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arXiv:2009.05693 · Communications in Theoretical Physics

General interaction quenches in a Luttinger liquid

You-Ming Wei, Hantao Lu

Published Nov 4, 2021• 4 citations
arXivHTMLPDF

Abstract

Abstract We discuss a general interaction quench in a Luttinger liquid described by a paired bosonic Hamiltonian. By employing su(1, 1) Lie algebra, the post-quench time-evolved wavefunctions are obtained analytically, from which the time evolution of the entanglement in momentum space can be investigated. We note that depending on the choice of Bogoliubov quasiparticles, the expressions of wavefunctions, which describe time-evolved paired states, can take different forms. The correspondence between the largest entanglement eigenvalue in momentum space and the wavefunction overlap in quench dynamics is discussed, which generalizes the results of Dóra et al (2016, Phys. Rev. Lett. 117, 010 603). A numerical demonstration on an XXZ lattice model is presented via the exact diagonalization method.

DOI
10.1088/1572-9494/ac3644

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Abstract We discuss a general interaction quench in a Luttinger liquid described by a paired bosonic Hamiltonian. By employing su(1, 1) Lie algebra, the post-quench time-evolved wavefunctions are obtained analytically, from which the time evolution of the entanglement in momentum space can be investigated. We note that depending on the choice of Bogoliubov quasiparticles, the expressions of wavefunctions, which describe time-evolved paired states, can take different forms. The correspondence between the largest entanglement eigenvalue in momentum space and the wavefunction overlap in quench dynamics is discussed, which generalizes the results of Dóra et al (2016, Phys. Rev. Lett. 117, 010 603). A numerical demonstration on an XXZ lattice model is presented via the exact diagonalization method.
Luttinger liquids (LLs) arise by coupling left- and right-moving particles through interactions in one dimension. This most natural partitioning of LLs is investigated by the momentum-space entanglement after a quantum quench using analytical and numerical methods. We show that the momentum-space entanglement spectrum of a LL possesses many universal features both in equilibrium and after a quantum quench. The largest entanglement eigenvalue is identical to the Loschmidt echo, i.e., the overlap of the disentangled and final wave functions of the system. The second largest eigenvalue is the overlap of the first excited state of the disentangled system with zero total momentum and the final wave function. The entanglement gap is universal both in equilibrium and after a quantum quench. The momentum-space entanglement entropy is always extensive and saturates fast to a time independent value after the quench, in sharp contrast to a spatial bipartitioning.
The dynamics of the Luttinger model after a quantum quench is studied. We compute in detail one- and two-point correlation functions for two types of quenches: from a noninteracting to an interacting Luttinger model and vice versa. In the former case, the noninteracting Fermi gas features in the momentum distribution and other correlation functions are destroyed as time evolves. In the infinite-time limit, equal-time correlations are power laws but the critical exponents are found to differ from their equilibrium values. In all cases, we find that these correlations are well described by a generalized Gibbs ensemble [M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007)], which assigns a momentum-dependent temperature to each eigenmode.
We study the relaxation dynamics of the one-dimensional Tomonaga–Luttinger model after an interaction quench, paying particular attention to the momentum dependence of the two-particle interaction. Several potentials of different analytical forms are investigated that all lead to universal Luttinger liquid (LL) physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the LL phenomenology. For generic regular potentials, the large time decay of the momentum distribution function toward the steady-state value is characterized by a power law with a universal exponent that depends only on the potential at zero momentum transfer. The commonly employed ad hoc procedure fails to give this exponent. In addition to quenches from zero to positive interactions, we also consider the abrupt changes in the interaction between two arbitrary values. Additionally, we discuss the appearance of a factor of two between the steady-state momentum distribution function and that obtained in equilibrium at equal two-particle interaction.
We study the relaxation dynamics of the one-dimensional Tomonaga–Luttinger model after an interaction quench, paying particular attention to the momentum dependence of the two-particle interaction. Several potentials of different analytical forms are investigated that all lead to universal Luttinger liquid (LL) physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the LL phenomenology. For generic regular potentials, the large time decay of the momentum distribution function toward the steady-state value is characterized by a power law with a universal exponent that depends only on the potential at zero momentum transfer. The commonly employed ad hoc procedure fails to give this exponent. In addition to quenches from zero to positive interactions, we also consider the abrupt changes in the interaction between two arbitrary values. Additionally, we discuss the appearance of a factor of two between the steady-state momentum distribution function and that obtained in equilibrium at equal two-particle interaction.
We study variable-rate linear quenches in the anisotropic Heisenberg (XXZ) chain, starting at the XX point. This is equivalent to switching on a nearest-neighbor interaction for hard-core bosons or an interaction quench for free fermions. The physical observables we investigate are the energy pumped into the system during the quench, the spin-flip correlation function, and the bipartite fluctuations of the zzz component of the spin in a box. We find excellent agreement between exact numerics (infinite system time-evolving block decimation) and analytical results from bosonization, as a function of the quench time, spatial coordinate, and interaction strength. This provides a stringent and much-needed test of Luttinger liquid theory in a nonequilibrium situation.
We study the relaxation dynamics of a nonequilibrium Tomonaga-Luttinger model after a sudden interaction switch-on (``quench''), focusing on a double-step initial momentum distribution function. In the framework of the nonequilibrium bosonization, the results are obtained in terms of singular Fredholm determinants that are evaluated numerically and whose asymptotics are found analytically. While the quasiparticle weights decay exponentially with time after the quench, this is not a relaxation into a thermal state, in view of the integrability of the model. The steady-state distribution emerging at infinite times retains two edges which support Luttinger-liquid-like power-law singularities smeared by dephasing. The obtained critical exponents and the dephasing length are found to depend on the initial nonequilibrium state.