(cid:12)(cid:12)(cid:12) πU Λ (cid:12)(cid:12)(cid:12)(cid:12) V (cid:33) (16)with exponents X j ( V j ) = −
12 Re (cid:20)(cid:0) s − β (cid:1) − (cid:0) s − j + 2 β ∗ (cid:1) − (cid:16) cs − β (cid:17) − (cid:16) cs − β ∗ (cid:17) ± ( c + s ) (cid:21) , j = 1 , , (17)where star denotes the complex conjugation. The decaylength κ − is equal to κ − = 4 uτ ϕ with τ ϕ given byEq. (12). The numerical prefactors ˜Γ j can be found inAppendix B, see Eq. (B1).Our results show that the limiting (long-time) station-ary state retains information about the system’s prehis-tory, i.e. about the initial state prior to the quench. In-deed, at long times after the quench the momentum dis-tribution function exhibits a double-step structure remi-niscent of the original nonequilibrium state (see Fig. 3).The initial sharp discontinuities in n ( p ) are replacedby power laws | p − p F | q and | p − p F − U/v F | q with q , = − (1 + X , ) which are smeared by nonequilibriumdephasing. The dependence of exponents q , and of theinverse dephasing length κ on the interaction strength K for a particular choice of the initial two-step distribution ( a = 0 .
8) is shown in Figs. 4 and 5.We stress that the exponents (17) differ from thosein the steady-state setup where the Luttinger liquid iscoupled to noninteracting reservoirs with nonequilibriumelectron distributions . Clearly, they also differ bothfrom the equilibrium exponent and from that emergingafter a quench in an equilibrium Luttinger liquid, seeSec. II. IV. SUMMARY
To summarize, we have studied the dynamics of anonequilibrium Luttinger liquid after a sudden interac-tion quench by employing the nonequilibrium bosoniza-tion formalism. At variance with a quench out of an K (cid:61) (cid:61) (cid:61) (cid:61) (cid:61) (cid:68) p q (cid:68) p q (cid:72) p (cid:45) p F (cid:76)(cid:144)(cid:64) U (cid:144) v F (cid:68) n (cid:43) (cid:72) p (cid:76) FIG. 3: Steady-state momentum distribution function for a =0 . K = 0.9 ( solid blue line ), 0.8, 0.7, 0.6, 0.5 ( dashed blueline ) obtained by a numerical evaluation of the determinantsin Eq. (5). At edges p = p F and p = p F + U/v F the initialdiscontinuities are replaced by power laws ∆ p q , smeared bydephasing. q q Γ(cid:142) Γ K q FIG. 4: Luttinger liquid exponents governing power-law sin-gularities of the momentum distribution functions: q at p = p F and q at p = p F + U/v F for quenched nonequi-librium with a = 0 .
8, ˜ γ for quenched equilibrium, and 2 γ forequilibrium setup. initial equilibrium state , the quasiparticle weights de-cay exponentially with time after the quench. This ex-ponential behavior is however not a relaxation into athermal state, which is related to the integrability of themodel. In particular, for an initial distribution with twoFermi edges, the distribution emerging at long times re-tains a double-step structure, with power-law singular-ities smeared by dephasing. The corresponding expo-nents as well as the dephasing rate depend on the initialnonequilibrium state.In conclusion we mention that the framework devel-oped here is also applicable to a more general situationwhen the interaction region possesses both spatial andtemporal boundaries. In this case the counting phaseswill be determined by the fractionalization processes onall the boundaries. K Κ (cid:144)(cid:64) U (cid:144) v F (cid:68) FIG. 5: Inverse decay length κ in units of U/v F as a func-tion of the interaction constant K shown for the double-stepdistribution with a = 0 . V. ACKNOWLEDGEMENT
A.D.M. thanks M.A. Cazalilla for an instructive dis-cussion. This work was supported by the collaborativeresearch grant SFB/TR12 of the DFG and by German-Israeli Foundation. D.B. is grateful to the TKM instituteat KIT for its hospitality.
Appendix A: Generalized Toeplitz determinants
In this appendix we summarize the main resultsof Ref. 39 for the asymptotic behavior of general-ized Toeplitz determinants. We consider a generalizedToeplitz matrix g j,k = (cid:90) Λ − Λ d(cid:15) e − i(cid:15)π/ Λ[ j − k − δ ( t j ) / (2 π )] ˜ g ( t j , (cid:15) ) (A1)which is defined via its symbol˜ g ( t, (cid:15) ) ≡ (cid:16) e iδ ( t ) − (cid:17) f ( (cid:15) ) . (A2)Let us focus on the special case when both the phase δ ( t )and the distribution function f ( (cid:15) ) are piecewise constantfunctions with jumps at times τ < τ < . . . < τ N τ andenergies µ < µ < . . . < µ N µ , respectively. They sat-isfy the boundary conditions δ ( t ) = 0 for t / ∈ [ τ , τ N τ ], f ( (cid:15) ) = 1 for (cid:15) < µ , and f ( (cid:15) ) = 0 for (cid:15) > µ N µ . Thediscontinuity points define a grid which subdivides thetime-energy plane in domains with different values of thesymbol. The domains can be labeled by the time indices j ∈ { , . . . , N τ } , and energy indices k ∈ { , . . . , N µ } .One associates with this set of domains a set of num-ber c jk , c jk = 12 πi ln ˜ g ( τ j + 0 , µ k + 0) + n jk , (A3) c j = δ ( t j + 0) / (2 π ) , c k = c N τ ,k = c j,N µ = 0 . where { n jk } is an arbitrary set of integers. In the aboveequation the logarithm ln ˜ g is understood as evaluatedat its principal branch, Im ln ˜ g ∈ ( − π, π ]. The summa-tion over integers n jk hence amounts to summing overdifferent branches of the logarithms. It was conjectured in Ref. 39 that the asymptotic be-havior of the (normalized) determinant ¯∆[ δ ( t ) , f ( (cid:15) )] =∆[ δ ( t ) , f ( (cid:15) )] / ∆[ δ ( t ) , T = 0] takes the form¯∆[ δ ( t ) , f ( (cid:15) )] = (cid:88) { n jk } ¯Γ { n jk } exp (cid:34) i (cid:88) ≤ j ≤ N t (cid:88) ≤ k ≤ N µ τ j β jk µ k (cid:35) (cid:89) ≤ j This appendix contains details of calculations of the asymptotic behavior of the determinants that lead to theresults presented in Sec.III B.Throughout the paper we use the nonequilibrium version of canonical bosonization developed in Refs. 37–40. Inthis framework nonequilibrium corrections to the right-movers’ equal-time Green’s function G ≷ + (¯ t ; ¯ x, 0) [see Eq.(5)]are expressed in terms of Fredholm determinants∆ µ ≡ Det (cid:2) + (cid:0) e iδ µ − (cid:1) f µ (cid:3) . with the counting phases δ µ [Eq.(10)] which were found in section III A.Let us consider first the equilibrium situation with distribution functions f + ( (cid:15) ) = f − ( (cid:15) ) = f ( (cid:15) ) = θ ( − (cid:15) ). Equa-tions (A4) and (A6) yield∆ + = G (1 − s ) G (1 + s ) G (1 − c ) G (1 + c ) e i Λ¯ x/v F (cid:12)(cid:12)(cid:12)(cid:12) Λ¯ xπv F (cid:12)(cid:12)(cid:12)(cid:12) − c − s (cid:12)(cid:12)(cid:12)(cid:12) (2 u ¯ t ) − ¯ x (2 u ¯ t ) (cid:12)(cid:12)(cid:12)(cid:12) s c , ∆ − = G (1 − cs ) G (1 + cs ) (cid:12)(cid:12)(cid:12)(cid:12) Λ¯ xπv F (cid:12)(cid:12)(cid:12)(cid:12) − c s (cid:12)(cid:12)(cid:12)(cid:12) (2 u ¯ t ) − ¯ x (2 u ¯ t ) (cid:12)(cid:12)(cid:12)(cid:12) s c . The prefactors containing the Barnes G-functions G do not directly follow from the asymptotic formulae for thegeneralized Toeplitz determinants. They can be found from the long-time limit where the factorization into simpleToeplitz determinants is applicable (we refer the reader to Ref. 38 for more details).The Green’s function following the equilibrium interaction quench is thus G < + (¯ t ; ¯ x, ∝ e i Λ¯ x/v F (cid:12)(cid:12)(cid:12)(cid:12) Λ¯ xπv F (cid:12)(cid:12)(cid:12)(cid:12) − ( c + s ) (cid:12)(cid:12)(cid:12)(cid:12) (2 u ¯ t ) − ¯ x (2 u ¯ t ) (cid:12)(cid:12)(cid:12)(cid:12) s c . Since 2 cs = ˜ γ and − ( c + s ) = − − ˜ γ the power laws are in perfect agreement with the exact result (3) of Ref. 8.In the following we will use the equilibrium quench as reference case to normalize our Fredholm determinants.The situation becomes more complicated when turning to nonequilibrium. We consider the double step distributionfunctions f ± ( (cid:15) ) = (1 − a ) θ ( − (cid:15) ) + aθ ( U − (cid:15) ) for right- and left-movers as the initial steady state of the noninteractingFermi sea before the quench. Using the asymptotic formulae (A4) and (A6) we obtain G < + (¯ t ; ¯ x, 0) = G < (0 , ¯ x ) ˜∆ + ˜∆ − for | ¯ x | , | u ¯ t − | ¯ x || , | u ¯ t | (cid:29) v F /U with equilibrium-normalized determinants˜∆ + = (cid:88) n ,n ,n ∈ Z ˜Γ + ( n , n , n ) e i ( β + β ) U | ¯ x | /v F e i ( n + n − n ) U | ¯ x | /v F e in U ¯ t u/v F × (cid:12)(cid:12)(cid:12)(cid:12) Λ πv F (cid:12)(cid:12)(cid:12)(cid:12) γ + + V + | ¯ x | X + (2 u ¯ t − | ¯ x | ) D + (2 u ¯ t + | ¯ x | ) S + (2 u ¯ t ) T + (cid:12)(cid:12)(cid:12)(cid:12) πU Λ (cid:12)(cid:12)(cid:12)(cid:12) V + , ˜∆ − = (cid:88) n ,n ,n ∈ Z ˜Γ − ( n , n , n ) e i ( β + β ) U | ¯ x | /v F e i ( n + n − n ) U | ¯ x | /v F e in U ¯ t u/v F × (cid:12)(cid:12)(cid:12)(cid:12) Λ πv F (cid:12)(cid:12)(cid:12)(cid:12) γ − + V − | ¯ x | X − (2 u ¯ t − | ¯ x | ) D − (2 u ¯ t + | ¯ x | ) S − (2 u ¯ t ) T − (cid:12)(cid:12)(cid:12)(cid:12) πU Λ (cid:12)(cid:12)(cid:12)(cid:12) V − for 2 u ¯ t > ¯ x > + = (cid:88) n ,n ,n ∈ Z ˜Γ (cid:48) + ( n , n , n ) e i ( β + β )2 U ¯ tu/v F e i ( n + n − n ) U | ¯ x | /v F e in U ¯ t u/v F × (cid:12)(cid:12)(cid:12)(cid:12) Λ πv F (cid:12)(cid:12)(cid:12)(cid:12) γ (cid:48) + + V (cid:48) + | ¯ x | X (cid:48) + ( − u ¯ t + | ¯ x | ) D (cid:48) + (2 u ¯ t + | ¯ x | ) S (cid:48) + (2 u ¯ t ) T (cid:48) + (cid:12)(cid:12)(cid:12)(cid:12) πU Λ (cid:12)(cid:12)(cid:12)(cid:12) V (cid:48) + , ˜∆ − = (cid:88) n ,n ,n ∈ Z ˜Γ (cid:48)− ( n , n , n ) e i ( β + β )2 U ¯ tu/v F e i ( n + n − n ) U | ¯ x | /v F e in U ¯ t u/v F × (cid:12)(cid:12)(cid:12)(cid:12) Λ πv F (cid:12)(cid:12)(cid:12)(cid:12) γ (cid:48)− + V (cid:48)− | ¯ x | X (cid:48)− ( − u ¯ t + | ¯ x | ) D (cid:48)− (2 u ¯ t + | ¯ x | ) S (cid:48)− (2 u ¯ t ) T (cid:48)− (cid:12)(cid:12)(cid:12)(cid:12) πU Λ (cid:12)(cid:12)(cid:12)(cid:12) V (cid:48)− for 0 < u ¯ t < ¯ x . Here, we left the n j -dependence of the exponents X ± , T ± , . . . implicit. ˜Γ ± , ˜Γ (cid:48)± are numericalprefactors which are not known in general. The determinants for ¯ x < u ¯ t ≷ | ¯ x | , which we consider below separately. Regime of separated phase pulses, u ¯ t > | ¯ x | Here the exponents are X + = (cid:0) − β + c − n (cid:1) (cid:0) β − c + n − n (cid:1) + ( − β − n + n ) ( β + n )+ ( − β − n + n ) ( β + n ) + (cid:0) − β − n − s (cid:1) (cid:0) β + n − n + s (cid:1) ,T + = (cid:0) − n + n − s − β (cid:1) (cid:0) c − n − β (cid:1) + (cid:0) − n − s − β (cid:1) (cid:0) c − n + n − β (cid:1) + ( n − n + β ) ( n + β ) + ( n + β ) ( n − n + β ) ,D + = ( n − n + β ) ( − n + n − β ) + (cid:0) − n + n − s − β (cid:1) (cid:0) − c + n − n + β (cid:1) ,S + = ( β + n ) (cid:0) c − β + n ) (cid:1) − s (cid:0) β − c + n (cid:1) ,γ + = − c − s ,X − = ( − n + n − β ) ( n + β ) + ( − n + cs − β ) ( n − n − cs + β )+ ( − n + n − β ) ( n + β ) + ( − n − cs − β ) ( n − n + cs + β ) ,T − = ( − n + n + cs − β ) ( − n − cs − β ) + ( − n + cs − β ) ( − n + n − cs − β )+ ( n − n + β ) ( n + β ) + ( n + β ) ( n − n + β ) ,D − = ( n − n + β ) ( − n + n − β ) + ( − n + n + cs − β ) ( n − n + cs + β ) ,S − = cs ( n + cs + β ) + ( n + β ) ( − cs − n + β )) ,γ − = − c s . In the long-time limit 2 u ¯ t (cid:29) | ¯ x | , the powers simplify to (2 u ¯ t − | ¯ x | ) D ± (2 u ¯ t + | ¯ x | ) S ± (2 u ¯ t ) T ± → (2 u ¯ t ) ˜ T ± with ˜ T + = − n ≤ T − = − n ≤ 0. Thus the correlation function relaxes to a stationary solution where solely terms with n = 0 = n contribute. The remaining powers simplify to X + = − (cid:18) n − − s − β (cid:19) − (cid:18) n − c − β (cid:19) − c + s ,V + = − (cid:18) n − − s − β (cid:19) − (cid:18) n − c − β (cid:19) + c + s ,X − = − (cid:18) n − cs − β (cid:19) − (cid:18) n − − cs − β (cid:19) − c s ,V − = − (cid:18) n − cs − β (cid:19) − (cid:18) n − − cs − β (cid:19) + c s . Since in the long-time limit the phases split into independent pulses, all Fredholm determinants factorize into Toeplitzdeterminants and the prefactors can be found in the closed analytical form using the generalized Fisher-Hartwigformula ˜Γ + ( n , n , n = 0) = G (1 − s − β − n ) G (1 + s + β + n ) G (1 + β + n ) G (1 − β − n ) G (1 − s ) G (1 + s ) × G (1 + c − β − n ) G (1 − c + β + n ) G (1 + β + n ) G (1 − β − n ) G (1 + c ) G (1 − c ) , ˜Γ − ( n , n , n = 0) = G (1 + cs − β − n ) G (1 − cs + β + n ) G (1 + β + n ) G (1 − β − n ) G (1 + cs ) G (1 − cs ) × G (1 − cs − β − n ) G (1 + cs + β + n ) G (1 + β + n ) G (1 − β − n ) G (1 − cs ) G (1 + cs ) . (B1)For moderate repulsive interaction √ − ≤ K ≤ 1, the dominant powers Re X ± are due to ( n , n ) = (0 , , (0 , n , n ) = (0 , ≡ ˜Γ + (0 , , − (0 , , ≡ ˜Γ + (0 , , − (0 , , a → 0, prefactors vanish for all n j but n = n = n = n = 0 forwhich one recovers the equilibrium exponents. Regime of overlapping phase pulses, u ¯ t < | ¯ x | Here the exponents are X (cid:48) + = (cid:0) − c + n − n − β (cid:1) (cid:0) c − n − β (cid:1) + (cid:0) − n − s − β (cid:1) (cid:0) n − n + s − β (cid:1) + ( − n + n + β ) ( n + β ) + ( n + β ) ( − n + n + β ) ,T (cid:48) + = ( n − n − β ) ( n + β ) + (cid:0) − n − s − β (cid:1) (cid:0) c − n + n + β (cid:1) + ( n − n − β ) ( n + β ) + (cid:0) c − n − β (cid:1) (cid:0) − n + n − s + β (cid:1) ,D (cid:48) + = ( − n + n + β ) ( n − n − β ) + (cid:0) − c + n − n − β (cid:1) (cid:0) − n + n − s + β (cid:1) ,S (cid:48) + = − s (cid:0) − c + n + β (cid:1) + ( n + β ) (cid:0) c − n + β ) (cid:1) ,X (cid:48)− = ( n − n − cs − β ) ( − n + cs − β ) + ( − n − cs − β ) ( n − n + cs − β )+ ( − n + n + β ) ( n + β ) + ( n + β ) ( − n + n + β ) ,T (cid:48)− = ( n − n − β ) ( n + β ) + ( − n − cs − β ) ( − n + n + cs + β )+ ( n − n − β ) ( n + β ) + ( − n + cs − β ) ( − n + n − cs + β ) ,D (cid:48)− = ( − n + n + β ) ( n − n − β ) + ( n − n − cs − β ) ( − n + n − cs + β ) ,S (cid:48)− = − cs ( n − cs + β ) + ( n + β ) ( cs − n + β )) . For long distances | ¯ x | (cid:29) u ¯ t the power-law dependence on distance simplifies to | ¯ x | X (cid:48)± ( − u ¯ t + | ¯ x | ) D (cid:48)± (2 u ¯ t + | ¯ x | ) S (cid:48)± → | ¯ x | ˜ X ± with the exponents˜ X + = − n + 1 / − n − n ) − , ˜ X − = − n − n − n ) . | ¯ x | → ∞ all terms vanish except for n = n + n or n = n + n − 1, and n = n + n . Then 1+ ˜ X + = 0 = ˜ X − , i.e.the normalized determinants ˜∆ ± are independent of ¯ x , and correlations drop off like G < + (¯ t ; ¯ x, ∼ G < (¯ t ; ¯ x, ∼ ¯ x − .The remaining exponents are T (cid:48) + ( V (cid:48) + ) = − (cid:18) n − − / − s − β (cid:19) − (cid:18) n − − / c − β (cid:19) + 14 ∓ c s (B2)for n = n + n , T (cid:48) + ( V (cid:48) + ) = − (cid:18) n − / − s − β (cid:19) − (cid:18) n − / c − β (cid:19) + 14 ∓ c s (B3)for n = n + n − T (cid:48)− ( V (cid:48)− ) = − (cid:18) n − − cs − β (cid:19) − (cid:18) n − cs − β (cid:19) ∓ c s (B4)for n = n + n . T. 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