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arXiv:1709.04796 · Nuclear Physics B

What is an integrable quench?

Lorenzo Piroli, Balázs Pozsgay, Eric Vernier

Published Oct 26, 2017• 200 citations
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Abstract

Inspired by classical results in integrable boundary quantum field theory, we propose a definition of integrable initial states for quantum quenches in lattice models. They are defined as the states which are annihilated by all local conserved charges that are odd under space reflection. We show that this class includes the states which can be related to integrable boundary conditions in an appropriate rotated channel, in loose analogy with the picture in quantum field theory. Furthermore, we provide an efficient method to test integrability of given initial states. We revisit the recent literature of global quenches in several models and show that, in all of the cases where closed-form analytical results could be obtained, the initial state is integrable according to our definition. In the prototypical example of the XXZ spin- s chains we show that integrable states include two-site product states but also larger families of matrix product states with arbitrary bond dimension. We argue that our results could be practically useful for the study of quantum quenches in generic integrable models.

DOI
10.1016/j.nuclphysb.2017.10.012

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Inspired by classical results in integrable boundary quantum field theory, we propose a definition of integrable initial states for quantum quenches in lattice models. They are defined as the states which are annihilated by all local conserved charges that are odd under space reflection. We show that this class includes the states which can be related to integrable boundary conditions in an appropriate rotated channel, in loose analogy with the picture in quantum field theory. Furthermore, we provide an efficient method to test integrability of given initial states. We revisit the recent literature of global quenches in several models and show that, in all of the cases where closed-form analytical results could be obtained, the initial state is integrable according to our definition. In the prototypical example of the XXZ spin- s chains we show that integrable states include two-site product states but also larger families of matrix product states with arbitrary bond dimension. We argue that our results could be practically useful for the study of quantum quenches in generic integrable models.
Abstract We consider quantum quenches in the integrable -invariant spin chain (Lai–Sutherland model) which admits a Bethe ansatz description in terms of two different quasiparticle species, providing a prototypical example of a model solvable by nested Bethe ansatz. We identify infinite families of integrable initial states for which analytic results can be obtained. We show that they include special families of two-site product states which can be related to integrable ‘soliton non-preserving’ boundary conditions in an appropriate rotated channel. We present a complete analytical result for the quasiparticle rapidity distribution functions corresponding to the stationary state reached at large times after the quench from the integrable initial states. Our results are obtained within a quantum transfer matrix (QTM) approach, which does not rely on the knowledge of the quasilocal conservation laws or of the overlaps between the initial states and the eigenstates of the Hamiltonian. Furthermore, based on an analogy with previous works, we conjecture analytic expressions for such overlaps: this allows us to employ the quench action method to derive a set of integral equations characterizing the quasi-particle distribution functions of the post-quench steady state. We verify that the solution to the latter coincides with our analytic result found using the QTM approach. Finally, we present a direct physical application of our results by providing predictions for the propagation of entanglement after the quench from such integrable states.
We consider quantum quenches in the integrable SU(3)SU(3)SU(3)-invariant spin chain (Lai-Sutherland model) which admits a Bethe ansatz description in terms of two different quasiparticle species, providing a prototypical example of a model solvable by nested Bethe ansatz. We identify infinite families of integrable initial states for which analytic results can be obtained. We show that they include special families of two-site product states which can be related to integrable ``soliton non-preserving'' boundary conditions in an appropriate rotated channel. We present a complete analytical result for the quasiparticle rapidity distribution functions corresponding to the stationary state reached at large times after the quench from the integrable initial states. Our results are obtained within a Quantum Transfer Matrix (QTM) approach, which does not rely on the knowledge of the quasilocal conservation laws or of the overlaps between the initial states and the eigenstates of the Hamiltonian. Furthermore, based on an analogy with previous works, we conjecture analytic expressions for such overlaps: this allows us to employ the Quench Action method to derive a set of integral equations characterizing the quasi-particle distribution functions of the post-quench steady state. We verify that the solution to the latter coincides with our analytic result found using the QTM approach. Finally, we present a direct physical application of our results by providing predictions for the propagation of entanglement after the quench from such integrable states.
Abstract. We present an identification of the spectra of local conserved operators of integrable quantum lattice models and the density distributions of their thermodynamic particle content. This is derived explicitly for the Heisenberg XXZ spin chain. As an application we discuss a quantum quench scenario, in both the gapped and critical regimes. We outline an exact technique which allows for an efficient implementation on periodic matrix product states. In addition, for certain simple product states we obtain closed-form expressions for the density distributions in terms of solutions to Hirota difference equations. Remarkably, no reference to a maximal entropy principle is invoked.
We analyze quantum quenches in integrable models and in particular the determination of the initial state in the basis of eigenstates of the post-quench hamiltonian. This leads us to consider the set of transformations of creation and annihilation operators that respect the Zamolodchikov-Faddeev algebra satisfied by integrable models. We establish that the Bogoliubov transformations hold only in the case of quantum quenches in free theories. In the most general case of interacting theories, we identify two classes of transformations. The first class induces a change in the S-matrix of the theory but not of its ground state, whereas the second class results in a "dressing" of the operators. As examples of our approach we consider the transformations associated with a change of the interaction in the Sinh-Gordon and the Lieb-Liniger model.
We analyze quantum quenches in integrable models and in particular we determine the initial state in the basis of eigenstates of the post-quench Hamiltonian. This leads us to consider the set of transformations of creation and annihilation operators that respect the Zamolodchikov–Faddeev algebra satisfied by integrable models. We establish that the Bogoliubov transformations hold only in the case of quantum quenches in free theories. For the most general case of interacting theories, we identify two classes of transformations. The first class induces a change in the S -matrix of the theory but not in its ground state, whereas the second class results in a 'dressing' of the operators. We consider as examples of our approach the transformations associated with a change of the interaction in the sinh–Gordon model and the Lieb–Liniger model.
We consider quantum quenches in the integrable SU(3)SU(3)SU(3)-invariant spin chain (Lai-Sutherland model), and focus on the family of integrable initial states. By means of a Quantum Transfer Matrix approach, these can be related to boundary transfer matrices in an appropriate transverse direction. In this work, we provide a technical analysis of such integrable transfer matrices. In particular, we address the computation of their spectrum: this is achieved by deriving a set of functional relations between the eigenvalues of certain fused operators that are constructed starting from the soliton-non-preserving boundary transfer matrices (namely the TTT- and YYY-systems). As a direct physical application of our analysis, we compute the Loschmidt echo for imaginary and real times after a quench from the integrable states. Our results are also relevant for the study of the spectrum of SU(3)SU(3)SU(3)-invariant Hamiltonians with open boundary conditions.