A Green’s Functions – Definitions and Identities

# Transport Across an Impurity in One Dimensional Quantum Liquids Far From Equilibrium

## Abstract

We study the effect of a single impurity on the transport properties of a one dimensional quantum liquid highly excited away from its ground state by a sudden quench of the bulk interaction. In particular we compute the time dependent dc current to leading order in the impurity potential, using bosonization, and starting both from the limit of a uniform system, and from the limit of two decoupled semi-infinite systems. Our results reveal that the nonequilibrium excitation of bulk modes induced by the global quench has important effects on the conductor-insulator quantum phase transition, turning it into a crossover for any small quench amplitude, and destroying the exact duality between the conducting and insulating fixed points. In addition, the current displays a faster decay towards the steady-state as compared to the equilibrium case, a signature of quench-induced decoherence.

## I Introduction

Transport phenomena in strongly correlated quantum systems are typically sensitive to interactions, inhomogeneities and low-dimensionality and often reveal intriguing and unexpected effects. A prominent example is the impact of a single potential barrier in a one dimensional interacting electron liquid. While the clean system would display, despite interactions, ideal conductance quantization in the presence of metallic leads Maslov and Stone (1995); Safi and Schulz (1995); Ponomarenko (1995); Rosch and Andrei (2000), the impurity at zero temperature is capable of turning the perfect conductor into an insulating link, giving rise to power law corrections to transport coefficients at finite temperature or finite voltage bias Kane and Fisher (1992a, b); Matveev et al. (1993). The physics of a single impurity in a Luttinger liquid has attracted enormous attention in the past decades Fendley et al. (1995a, b); Chang (2003) and still finds beautiful applications in systems such as the edges of a quantum spin Hall insulator Wu et al. (2006); Ilan et al. (2012) or in electronic quantum circuits Safi and Saleur (2004); Jezouin et al. (2012); Souquet et al. (2013). Traditional condensed matter transport settings involve a system initially in thermal equilibrium which is perturbed away from it by the action of some external field conjugate to a conserved current. Within linear response theory one then obtains transport coefficients, which give information on the structure of the underlying ground-state and its low-energy excitations.

Recently, experimental advances in controlling and probing ultra-cold atomic gases has offered a new platform to study nonequilibrium time-dependent phenomena in a fully tunable setting Bloch et al. (2008); Langen et al. (2014). First generation of experiments studied the dynamics induced by rapidly changing in time some system parameter Greiner et al. (2002); Chen et al. (2011); Trotzky et al. (2012) in an otherwise isolated system, a so called quantum quench. More recently, the experimental focus shifted towards realizing genuine transport experiments with cold atoms Brantut et al. (2012); Krinner et al. (2014). It is important to realize that these systems are very well isolated from their environment and hence intrinsically out of equilibrium, therefore the standard condensed matter idealizations do not necessarily apply.

Motivated by these experimental developments, in this paper we investigate the transport properties of nonequilibrium quantum many body states which are thermally isolated and highly excited above their ground state. While for generic, non-integrable and ergodic, quantum many-body systems one may expect the excitation energy to be effectively converted into temperature at long times Lux et al. (2014); Tavora et al. (2014), and hence to induce decoherence, the situation is less clear at intermediate time scales where long-lived prethermal states may emerge Berges et al. (2004); Moeckel and Kehrein (2008); Schiró and Fabrizio (2010); Gring et al. (2012); Karrasch et al. (2012); Sandri et al. (2012); Mitra (2013); Tavora et al. (2014), especially parametrically close to an integrable point. The idea we pursue in this work is to use transport as a probe to unveil the structure of these prethermal excited states, to understand the relevant excitations and whether non-trivial quantum phenomena survive at these high-energies, and if so, in which form. It should be noted that there has been a recent spur of interest towards understanding similar dynamical quantum correlations in isolated many body systems excited after quantum quenches Foini et al. (2011); Essler et al. (2012); Foini et al. (2012); Rossini et al. (2014). We have considered an example along these lines in a recent work on a time-dependent orthogonality catastrophe problem Schiró and Mitra (2014) where we introduced a novel dynamical Loschmidt echo, encoding the response of a highly excited state to a local perturbation. Here we study a similar problem from the point of view of transport.

Specifically, in this paper we compute the time-dependent current of a one dimensional system of spinless electrons described by the Luttinger model, which is excited by a sudden change of the two-particle interaction and a simultaneous switching-on of a local scattering potential. These two perturbations have very different effects, the former injecting extensive energy into the bulk modes, the latter creating a non-linear channel for local dissipation. We discuss the physics both in the limit of a uniform system, where the local potential induces a weak backscattering term, and in the limit of two decoupled semi-infinite systems where we switch on a local tunneling. In both cases, using bosonization, we formulate the problem in terms of a boundary sine-Gordon model in a nonequilibrium transient environment. Using perturbative approaches we discuss the fate of the conducting-insulating zero temperature quantum phase transition in the presence of a nonequilibrium bulk excitation, thus complementing and expanding previous results Perfetto et al. (2010); Kennes and Meden (2013).

The results for the steady state current reveal the emergence of a novel energy scale associated with a quench-induced decoherence effect, which turns the sharp equilibrium transition into a smooth crossover. Interestingly, the very same energy scale has been shown to play a key role in the transient orthogonality catastrophe problem, cutting off the renormalization group flow of the backscattering and resulting in an exponential decay of the Loschmidt echo Schiró and Mitra (2014).

Our analysis, although perturbative in nature, ultimately suggests that the steady state impurity problem in a quenched Luttinger model has very different properties than in equilibrium at zero temperature. In particular the quench acts as a relevant perturbation on both sides of the equilibrium transition, even when the local non-linearity would be irrelevant at the tree level, generating an effective decoherence which drives the system away from the uniform and open-chain fixed points. While this behavior is reminiscent of an effective temperature, we will see that this analogy is only qualitative rather than quantitative, as the scaling of physical quantities in the steady state with respect to this emergent energy scale do not show signature of non-trivial power-laws like the ones encountered at non-zero temperatures. At the same time we find that deviations from the asymptotic steady state regime, due to transient effects at finite time, display nonequilibrium power laws with characteristic Luttinger liquid exponents which may be a signature of a sort of Luttinger liquid universality out of equilibrium, as recently discussed Karrasch et al. (2012); Kennes and Meden (2013); Kennes et al. (2014).

Note that the problem of an impurity in a quenched Luttinger model has been recently addressed in Refs. Perfetto et al., 2010 and Kennes and Meden, 2013. In the former, the dynamics of the current in the tunneling regime was investigated after a global quench of the bulk interaction and a faster decay to the stationary state was found. In the latter, results for the temperature scaling of the conductance in the steady state after the quench have been obtained using different approaches such as bosonization and functional RG. Although we do not address temperature dependence of transport coefficients in this paper, and we work always at zero temperature, it is useful to comment on the relation between the results of Ref. Kennes and Meden, 2013 and the physical picture emerging from this work. We will do this in the discussion section.

Finally, it is worth mentioning that the question of nonequilibrium effects on local quantum criticality has been addressed recently also in the context of driven quantum systems Ribeiro et al. (2013) and specifically for single impurity in Luttinger liquids in the context of noisy driven Josephson junction circuits Dalla Torre et al. (2012) or tunneling in biased quantum wires Ngo Dinh et al. (2010) using the out-of-equilibrium bosonization framework Gutman et al. (2010); Ngo Dinh et al. (2013). The picture emerging in these cases, where the nonequilibrium perturbation induces a decoherence mechanism which eventually cuts the quantum critical power-law behavior, is consistent with our results for the isolated quenched problem.

The paper is organized as follows. In section II we introduce the system, the nonequilibrium protocol, and a derivation of an expression for the time dependent current in terms of a bosonic Green’s function. We then evaluate the current in the weak-backscattering limit in section II.1. Such a perturbative approach eventually breaks down in a certain region of parameter space and this will lead us in section III to formulate the problem in the limit of two disconnected 1D systems coupled by a local tunneling term. We will compute perturbatively the tunneling current in section III.1 and thus obtain a complete picture of the problem.

## Ii Transport in the Weak-Backscattering Regime

We start by discussing the model and the nonequilibrium set up. We consider a one dimensional system of interacting spinless fermions described by the Tomonaga-Luttinger (TL) model. We assume the system to be initially (at time ) in the ground-state of the TL Hamiltonian

 H0=Hfree+g042∫dx(ρ2L(x)+ρ2R(x)) +g02∫dxρL(x)ρR(x) (1)

where with , is the fermion density with given chirality , and are the strengths of the intra (inter)-branch scattering processes.

At time we assume a sudden change (quench) in the value of the bulk interactions to , so that the system will evolve with the Hamiltonian

 H=Hfree+g42∫dx(ρ2L(x)+ρ2R(x)) +g2∫dxρL(x)ρR(x) (2)

This global quench injects extensive energy into the system and triggers a nonequilibrium occupation of the bulk modes. Dynamics arising simply due to this quench has been discussed extensively in the literature Cazalilla (2006); Iucci and Cazalilla (2009); Mitra and Giamarchi (2011). In addition to the bulk excitation, we assume that a static impurity potential has also been suddenly switched on at time . This static impurity can induce an intra-branch as well as inter-branch scattering of fermions, that we write as the sum of two contributions

 Vloc=Vfs(ψ†L(0)ψL(0)+L↔R)+ (3) +Vbs(ψ†L(0)ψR(0)+h.c.). (4)

the first term representing the forward scattering while the latter the backward scattering. As a result the wave function at time is , with .

We employ bosonization to describe the system, thus introducing the bosonic fields and describing collective density and current excitations respectively,

 ψL/R(x)∼1√2παei(θ(x)±ϕ(x)) (5) ρL/R(x)=−∂x(ϕ(x)±θ(x))/2π (6)

The Hamiltonians before and after the quench in terms of these bosonic fields are

 H0=u02π∫dx[K0(∂xθ(x))2+1K0(∂xϕ(x))2] (7)

and

 H+=u2π∫dx[K(∂xθ(x))2+1K(∂xϕ(x))2]+Vloc (8)

Within bosonization, the impurity potential in Eq. (3) can be written as

 Vloc=gfs∂xϕ(0)+gbscos2ϕ(0) (9)

where the effective impurity couplings read respectively Giamarchi (2003)

 gfs=−Vfsπgbs=Vbsπα (10)

The Luttinger parameter and the sound velocities are related to the Fermi velocity and the interaction parameters as

 u0=vF√(1+g04/2πvF)2−(g02/2πvF)2 (11) K0= ⎷1+g04/2πvF−g02/2πvF1+g04/2πvF+g02/2πvF (12)

with similar relations holding for as a function of . In order to preserve Galilean invariance we choose the values of such that . This amounts to simply requiring that .

In order to probe transport through the system we will study linear response to a weak electric field applied after the quench,

 E(x,t)=−∂tA(x,t) (13)

with being the vector potential. In the presence of the electric field, the LL Hamiltonian is modified according to the minimal substitution which amounts to the shift

 ∂xθ(x)→∂xθ(x)−eA(x,t) (14)

The current operator can be obtained as the functional derivative of with respect to the vector potential

 J(x,t)=−δHδA(x,t) (15)

As usual, the current has two contributions, , the diamagnetic () and the paramagnetic () one. The former is given by (hereafter )

 Jd(x,t)=−e2uKπA(x,t)≡−DA(x,t) (16)

where we have introduced the diamagnetic term . The latter is

 Jp(x,t)=euKπ∂xθ(x,t)=eπ∂tϕ(x,t) (17)

Finally, we notice that the total current can be equivalently obtained from the continuity equation

 ∂tρ(x,t)+∂xJ(x,t)=0

by computing the time derivative of the density

 ∂tρ(x,t)=−ieπ[H(A),∂xϕ(x,t)]

using the LL Hamiltonian with the minimal substitution. The result of this calculation obviously recovers the expression for the current .

We can now compute the average current within linear response theory, by doing perturbation theory in . Since the diamagnetic part of the current is already linear in the vector potential, we only need to take into account the paramagnetic contribution which gives

 ⟨Jp(x,t)⟩=−∫∞0dt′∫dx′χRJJ(xt;x′t′)A(x′,t′) (18)

where we have defined the retarded current-current correlation function

 χRJJ(xt;x′t′)=−iθ(t−t′)⟨[Jp(x,t),Jp(x′,t′)]⟩ =−ie2π2θ(t−t′)⟨[∂tϕ(x,t),∂t′ϕ(x′,t′)]⟩ (19)

and extended the time integral up to infinity. We stress here that the average in Eq.(II) is taken over the initial density matrix (ground state of ) while the operators are evolved with the Hamiltonian . The resulting correlator is not the usual equilibrium one and in particular it is not time-translational invariant. This reflects the effect of the global and local quantum quenches that have been performed on the system at time . Finally we have for the current

 ⟨J(x,t)⟩=−DA(x,t)+⟨Jp(x,t)⟩ (20)

We now use an important identity relating the current-current correlation (II) to the retarded Green’s function of the field defined as

 GR(xt;x′t′)=−iθ(t−t′)⟨[ϕ(x,t),ϕ(x′,t′)]⟩ (21)

We have

 χRJJ(xt;x′t′)=e2π2∂t′∂tGR(xt;x′t′)−2Dδ(x−x′)δ(t−t′) (22)

Substituting this result into Eq. (20) and using Eq. (18), we find that the last term cancels the diamagnetic term exactly, and after an integration by parts we obtain

 ⟨J(x,t)⟩=−e2π2∫t0dt′∫dx′∂tGR(xt;x′t′)E(x′t′) (23)

This is in principle an exact result for the time dependent current through a finite-size LL and we can now further specify the electric field profile. We assume a sudden switching of the field, and in addition, since we are ultimately interested in the current due to the impurity (3), we assume that the potential drop occurs around , , so that the electric field is effectively a delta-function . Then, the current at , can be written only in terms of the exact retarded Green’s function of the local field at the impurity site,

 GR(t,t′)=−iθ(t−t′)⟨[ϕ(t),ϕ(t′)]⟩ (24)

 I(t)=−e2Vπ2∫t0dt′∂tGR(t,t′) (25)

This equation, which gives the time-dependent current in terms of an exact dynamical correlator of the local field, is the main result of this section. It generalizes to the time-dependent quench problem the well known equilibrium result Giamarchi (2003). From this we can compute the current for a pure LL, using the expression for the bare retarded local Green’s function

 GR0(t>t′)=−KarctanΛ(t−t′) (26)

where is an ultra-violet cut-off. We obtain

 I0(t)=e2KVπ2arctanΛt (27)

which approaches in the long time limit as expected. In the next section we will evaluate the corrections to the time dependent current due to the local potential. The forward scattering term in Eq. (9) turns out to not contribute to the current, as we discuss explicitly in appendix B, therefore in the next section we will focus our attention on the backward scattering term which is responsible for the interesting physical effects with .

### ii.1 Weak-Backscattering Corrections to Linear Conductance

We are now in the position to evaluate the weak back-scattering correction to the conductance. We just need to evaluate the local Green’s function for the field

 Gab(t,t′)=−i⟨ϕa(t)ϕb(t′)⟩ (28)

where are Keldysh indices. We have defined the various Green’s functions and the relations between them in appendix A. To lowest order in the backscattering we obtain Tavora and Mitra (2013)

 δGab(t,t′)=−g2bs2∫dt1dt2∑cd(cd)Fcd(t1,t2)Λcdab(t1,t2) (29)

where we define ,

 Fcd(t1,t2)=−i⟨cos2ϕc(t1)cos2ϕd(t2)⟩

and

The retarded component is given by

 GR(t,t′)=−12∑abbGab(t,t′), (31)

Substituting the resulting expression into Eq. (25), and after some algebra we obtain for the current

 δI(t)=−C∫t0dt1∂tGR0(t,t1)∫t10dt2FR(t1,t2)Φ(t1,t2) (32)

with and

 Φ(t1,t2) = K∫t1t2dτarctanΛτ=KΛt1arctanΛt1+ −KΛt2arctanΛt2−K2log(1+Λ2t211+Λ2t22)

The retarded correlator entering the expression for the current is found to be

 FR(t1>t2)=−sin(2KarctanΛ(t1−t2))(1+Λ2(t1−t2)2)Kneqf(t1,t2) (33)

where we have introduced

 f(t1,t2)=[(1+4Λ2t21)(1+4Λ2t22)(1+Λ2(t1+t2)2)]Ktr/2 (34)

and the exponents

 Kneq=K02(1+K2K20)Ktr=K02(1−K2K20) (35)

We can further simplify Eq. (32) by noticing that for

 ∂tGR0(t−t1)≃−πKδ(t−t1)

Then we can write the correction to the time dependent current as

 δI(t)=−~C∫t0dτsin[2KarctanΛτ](1+Λ2τ2)Kneqf(t,t−τ)Φ(t,t−τ) (36)

with . In the next section we will discuss the behavior of this quantity as a function of time and quench amplitude.

### ii.2 Discussion

We start by discussing the equilibrium zero temperature case corresponding to . In this case we have and and the integral in Eq. (36) simplifies. We obtain for the transient current at long times ()

 δI(t)/V∼1t2(K−1) (37)

i.e., for the weak-backscattering correction to the current vanishes in the long time limit as a power law, the junction is perfectly conducting and the backscattering is irrelevant, this is the well known result from Kane and Fisher Kane and Fisher (1992b). As soon as the quench amplitude is non-zero, , we find a number of interesting differences. We first focus on the long time steady state value of the backscattering current, , this is

 δIss/V=−4πK2g2bsG0∫∞0dττsin(2KarctanΛτ)(1+Λ2τ2)Kneq (38)

the quantum of conductance, which we set to one from now on, .

We first notice that the integral is well defined as long as , as expected from perturbative RG calculations Schiró and Mitra (2014), since only in this regime the local backscattering is an irrelevant perturbation for the steady state. We plot in figure 1 the region of parameters where this condition is satisfied, and our calculations are therefore controlled. In figure 2 we plot the backscattering correction to the steady state current for different values of in region I, as a function of . Quite interestingly, the correction to the steady state current is different from zero at any finite quench amplitude , in other words the systems deviates from the perfect conduction limit. For a small quench amplitude one finds that the steady state correction to the conductance is proportional to ,

 δIss/V∼−g2bs(K−K0)2 (39)

We now discuss the transient behavior and the approach to the steady state. To this extent we evaluate numerically the integral in Eq. (36) and plot the result in figure 3. We find that the current decays to the steady state in a power law fashion,

 δI(t)−δIss∼1tα (40)

with an exponent whose dependence on the quench parameters is shown in the bottom panel of figure 3, as a function of at fixed in the region I. We notice that the exponent behaves non-monotonically with and reaches a minimum value for , i.e., the decay of the current is stronger in the presence of a finite quench amplitude. We can get an analytical understanding of this by looking at the integral expression for , Eqn. (36). Using the fact that for large we have while , we obtain . We notice that this is indeed confirmed by the numerical evaluation of the integral, although for larger values of the bulk quench, deviations start to appear that we attribute to the finite time resolution of the numerical integration.

We have analyzed only a few values of while tuning , yet we expect this behavior to hold throughout the region I in figure 1. We however expect the perturbative expansion to eventually break down as we approach the regime and enter region II, where the backscattering becomes a relevant perturbation, i.e., its strength grows under RG. In equilibrium, a complementary approach to explore this strong coupling regime amounts to starting from the limit of two decoupled semi-infinite systems and switching on a local tunneling. In the next section we will suitably generalize this approach to the quench case and discuss to what extent it can be used to extract useful information about the strong coupling regime.

## Iii Transport in the Tunneling Regime

We now consider a different quench protocol which will allow us to access the strong coupling regime where the back-scattering is relevant. Let us suppose at we have two disconnected and identical semi-infinite TL models describing 1D interacting spinless fermions. By definition, each lead contains left and right moving fermions interacting in the bulk through intra-branch () and inter-branch () scattering processes, as in the previous case. A major difference however arises due to the presence of a sharp edge in each lead, which imposes open boundary conditions for the fermionic field. As a result, and differently from the translational invariant case of the previous section, the two fermionic species are not independent of each other. This suggests an equivalent and more convenient representation of each lead in terms of a single chiral fermionic field defined on an infinite system Fabrizio and Gogolin (1995) (i.e., obeying periodic boundary conditions). In terms of this field, the initial Hamiltonian in each lead becomes

 H0i=Hfree+g042∫dxρi(x)ρi(x)+g022∫dxρi(x)ρi(−x) (41)

where , while , and one should notice the non-local interaction that results from the single chiral field representation.

We prepare the system in the ground state of and then, for , we evolve the system with a different Hamiltonian

 H=∑i=1,2Hi+∑i=1,2μiNi+HT (42)

where we have (a) quenched the bulk interactions,

 Hi=Hfree+g42∫dxρi(x)ρi(x)+g22∫dxρi(x)ρi(−x) (43)

(b) switched on a tunneling term coupling the two wires at the edge

 HT=λ0(ψ†1(0)ψ2(0)+h.c.) (44)

and finally (c) switched on a bias voltage that couples to the charge imbalance. We are interested in computing the time dependent current

 I(t)=⟨Ψ(t)|^I|Ψ(t)⟩=⟨Ψ0|eiHt^Ie−iHt|Ψ0⟩ (45)

where the current operator is

 ^I=ie[H,(N1−N2)]=−2ieλ0(ψ†1(0)ψ2(0)−h.c.) (46)

In order to proceed, it is useful to perform a unitary transformation to eliminate the bias voltage in , which amounts to going to the rotating frame defined by an operator . Inserting into the expression for the current (45) we obtain

 I(t)=⟨~Ψ(t)|^I(t)|~Ψ(t)⟩ (47)

where the new state

 |~Ψ(t)⟩=Ω†(t)|Ψ(t)⟩ (48)

satisfies the Schrodinger equation

 i∂t|~Ψ(t)⟩=~H(t)|~Ψ(t)⟩ (49)

with a rotated-frame Hamiltonian

 ~H(t)=(i∂tΩ†)Ω+Ω†HΩ≡∑iHi+HT(t) (50)

While the bias voltage has disappeared, the tunneling has become explicitly time-dependent

 HT(t)=Ω†(t)HTΩ(t)=λ0(eieVtψ†1(0)ψ2(0)+h.c.) (51)

where we have used the fact that

 Ω†(t)ψi(x)Ω(t)=e−iμitψi(x) (52)

The current operator in (45) also acquires an explicit time dependence

 ^I(t)=Ω†(t)^IΩ(t)=−2ieλ0(eieVtψ†1(0)ψ2(0)−h.c.) (53)

The time dependent state can be written in the interaction picture with respect to the unperturbed Hamiltonian of the uncoupled wires as

 |~Ψ(t)⟩=e−i∑iHitTe−i∫t0dt′HT,I(t′)|Ψ0⟩ (54)

We can now evaluate the time dependent current to the lowest order contribution in the tunneling, which gives

 I(t)=−i∫t0dt1⟨Ψ0|[^II(t),HT,I(t1)]|Ψ0⟩ (55)

where all the operators are explicitly time dependent as in previous equations and evolved with the Hamiltonian of the uncoupled wires, after the global quench.

It is now useful to use bosonization, in its open-boundary formulation Fabrizio and Gogolin (1995) to proceed with the evaluation of the tunneling current. We first write the fermionic operator in each lead in terms of a single chiral bosonic mode

 ψi(x)=1√αeiφi(x) (56)

as well as the electron density in the lead as

 ρi(x)=∂xφi(x)/2π (57)

The Hamiltonian before and after the global quench of the interactions (Eqns 41 and 43) become now quadratic in this bosonic field and can be easily diagonalized with a Bogoliubov rotation (see appendix C). For example we have for

 H0i=u04(1K0+K0)∫dx2π(∂xφi(x))2+ −u04(1K0−K0)∫dx2π∂xφi(x)∂xφi(−x) (58)

The resulting harmonic theory is again described in terms of two parameters, the sound velocity and the Luttinger parameter , which by construction are identical in the two leads, and whose expressions in terms of the coupling constant and are the same as in the translational invariant case (see equations (11)). We rewrite them here for convenience

 u0=vF√(1+g04/2πvF)2−(g02/2πvF)2 (59) K0= ⎷1+g04/2πvF−g02/2πvF1+g04/2πvF+g02/2πvF (60)

with similar relations holding for as functions of . In addition we need to write the tunneling and current operator in the bosonic language. To this extent it is useful to introduce the combinations

 φ±(x)=φ1(x)±φ2(x)√2 (61)

and to notice that only the combination enters both the tunneling and the current. Indeed we have disregarded the Klein factors which can be shown to be unimportant. Thus,

 HT(t)=2λcos(√2φ−(t)−eVt) (62) ^I(t)=−4eλsin(√2φ−(t)−eVt) (63)

where we have defined and .

### iii.1 Tunneling Correction To Current

In order to proceed further, we plug Eqns. (62,63) in the expression for the tunneling current to obtain, after some simple algebra,

 I(t)=−8eλ2∫t0dτsin(eVτ)FR(t,t−τ) (64)

where is the retarded component of the correlator

 Fab(t,t′) = −i⟨cos(√2φ−a(t))cos(√2φ−b(t′))⟩= (65) = −i2⟨ei√2φ−a(t)e−i√2φ−b(t′)⟩

It is convenient to express this correlator in terms of the fields in the two semi-infinite leads. We have

 Fab(t,t′)=−i2⟨eiφ1a(t)e−iφ1b(t′)⟩⟨eiφ2b(t′)e−iφ1a(t)⟩

where we have used the fact that in the absence of tunneling the two leads are decoupled. To compute this correlator we need therefore the local Green’s function of the field after a quench of the bulk interaction parameter in a semi-infinite chain, which we give in appendix C. The result for the retarded component, is found to be

 FR(t1>t2)=−sin(2KarctanΛ(t1−t2))(1+Λ2(t1−t2)2)Kdualneqf(t1,t2) (66)

where we have introduced

 f(t1,t2)=[(1+4Λ2t21)(1+4Λ2t22)(1+Λ2(t1+t2)2)2]Kdualtr/2 (67)

with the exponents

 Kdualneq=12K0(1+K20K2)Kdualtr=12K0(1−K20K2) (68)

We stress that Eq. (64) is perturbative in the tunneling but contains the bias voltage to all orders. In fact in the absence of the bulk quench, , we can recover from this the well known result Kane and Fisher (1992b) for the non-linear I-V characteristic of the wire, . In the following we will instead consider mostly the low bias regime, for a finite quench amplitude and to this extent we will evaluate the current in the low bias regime and equivalently the non-linear differential conductance at zero bias. Note that the latter in the absence of quench.

After some simple manipulations, we can write the time dependent current at small bias voltage as

 I(t)=8e2Vλ2∫t0dττsin(2KarctanΛτ)(1+Λ2τ2)Kdualneqf(t,t−τ) (69)

We notice the result we have obtained for the current in the tunneling regime is dual to the one in the weak-backscattering case in the sense that the two currents are related by a transformation of the Luttinger parameters, before and after the quench, into

 K,K0⟶1K,1K0 (70)

This is the natural generalization of the equilibrium duality Giamarchi (2003). Yet, as one can immediately see from the expressions (35) for and (68) for , the exponents controlling the decay of the correlation in the tunneling and weak-backscattering regime are not related by this simple duality, i.e.,

 Kdualneq≠1/Kneq (71) Kdualtr≠1/Ktr (72)

unless of course . This will have important consequences as we are going to discuss in the next section. A simple way to understand the origin of this result is to notice that while for the Hamiltonian after the quench the duality still holds, the initial condition of the problem (the ground state of the Luttinger model with interaction parameter ) is not dual when written in the basis of the natural eigenmodes of the system after the quench (Luttinger model with interaction parameter ). The fact that the Luttinger model is integrable and therefore never fully loses memory of this initial condition implies the breakdown of duality in the stationary state.

### iii.2 Discussion

As in the weak-backscattering case, we start discussing the equilibrium zero temperature result corresponding to . In this case we have and , and the integral in Eq. (69) simplifies. For the transient current at long times () we obtain,

 I(t)/V∼1t1/K−1 (73)

i.e. for the correction to the current vanishes in the long time limit as a power law, the junction is perfectly insulating and the tunneling is irrelevant Kane and Fisher (1992b).

We now consider the finite quench case, , and first focus on the long time steady state value of the tunneling current which reads

 Iss/V=8e2λ2∫∞0dττsin(2KarctanΛτ)(1+Λ2τ2)Kdualneq (74)

We notice that the integral is well defined as long as , a condition which is satisfied in the region of parameters plotted in figure 4 (region I). Interestingly, this region is not simply the dual of the region I for the weak-backscattering case, (light shaded line inside region I in figure  4) as it would be in the equilibrium case. This is a consequence of Eqns (71,72). Therefore we conclude that the quenched impurity model does not display a full duality, contrarily to other nonequilibrium realizations such as the driven noisy one Dalla Torre et al. (2012). We will come back to this point in the next section. Let us now discuss the behavior of the steady state current. Quite interestingly, the steady state current is different from zero at any finite quench amplitude , in other words the system deviates from the perfect insulating limit. In figure 5 we plot the steady state value of the tunneling current at fixed and as a function of . As in the weak-backscattering limit we find that for small quench amplitudes the steady state current is

 Iss/V∼λ2(K−K0)2 (75)

We now discuss the transient behavior and the approach to the steady state. We evaluate numerically the integral in eq. (69) and plot the result in figure 6. We find that the current decays to the steady state as a power law

 I(t)−Iss∼1tα (76)

with an exponent whose dependence on the quench parameters is shown in figure 6. As in the weak-backscattering case previously discussed, we find that the decay of the current is in general faster for a finite quench amplitude, in accordance with the analysis of Ref. Perfetto et al., 2010. In particular the exponent reaches its minimum for and behaves as .

## Iv Discussion

Let us summarize the results of previous sections and discuss their physical consequences. We started from the limit of a uniform system and found that for , corresponding to region I of fig. 1, perturbation theory in the backscattering is well behaved. While this means that in equilibrium, , the conductance approaches the unitary limit, a bulk quench gives rise to a finite deviation from this perfect conducting regime. This is consistent with the RG analysis we developed in Ref. Schiró and Mitra, 2014. There we showed that although for the backscattering is an irrelevant perturbation for the steady state, it nevertheless gives a sizeable effect, generating an effective temperature for the local degree of freedom, which explains, at least qualitatively, the deviation from the unitary limit. When , the backscattering becomes relevant, and perturbative approaches in the back-scattering breaks down, and one enters the strong coupling regime.

A possible approach to grasp the behavior of the system in this limit of strong backscattering is to start from the strong coupling fixed point, assuming the effective temperature is not enough to cut the growth of the backscattering under RG flow, and perturb this system of two decoupled wires with the sudden switching on of a local tunneling. We have considered this regime in section III and found that for , corresponding to region I in fig. 4, perturbation theory in the tunneling is well behaved. While this could naively suggest the strong coupling fixed point remains stable as in equilibrium, we have found that a finite tunneling current appears in the steady state for any finite quench amplitude . This result is again reminiscent of a thermal behavior and we may speculate that an effective temperature would indeed be generated under RG by the local tunneling, in analogy with the backscattering case.

Taken together, these results suggest that a global bulk quench has a dramatic effect on the problem, smearing out the sharp distinction between the conducting and the insulating phase that exists in equilibrium at zero temperature.

To further appreciate the role played by the quench amplitude in the problem, it is useful to look again at the steady state current in both regimes. In figure 7 we plot the weak-tunneling and weak-backscattering correction to the current as a function of and at fixed . At zero quench amplitude this would give the usual sharp transition from an ideal insulator for to a perfect conductor for . However as we clearly see, any small finite quench amplitude turns this into a smooth crossover. Finally, it is worth noticing that, as we mentioned earlier, the two regions and are not dual to each other. This means that there exist regions of the parameter space where neither the backscattering nor the tunneling are relevant, i.e., perturbation theory is well behaved. We interpret this result as a further signature of the effective temperature behavior of the problem.

These results for transport offer therefore another example of the quench-induced decoherence mechanism that we had previously identified in the orthogonality catastrophe problem Schiró and Mitra (2014). As in that case, the relevant energy scale, in the limit of small quenches, is proportional to the quench amplitude squared

 γ⋆∼g2(K−K0)2 (77)

where in the equation above indicates any source of local non-linearity, either due to the backscattering potential in the weak impurity limit, or due to the tunneling term in the limit of two decoupled chains. While the qualitative behavior is suggestive of an effective thermal behavior, it is important to stress that such an analogy does not fully carry over. In particular, the characteristic power law structure exhibited by the finite temperature current in the equilibrium problem Kane and Fisher (1992a, b); Giamarchi (2003), on both sides of the transition, appears to be washed out by nonequilibrium effects. Indeed, as we have discussed earlier, the scaling of the current in the steady state does not show signature of non-trivial power laws, even in the small quench amplitude regime where transport is essentially set by the quench-induced decoherence scale , with very weak additional dependence from the Luttinger parameters. This suggests that the analogy between this scale and an effective temperature, meant here as an infrared cutoff, cannot be pushed too far.

Finally it is worth comparing our results for the steady state impurity physics with the results and the predictions of Ref. Kennes and Meden, 2013. Here the authors computed the zero frequency charge susceptibility in a uniform LL after an interaction quench starting from a free system. For repulsive interactions they found it to diverge at the backscattering wave vector and concluded that, as in equilibrium, even a weak single impurity strongly disturbs the homogeneous LL or, in an RG language, the perfect chain fixed point is unstable. Then, they computed the local density of states close to an open boundary and found that it saturates to a non-zero value, with power-law corrections at low frequency. From the scaling of these corrections they conclude that the steady state analog of the open-chain fixed point is stable, with a power-law finite temperature conductance, which eventually crosses over to a non-zero value at low enough temperature. Based on their results the authors argue that the fixed point structure of a single impurity in a nonequilibrium steady state LL is similar to the one in equilibrium Kennes and Meden (2013).

As we have seen, our transport results combined with our analysis of the transient orthogonality catastrophe problem Schiró and Mitra (2014), suggest a rather different picture and highlight the importance of inelastic effects for the steady state impurity physics after the quench. In particular in the back-scattering case we found that, due to the global quench, a local non-linearity generates an effective decoherence in the problem, even in the regime where it would nominally be irrelevant at the one loop level. This quench-induced decoherence energy scale reduces the value of the zero temperature conductance away from the perfect uniform limit and, more importantly, competes with the growth of the backscattering as one enters the strong coupling phase, corresponding to for the case considered in Ref. Kennes and Meden, 2013. The result of this competition is clearly a subtle issue to establish firmly. Our results for the Loschmidt echo Schiró and Mitra (2014) suggested that well within the putative strong coupling phase this new energy scale acts as an effective infrared cut off, resulting in an exponential decay of the echo. This result would suggest that, quite differently from the equilibrium case, a weak single impurity does not disturb substantially the uniform system, i.e., one should not trust the perturbative break down for .

To further confirm this intuition, we have considered the opposite limit of an open chain, perturbed by an interaction quench and by a sudden switching on of a local tunneling. For any finite bulk quench amplitude we find a non-vanishing zero temperature conductance essentially set by the quench-induced decoherence scale. This result suggests that, differently from the equilibrium case, one never reaches the open-chain fixed point when the bulk has been quenched because the weak tunneling is always effective in generating a non-zero conductance. We may interpret this in an RG picture for the tunneling problem, dual to what we discussed in Ref. Schiró and Mitra, 2014 for the backscattering case, where the local non-linearity generates an effective decoherence that cuts off the RG flow. While a non-zero tunneling conductance was also found in Ref. Kennes and Meden, 2013, the physics behind this effect was not discussed. We now fully clarify its origin as a quench-induced decoherence phenomenon.

## V Conclusions

In this paper we have studied the problem of transport through a localized impurity in a Luttinger model far from equilibrium due to a bulk interaction quench. This model in its ground state has a familiar and well studied zero temperature transition between a conducting and an insulating phase, depending on the value of the bulk interaction parameter , and has an associated finite-temperature and finite-voltage correction which shows universal power law behavior.

Here we have presented perturbative calculations in the strength of the impurity potential starting both from the limit of a uniform liquid and from the one of two decoupled semi-infinite systems. In the former case a standard bosonization approach gives rise to a local backscattering term (described by a boundary sine-Gordon problem in a time-dependent nonequilibrium bath) that we treat to leading order in perturbation theory. In the latter, we use open boundary bosonization to treat the problem and obtain a dual formulation again in terms of a boundary sine-Gordon model where the non-linearity comes from a local tunneling between the isolated wires. Our results quite generically reveal that the nonequilibrium excitation of bulk modes induced by the global quench has important and peculiar effects on the conducting-insulating transition, which gets smeared out into a crossover for any small quench amplitude