A random phase approximation study of one-dimensional fermions after a quantum quench
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug A random phase approximation study of one-dimensional fermions after a quantumquench
Jarrett Lancaster, Thierry Giamarchi, and Aditi Mitra Department of Physics, New York University, 4 Washington Place, New York, New York 10003, USA DPMC-MaNEP, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland (Dated: August 19, 2018)The effect of interactions on a system of fermions that are in a non-equilibrium steady state dueto a quantum quench is studied employing the random-phase-approximation (RPA). As a resultof the quench, the distribution function of the fermions is highly broadened. This gives rise toan enhanced particle-hole spectrum and over-damped collective modes for attractive interactionsbetween fermions. On the other hand, for repulsive interactions, an undamped mode above theparticle-hole continuum survives. The sensitivity of the result on the nature of the non-equilibriumsteady state is explored by also considering a quench that produces a current carrying steady-state.
PACS numbers: 05.70.Ln,37.10.Jk,71.10.Pm,03.75.Kk
I. INTRODUCTION
Recent remarkable experiments with cold atoms havemotivated an explosion of theoretical interest in the areaof non-equilibrium quantum dynamics with a focus onaddressing fundamental questions about thermalization,chaos and integrability, issues that are very relevant tothese experimental systems. Without many general re-sults on generic non-equilibrium phenomena, the analysisof specific, tractable models is a common way to makeprogress. One hopes clues gathered from these specificsystems will lead to more general predictions.One-dimensional (1D) systems are where much of thetheoretical work has taken place since a wide array oftools is available for investigating dynamics. An inter-esting class of these systems is integrable models, whereconserved quantities tightly constrain the time evolu-tion. While a consensus lacks on a rigorous definitionof quantum integrability, progress has been made us-ing many quantum models satisfying classical notions ofintegrability. Fruitful studies have investigated dynam-ics of Bethe-ansatz solvable models, but the simplestintegrable models are the quadratic ones. These effec-tively non-interacting theories, including those consid-ered in this paper, allow for exact analytical treatmentof the non-trivial dynamics. In 1D, efficient numericalstudies are also possible with the time-dependent densitymatrix renormalization group (tDMRG), and exactdiagonalization studies of finite systems.
Some of the analytical and numerical studies haverevealed that 1D systems after a quantum quench of-ten reach athermal steady states which can be char-acterized by a generalized Gibbs ensemble (GGE) con-structed from identifying the conserved quantities of thesystem.
There are also many counter-exampleswhere such a description fails, as not all physical quanti-ties can be described using the GGE.
One important question concerns the stability of theseathermal steady states generated after a quantum quenchto other perturbations such as non-trivial interactions that introduce mode-coupling and/or the breaking ofintegrability. Precisely this question was addressed re-cently in Ref 21. In particular an initial interactionquench in a Luttinger liquid gives rise to an ather-mal steady state characterized by new power-law expo-nents, which can also be captured using a GGE.The effect of mode-coupling arising due to a periodicpotential on this non-equilibrium state was studied inRef. 21 using perturbative renormalization group. Theanalysis revealed that infinitesimally small perturbationscan generate not only an effective-temperature but alsoa dissipation or a finite lifetime of the bosonic modes.While the appearance of an effective-temperature, al-though highly non-trivial in itself, can be rationalized onthe grounds that a system after a quench is in a highlyexcited state, and that interactions between particles willsomehow cause the system to “thermalize”, the appear-ance of dissipation is an unexpected and non-trivial re-sult. Thus one of the motivations of the current paperis to identify other physical situations where this dissi-pation might appear, and to try to investigate the phys-ical mechanisms that could be behind it. Due to theclose parallels between interacting bosons and fermionsin 1D, a natural candidate for analyzing this question is aone-dimensional system of free fermions that is in a non-equilibrium steady state after a quench. We analyze theeffect of weak interactions on this system by employingthe random-phase-approximation (RPA).In equilibrium, 1D systems are the ideal playgroundfor invoking the RPA. It is an exact low-energy treat-ment of weak interactions in 1D.
In particular, byapplying RPA to a 1D system of electrons one re-covers the standard bosonization of the model, describedby a Luttinger liquid. Note that this direct equivalenceonly holds for the long wavelength properties, while otherexcitations require more sophisticated methods such asbosonization. While the accuracy of RPA in 1D is knownin equilibrium, its applicability out of equilibrium is notguaranteed. In the type of quench problem consideredhere, the initial state has nonzero overlap with excitedeigenstates of the Hamiltonian generating time evolution.It is far from certain that a low-energy description cap-tures all the important physics. While this caveat leadsto intriguing, unanswered questions, we will use in thispaper RPA as an approximation scheme and will not ad-dress the deeper question of its potential breakdown outof equilibrium.In this paper, we thus apply the RPA to a non-equilibrium state in the
XXZ spin-chain. This stateis prepared as follows. The system is initially in theground state of an exactly solvable Hamiltonian H i . Wechoose two different models for H i , one corresponding tothe transverse-field Ising model with the magnetic fieldtuned to the critical value where the spectrum is gapless,and the second is the same as above but with an addi-tional Dzyaloshinskii-Moriya interaction added. A quan-tum quench is then performed by switching off the fieldand changing the exchange anisotropy so that the timeevolution is due to the XX model. Since this model is de-scribed by free fermions, at long times after the quench,the system reaches an athermal steady state character-ized by a GGE. For H i that has Dzyaloshinskii-Moriyainteractions, the steady state is qualitatively different inthat it carries a net current. We then ask how theseathermal steady states are affected by weak Ising inter-actions of the XXZ chain ( J z P j ˆ S zj ˆ S zj +1 ) which are as-sumed to have been switched on very slowly. The effectsof the Ising interactions are treated using RPA.We demonstrate the existence of a single undampedcollective mode for repulsive interactions ( J z >
0) whichis qualitatively similar to the predictions of the RPA inequilibrium, but with some quantitative changes to themode-velocity. On the other hand for attractive interac-tions ( J z < TheRPA analysis shows that for attractive interactions thecollective modes lie in the particle-hole continuum andare therefore overdamped. The analysis of the presentpaper thus allows one to interpret the generation of thefriction that was put in evidence by the RG analysis ofRef 21 as due to a generalization to an out of equilibriumcase of Landau damping. The paper is organized as follows. In section II themodels that will be studied and the notations and con-ventions are defined. In section III the RPA analy-sis where the quench is from the gapless phase of thetransverse-field Ising model to the XX model is consid-ered. In section IV RPA involving fermions in a currentcarrying steady state is presented, and in section V wesummarize our results. II. MODEL
Below we describe the two different quenches whichlead to non-equilibrium steady states without (sub-section II A) and with (sub-section II B) currents.
A. Quench from ground state
The XY spin chain in a magnetic field is defined asˆ H i = − J X j h (1 + γ ) ˆ S xj ˆ S xj +1 + (1 − γ ) ˆ S yj ˆ S yj +1 i + h X j ˆ S zj . (1)where γ = 1 corresponds to the transverse-field Isingmodel. The XY model has been extensively studied, and its equilibrium properties are well understood. It isalso a popular model for studying non-equilibriumsituations due to its simple mapping to free fermions.Writing this Hamiltonian in terms of Jordan-Wignerfermions, ˆ S + j = c † j exp iπ X n 2. As we shall seelater, since the system is out of equilibrium, this temper-ature is not universal, but depends on the quantity beingstudied.Once the above steady state has been reached, we con-sider the effect of nearest-neighbor Ising interactions inthe XXZ modelˆ H f = ˆ H XX + J z X j ˆ S zj ˆ S zj +1 , (20)where we assume that J z was switched on very slowly, sothat in the absence of a quench, the fermions will evolveinto the ground state of the XXZ chain. The effects ofthis interaction term will be treated within the RPA.The basic fermionic Green’s functions defined by G Rf ( k ; t, t ′ ) = − iθ ( t − t ′ ) h{ c k ( t ) , c † k ( t ′ ) }i , (21) G Kf ( k ; t, t ′ ) = − i h h c k ( t ) , c † k ( t ′ ) i i , (22)are found to be G Rf ( k ; t, t ′ ) = − iθ ( t − t ′ ) e − iǫ k ( t − t ′ ) , (23) G Kf ( k ; t, t ′ ) = − ie − iǫ k ( t − t ′ ) (cid:12)(cid:12)(cid:12)(cid:12) sin k (cid:12)(cid:12)(cid:12)(cid:12) sign( ǫ ik ) . (24)Within the RPA, the particle-hole bubbles are Π R (1 , 2) = − i (cid:2) G Rf (1 , G Kf (2 , G Kf (1 , G Af (2 , (cid:3) , (25)Π K (1 , 2) = − i (cid:2) G Kf (1 , G Kf (2 , 1) + G Rf (1 , G Af (2 , G Af (1 , G Rf (2 , (cid:3) . (26)which in frequency-momentum space are given by,Π R ( q, ω ) = − i Z dk π d Ω2 π (cid:2) G Rf ( k + q, ω + Ω) G Kf ( k, Ω)+ G Kf ( k + q, ω + Ω) G Af ( k, Ω) (cid:3) (27)Π K ( q, ω ) = − i Z dk π d Ω2 π (cid:2) G Kf ( k + q, ω + Ω) G Kf ( k, Ω)+ G Rf ( k + q, ω + Ω) G Af ( k, Ω)+ G Af ( k + q, ω + Ω) G Rf ( k, Ω) (cid:3) . (28)The collective mode dispersion is defined by the roots ofthe complex dielectric function, ǫ RPA ( q, ω q ) = 1 − V q Π R ( q, ω q ) = 0 , (29)where we neglect the q -dependence of V q and take it to be V q = J z ≡ V . The RPA analysis is given in Section III.As mentioned above, it is not critical to work withthe diagonal ensemble. If we had retained the full time-dependence in Eq. (16), the integration over the internal k variable in the evaluation of the RPA bubbles wouldresult in terms that decay with time. Since we are ulti-mately interested in the long-time limit, these contribu-tions are not important for us. B. Quench resulting in a current-carrying state We will also be interested in how the collective dy-namics change when the athermal steady state is char-acterized by a net current. This is generated by addinga Dzyaloshinskii-Moriya interaction term to the XY model, ˆ H i → ˆ H i + ˆ H DM , whereˆ H DM = − λ X j h ˆ S yj ˆ S xj +1 − ˆ S xj ˆ S yj +1 i . (30)This new Hamiltonian is diagonalized by the same Bo-goliubov rotation that diagonalizes the pure XY -model. In the isotropic case ( XX chain), this can beinterpreted as a spatially dependent, physical rotation ofthe spins. For the more general anisotropic chain, thespectrum is similarly modified ˆ H i ( λ ) = X k ǫ ′ k η † k η k , (31) ǫ ′ k = ǫ ik − λ sin k. (32)with ǫ ik given in Eq. (7). λ has the effect of raising theenergies of states with k < k . The occupation number isnow nonzero for λ > J , and the η -fermion occupationis n k ≡ D η † k η k E = θ ( k ) θ ( k − k ), with k = 2 cos − Jλ . (33)We will see in section IV that the presence of thisnon-zero “Fermi momentum” will give rise to multipledamped modes within the particle-hole continuum thatare not present for the zero-current steady-state. Fur-thermore, above a certain critical filling factor the singleundamped collective mode will cease to exist.The asymmetry in momentum space drives a currentin the modified ground state given by h j n i = J Im D ˆ S + n +1 ˆ S − n E = ( λ < J ) Jπ (cid:16) − (cid:0) Jλ (cid:1) (cid:17) ( λ > J )(34) −π −π/2 0 π/2 π00.51 k n k k = 0k = π /2k = π FIG. 1: Initial c -fermion distributions for (a) no current ( k =0), (b) non-zero current ( k = π/ k = π ). Note the sharp discontinuity for the case of non-zero current. It should be noted that this operator can be interpretedas the current operator only within the XX -model wherethe total magnetization commutes with the Hamiltonian.Performing a quench where λ , h and γ are switched offallows this state to evolve under the XX model, obtain-ing a non-equilibrium momentum distribution D c † k c k E = 12 (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) sin k (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + θ ( k ) θ ( k − k )2 (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) sin k (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) − θ ( − k ) θ ( k + k )2 (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) sin k (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (35)and a current given by Eq. (34). The distribution func-tion for several different current strengths is shown inFig. 1.After the decay of transients h c k c − k i , D c † k c †− k E , we in-vestigate the collective modes by employing the RPAanalysis outlined in sub-section II A. The RPA requiresknowledge of the single-particle Green’s functions. Thepresence of a current does not affect the retarded Green’sfunction (Eq. (23)), but modifies the Keldysh Green’sfunction as follows G Kf ( k ; t, t ′ ) = − ie − iǫ k ( t − t ′ ) (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) sin k (cid:12)(cid:12)(cid:12)(cid:12) (1 − n k − n − k ) − ( n k − n − k )] , (36)where n k = θ ( k ) θ ( k − k ) is the occupation number ofthe η -fermions in the initial state. III. RPA FOR QUENCH FROM GROUNDSTATE In this section, we investigate the effect of interactionson the athermal steady state (sub-section II A) obtainedfrom quenching from the ground state of the transverse-field Ising model. The RPA particle-hole bubbles areΠ R ( q, ω ) = − Z dk π (cid:20) cos θ k (1 − n k )( ω + iδ ) + ǫ k − ǫ k + q − cos θ k + q (1 − n k + q )( ω + iδ ) + ǫ k − ǫ k + q (cid:21) . (37)Π K ( q, ω ) = i π ) Z dk π δ ( ω + ǫ k − ǫ k + q ) × [cos θ k cos θ k + q (1 − n k )(1 − n k + q ) − . (38)For the case of interest, γ = h/J = 1, we have cos θ k = (cid:12)(cid:12) sin k (cid:12)(cid:12) , and the distribution of the η fermions n k = 0.There are some basic symmetries of the po-larization bubbles that are worth mention-ing. Firstly Π R,K ( q, ω ) = Π R,K ( − q, ω ), whileRe[Π R ]( q, − ω ) = Re[Π R ]( q, ω ) , Π K ( q, − ω ) = Π K ( q, ω )and Im[Π R ( q, − ω )] = − Im[Π R ]( q, ω ). Therefore in whatfollows we will assume q > , ω > 0, and the results forthe other regimes can be extrapolated from the abovesymmetries.There are two regimes which we will study separately.One is ω > J sin q where Im[Π R ] = Π K = 0, and theother is ω < J sin q where Im[Π R ] = 0 , Π K = 0. A. Evaluation for ω > J sin q In this regime, the integrand contains no poles, andthe result is purely real: Re (cid:2) Π R ( q, ω ) (cid:3) = Π R ( q, ω ) andIm (cid:2) Π R ( q, ω ) (cid:3) = 0. One may safely take δ → R ( q, ω ) = − cos q πi p ω − (2 J sin q ) × z + ln sin q z + − sin q z + − z − ln sin q z − − sin q z − − sin q πi p ω − (2 J sin q ) × z + ln cos q z + − cos q z + − z − ln cos q z − − cos q z − , (39)with z ± = 12 ± s − (cid:18) ω J sin q (cid:19) ∀ ω < J sin q 2= 12 ± i s(cid:18) ω J sin q (cid:19) − ∀ ω > J sin q ω > J sin q there are no roots to the argument of thedelta function in the Keldysh component, and we haveΠ K ( q, ω ) = 0. With Π R ( q, ω ) purely real, this regime liesoutside the particle-hole continuum. In sub-section III Cwe will demonstrate the existence of an undamped col-lective mode lying just above the particle-hole continuumfor repulsive interactions only. B. Evaluation for ω < J sin q For ω < J sin q the integrand generically containspoles. We extract the real and imaginary parts in theusual way by writing, Z dk π f ( k ) ω + iδ + ǫ k − ǫ k + q = Z dk π P (cid:18) f ( k ) ω + ǫ k − ǫ k + q (cid:19) − iπ Z dk π f ( k ) δ ( ω + ǫ k − ǫ k + q ) (41) ≡ Re (cid:2) Π R (cid:3) + i Im (cid:2) Π R (cid:3) (42)where P denotes taking the principal value of the inte-gral. We obtainRe (cid:2) Π R ( q, ω ) (cid:3) = − cos q π p (2 J sin q ) − ω × z + ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin q z + − sin q z + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − z − ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin q z − − sin q z − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − sin q π p (2 J sin q ) − ω × z + ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos q z + − cos q z + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − z − ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos q z − − cos q z − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (43)with z ± defined in Eq. (40). The results for the imagi-nary part, Im (cid:2) Π R (cid:3) = Π R − Π A i , and the Keldysh compo-nent subdivide the particle-hole continuum into two sub-regions, 0 < ω < J sin q and 2 J sin q < ω < J sin q .For 0 < ω < J sin q one findsIm (cid:2) Π R ( q, ω ) (cid:3) = − p (2 J sin q ) − ω h(cid:16) cos q q (cid:17) × sin (cid:20) 12 sin − ω J sin q (cid:21)(cid:21) (44)Π K ( q, ω ) = i p (2 J sin q ) − ω × (cid:20) cos (cid:18) sin − (cid:18) ω J sin q (cid:19)(cid:19) − (cid:21) . (45)whereas in the region 2 J sin q < ω < J sin q , the resultis Im (cid:2) Π R ( q, ω ) (cid:3) = − p (2 J sin q ) − ω × (cid:20) sin q (cid:18) sin (cid:20) 12 sin − ω J sin q (cid:21) + cos (cid:20) 12 sin − ω J sin q (cid:21)(cid:19)(cid:21) (46)Π K ( q, ω ) = i p (2 J sin q ) − ω h cos q − i . (47)The particle-hole continuum, which is the region in ω, q space where ImΠ R = 0 is indicated as the shaded regionin Fig. 2 and compared with the equilibrium (no quench)result (inset). The two different shadings refer to the dis-continuity in the functional forms of Im[Π] R , Π K across ω = 2 J sin q . The consequences of these results are dis-cussed below. C. Undamped mode for ω > J sin q and V > The undamped mode is obtained from the solution of1 − V Π R ( q, ω ) = 0 (48)with Π R ( q, ω ) given in Eq. (39). For sufficiently small V , we need to identify the points where Π R diverges.This occurs for ω = 2 J sin( q/ ω → J sin q , we can write z ± = q (cid:16) ± iǫ J sin q (cid:17) with ǫ = q ω − (cid:0) J sin q (cid:1) . The dominant contribution as-suming that q → q ≈ 1) is given byΠ R ( q, ω ) ≃ − sin q πi q ω − (cid:0) J sin q (cid:1) × z + ln √ i ǫ √ J sin q − √ i ǫ √ J sin q − z − ln √ − i ǫ √ J sin q − √ − i ǫ √ J sin q . (49)Due to the branch-cut in the logarithm (chosen to be onthe negative real axis), in the limit ǫ → 0, the aboveexpression can be further simplified to giveΠ R ( q, ω ) ≃ sin q √ p ω − (2 J sin q ) , (50)Thus from Eq. (48) we find a single undamped mode provided V is positive with a dispersion ω q ≃ θ ( V ) J | q | r V J . (51) q ω q ω FIG. 2: Undamped mode (dashed line) above the extendedparticle-hole continuum (shaded region). Inset: equilibriumcontinuum and undamped mode. This can be compared to the undamped mode in theequilibrium problem, ω eq q ≃ J | q | r V πJ , (52)which exists for both attractive and repulsive interac-tions. Thus, for repulsive interactions, the obtainedsound wave is qualitatively similar to the equilibriumcase, but with a slightly modified velocity of propaga-tion, whereas for attractive interactions, no undampedmodes exist. D. Enhanced particle-hole continuum and effectivetemperature The highly broadened initial fermion distribution givesone way to define an effective temperature in this non-equilibrium state (cf. Eq. (19)). By analogy with theequilibrium properties of the particle-hole bubbles, onemay define an effective temperature in terms of the col-lective degrees of freedom. Since the system is out ofequilibrium, this temperature will in general depend on ω, q and also the chosen correlation functionΠ R ( q, ω ) − Π A ( q, ω )Π K ( q, ω ) = tanh (cid:18) ω T ′ eff ( q, ω ) (cid:19) . (53)For small frequencies, this ratio yieldsΠ R − Π A Π K ω → −−−→ ω J sin q (cid:16) sin q q (cid:17) , (54)so that for ω → 0, we obtain an effective-temperature T ′ eff ( q, ω → 0) = J sin q cos q + sin q . (55)We argue below that T ′ eff ( q, ω → 0) is responsible forsmearing out the particle-hole continuum in much thesame way that temperature in an equilibrium systemdoes. To see this recall that the particle-hole continuumrepresents the region in the ( q, ω )-plane where a collec-tive mode of frequency ω and wave-vector q is unstableto decay into single particle-hole excitations. The upperand lower continuum boundaries in equilibrium at zerotemperature are given by ω L ( q ) = J sin q, (56) ω U ( q ) = 2 J sin q . (57)A simple argument makes these boundaries plausible:consider the energy of a single particle-hole excitationwhich is created by removing a particle of momentum k and creating a particle of momentum k + q , ω ( k, q ) = ǫ k + q − ǫ k = 2 J sin q (cid:16) k + q (cid:17) . (58)This excitation energy depends not only on the momen-tum of the excitation, but also on the momentum ofthe original particle, k . For a half-filled band at zerotemperature, the occupation of fermions is D c † k c k E eq = θ (cid:0) π − | k | (cid:1) . The only momenta available for hole cre-ation are those with | k | < k F = π . Because the cosinedispersion has maximal slope at k = π , the maximumexcitation energy occurs for a given q with k = π − q .The smallest excitation energy for a given q at zero tem-perature occurs at k = π or k = π − q . Thus ω max ( q ) = ω (cid:16) π − q , q (cid:17) = 2 J sin q , (59) ω min ( q ) = ω (cid:16) π , q (cid:17) = J sin q. (60)which are just the upper and lower boundaries of theparticle-hole continuum (inset Fig. 2). Now, considerlowering k F . Excitations of smaller energy for a given q are now possible, and in the limit k F → 0, we have ω min → J sin q . The result is the same if one considersthe opposite limit of k F → π at zero temperature.In the present non-equilibrium situation, we find aparticle-hole continuum (Im (cid:2) Π R ( q, ω ) (cid:3) = 0) that ex-tends below this lower-bound all the way to ω = 0.A finite temperature is known to smear out this lowerboundary, due to the smoothing out of the zero-temperature step function for the occupation proba-bility. It is interesting to note that the expressionsfor Im (cid:2) Π R ( q, ω ) (cid:3) and Π K ( q, ω ) are actually continuousacross the line ω = 2 J sin q , with discontinuities appear-ing in their derivatives. In Fig. 2 we plot the undampedcollective mode dispersion with the particle-hole contin-uum represented by the shaded region. The two differentshadings are separated by the line ω = 2 J sin q . Theanalogous plot for the equilibrium situation is shown inthe inset. IV. RPA FOR CURRENT CARRYING STATE We now apply RPA to study the current carrying non-equilibrium steady state described in section II B. TheKeldysh component of the fermion Green’s function is, iG Kf ( k ; t, t ′ ) = [cos θ k (1 − n k − n − k ) − ( n k − n − k )] e − iǫ k ( t − t ′ ) . (61)while the retarded Green’s function is given in Eq. (23),and n k = θ ( k ) θ ( k − k ). Eq. (61) implies that the dis-tribution function for the Jordan-Wigner fermions in thecurrent carrying post-quench state is not only broad asfor the zero current case, but is also asymmetric in k ,with sharp discontinuities superimposed on it (see Fig. 1).Thus we will find that as for the zero-current case, theparticle-hole continuum here too is broadened (extend-ing everywhere below the line ω max = 2 J sin q ), whilethe sharp structure in the distribution gives rise to somediscontinuities in the expression for Im[Π R ], and the ap-pearance of additional damped modes.The particle-hole bubbles are now given byΠ R ( q, ω ) = − Z dk π " cos θ k (1 − n k − n − k ) ω + iδ − J sin q sin (cid:0) k + q (cid:1) − ( n k − n − k ) ω + iδ − J sin q sin (cid:0) k + q (cid:1) − cos θ k + q (1 − n k + q − n − k − q ) ω + iδ − J sin q sin (cid:0) k + q (cid:1) + ( n k + q − n − k − q ) ω + iδ − J sin q sin (cid:0) k + q (cid:1) (62)Π K ( q, ω ) = i π ) Z dk π δ ( ω + ǫ k − ǫ k + q ) × { [cos θ k (1 − n k − n − k ) − ( n k − n − k )] × [cos θ k + q (1 − n k + q − n − k − q ) − ( n k + q − n − k − q )] − } . (63)As before two regions appear, one where ω > J sin q for which Im[Π R ] = Π K = 0, and the second being ω < J sin q where a particle-hole continuum is found to exist.We discuss these two regions separately. A. Evaluation for ω > J sin q In this regime, as before, the result is entirely realand we let δ → 0. We find it convenient to writeΠ R = Π (1) + Π (2) , where Π (1) depends on the Bogoli-ubov angle, cos θ k = (cid:12)(cid:12) sin k (cid:12)(cid:12) , while Π (2) contains therest. As before it is convenient to summarize the symme-tries of the polarization bubbles. We find Π (1) ( − q, ω ) =Π (1) ( q, ω ), however due to current flow, Π (2) ( − q, ω ) = − Π (2) ( q, ω ). Similarly, Re[Π (1) ]( q, − ω ) = Re[Π (1) ]( q, ω ),while Re[Π (2) ]( q, − ω ) = − Re[Π (2) ]( q, ω ). In the discus-sion that follows, we take q > , ω > (1) ( q, ω ) = − cos q πi p ω − (2 J sin q ) × (cid:26) F (cid:18) sin (cid:20) k q (cid:21)(cid:19) + F (cid:18) sin (cid:20) − k q (cid:21)(cid:19)(cid:27) − sin q πi p ω − (2 J sin q ) × (cid:26) F (cid:18) cos (cid:20) k q (cid:21)(cid:19) + F (cid:18) cos (cid:20) − k q (cid:21)(cid:19)(cid:27) , (64)where F ( z ) = z + ln (cid:16) zz + (cid:17)(cid:16) − zz + (cid:17) − z − ln (cid:16) zz − (cid:17)(cid:16) − zz − (cid:17) . (65)In the limit k → 0, we recover the results of section III.The remaining terms can be collected asΠ (2) = 14 π (Z k + q/ k − q/ − Z q/ − q/ ) × dk (cid:26) ω − J sin q sin k + 1 ω + 2 J sin q sin k (cid:27) (66)= − πi p ω − (2 J sin q ) × ln ω tan ( k + q ) J sin q z − ω tan ( k + q ) J sin q z − ln ω tan ( k − q ) J sin q z − ω tan ( k − q ) J sin q z − ln ω tan ( k + q ) J sin q z − − ω tan ( k + q ) J sin q z − + ln ω tan ( k − q ) J sin q z − − ω tan ( k − q ) J sin q z − − ω tan ( q ) J sin q z − ω tan ( q ) J sin q z + 2 ln ω tan ( q ) J sin q z − − ω tan ( q ) J sin q z − , (67)The consequence of the above expressions for Π (1 , will be discussed in section IV C. B. Evaluation for ω < J sin q In this regime, the real part of Π R ( q, ω ) is given bythe principal value of the integral in Eq. (62). Writing Re (cid:2) Π R ( q, ω ) (cid:3) = Re[Π (1) ] + Re[Π (2) ], whereRe[Π (1) ]( q, ω ) = − cos q π p (2 J sin q ) − ω × (cid:26) ˜ F (cid:18) sin (cid:20) k q (cid:21)(cid:19) + ˜ F (cid:18) sin (cid:20) − k q (cid:21)(cid:19)(cid:27) − sin q π p (2 J sin q ) − ω × (cid:26) ˜ F (cid:18) cos (cid:20) k q (cid:21)(cid:19) + ˜ F (cid:18) cos (cid:20) − k q (cid:21)(cid:19)(cid:27) , (68)where˜ F ( z ) = z + ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) zz + (cid:17)(cid:16) − zz + (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − z − ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) zz − (cid:17)(cid:16) − zz − (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (69)and Re[Π (2) ]( q, ω ) = − π p (2 J sin q ) − ω × ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω tan ( k + q ) J sin q z − ω tan ( k + q ) J sin q z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω tan ( k − q ) J sin q z − ω tan ( k − q ) J sin q z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω tan ( k + q ) J sin q z − − ω tan ( k + q ) J sin q z − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω tan ( k − q ) J sin q z − − ω tan ( k − q ) J sin q z − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω tan ( q ) J sin q z − ω tan ( q ) J sin q z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω tan ( q ) J sin q z − − ω tan ( q ) J sin q z − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (70)We do not give expressions for Im (cid:2) Π R (cid:3) , Π K as theboundaries of the particle-hole continuum are the sameas in Fig. 2, though there are additional discontinu-ities within the continuum besides the one along ω =2 J sin q . Instead in the subsequent sections, by study-ing the divergences in Re[Π R ] we will identify a singleundamped mode for repulsive interactions and k < π/ C. Undamped mode for ω > J sin q and V > In this section we demonstrate that an undamped col-lective mode survives provided the current is not toolarge. As before, define ǫ = q ω − (cid:0) J sin q (cid:1) . In thelimit q → k , the most divergent termsin Π R ( q, ω ) areΠ R ( q, ω ) ≃ − sin q π q(cid:0) J sin q (cid:1) − ω (cid:26) F (cid:20) cos (cid:18) k (cid:19)(cid:21) + F (cid:20) cos (cid:18) − k (cid:19)(cid:21)(cid:27) (71) ≃ sin q πi √ p ω − (2 J sin q ) × ( ln " − √ k i ǫ √ k J sin q − ln " − √ k − i ǫ √ k J sin q (72)The above expression shows that provided k < π , whichcorresponds to the logarithms having a branch-cut, weobtain Π R ( q, ω ) = sin q √ p ω − (2 J sin q ) , (73)Thus by setting 1 − V Π R = 0, we recover the same dis-persion as in the absence of current ω q ≃ J | q | r V J θ ( V ) θ ( π/ − k ) (74)Thus, the undamped mode is unchanged for a currentwhich is below the threshold value of k < π/ 2. On theother hand, for currents larger than this value ( k > π/ q ≪ k , no undamped modes exist. D. Damped modes for ω < J sin q In this regime, all modes are damped. We iden-tify these damped modes by looking for solutions to1 − V Re[Π R ]( q, ω q ) = 0. For V → 0, all we need todo is identify where Re (cid:2) Π R (cid:3) → ±∞ . Then positive di-vergences correspond to damped modes with repulsiveinteractions, while negative divergences correspond todamped modes with attractive interactions.Upon examining Eqns. (68) and (70), we find logarith-mic divergences in Re (cid:2) Π R ( q, ω ) (cid:3) along the characteristiclines (for ω, q > ω ( q ) = 2 J sin q , (75) ω ( q ) = 2 J sin q (cid:16) k + q (cid:17) , (76) ω ± ( q ) = ± J sin q (cid:16) q − k (cid:17) , (77)Note that ω , coincide with the characteristic lines in theequilibrium problem with an arbitrary Fermi momentum k . In the equilibrium problem, these lines represent q ω FIG. 3: Characteristic lines along which Re (cid:2) Π R ( q, ω ) (cid:3) di-verges, giving rise to damped modes for repulsive (solid lines)and attractive (dashed lines) interactions for k = π . Onlythe top line ω = 2 J sin q corresponds to an undamped modeexisting above the particle-hole continuum. boundaries across which Im (cid:2) Π R (cid:3) undergoes a jump dis-continuity. This is also the case here, though we willfocus our attention on the behavior of the real part.One finds Re (cid:2) Π R (cid:3) → + ∞ along the line ω +3 ( q ) for2 k < q < π and along the line ω ( q ) for q < π − k .These correspond to damped collective modes for repul-sive interactions. Furthermore, Re (cid:2) Π R (cid:3) → −∞ alongthe line ω ( q ) for all q ∈ (0 , π ), along ω − ( q ) for q < k ,and along ω ( q ) for q > π − k . These negative diver-gences represent collective modes created by attractiveinteractions. We plot these characteristic lines in Fig. 3and indicate whether the mode exists for attractive orrepulsive interactions.Such damped modes are usually considered physicallyuninteresting compared to any undamped excitationsin the system, as the damping makes these modes ex-perimentally unobservable. The divergences in Π R ( q, ω )that give rise to these damped modes are of a differentnature than those giving rise to the undamped mode. Tosee this consider the case of ω ≃ ω = 2 J sin q (cid:0) sin q ± ǫ (cid:1) for small ǫ . The dominant contribution is given byRe (cid:2) Π R ( q, ω ) (cid:3) ≃ − πJ sin q (cid:12)(cid:12) cos q (cid:12)(cid:12) (cid:20) A q ǫ (cid:21) . (78)where A q is a q -dependent factor.Solving 1 = V ReΠ R [( q, ω q )] to leading order in ǫ , onefindsRe [ ω q ] ≃ J sin q (cid:16) sin q ± A q e − (2 πJ sin q ) / | V | (cid:17) , (79)where q > ω = ω c ± Jǫ sin q leads to a divergence of the form Π R ( q, ω ) ∼± log ǫ . Each logarithmic divergence corresponds to twodamped modes lying exponentially close to each charac-teristic line.0One may study the modes near ω ≈ ω ( q ) in a waysimilar to our analysis for the modes near ω . For ω ≈ ω ( q ) ± Jǫ sin q , one findsRe (cid:2) Π R ( q, ω ) (cid:3) ≃ k πJ sin q (cid:12)(cid:12) cos (cid:0) k + q (cid:1)(cid:12)(cid:12) log A ′ q ǫ , (80)which gives rise to damped modes for repulsive interac-tions ( V > 0) when q < π − k withRe [ ω q ] ≃ J sin q (cid:16) sin (cid:16) q k (cid:17) ± A ′ q exp " − πJ sin q (cid:12)(cid:12) cos (cid:0) k + q (cid:1)(cid:12)(cid:12) V (cid:0) k (cid:1) . (81)The results for the other characteristic line ω +3 is similarand we do not discuss it further. V. SUMMARY AND CONCLUSIONS In this paper, we have applied the RPA to study the ef-fect of weak Ising interactions in a non-equilibrium steadystate of the XXZ spin chain. This non-equilibrium statewas created in two different ways. One is by quenchingfrom the ground state of the transverse-field Ising modelat critical magnetic field to the XX -model. The sec-ond was to modify the Hamiltonian before the quench byadding Dzyaloshinskii-Moriya interactions. This had theeffect of creating a current carrying state.The RPA for both the steady-states shows the exis-tence of a single, undamped, collective mode for repulsiveinteractions which is qualitatively similar to the soundmode in equilibrium, but with quantitative changes to the mode velocity (c.f. Eq. (51)). However if the cur-rent is larger than a threshold value, this undampedmode ceases to exist in the long-wavelength limit (c.f.Eq. (74)). The primary effect of the quench is to give riseto a highly broadened distribution function (c.f. Fig. 1,Eq. (35)) which results in an enhanced particle-hole con-tinuum. The boundaries of the particle-hole continuumare shown in Fig. 2. Thus for attractive interactions ei-ther no modes are found for the first steady-state, or somedamped collective modes are found for the steady-statewith current.These results, and in particular the generation of afinite friction due to an out of equilibrium situation,are rather generic and do not depend on the detailsof the non-equilibrium steady-state. Further, the up-per boundary of the particle-hole continuum occurs at ω max = 2 J sin q and is related to the fact that the sys-tem is on a lattice, and therefore the excitations have amaximum velocity. If instead a quadratic dispersion forthe fermions is adopted, then there is no upper-limit tothe velocity of excitations. 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