Steady States and Universal Conductance in a Quenched Luttinger Model

Abstract

We obtain exact analytical results for the evolution of a 1+1-dimensional Luttinger model prepared in a domain wall initial state, i.e., a state with different densities on its left and right sides. Such an initial state is modeled as the ground state of a translation invariant Luttinger Hamiltonian H λ with short range non-local interaction and different chemical potentials to the left and right of the origin. The system evolves for time t > 0 via a Hamiltonian H λ ′ which differs from H λ by the strength of the interaction. Asymptotically in time, as t → ∞, after taking the thermodynamic limit, the system approaches a translation invariant steady state. This final steady state carries a current I and has an effective chemical potential difference µ + − µ − between right-(+) and left-(−) moving fermions obtained from the two-point correlation function. Both I and µ + − µ − depend on λ and λ ′ . Only for the case λ = λ ′ = 0 does µ + − µ − equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, G = I (µ + − µ − ), has a universal value equal to the conductance quantum e 2 h for the spinless case.

Higher-order generalized hydrodynamics in one dimension: The noninteracting test

“…There, under the assumption of quasistationarity, a semiclassical picture applies where the information about the initial state is carried by free stable quasiparticles moving throughout the system. Similar results were obtained in the framework of conformal field theory and Luttinger liquid descriptions [25][26][27][28][29][30][31][32][33][34][35][36][37]. In the presence of interactions the situation was less clear [38][39][40][41][42][43][44][45][46][47], but, eventually, Refs [48,49] have shown that the continuity equations satisfied by the (quasi)local conserved quantities are sufficient to characterize the late-time behavior.…”

Finite-Time Universality in Nonequilibrium CFT

“…and the plasmon velocity v = v F 1 + 2λ/(πv F ) [36,39]. We found in [33] the following exact expectation values of the energy density and the heat current in the thermodynamic limit marked by the superscript ∞ on the expectations: 3…”

“…The evolution of the energy density and the heat current obtained in [33] permits then to better understand the shortcomings of the partitioning protocol. It also sheds a new interesting light on transport in integrable systems and, in particular, on how its universal features [35] emerge at long times for a large class of nonequilibrium initial states, see [36] and also Sect. 4.3 below for a related discussion of charge transport in the Luttinger model.…”

“…Suppose that our CFT possesses a U(1) symmetry generated by a conserved current with components ρ and j satisfying ∂ t ρ + ∂ x j = 0. For instance, ρ = ρ + + ρ − and j = Kv(ρ + − ρ − ) for the local Luttinger model, see, e.g., Appendix B in [36] (we recall that K is the Luttinger parameter). One may then consider states with both temperature and chemical-potential 11 profiles by taking…”

Transport in the sine-Gordon field theory: From generalized hydrodynamics to semiclassics

“…A particularly simple, yet non-trivial, setting which has been widely investigated is the so-called bipartitioning protocol, where two semi-infinite systems, initially held at different temperatures, are suddenly joined together and left to evolve unitarily under a homogeneous Hamiltonian. While this protocol is of obvious interest for the study of transport, until recently its analytic understanding was limited to the cases of free [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] or conformally invariant models [51][52][53][54][55][56][57][58][59][60][61][62]. A major breakthrough came in Refs.…”