On computing non-equilibrium dynamics following a quench

Computing the non-equilibrium dynamics that follows a quantum quench is difficult, even in exactly solvable models. Results are often predicated on the ability to compute overlaps between the initial state and eigenstates of the Hamiltonian that governs time evolution. Except for a handful of known cases, it is generically not possible to find these overlaps analytically. Here we develop a numerical approach to preferentially generate the states with high overlaps for a quantum quench starting from the ground state or an excited state of an initial Hamiltonian. We use these preferentially generated states, in combination with a "high overlap states truncation scheme" and a modification of the numerical renormalization group, to compute non-equilibrium dynamics following a quench in the Lieb-Liniger model. The method is non-perturbative, works for reasonable numbers of particles, and applies to both continuum and lattice systems. It can also be easily extended to more complicated scenarios, including those with integrability breaking.Computing the non-equilibrium dynamics that follows a quantum quench is difficult, even in exactly solvable models. Results are often predicated on the ability to compute overlaps between the initial state and eigenstates of the Hamiltonian that governs time evolution. Except for a handful of known cases, it is generically not possible to find these overlaps analytically. Here we develop a numerical approach to preferentially generate the states with high overlaps for a quantum quench starting from the ground state or an excited state of an initial Hamiltonian. We use these preferentially generated states, in combination with a "high overlap states truncation scheme" and a modification of the numerical renormalization group, to compute non-equilibrium dynamics following a quench in the Lieb-Liniger model. The method is non-perturbative, works for reasonable numbers of particles, and applies to both continuum and lattice systems. It can also be easily extended to more complicated scenarios, including those with integrability breaking.

arXiv:1911.11101SciPost PhysicsDec 9, 20211 citation

On non-equilibrium dynamics in low-dimensional strongly correlated quantum systems

Benedikt Schoenauer

In this work, we study the non-equilibrium dynamics of low-dimensional strongly correlated quantum systems, which in our case are generated by sudden and time-dependent quantum quenches. Such non-equilibrium dynamics are a particularly difficult problem, because the time evolution of the initial many-body quantum state, which is governed by the post-quench Hamiltonian, is highly non-trivial. On top of that, applying an operator to this many-body quantum state in order to calculate its expectation values, typically creates an intricate set of excitations. Throughout the thesis, we furthermore contrast the results that we have obtained using exact numerical methods for two microscopic models, namely the Heisenberg and Fermi-Hubbard chains, with results derived from the Luttinger liquid theory, which, in equilibrium, describes the universal behavior of the two models.In the second part of the thesis, we reduce the dimensionality even further and employ complementary numerical and analytical tools to examine the unusual non-equilibrium behavior of a ring-shaped quantum impurity hosting interacting spin-less fermions.In this work, we study the non-equilibrium dynamics of low-dimensional strongly correlated quantum systems, which in our case are generated by sudden and time-dependent quantum quenches. Such non-equilibrium dynamics are a particularly difficult problem, because the time evolution of the initial many-body quantum state, which is governed by the post-quench Hamiltonian, is highly non-trivial. On top of that, applying an operator to this many-body quantum state in order to calculate its expectation values, typically creates an intricate set of excitations. Throughout the thesis, we furthermore contrast the results that we have obtained using exact numerical methods for two microscopic models, namely the Heisenberg and Fermi-Hubbard chains, with results derived from the Luttinger liquid theory, which, in equilibrium, describes the universal behavior of the two models.In the second part of the thesis, we reduce the dimensionality even further and employ complementary numerical and analytical tools to examine the unusual non-equilibrium behavior of a ring-shaped quantum impurity hosting interacting spin-less fermions.

Jun 12, 20190 citations

Non-equilibrium quantum dynamics of interacting integrable models by Monte Carlo sampling Lehmann representations

Riccardo Senese, Fabian H. L. Essler

Determining the dynamics of interacting integrable many-particle quantum systems at finite times after homogeneous quantum quenches is a long-standing challenge. We present a Monte Carlo sampling scheme that numerically evaluates the Lehmann representation for time-dependent expectation values of local operators, allowing us to access system sizes and times significantly beyond the reach of existing methods. The approach accommodates both the full Lehmann sum and the Quench Action formalism. We benchmark against exact results for non-interacting lattice and continuum models and short-time results at weak interactions, finding excellent agreement. We apply the method to quantum quenches from a Bose-Einstein condensate in the repulsive Lieb-Liniger model and determine the time evolution of the order parameter for a wide range of interaction strengths. We discuss the emergence of a "sign problem" for more general dynamical correlators and setups.Determining the dynamics of interacting integrable many-particle quantum systems at finite times after homogeneous quantum quenches is a long-standing challenge. We present a Monte Carlo sampling scheme that numerically evaluates the Lehmann representation for time-dependent expectation values of local operators, allowing us to access system sizes and times significantly beyond the reach of existing methods. The approach accommodates both the full Lehmann sum and the Quench Action formalism. We benchmark against exact results for non-interacting lattice and continuum models and short-time results at weak interactions, finding excellent agreement. We apply the method to quantum quenches from a Bose-Einstein condensate in the repulsive Lieb-Liniger model and determine the time evolution of the order parameter for a wide range of interaction strengths. We discuss the emergence of a "sign problem" for more general dynamical correlators and setups.

cond-mat.stat-mechCross-listedarXiv:2605.20065May 20, 20260 citations

Failure of the Quench Action Formalism for Mott Insulator Initial States

Garry Goldstein

The quench action formalism relies on the assumption that the overlap between a generic initial state $\left|Ψ_{0}\right\rangle $ and an eigenstate of an integrable model - defined through the rapidities $\left|k_{1},...k_{N}\right\rangle $ - can be written as: \begin{equation} \left\langle k_{1},...k_{N}\midΨ_{0}\right\rangle =\exp\left(-S_{Ψ_{0}}\left(ρ\left(k\right)\right)\right),\label{eq:Exponential} \end{equation} where $ρ\left(k\right)$ is the quasiparticle density of the state $\left|k_{1},...k_{N}\right\rangle $ and $S_{Ψ_{0}}$ is some smooth function of $ρ\left(k\right)$ that depends on $Ψ_{0}$ . In particular the quench action formalism assumes the overlap depends smoothly on the quasiparticle density $ρ\left(k\right)$ . In this work, by explicit counter example, we show that this is not the case. We consider the quench between a Mott insulator and a Lieb Liniger gas. We show that the overlap between the ground state of the Mott insulator and arbitrary eigenstates of the Lieb Liniger gas has a highly singular behavior and no expression like Eq. (1) applies. We do so within the Tonks Girardeau limit of the Lieb Liniger gas and to leading order in the $1/c$ expansion for the overlap (with $c$ being the coupling constant of the Lieb Liniger gas). In the Appendix we show similar results for overlaps in the XXZ model with crystal states.The quench action formalism relies on the assumption that the overlap between a generic initial state $\left|Ψ_{0}\right\rangle $ and an eigenstate of an integrable model - defined through the rapidities $\left|k_{1},...k_{N}\right\rangle $ - can be written as: \begin{equation} \left\langle k_{1},...k_{N}\midΨ_{0}\right\rangle =\exp\left(-S_{Ψ_{0}}\left(ρ\left(k\right)\right)\right),\label{eq:Exponential} \end{equation} where $ρ\left(k\right)$ is the quasiparticle density of the state $\left|k_{1},...k_{N}\right\rangle $ and $S_{Ψ_{0}}$ is some smooth function of $ρ\left(k\right)$ that depends on $Ψ_{0}$. In particular the quench action formalism assumes the overlap depends smoothly on the quasiparticle density $ρ\left(k\right)$. In this work, by explicit counter example, we show that this is not the case. We consider the quench between a Mott insulator and a Lieb Liniger gas. We show that the overlap between the ground state of the Mott insulator and arbitrary eigenstates of the Lieb Liniger gas has a highly singular behavior and no expression like Eq. (1) applies. We do so within the Tonks Girardeau limit of the Lieb Liniger gas and to leading order in the $1/c$ expansion for the overlap (with $c$ being the coupling constant of the Lieb Liniger gas). In the Appendix we show similar results for overlaps in the XXZ model with crystal states.

arXiv:2606.07111Jun 5, 20260 citations

Non-Equilibrium Steady State of the Lieb-Liniger model: exact treatment of the Tonks Girardeau limit

Spyros Sotiriadis

Aiming at studying the emergence of Non-Equilibrium Steady States (NESS) in quantum integrable models by means of an exact analytical method, we focus on the Tonks-Girardeau or hard-core boson limit of the Lieb-Liniger model. We consider the abrupt expansion of a gas from one half to the entire confining box, a prototypical case of inhomogeneous quench, also known as "geometric quench". Based on the exact calculation of quench overlaps, we develop an analytical method for the derivation of the NESS by rigorously treating the thermodynamic and large time and distance limit. Our method is based on complex analysis tools for the derivation of the asymptotics of the many-body wavefunction, does not make essential use of the effectively non-interacting character of the hard-core boson gas and is sufficiently robust for generalisation to the genuinely interacting case.Aiming at studying the emergence of Non-Equilibrium Steady States (NESS) in quantum integrable models by means of an exact analytical method, we focus on the Tonks-Girardeau or hard-core boson limit of the Lieb-Liniger model. We consider the abrupt expansion of a gas from one half to the entire confining box, a prototypical case of inhomogeneous quench, also known as "geometric quench". Based on the exact calculation of quench overlaps, we develop an analytical method for the derivation of the NESS by rigorously treating the thermodynamic and large time and distance limit. Our method is based on complex analysis tools for the derivation of the asymptotics of the many-body wavefunction, does not make essential use of the effectively non-interacting character of the hard-core boson gas and is sufficiently robust for generalisation to the genuinely interacting case.

arXiv:2007.12683Jul 24, 20202 citations

Relaxation dynamics of local observables in integrable systems

Jacopo De Nardis, Lorenzo Piroli, Jean-Sébastien Caux

We show, using the quench action approach (Caux and Essler 2013 Phys. Rev. Lett. 110 257203 ), that the whole post-quench time evolution of an integrable system in the thermodynamic limit can be computed with a minimal set of data which are encoded in what we denote the generalized single-particle overlap coefficient This function can be extracted from the thermodynamically leading part of the overlaps between the eigenstates of the model and the initial state. For a generic global quench the shape of in the low momentum limit directly gives the exponent for the power law decay to the effective steady state. As an example we compute the time evolution of the static density–density correlation in the interacting Lieb–Liniger gas after a quench from a Bose–Einstein condensate. This shows an approach to equilibrium with power law t −3 which turns out to be independent of the post-quench interaction and of the considered observable.We show, using the quench action approach (Caux and Essler 2013 Phys. Rev. Lett. 110 257203 ), that the whole post-quench time evolution of an integrable system in the thermodynamic limit can be computed with a minimal set of data which are encoded in what we denote the generalized single-particle overlap coefficient This function can be extracted from the thermodynamically leading part of the overlaps between the eigenstates of the model and the initial state. For a generic global quench the shape of in the low momentum limit directly gives the exponent for the power law decay to the effective steady state. As an example we compute the time evolution of the static density–density correlation in the interacting Lieb–Liniger gas after a quench from a Bose–Einstein condensate. This shows an approach to equilibrium with power law t −3 which turns out to be independent of the post-quench interaction and of the considered observable.

arXiv:1505.03080Journal of Physics A Mathematical and TheoreticalOct 5, 201561 citations

ρ\left(k\right)

is the quasiparticle density of the state $\left|k_{1},...k_{N}\right\rangle $ and

S_{Ψ_{0}}

is some smooth function of

ρ\left(k\right)

that depends on

Ψ_{0}

. In particular the quench action formalism assumes the overlap depends smoothly on the quasiparticle density

ρ\left(k\right)

. In this work, by explicit counter example, we show that this is not the case. We consider the quench between a Mott insulator and a Lieb Liniger gas. We show that the overlap between the ground state of the Mott insulator and arbitrary eigenstates of the Lieb Liniger gas has a highly singular behavior and no expression like Eq. (1) applies. We do so within the Tonks Girardeau limit of the Lieb Liniger gas and to leading order in the

1/c

expansion for the overlap (with

c

being the coupling constant of the Lieb Liniger gas). In the Appendix we show similar results for overlaps in the XXZ model with crystal states.