Quench dynamics of noninteracting fermions with a delta impurity
Abstract
We study the out-of-equilibrium dynamics of noninteracting fermions in one dimension and in continuum space, in the presence of a delta impurity potential at the origin whose strength is varied at time . The system is prepared in its ground state with , with two different densities and Fermi wave-vectors and on the two half-spaces and respectively. It then evolves for as an isolated system, with a finite impurity strength . We compute exactly the time dependent density and current. For a fixed position and in the large time limit , the system reaches a non-equilibrium stationary state (NESS). We obtain analytically the correlation kernel, density, particle current, and energy current in the NESS, and characterize their relaxation, which is algebraic in time. In particular, in the NESS, we show that, away from the impurity, the particle density displays oscillations which are the non-equilibrium analog of the Friedel oscillations. In the regime of “rays”, fixed with , we compute the same quantities and observe the emergence of two light cones, associated to the Fermi velocities and in the initial state. Interestingly, we find non trivial quantum correlations between two opposite rays with velocities and which we compute explicitly. We extend to a continuum setting and to a correlated initial state the analytical methods developed in a recent work of Ljubotina, Sotiriadis and Prosen, in the context of a discrete fermionic chain with an impurity. We also generalize our results to an initial state at finite temperature, recovering, via explicit calculations, some predictions of conformal field theory in the low energy limit.
1 Introduction
There is a recent revived interest in noninteracting fermions in continuum inhomegeneous settings as analytically tractable models for studying equilibrium and out-of-equilibrium quantum correlations. For one-dimensional fermions at equilibrium in an external potential, there are interesting connections to random matrix theory (for a review see [1]). These relations allow to compute the density, the full counting statistics and the entanglement entropy in a variety of potentials [2, 3] using the tools of determinantal point processes [4, 5] in agreement with other approaches using inhomogeneous bosonization [6]. In particular, the correlation functions can be expressed as determinants built from a central object called the kernel [7, 8]. In the limit of a large number of fermions the kernel in the bulk of the Fermi gas becomes universal at the scale of the inter-particle distance , where is the local Fermi wave-vector. It is given, at zero temperature, by the celebrated sine kernel [7]. This universal behavior holds for smooth potentials [1] but breaks down for more singular potentials which vary over a region of size . In this case, the kernel differs from the sine kernel and has been calculated, for instance, for the hard wall [9, 10, 11, 12], the step potential [13], as well as for static impurities modeled by a delta potential [14]. Far from the singular region the kernel exhibits Friedel type oscillations [15, 16, 17], which have been characterized using reflection and transmission coefficients through [13, 18].
It was shown that the bulk universality for smooth potentials can be extended to equilibrium dynamics [19] (in terms of the so-called extended sine kernel [20]). It is natural to ask similar questions in the case of non-equilibrium dynamics. In the case of fermions trapped in a confining potential it was found for some time-dependent potentials (such as the harmonic oscillator or the potential) and some initial conditions, that the kernel keeps its equilibrium form up to time dependent factors [21, 22, 23, 24]. For more general potentials and initial conditions the situation is more complicated [24, 25, 26]. The large time limit of the kernel can be obtained from the so-called diagonal ensemble and coincides with the prediction from the generalized Gibbs ensemble (GGE) [27]. However the large time limit of the multi-point correlations exhibits a more complicated behavior [24].
One can also consider the out of equilibrium dynamics in the absence of a confining potential. A seminal example is the Landauer-Büttiker theory for transport between two reservoirs [28, 29]. Another important question, much studied in the context of Luttinger liquids, is to characterize the transport that takes place when two systems are connected via a contact [30, 31, 32, 33, 34, 35]. This problem has attracted a renewed interest in the context of quantum quenches where the system is translationally invariant but the initial condition is inhomogeneous. Important questions concern the large time behavior of the system, in particular the local convergence to a non equilibrium steady state (NESS), characterized by stationary currents, density profiles and counting statistics. Another question concerns non-stationary fluctuations and moving fronts. These questions have been addressed using conformal field theory (CFT) [36, 37, 38], generalized hydrodynamics [39, 40], as well as exact solutions of free fermions either on a lattice [41, 42, 43, 44, 45, 46, 47, 48] or in the continuum [49, 50].
It is natural to ask how these results for noninteracting fermions will be modified in the presence of a (non-confining) external potential. The simplest example is the delta function impurity model in the continuum which was solved at equilibrium in [14, 15, 16, 17]. In this paper we explore the out-of-equilibrium counterpart of this model, starting from an inhomogeneous initial condition. It is inspired by a recent work of Ljubotina, Sotiriadis and Prosen [51] where a similar model with an impurity was studied for a discrete fermionic chain. In that work the initial state was chosen fully decorrelated. On the other hand it is known that fermionic correlations can affect the transport, see e. g. [52]. In the present work, we consider an initial state that consists in two independent Fermi gases on semi-infinite lines separated by an infinite wall which are each in their ground state, and hence both exhibit the sine kernel bulk correlations with different Fermi wavevectors . They are joined at and evolve freely in the presence of a delta impurity at the origin. We find that the system reaches a NESS at large time. We calculate exactly the asymptotic kernel in the NESS, from which we obtain the asymptotic particle and energy current as well as the density profile. These quantities depend only on and the strength of the impurity. In the case where and when the impurity is repulsive , one finds that the NESS coincides with the ground state of the system on the infinite line, although the initial state is far from equilibrium. In this case one thus recovers the equilibrium kernel obtained in [14]. This can be understood since all the excitations move to infinity. On the contrary when , the Hamiltonian possesses a bound state and one finds that the NESS always differs from the ground state (even when ). In this case, the mean occupation number of the bound state does not converge to unity at large time. Our methods also allow us to study the relaxation towards the NESS, which we find to be algebraic in time. We also compute the kernel in the vicinity of space-time “rays” in the plane with fixed velocities . It exhibits a change of behaviors on the two light cones and . Interestingly we find nontrivial correlations between points belonging to rays with opposite velocities , an effect not discussed in [51], although we expect this effect to be also present in the discrete setting. An important difference with Ref. [51] is the existence of two light cones which is due to the presence of local correlations in the initial state which are different on both sides of the impurity. Outside these light cones one recovers the sine kernel, but inside the light cones (including in the NESS for ) the asymptotic currents are non zero and the kernel is different from the sine kernel.
Our results are then extended to an initial condition with two different nonzero temperatures and . In the low temperature and weak impurity strength limit and equal Fermi wavevectors , we recover, via explicit computations, the result for the energy current in the NESS obtained from CFT given in [32]. Finally we show that the asymptotic kernel, hence the correlations, along rays at fixed velocity can be recovered by a semi-classical argument. However this is not the case for correlations between points belonging to rays with opposite velocities . Also, the semi-classical method does not allow to recover the kernel inside the NESS. Finally our main results are checked versus numerical evaluation of the exact starting formulas.
Note that other recent works have addressed different effects of an impurity on the dynamics. In particular, full counting statistics and entanglement growth in fermionic chains [53, 54, 55, 56], interactions localized at a contact [57], dynamics starting from a homogeneous state [58] and moving and time dependent defects [59, 60, 61, 62] have been studied.
The paper is organized as follows. In Section 2 we define the model, the initial conditions and the observables that we study here. In Section 3 we give a detailed and pedagogical presentation of our main results. In Section 4 we derive the expression of the time dependent kernel in a hard box of finite size . In Section 5 we obtain the finite time kernel in the limit . To this aim we have adapted a trick based on contour integrals used in [51] (see also [63, 64]). As we show in that section, the study of the limit turns out to be more involved in the present continuum setting with a correlated initial condition. In Section 6 we obtain the large time limit of the kernel in the regime which characterizes the NESS. From it we compute the density and currents. In Section 7 we study the large time limit of the kernel along rays. In Section 8 we apply a semi -classical method which allows us to recover some of the above exact results in the regime of rays. Finally, we conclude in Section 9. The appendices contain many of the technical details of the calculations.
2 Model, quench protocol and observables
2.1 Model
We consider noninteracting fermions in one dimension and in the presence of a delta impurity at the origin described by the single particle Hamiltonian
| (1) |
We work here in units where the fermion mass is unity and . In these units, denotes the strength of the impurity, which can be repulsive () or attractive (). To fully specify the model we define it on the interval with a hard wall boundary condition (i.e., vanishing wave-function at ). We will be eventually interested in the problem on the full line obtained by taking the limit , with fixed fermion densities.
2.2 Initial state
The system is prepared at in the ground state of the many body Hamiltonian with , associated to the single particle Hamiltonian . This corresponds to imposing a hard-wall at , so that the system is cut into two independent halves and at . We will denote by and respectively the number of fermions in the left () and right () halves. Let us introduce and , with , the normalized eigenfunctions of the single particle Hamiltonian
| (2) |
where denotes the Heaviside theta function and
| (3) |
The corresponding energy levels are which are doubly degenerate ().
The ground state many-body wave function is a Slater determinant built from the eigenstates and with and . The -point correlation function (see below for a precise definition) can be expressed as a determinant built from the so-called correlation kernel
| (4) |
where
| (5) | |||
| (6) |
In particular, the mean fermion density is given by with , where and denote respectively the average fermion density to the left and to the right of the origin. We also define the Fermi momenta of the left and right half-spaces associated to the initial condition
| (7) |
and the corresponding Fermi energies
| (8) |
which will be useful in the following.
2.3 Dynamical evolution and observables
We now consider the time evolution for described by the single particle Hamiltonian (1) with a finite strengtgh of the impurity at . We denote the solution of the Schrödinger equation with initial condition , and similarly with where are given in Eq. (2). Under this evolution, the time dependent many-body wave function for the fermions, , remains a Slater determinant at all times, built from the time-dependent wave-functions and with and . The observables of interest are the time dependent -point correlations defined as
| (9) |
Using standard manipulations, can be written as (see e.g. [24])
| (10) |
where is the time dependent correlation kernel. In the present case it reads
| (11) |
where
| (12) |
Of particular interest is the time-dependent density given by
| (13) |
Another important observable is the total particle current defined as
| (14) | |||
which can be rewritten in terms of the kernel as
| (15) |
where . From the Schrödinger equation, the current satisfies the fermion number conservation equation
| (16) |
In this paper we compute the time evolution of the density , of the kernel and of the current . Since we are interested in the large time behavior of the bulk of the system, we will take the limit before taking . More precisely we will take the limit , with fixed and , i.e. with fixed mean densities
| (17) |
or equivalently with fixed Fermi energies (see Eq. (8)).
3 Main results
In this section we present our main results. The first one is the expression for the kernel in the thermodynamic limit at any fixed time . It is a lengthy although fully explicit expression which is given in (87) as a sum of terms which are given respectively in (5), (5), (95), (96) and (98). From this expression one can read the time dependent density from (13) and the current from (15).
The subsequent results concern the large time behavior obtained from the kernel, once the thermodynamic limit is taken. We have found that there are actually two different scaling regimes. The first one is the NESS where and the second one is the regime of rays where both . Since the analytical computations of the asymptotic behaviors are quite tricky, we have carefully checked numerically our main predictions, which are shown in Figs. 1, 2 and 4 below.
Finally, we extend our study to an initial state at finite temperature, with two different temperatures and to the left and to the right of the origin. We also obtain the heat current as a function of these two temperatures.
As we will see below some of these results (but not all) can also be obtained from a heuristic semi-classical method which relies on the momentum dependent transmission and reflection coefficients and , which for a delta function impurity are given by
| (18) |
The results presented here are derived from first principles, starting from from an exact expansion over the eigenfunctions of many-body system.
3.1 Non equilibrium stationary state (NESS)
The first regime corresponds to fixed spatial positions with . In this case the kernel, the density and the particle current reach a stationary limit which we compute explicitly, namely
| (19) |
Note that from the fermion number conservation in Eq. (16) the current is constant in space in the large time limit. From the symmetry of the problem under the change , these observables satisfy the following relations
| (20) |
3.1.1 The case of a repulsive impurity ()
We first give the results in the case of a repulsive impurity , and consider the case in the next section.
Density. For the density we find, for
| (21) |
Note that is obtained from this expression Eq. (3.1.1) together with the symmetry relation (20). In the second line of Eq. (3.1.1), denotes the imaginary part and denotes the exponential integral (see [14] for details on obtaining the second line from the first one). The function is also denoted in Mathematica, the incomplete gamma function of index , and the contour of integration defining it should not cross the negative real axis. This result for is shown in Fig. (1) and compared with a numerical evaluation of the exact formula for (from (86) with ) at a relatively large time. It is also plotted in Fig. 3 for smaller values of the time. As one can see on these figures, the convergence to our prediction within the NESS is rather fast.
Let us now discuss a few salient features of this result. Far from the impurity, which is located at , the stationary density profile approaches constant which is different on both sides and given by
| (22) |
These asymptotic values can also be predicted by the semi classical method (see Section 8) and written in the equivalent form
| (23) |
in terms of the transmission coefficient given in Eq. (18). The mean density is continuous at but exhibits a cusp, with different left and right derivatives given by
| (24) |
At variance with the equilibrium case (see below), this cusp is asymmetric.
Finally, the result for can be compared to the result obtained in [14] for the mean density of the equilibrium problem, i.e. in the ground state with Fermi energy in the presence of a repulsive delta impurity of strength (see formula (60) and (138) there with )
| (25) | |||||
We see that for the NESS density coincides with the equilibrium (ground state) density
(as discussed below this holds only for ).
This is quite interesting since the initial state in the present work is far from the ground state of the system in the presence of the impurity.
An explanation for this property is that the components of the initial state on the excited states correspond to fermionic
waves (quasi particles) propagating towards the edges of the system. If the observation time is smaller than ,
such that the fastest fermions traveling at speed have not been reflected yet, i.e. ,
we expect relaxation to the equilibrium state for (and to the NESS for ).
This will always occur if the limit is taken first.
For a finite size system, oscillations will take place on larger time scales.
Current. In addition, in the NESS, we show that there is a non zero particle current given by
| (26) |
which can alternatively be expressed, using the transmission coefficient given in Eq. (18), as
| (27) |
This result can also be obtained by a semi-classical method, see Section 8. The current is shown in Fig. 2 for , (note that it is negative in that case). Its maximal (absolute) value is reached for when there is no defect. In that case one finds which corresponds to a unit conductance . In the limit it vanishes as
| (28) |
which shows, as expected, that for a very strong defect the system is effectively cut into two almost independent halves.
Kernel. Finally we also obtain the kernel in the NESS which reads explicitly for
while for it reads
| (30) | |||
the other cases being obtained by symmetry (see (20)). For equivalent expressions are given in (117). The kernel simplifies in the limit where the points . In this limit, keeping one finds the asymptotic behavior
| (31) |
On the other hand, if and with one finds
| (32) |
while the asymptotic kernel, for , vanishes for generic value of different from .
It is interesting to note that although the semi-classical method cannot obtain the full expressions in Eqs. (3.1.1), (30), it does predict the asymptotic form in (31), see Section 8. However, we will see that the result (32) cannot be predicted by this semi-classical method.
Comparison with the GGE. It is interesting to compare our result for the kernel in the NESS to the prediction from the GGE. This is done in details in the Appendix Appendix G: Remarks on NESS and GGE. The main result is that the kernel contains only ”diagonal” terms (which are time independent already at finite ) and does not lead to any current. However we show that there are non-diagonal terms (called below) which carry current and contribute to the NESS if the limit is carried first.
3.1.2 The case of an attractive impurity ()
If is negative there is an additional eigenstate of , denoted with eigenenergy , which is a bound state. All other eigenstates have positive energies and hence propagate over the whole system. One finds that the kernel for has the same expression as above [see Eqs. (3.1.1) and (30)], up to an additive term denoted which takes the form
| (33) |
Because of the exponential factor , this additional contribution in the kernel is localised around .
From the result for the kernel one obtains that for the density in the NESS is given for
| (34) | ||||
For the formula for is again obtained from (3.1.2) simply by permuting and . Finally the current in the NESS is still given by formula (26), which is invariant under the change , and is thus not affected by the bound state.
In the case one can compare with the equilibrium result for obtained in [14]. For the density it was found there that at equilibrium for
| (35) | |||||
which is compatible with (25) because the function has a branch cut on the negative real axis. Hence, by comparing Eq. (3.1.2) and (35), we see that, at variance with the case , the NESS for differs from the equilibrium ground state. This is because the mean occupation number of the bound state which is unity in the ground state, and which is defined in the NESS from is given from 33 as
| (36) | |||
| (37) |
is strictly smaller than unity (including in the case ). Hence the last term in the last line of Eq. (3.1.2) is proportional to . Therefore the post-quench bound state is always partially empty in the NESS, see [65] for a related effect in a similar model based on a GGE calculation.
Note that in the present model there is a single bound state. In the case of multiple bound states it has been found in related models that the NESS can present persistent oscillations in time [51, 55, 65]. To obtain such an effect in the present model would require to consider two delta impurities.
3.1.3 Relaxation to the NESS for
We have also obtained the large time decay of the kernel towards its value in the NESS for . The result for is given in (B22). From it one extract the density and the current. The decay of the density is found to be of the schematic form
where the functions can be read off from Eq. (B23) in the Appendix Appendix B: Convergence to the NESS at large time (). Hence the leading decay is modulated by oscillations, together with a term which is non oscillating. We also find that the current has also a leading algebraic decay as modulated by oscillations. Note that power law decays with oscillations have been obtained in other systems of noninteracting fermions [25, 43].
3.2 Large time regime with fixed (ray regime)
The second regime corresponds to both with a fixed ratio , i.e. along rays. In this case the density and the current reach finite limits, which are only functions of the scaling variable
| (39) |
Note that the fermion number conservation Eq. (16) implies that these two functions must be related via
| (40) |
All the results below in that regime hold for any (positive or negative) since the bound state (that exists for ) does not contribute in that limit.
Density. We find through explicit calculation of the large time limit, that the scaling function for the density reads
| (41) |
This gives, using the explicit formulae for and in (18), for ,
| (48) |
A plot of is shown in the right panel of Fig. 4, together with an exact evaluation at finite time illustrating the convergence. Note that is a continuous function of except at where it has a jump discontinuity. The values on each side of the jumps at are found to agree with the large distance limit of the density obtained the NESS regime, i.e.
| (49) |
This matching shows that there is no additional intermediate regime between the NESS , and the regime of rays .
Current. For the scaling function of the current we obtain, again through explicit calculation of the large time limit
| (50) |
where is given in (18). More explicitly, for it gives
| (51) |
where we recall that the Fermi energies are . A plot of is shown in the right panel of Fig. 4, together with an exact evaluation at finite time illustrating the convergence. The function is a continuous function of everywhere. One can check that the conservation equation Eq. (40) is obeyed, including at the point (the delta function in is cancelled by the factor in Eq. (40)). Note that in this model there are two pairs of light cones at and respectively. Outside these two light cones () the current vanishes at large time. Inside these two light cones () the current is constant and equal to its value in the NESS (and so is the density).
Kernel. We have found by explicit calculation that in this ray regime the kernel vanishes unless or . More precisely one obtains the limiting scaling forms for
| (52) |
The expression for is obtained as
where we recall that . Note that is a function of only. From this expression (3.2) in the limit of coinciding points one recovers the density
in Eq. (41) (using ) and the current in Eq. (50) using (15). It is important to note that
the kernel (3.2) matches exactly in the limit with the result
(31) for the
kernel of the NESS in the large distance limit .
The expression for is given by
| (54) |
This measures the quantum correlation between opposite rays. Once again the kernel (54) matches exactly in the limit with the result (32) for the kernel of the NESS in the limit of very separated points .
It is important to remark that the results for the density (41), the current (50) and the kernel at coinciding rays (3.2) can be also obtained using the semi-classical method as we will see in Section 8. However the result for for opposite rays in Eq. (54) cannot be obtained from this semi-classical method. Indeed, it contains additional information about correlations at a finite distance from the two opposite rays which correspond to fermion wave-functions which are split between reflected and transmitted waves by the defect.
3.3 Finite temperature generalisation
The previous result can be generalised to an initial state with non zero temperature different on both sides of the impurity and with associated chemical potential . This amounts to take as an initial kernel at
| (55) | |||
| (56) |
where
| (57) |
is the Fermi factor with . The density of fermions in the initial state is now related to the chemical potentials via
| (58) |
The calculation proceeds in the same way as for with a few additional details which are given in Appendix Appendix D: Details for finite temperature. We present here only the result for .
In the NESS regime the full kernel is presented in (D11) and from it we find the asymptotic density at finite temperature, for :
| (59) |
and the expression for can be obtained from the symmetry (20) with here and . Note that the density retains a cusp at even at finite temperature. The current in the NESS at finite temperature is obtained as
| (60) |
Its low temperature expansion is performed in Appendix Appendix F: Low temperature expansions and we display it here in the case
| (61) |
Note that in the absence of impurity, for , one has instead (for any )
| (62) |
which generalizes the standard zero temperature result. Note that the case has been studied for the same model
in a high temperature limit where interesting equilibration properties were found [50].
In our case (), one can show that the current is proportional to at high temperature.
In the ray regime we find the density
| (63) |
and the current
| (64) |
These results are plotted in Fig. 5.
3.4 Energy current
Let us a recall the definition of the heat current in quantum mechanics for a single particle with a Hamiltonian and described by the wavefunction , see e.g. [66]. The local energy and heat current are given by
| (65) | |||
| (66) |
The total averaged energy is recovered from while and obey the conservation equation (from the Schrödinger equation)
| (67) |
For noninteracting fermions with the initial condition considered here (see Section 2.3), each state evolves independently, hence the total local energy and energy current are given by the corresponding sums of the one-body contributions weighted by the Fermi factors. The time dependent energy current is thus given by
| (68) |
where and is the single-particle Hamiltonian with a delta-impurity in Eq. (1).
The explicit calculation is performed using methods which are similar the ones used for the density and the particle current. They are summarized in Appendix Appendix E: Energy current. One finds that in the large time limit the energy current converges to an asymptotic constant value which reads
| (69) |
In the low temperature limit and in the case the current has the following expansion (see Appendix Appendix F: Low temperature expansions)
| (70) |
It is interesting to compare the first term in this low-temperature expansion with a result for the energy current obtained in [32]
| (71) |
where is the central charge. A simple derivation of this result was given in [32] for free Majorana fermions (corresponding to ). In that work is the transmission coefficient of the defect, i.e. the analog of here. However the coefficient of that we obtain here in (70) is not equal to as would be predicted by the model of [32] taking into account that here . The discrepancy can be understood as follows. From similar methods as in Appendix Appendix F: Low temperature expansions one can show that for a general dispersion relation our formula (70) becomes to leading order for small and
| (72) |
Performing the derivative with respect to in (72) we obtain two terms. The first one is and coincides with the
result of [32]. The additional term is non zero when the transmission
coefficient depends on the momentum , as it is the case here. This additional term is negligible compared to the first one only when the impurity strength is small, i.e., .
Note that in the absence of the impurity, i.e. for , the low-temperature expansion of reads for arbitrary
| (73) |
The expansion formula for and are given in the Appendix Appendix F: Low temperature expansions.
4 Exact calculation at finite time and finite size
In this section we derive an exact formula for the kernel at any time in a system with of size . This will be the starting point for the asymptotic analysis for and subsequently for the computation of the large time limit performed in the following sections.
4.1 Eigenbasis of
We first consider here , and a finite interval and specify further by imposing the vanishing of the wavefunctions at . The eigenfunctions of are either even or odd in , respectively labelled by a subscript ’’ or ’’. The odd eigenfunctions do not feel the delta impurity (since they vanish at the location of the impurity) hence they read
| (74) |
where we denote the lattice of possible values for the wavevector .
The even eigenfunctions are also plane vaves, and denoted by , but with a different quantization condition on . They read
| (75) |
with the quantification condition (see Fig. (6))
| (76) |
i.e., , which defines the lattice of possible wavevectors
| (77) |
Note that and correspond to the same state, hence the condition . Equivalently the states can be labeled by the strictly positive integers (see Fig. (6)).
Finally, both the odd and even eigenstates are associated to the eigenenergy
| (78) |
4.2 Time dependent kernel
As discussed above, see Eq. (11), the time dependent kernel splits into two parts
| (79) |
where each component evolves independently
| (80) |
Since the ’s evolve according to the Schrödinger equation with Hamiltonian in Eq. (1) these components can be rewritten using the real time Green’s function
| (81) |
where the eigenfunctions of are given in (74) and (75). This leads to
| (82) | |||
| (83) |
The time evolution of the total kernel is thus obtained as the sum as in (79).
Let us first study in Eq. (83). Inserting the decomposition (81) of the Green’s function together with the explicit expression of in Eq. (6) we obtain
| (84) |
where, in the third and fourth lines, we have just reorganized the discrete sums and the integrals over and . The same manipulations can be performed for . The overlap integrals over and can be performed explicitly, which gives
| (85) |
where, in the last two factors in brackets, the sign refers to and the to . To compute the full kernel we add the two halves and develop the product to get
| (86) | |||
which is the sum of four terms denoted as indicated in the equation. These terms satisfy the symmetries: is real and symmetric, and .
Remarks:
(i) Symmetries: Since the initial kernel is unchanged under the simultaneous transformation and because of the invariance of under parity transformation , it follows that is also unchanged under . The density has thus the same invariance and the current satisfies . This property is the finite time analog of the relations valid for the NESS stated in Eq. 20.
(ii) Contribution to the current: Since the term is real, it does not contribute to , i.e., (see Eq. (15)). It is easy to see that the term gives only an odd contribution to the current, i.e. . Hence the term cannot contribute to , the current in the NESS, which is uniform. It does however contribute to . The current in the NESS is thus only determined by which gives an even contribution at all time , i.e., .
5 Thermodynamic limit
Our first goal is to obtain a formula for the kernel for the infinite system, i.e., for fixed , which, for notational convenience, we will also denote by . Here we consider the case of a repulsive impurity – the analysis of an attractive impurity is performed in Appendix Appendix C: Details for . Let us recall that we take the limit while fixing the left and right initial densities . This is possible thanks to a contour integration trick that we explain below. From the exact expression for the kernel in (86), one can write
| (87) |
where we have used the relation . We now give the expressions of the limits when of the terms and separately.
The term : From (86) we see that it is time independent and, in the large limit, it is given by the reflected sine-kernel (see e.g. [10, 11])
| (88) |
The term : From (86) the term is given by a double sum
| (89) |
with , and and we recall that the lattices and are defined in Eqs. (74) and (77) respectively. We have also defined the function 111Since we are eventually interested in the large limit we omitted in (90) the unimportant extra factor in the denominator in (86), which should be restored to obtain finite formula.
| (90) |
Taking the limit of (89) to obtain a double integral is very delicate due to the presence in the sum of a pole and the fact that the two lattices and are intertwined (see Fig. 6). Let us first state the result and give its derivation below. One finds
| (91) |
The first line of (5) is an integral over a contour that we denote consisting in straight lines in the complex -plane, forming a half-rectangle as represented in Fig. 7. Its value does not depend on the parameter since as a function of does not have poles in the strip . In addition to the derivation given below we have carefully checked numerically this formula (5) for a variety of function which share the same properties.
Let us now give the derivation of this result in Eq. (5). The idea is inspired from [51], although some details are different here. The trick is to replace the discrete sum over in (89) by a contour integral as follows
| (92) |
where the contour is a union of very small circles centered around the points with and oriented counterclockwise (see Fig. 8). The circles should be small enough so that the contour does not enclose any point . We now deform the contour into the closed counterclockwise contour which is the rectangle with the four corners , , , , represented in Fig. 8. The parameter is chosen small enough so that the contour does not contain any point of located to the left of and to the right of . During this deformation one encounters only the poles at . Taking into account the residues associated to these poles one obtains
| (93) |
Until now this is an exact rewriting of in Eq. (89) valid for any . For any , using the second relation in (76), one can evaluate the factor
| (94) |
We now take the large limit on Eq. (93). The factor for and for . Hence the integral in (93) over the counterclockwise closed contour becomes the integral over the clockwise half-rectangle in (5) 222Proving the convergence for the vertical parts of the contour is non-trivial, which is why we did careful numerical checks of (5). This problem does not occur in [51, 63]. In addition the sum over becomes an integral in the large limit. This leads to the result (5).
The term : We now give the large limit of and defined in (86). Each term can be decomposed as the sum of two terms
| (95) |
where the first term comes from the terms in the triple sum in (86), while the second one comes from the terms and does not depend on time. As detailed in the Appendix Appendix A: Large limit, term , we obtain the first term in (95) as
| (96) | |||
where we have defined the function
| (97) |
The second term in (95) is obtained in the large limit as
| (98) |
The terms and are simply obtained from and by substituting in Eqs. (96) and (98).
6 Large time limit: stationary state at fixed position
In the previous section we have obtained the expression of the kernel in the thermodynamic limit in (87) together with (95), as a sum of several terms. Now we can take the limit of each term in this expression for fixed positions and to obtain
| (99) |
In (87) and (95) the terms and are independent of time. As shown in Appendix Appendix B: Convergence to the NESS at large time (), the terms decay to zero at large time as power laws in time. Finally goes to a finite limit where only the last term in (5) (coming from the residues) survives at large time, as discussed in Appendix Appendix B: Convergence to the NESS at large time () where the time decay is studied. From the last term in (5) we obtain
| (100) |
As a result the kernel at infinite time, i.e., in the NESS, is obtained as the sum
| (101) |
where the terms are given in (5),
| (102) |
where is given in (100), and we recall
| (103) |
Stationary density. The stationary density can be obtained from the kernel using the relation . It can be written as the sum of the contributions of the terms in (101) with
| (104) | ||||
| (105) | ||||
| (106) |
Let us rewrite using the identity
| (107) |
Summing the different contributions to , one obtains
| (108) | |||
| (109) |
In that expression the constant parts can be rewritten for and respectively as
| (110) |
| (111) |
as in Eq. (3.1.1). Next, regrouping the terms proportional to and in the integrals one can check that it is identical to the expression given in Eq. (3.1.1).
Stationary current. For the current, using (15) one obtains in the large time limit
| (112) | ||||
| (113) | ||||
| (114) |
where the first two contributions vanish since and are both real, and the third simplifies. The total result for the current in the NESS regime can thus be written as
| (115) |
Using one can rewrite this result as in (26) in terms of the Fermi energies on both sides.
Stationary kernel. Putting all terms together in (101) and using standard trigonometric relations we find for the result for given in Eq. (3.1.1). Note that the first term in (3.1.1) is the sine-kernel associated to the right side of the system, namely . One can rewrite this result as (we recall that here )
| (116) | |||||
Similarly, for and one finds the result for given in Eq. (30). Once again, it is equivalent to the following expression
| (117) |
Large distance limits. Interestingly this stationary kernel has several non-trivial limits far from the defect. Indeed consider first the expression (3.1.1) for with fixed. Let us use the following property. For any smooth function with bounded , as one has
| (118) | |||
| (119) |
This can be shown by performing two successive integrations by parts with respect to . This shows that the last term in (3.1.1)
| (120) | |||
decays to zero at large distance. The asymptotic behavior is thus given by (31).
7 Large time limit with fixed
In this section we study the large time limit of the kernel in the regimes of rays, i.e.
with and fixed and . We start from the expression (87) for the kernel
in the limit.
It is a sum of terms of type which are given in Eq. (5) for ,
in Eq. (5) for and in Eqs. (95),
(96) and
(98) for . In these expressions, we will set and and study the large limit at fixed
and fixed . As we show below the terms and are simple to analyze.
The study of the term requires a modification of the contour integral trick used for the NESS, as pointed
out in [51]. We now examine
these terms independently.
The term . From (5) this term decays to zero unless . In the first case one finds
| (121) |
while in the second case one finds
| (122) |
The term . Inserting and in (5) we will treat separately the contour integral in the first line and the residue part in the second line of that equation. In the contour integral over in (5) one must be careful with the factor which comes from the factor containing in the function in (90), and which reads schematically
| (123) |
Previously, for , this term was decaying to zero at large since along the contour. For let us rewrite (123) setting , as
| (124) |
Let us recall that on the contour in (5) and . Consider first . In that case the first term in (124) can be discarded at large time, and the second term diverges at large time if . Since , if we can still discard the contribution of the contour integral at large time. If we need to deform the contour to be able to handle the large time limit of (5). Consider first the case . Denoting the original contour, one writes where the contours are represented in Fig. 9.
One has
| (125) |
and, as we argue below, the contribution of vanishes at large time. On the other hand, since is a closed contour which surrounds the interval , its total contribution to is given by a sum of residues at and reads (it involves only the second term in (123) since the first term can be discarded for any )
| (126) |
where
| (127) |
It thus adds to the residue term on the second line of (5), noting however that in (5) .
Let us now discuss the contribution of the contour in Fig. (9) denoting . The general idea is that this new contour has been chosen such that its contribution decays to zero at large time. For the first and last term in (125), it is clear that the large time limit vanishes since in both cases on the new contour. It is convenient to take again the limit which makes the contributions of the horizontal parts (second and fourth terms in (125)) vanish. Let us analyze the contribution to of the remaining integral (the third integral in the right hand side of Eq. (125)) along the vertical axis at , setting the conclusion remains unchanged for the general case. Writing , it reads (retaining only the second term in Eq. (124))
| (128) |
One can argue that this integral vanishes at large time. By expressing the sine and cosine terms in (128) as the sum of exponentials, the resulting exponential terms have the form which have a saddle point at , . By expanding around these saddle points one finds that the integral decays algebraically at large .
For it is the second term in (124) which vanishes at large time. For the first term in (124) one can perform the same manipulations as above with and the right hand side in (126) is changed to
| (129) |
where we have used that the shift by in the bounds of the integral is irrelevant in the following.
Discarding all the terms which vanish at large time, we are left with the sum of (124) and the contribution from the second line in (5) which leads to
| (130) | |||
It vanishes at large time unless or . For we find using standard trigonometric identities
| (131) | |||
from which one computes the combination which is needed for the kernel (see Eq. (87)). For we find similarly
| (132) | |||
from which one computes the combination which is needed for the kernel (see Eq. (87)).
The term . Let us start with . From (98) one has
| (133) |
where
| (134) |
Using trigonometric identities, one finds that it vanishes at large time except for . One finds for
| (135) |
and for
| (136) |
Finally note that the term , which was shown to vanish in the NESS
regime (see previous Section 6), is expected to also vanish in the present ray regime
(although we will not study here its decay in detail).
Final result. Adding all the terms computed above we obtain the final result for the kernel . For we obtain, recalling the identity ,
| (137) | |||
This can be reorganized, leading to the result given in the text in (3.2). From this formula we obtain the density , which is independent of , and given in the text in (41). Similarly one obtains the current leading to the expression (50).
8 Wigner function and semi-classical approach
It turns out that the results for the density and current obtained here by an exact computation in the regime of large time and fixed, agree with the prediction of a semi-classical approach. A similar agreement was observed for the discrete model studied in [51]. Here in addition we show that the agreement extends to the full kernel at large time with , obtained here in Section 7 with and from an exact computation. However, as we discuss below, this semi-classical approach does not allow to predict the non trivial correlations obtained here for . Let us present here this approach.
The many-body Wigner function was defined and studied in e.g. [67] in the case of noninteracting fermions (see also [25]). It was shown to be related to the kernel via (we recall that we set here )
| (139) |
and we recall that the time-dependent mean fermion density is simply
| (140) |
The Wigner function for fermions in an external potential obeys an exact time evolution equation, see e.g. [24]. For smooth potentials one can define a semi-classical limit (, large and fixed see e.g. [25]) and this equation simplifies into the Liouville equation . For a delta function potential this limit cannot be defined in that way, but one can define instead a semi-classical form for the Wigner function by introducing transmission and reflection coefficients, as was done in [51]. Let us now recall this approach.
Consider first the easier case with no defect. The fermions evolve freely with speed , and in the present model . The Liouville equation applies to this free evolution with and the Wigner function is simply transported along the classical trajectories. It is thus obtained from the trajectories run backward in time as
| (141) |
where is the Wigner function at time . As described in Section 2.2, here we consider an initial condition which is the product of two independent fermionic ground states, separated by an infinite hard wall at . On each side of the wall one considers the ground state at Fermi energy for , respectively for , of noninteracting fermions in the absence of an external potential, with vanishing wave function at . The corresponding kernel is given in Eqs. (4)-(6). As a consequence
| (142) |
where the functions are obtained from in (5),(6) by the same transformation as in (139). The exact calculation of these functions is performed in Appendix C.1 of [68]. Away from a small layer of width near the wall, it takes the form predicted by the local density approximation (LDA)
| (143) |
which is discontinuous at , as represented in Fig. 10 (left panel). We will use this form (143) from now on.
Now we add the defect at . Consider a fermion at position with and with momentum . If the condition holds, then the fermion has traveled to position without crossing the defect, as shown by the red solid line in Fig. 11. On the other hand, if the condition holds, there are two possibilities (indicated by the blue solid lines in Fig. 11): (i) either the fermion was at on the half-line to the left of the defect, and it has been transmitted, or (ii) it was at on the half-line to the right of the defect, and it has been reflected, see Fig. 11. The picture is similar for . Setting this leads to the prediction, for
| (144) |
and for
| (145) |
where and are the transmission and reflection coefficients of the defect. We have assumed here for simplicity that . Although this applies to any defect which is symmetric under , for the present delta function potential one has
| (146) |
Note that the Wigner function is invariant under the symmetry and .
It is useful to rewrite the Wigner function using in the following form, with , for
| (147) |
and for
| (148) |
The density is then obtained from (140). Using (147) and (148) we obtain, with fixed
| (149) | |||
| (150) |
where in the second equation we have changed and used . One can check
upon rearranging and using that this prediction coincides with the result of the exact calculation obtained
in (41).
Let us now extend this prediction to the full kernel using the semi-classical Wigner function. By inverse Fourier transform of (139) one has
| (151) |
Let us consider the large time limit in the ray regime with and , where . In that case one can approximate (151) as
| (152) |
Using again (147), for and one obtains
| (153) |
and using (148) for
| (154) |
where we have used and the symmetry . Remarkably, this semi-classical prediction coincides with the exact limiting kernel in (3.2) obtained by explicit calculation for the delta function potential model in the ray regime for any , including and . It is natural to expect that this semi-classical method will predict the correct behavior for more general potential, although this remains to explored. However, as discussed in the previous section, the semi-classical approach, in the present form, does not predict the kernel in the NESS regime (except in the limit with ) nor the kernel which is obtained for .
9 Conclusion
In summary, we have studied a quantum quench of noninteracting fermions in one dimension in the presence of a delta impurity at the origin whose strength is changed at from to a finite value . The initial state consists in two Fermi gases separately at equilibrium on the two half-axis, with different bulk densities and temperatures . We have obtained exact results in the thermodynamic limit (i.e., for an infinite size system) for the correlation kernel, density and current (particle and energy) for any time after the quench. This allowed us to study analytically the large time limit of these quantities and their relaxation properties. We found that there are two distinct regimes:
(i) For fixed position and large time, the system relaxes to a non equilibrium steady state (NESS) characterized by a uniform particle transport current and energy current . We have obtained explicit expressions for the kernel, density and in this regime as a function of . In particular, we showed that the density in the NESS exhibits a non-trivial spatial dependence near the defect, with a jump in the first derivative at , oscillations which extend far from the defect (which are non equilibrium analogs of Friedel oscillations) and different plateau values at . We obtain the temperature dependence of the currents and . In particular, in the low temperature limit, we find , with a prefactor that matches with the conformal field theory universal prediction in the small limit.
(ii) The ray regime , at fixed , where the density and current reach asymptotic finite values which depend only on . At zero temperature, their dependence on shows slope discontinuities along the two pairs of light cones at . We also obtain the spatial dependence of the asymptotic kernel, denoted , in the vicinity of the ray . We also obtain , which describes the correlations between two points with two opposite rays and .
It is important to note that these two regimes match smoothly, as we have shown, which indicates the absence of possible intermediate regime. This means that in the ray regime, as one recovers the asymptotic densities of the NESS, as well as the current . Finally, in the case (i) we have studied in detail the convergence in time towards the NESS. We have found that at zero temperature the kernel and the density decay as modulated by oscillations, superposed to a non-oscillating part, towards their values in the NESS.
The present study is the analog for the non-equilibrium dynamics of our recent work in [14] where we studied the ground state of noninteracting fermions in the presence of a delta impurity. The form obtained here for the kernel and density in the NESS can be compared with the one obtained at equilibrium in [14]. Although they have some common features (such as the cusp in the density and Friedel-like oscillations) their overall form is different. In the limit (at zero temperature) they become identical, however only in the case . This can be qualitatively understood since for the excitations which are present in the initial state can propagate to infinity, while for there is a bound state which cannot carry propagating excitations.
Our work has close relations to the work of Ljubotina, Sotiriadis and Prosen [51], which has inspired the analytical methods used here. However, in addition to being defined on a lattice, the initial state considered there is completely uncorrelated, while in our case there are the correlations of a Fermi gas with two Fermi momenta and . This leads to a different structure in the ray regime with two distinct pairs of light cones. As pointed out in Ref. [51], the results for the density and the current in the ray regime can be obtained from a semi-classical method. However the density in the NESS (at finite ) cannot be obtained from this method. We have shown that the same properties hold in the present continuum model. In addition we have computed the kernels : while can indeed be predicted by the semi-classical method, the kernel which measures correlations in opposite rays cannot be predicted by this approach, and requires exact methods.
The present study unveils a number of correlation kernels, which have a non trivial form at the Fermi scale , different from the universal sine kernel (which is recovered here only outside the light cones). At variance with standard kernels of RMT, which also describe trapped fermions at equilibrium, the present ones found here carry currents, hence they are not purely real. It would be interesting to find analog in the context of RMT.
Finally, we obtained here the kernel in the NESS, but it would be interesting to obtain also the full density matrix and compute other observables such as the entanglement entropy. This is left for future studies.
Acknowledgements
We thank D. S. Dean for suggesting this calculation and helpful comments. We thank D. Bernard, A. Krajenbrink, S. N. Majumdar and L. Zadnik for useful discussions. We acknowledge support from ANR grant ANR-17-CE30-0027-01 RaMaTraF.
Appendix A: Large limit, term
In this appendix we give some more details on the evaluation of the large limit of the term defined in the text in (86). As discussed in the text [see Eq. (95)] each is split into two parts, an off-diagonal one with and a diagonal one with . Below, we discuss them separately.
A.1 The off-diagonal part
Let us focus on the off-diagonal term from (86) (the term is obtained similarly by changing )
| (A1) |
where the discrete sum has been restricted to (and the term which is subdominant at large has been removed). As in the text we denote
| (A2) |
The large analysis is very similar to the one presented in the text for the term in Section 5. Namely we replace the discrete sum over in (A.1) by a contour integral using the same trick as in (93). The contour in Fig. 8 is now a rectangle with replaced by , which in the large limit becomes again a half-rectangle, similar to in Fig. 7 (with set to zero). In addition to this contour integral there are residues which we need to take into account. Since one sees that there are only simple poles at either or and their residues must be summed up. Hence we arrive at the following formula in the large limit
| (A3) | |||
Let us now compute each term separately in (A.1). The last line (the residue part) is found to be equal to
| (A4) |
On the other hand the contour integral can be calculated explicitly, namely
| (A5) | |||
which holds for any . When adding to (A4) we see that the imaginary part cancels and we are left with the result given in the text in (96).
A.2 The diagonal part
We now turn to the diagonal part, i.e., the term with in (86), focusing on (the term is obtained similarly by changing ). It reads
| (A6) |
where is given in (A2). To take the large limit, we again use the contour integral trick used above. We convert the discrete sum over into a contour integral, denoted by , plus a residue term, denoted by . However, instead of having two simple poles as in the off-diagonal terms, we now have a double pole at , which changes the result for the residue.
Let us first consider the contour integral term. It has the form, in the large limit
| (A7) |
where the expression of the integral is obtained from (A5) by setting . We get
| (A8) | |||
The remaining integral over is well defined (in the principal value sense around ). Hence this term clearly vanishes in the large limit since it is of order . This is because the initial triple sum in has become a double integral, and hence an extra factor .
Let us now consider the residue term. It can be read off from (A6) as
| (A9) |
where is a contour in the complex -plane consisting of the union of small circles oriented clockwise centered around with . We now use the residue formula for a double pole with a counterclockwise contour around , namely
| (A10) |
which, applied to the integral over in Eq. (A9), leads to
| (A11) | |||||
where, in the last line, we have computed the derivative with respect to , set and used the quantification condition (76) . In the large limit, the first term in (A11) dominates and plugging it into (A9), we finally obtain the residue part as
| (A12) |
Since we found that vanishes, we finally obtain the result for the limit given in the text for in (98).
Appendix B: Convergence to the NESS at large time ()
In this appendix we extract the leading decay in time of the kernel to its stationary value in the NESS. We start from the result for the kernel at infinite given as a sum of terms in (87), where the terms are themselves decomposed in (95). In this sum only the terms and depend on time. Below we study them separately.
B.1 Time decay of the term
Let us define . We start from the expression (5) for . Only the contour integral in the first line depends on time. As discussed above it is independent of so we can choose here , in which case the contribution of the second integral in the first line of (5) vanishes (since it contains a factor where with on this contour). Hence we obtain
| (B1) |
where is defined in (90). We now notice that it is convenient to take the time derivative of this expression so that the two integrals decouple, leading to
| (B2) | |||
| (B3) | |||
| (B4) |
At large , the integral in is dominated by the vicinity of , while the integral in is dominated by the vicinity of . One finds, assuming and fixed
| (B5) |
Putting all together and integrating (up to subdominant terms) we obtain the estimate
| (B6) |
which we have also checked numerically.
Remark. One can also start from the discrete double sum expression (89) at finite for and take a time derivative, which leads to a product of two decoupled discrete sums
| (B7) | |||
| (B8) | |||
| (B9) |
For each term the large limit is straightforward, hence here there is no need for the contour integral trick. The formula for gives immediately the formula for in Eq. (B3) in the large limit. The formula for becomes, in the large limit
| (B10) |
which can be shown to be equal to upon changing the contour of integration. Note that using the integration by part method described in (118) one shows that for any smooth function with bounded
| (B11) |
from which we can obtain the asymptotics of from (B10), recovering
the result in (B5).
B.2 Time decay of the term
We start from the expression for in (96) together with the definition of in (97). The same analysis can be made for . As we show below, the large time limit is dominated by two contributions: one where in the double integral in (96) are both close to zero, and one where one of these momenta is close to . Before presenting the detailed computation, let us show how the first contribution can be obtained by a simple argument. Upon rescaling and , this gives straightforwardly by estimating the behavior of the integrand near ,
| (B12) | |||||
In this calculation the terms in (96) containing the functions do not contribute since they vanish when both . However the full calculation, to which we now turn, shows that this part does produce an additional contribution.
Let us now again consider the time derivative of which allows to decouple the two integrals over and . It can be written as a sum of two parts. The first part is
| (B13) |
where we have defined
| (B14) | |||
| (B15) |
It is easy to analyze the large time behavior of these functions, and one obtains
| (B16) |
This leads to
| (B17) |
which agrees with the previous result (B12). The second part reads
| (B18) |
where we have defined
| (B19) |
Using linear combinations of (B11) (and the fact that as , we obtain at large time
| (B20) |
This leads to
Integrating with respect to time, and putting together the two terms, we finally obtain the large time behavior
| (B21) | |||
which is the sum of term and an oscillating term.
B.3 Summary
One can put together the terms computed above and obtain the convergence of the kernel towards its asymptotic value in the NESS. Denoting we obtain
| (B22) |
where is given in (B6) and is given in (B21) (with the same formula for with replaced by . We display here the large time behavior of the density
| (B23) | |||
We note that these formulae are valid for large time at fixed and for . We expect that this expansion breaks down when becomes large simultaneously with . Note also that this formula does not apply to the case . Indeed, consider e.g. the in Eq. (B12): this decay was obtained by rescaling and , and approximating the denominator by . Obviously, this estimate and the decay holds only if .
Appendix C: Details for
If is negative there is an additional eigenstate of , namely of energy . The kernel then has an additional term which we call .
Before we start our computation we recall the overlap of the eigenstate and with the initial state
| (C1) |
and we also introduce the new overlaps of with the left and right initial states
| (C2) | ||||
| (C3) |
Let us first consider which for is given by (4.2) and reads
| (C4) |
For simplicity let us introduce the schematic notation
| (C5) |
Now for , we must add one additional term to the sum over corresponding to the bound state. This means that we substitute , and the overlap
| (C6) |
In our schematic notation (C5) this becomes . The same procedure is applied to the sum over . The product of sums have to be expanded leading to three additional terms
| (C7) |
with in terms of our schematic notations. Each term is given explicitly below. Similarly, one can obtain . The full expression of is then obtained by summing the right and left contributions, namely . Among those three terms, the only one remaining in the stationary state is which is time independent and computed as follows.
| (C8) | ||||
| (C9) | ||||
| (C10) | ||||
| (C11) | ||||
| (C12) |
This term corresponds precisely to given in Eq. (33) in the text. Now we show that the other terms do not contribute to the large time limit. For the other term, we have (here implicitly, in all the discrete sums, and )
| (C13) | ||||
| (C14) | ||||
| (C15) |
in terms of the quantities
| (C16) | |||
| (C17) |
where we have used the explicit expressions of the overlaps and in Eq. (Appendix C: Details for ).
It is easy to take the large limit of the second piece (C17), which gives
| (C18) |
At large times, this term decays to zero as modulated by oscillations at and . To take the large limit of the first piece (C16) requires again to introduce a contour integral because of the pole at . The contours and are defined such that . Let us introduce the notations
| (C19) | |||
| (C20) | |||
where is a continuous function except for where is has a logarithmic divergence. Using the usual arguments we obtain
| (C21) |
and one can argue that the large time limit of this term vanishes. Finally the last term can be computed using with a similar conclusion.
Appendix D: Details for finite temperature
In this appendix we give some details of the finite temperature calculation. One starts from the initial kernel defined in Section 3.3. The formula (4.2) for becomes
| (D1) |
where the only difference with (4.2) is the introduction of the Fermi factors and the fact that the sum over extends over the entire lattice . Expanding the terms in parenthesis one finds an expression similar to (86). The obtained expression defines the terms as in (87).
D.1 Large limit
The large limit is easy to perform on the term and one obtains
which is the finite temperature generalization of (5).
For the term one needs again to use the contour integral trick. The contour in Fig. 8 is now replaced by the semi-infinite rectangular contour with horizontal width shown in Fig. D1 (top right panel). A formula analogous to (93) can be written where the integrand in the first term now contains the additional factor while the sum in the second term contains the additional factor and is now summed over the whole lattice . This formula is valid however only provided the contour does not enclose a pole of the Fermi factors. Recall that the Fermi factor has poles at with
| (D2) |
Hence we need to choose
| (D3) |
which we will assume from now on. The limit can now be performed and as before (see Section 5) the contribution of the lower half of (i.e., for ) vanishes in that limit. This leads to the finite temperature formula for the term at infinite
| (D4) |
where we denote the integration contour ,
which is the finite temperature analog of (5) and is defined in (90).
The analysis of the term proceeds along the same line as in the zero-temperature case (see Section Appendix A: Large limit, term ). The term is the sum of two parts as in (95). The starting formula for (and similarly for ) is (A.1) where the sum over is over the whole lattice and the Fermi factor has been inserted. One then uses the contour integral trick with the same modifications of the contour as explained above (see Fig. D1). This leads to the finite temperature analog of the formula (A.1) which reads
| (D5) | |||
where is defined in (97) and is obtained by changing . The starting formula for is (A6) where the sum over is over the whole lattice and the Fermi factor has been inserted. The same argument shows, as in Appendix Appendix A: Large limit, term , that the first part vanishes at large . The second part is given by formula (A9) where the factor has been inserted in the contour integral which is now over which is the union of small circles oriented clockwise centered around with . When computing the residues of the double pole one must be careful that the Fermi factor has been inserted. Hence Eq. (A11) is replaced by
| (D6) | |||||
This finally leads to the large limit of the term as
| (D7) |
and is obtained by changing .
To summarize, the infinite limit of the kernel at finite temperature is equal to the following sum
| (D8) | |||
where is given in (D.1), is given in (D.1), is given in (D.1) and
is given in (D7). Note that and are time independent.
Remark. In the above formula we have assumed that satisfies the bound in (D3) such that the contour does not enclose any pole of the Fermi factors. However these poles get closer to the real axis when temperature goes to zero. In other words the bound (D3) becomes at low . So one can ask how is the recovered. The answer is illustrated in the Fig. D1. The contour can be deformed into the contour as shown in the third panel in Fig. D1. Consider for instance the term . One notes that as the contribution of the part of the contour to the left of and the part of the contour to the right of vanishes. Hence the result is indentical in that limit to the one obtained from the previously considered contour (see first panel in Fig. D1), and to the result.
D.2 Large time limit: regime of the NESS
In the large time limit (once the infinite limit has been taken) the terms in (D8), as well as the contour integral part of are found to vanish, by similar arguments as for . One is thus left with
| (D9) |
with
| (D10) |
given by the residue part in (D.1).
Putting all terms together, using also (D.1) and (D7) we obtain the kernel in the NESS at finite temperature for as
| (D11) | |||
We have checked that, in the limit , this formula (D11) coincides with the expression obtained for the zero-temperature kernel in the NESS given in Eqs. (3.1.1) and (30).
From the kernel (D11),
we can recover the result for the density and the current at finite temperature given in (59) and (60).
Large time decay to the NESS. The decay in time to the NESS is different at finite . We will not present the analysis for the full kernel but only show the decay of the term . For fixed let us compute the large time behaviour of . One has, from (D.1), where the time independent residue part is cancelled,
| (D12) |
Taking the time derivative leads to the following decoupling of integrals
| (D13) |
where is defined in (B3) and
| (D14) |
The large time behavior of was obtained before, and now the behavior of is dominated by the vicinity of on the vertical part of the contour, leading to
| (D15) | |||
In summary we obtain the decay (provided )
| (D17) |
It is slower that the result obtained at zero temperature, where we found an oscillating decay dominated
by (see Eq. (B6)).
It is also interesting to check how to recover the decay in the zero temperature limit. To this aim we first send to infinity in the contour . When moving the contour it will cross the poles from the Fermi function (in the upper half-plane) as in Fig. (D1). Taking into account the series of residues at these poles we obtain
| (D18) | |||||
where we recall that ’s are defined in Eq. (D2). In the zero temperature limit the integral part vanishes since decays to zero. Using that in that limit the two series converges towards the already known integrals and defined in (B2), leading to
| (D19) |
which leads to the zero temperature result and allows to match
the finite temperature decay to the zero temperature decay.
D.3 Large time limit: ray regime at fixed
Appendix E: Energy current
In this section we give some details about the calculation of the energy current. We recall its definition
| (E1) |
where and is the single-particle Hamiltonian with a delta-impurity in Eq. (1). We recall that the evolution of a given state is given by
| (E2) |
and similarly for . Substituting this expression in Eq. (E1), we get the time evolution of the heat current
where, here and below, and . Injecting the overlaps as in the calculation of the kernel (see Eq. 4.2) we obtain
As before for the computation of the kernel, we decompose in four different contributions
| (E5) | |||
The term is trivially zero since the summand is real. The term can again be split into off-diagonal and diagonal part. The off-diagonal part vanishes in the large time limit by analogy with the calculation of the kernel (see Section B.2). The diagonal part, i.e. keeping only in the sum is non zero but becomes zero when taking the imaginary part.
Let us now consider the crossed terms ( and ):
| (E6) | |||
| (E7) | |||
Here is the contour such that . In the large time limit the contour integral vanishes and
| (E8) |
A similar method gives .
| (E9) | |||
| (E10) |
After summing these contributions, we obtain the result for the asymptotic energy current in the NESS (69).
Appendix F: Low temperature expansions
In this appendix we perform the low temperature expansions for the energy and particle currents.
F.1 Energy current
Let us compute the energy current as given by formula (69), recalling that here . Using the variable it takes the form with
| (F1) |
where . At low temperature one one can use the Sommerfeld expansion
| (F2) |
and one obtains
| (F3) |
In the case it simplifies (using (F2) up to order ) into the formula (70) given in the text. In the absence of impurity, for , one has and all terms beyond in the Sommerfeld expansion vanish. To obtain the subleading terms which are exponential in one uses the low temperature expansion
| (F4) |
where is the di-logarithm function. This expansion (F7) leads to the formula (73) in the text.
F.2 Particle current
Appendix G: Remarks on NESS and GGE
For non-interacting fermions the prediction from the GGE takes the form for the density matrix
| (F8) |
where here labels the eigenstates of the single particle post-quench Hamiltonian and the are the corresponding creation operators. Since it has a Gaussian form it leads to the kernel
| (F9) |
In the GGE the coefficients (which should not be confused with the Fermi factors ) are determined so that the occupation numbers are equal to those in the initial state (which has density matrix ). Hence one has
| (F10) |
In the present problem, for any finite size (i.e. before taking the thermodynamic limit), the post-quench eigenstates are denoted , where for respectively even and odd eigenstates given explicitly in (75) and (74), and where belongs to either the even or odd lattices respectively, , see (77) and (74) respectively. The initial density matrix reads
| (F11) |
where the are defined in (2) and are the Fermi factors for the left and right sides of the system. Hence the GGE prediction (for any ) takes the form
We can now compare (Appendix G: Remarks on NESS and GGE) to the exact formula for at finite given in (4.2) in the particular case of an initial condition at zero temperature. Clearly is obtained by retaining only the terms and in the double sum over post-quench eigenstates. This is also called the diagonal approximation and sometimes denoted (the fact that the two coincide quite generally for a finite size system or for trapped fermions was discussed in [24]). In the expression (86) it corresponds to keeping only the terms (which correspond to ) and the term denoted in (95) (which correspond to in the sum ). Hence one has, for any ,
| (F13) |
which is time independent by construction. This has a well defined limit, which is studied in this paper.
The important remark is that the GGE prediction (F13) in the limit is different from the result that we obtained for the NESS, i.e. . Indeed one has
| (F14) |
While is real and does not carry any current, the additional terms lead to a non zero current in the NESS. Although they are not strictly diagonal for finite they contain ”almost diagonal” oscillating terms of the form where and do not belong to the same lattice. Since can be of order at large (for some couples ) they do lead to a finite contribution in the NESS where one takes first.
The above result also implies that . One can in principle obtain from our result for the kernel . Let us just indicate it in the simpler case . In that case one has from (D11)
| (F15) |
Using that we obtain
| (F16) |
Hence , i.e. the fermions with positive momentum as those coming from the left and reciprocally.
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