Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1211.1195 · Physical Review B

Linear quantum quench in the Heisenberg XXZ chain: Time-dependent Luttinger-model description of a lattice system

Frank Pollmann, Masudul Haque, Balázs Dóra

Published Jan 31, 2013• 39 citations
arXivHTMLPDF

Abstract

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 $z$ 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.

DOI
10.1103/physrevb.87.041109

Related papers

Slow quenches in XXZ spin-chains -- the role of Galilean invariance breaking

Piotr Chudziński

We study a XXZ spin-chain in a gapless Tomonaga-Luttinger liquid (TLL) phase with time dependent anisotropy of spin exchange interactions. To begin we focus on a linear ramp of $J_z$, starting at XX point and slowly increasing towards the anti-ferromagnetic Heisenberg point. Although the problem of a linear ramp in the TLL has been recently under intense scrutiny in a perturbative \emph{g-ology} framework, an aspect that has been overlooked so far is the role of the Galilean invariance breaking. We find that, although the differential equation that needs to be solved to find time evolution of the system is substantially more complicated, in some cases exact analytic solutions can be given. We obtain them for the linear ramp in the limit of small $J_z$ as well as $J_z\rightarrow 1$, and for such protocols that are tailored to keep the Galilean invariance breaking term constant for every $J_z$. We point out the features of dynamics during the quench that stays unaltered, and those that need to be taken with care when Galilean invariance breaking is present. We are able to deduce that it is the shape of the propagating front that is affected in the most pronounced way.

arXiv:1607.00995•Jul 4, 2016•3 citations

Quantum quench within the gapless phase of the spin−12 Heisenberg XXZ spin chain

Mario Collura, Pasquale Calabrese, Fabian H. L. Eßler

We consider an interaction quench in the critical $\text{spin}\ensuremath{-}\frac{1}{2}$ Heisenberg XXZ chain. We numerically compute the time evolution of the two-point correlation functions of spin operators in the thermodynamic limit and compare the results to predictions obtained in the framework of the Luttinger liquid approximation. We find that the transverse correlation function $\ensuremath{\langle}{S}_{j}^{x}{S}_{j+\ensuremath{\ell}}^{x}\ensuremath{\rangle}$ agrees with the Luttinger model prediction to a surprising level of accuracy. The agreement for the longitudinal two-point function $\ensuremath{\langle}{S}_{j}^{z}{S}_{j+\ensuremath{\ell}}^{z}\ensuremath{\rangle}$ is found to be much poorer. We speculate that this difference between transverse and longitudinal correlations has its origin in the locality properties of the respective spin operator with respect to the underlying fermionic modes.

arXiv:1507.03492•Physical Review B•Sep 17, 2015•69 citations

Quantum quenches in the anisotropic spin-\frac{1}{2} Heisenberg chain: different approaches to many-body dynamics far from equilibrium

Peter Barmettler, Matthias Punk, Vladimir Gritsev +2 more

Recent experimental achievements in controlling ultracold gases in optical lattices open a new perspective on quantum many-body physics. In these experimental setups, it is possible to study coherent time evolution of isolated quantum systems. These dynamics reveal new physics beyond the low-energy properties that are usually relevant in solid-state many-body systems. In this paper, we study the time evolution of antiferromagnetic order in the Heisenberg chain after a sudden change of the anisotropy parameter, using various numerical and analytical methods. As a generic result, we find that the order parameter, which can show oscillatory or non-oscillatory dynamics, decays exponentially except for the effectively non-interacting case of the XX limit. For weakly ordered initial states, we also find evidence for an algebraic correction to the exponential law. The study is based on numerical simulations using a numerical matrix product method for infinite system sizes (iMPS), for which we provide a detailed description and an error analysis. Additionally, we investigate in detail the exactly solvable XX limit. These results are compared to approximative analytical approaches including an effective description by the XZ model as well as by mean-field, Luttinger-liquid and sine-Gordon theories. The comparison reveals which aspects of non-equilibrium dynamics can, as in equilibrium, be described by low-energy theories and which are the novel phenomena specific to quantum quench dynamics. The relevance of the energetically high part of the spectrum is illustrated by means of a full numerical diagonalization of the Hamiltonian.

arXiv:0911.1927•New Journal of Physics•May 28, 2010•128 citations

Quantum quenches in Luttinger liquids

D. Loupas

It is common in physics to treat 1D system as a toy model in order to test a physical theory because it is easier to work with low dimensional systems and find analytical solutions. One naively will say that one-dimensional models serve as toy models and they do not apply in the real world. But, one-dimensional systems, e.g quantum wires, exists in nature and have quite different physics from higher dimensional systems. It is remarkable that, in 1D, an interacting fermionic problem can be turned to a free bosonic theory. An interesting question to ask is how the system evolves after a change in the systems internal parameters. This is called a quantum quench. In this thesis we investigate quantum quenches in the Luttinger liquid model, which is a 1D model and serves as the analog of the Fermi liquid theory in one dimension. More specifically, we study the kinetic energy, when we periodically drive the internal parameters of the system, and the two point correlation function, after a periodic quench of finite duration.

•Jan 1, 2012•0 citations

Finite-time quantum quenches in the XXZ Heisenberg chain

Benedikt Schoenauer, Dirk Schuricht

We study the time evolution of the two-point correlation functions in the XXZ Heisenberg chain after a finite-time quantum quench in the anisotropy. We compare results from numerical simulations to ones obtained in the Luttinger model and find good agreement. We analyze the spreading of the correlations and the associated light-cone features. We observe a delay in the appearance of the light cone as compared to the sudden-quench setup, and link this delay to the properties of the quench protocol.

arXiv:1905.02678•Physical review. B./Physical review. B•Sep 12, 2019•3 citations

Quantum quench spectroscopy of a Luttinger liquid: Ultrarelativistic density wave dynamics due to fractionalization in an XXZ chain

Matthew S. Foster, Timothy C. Berkelbach, David R. Reichman +1 more

We compute the dynamics of localized excitations produced by a quantum quench in the spin-1/2 XXZ chain. Using numerics combining the density-matrix renormalization group and exact time evolution, as well as analytical arguments, we show that fractionalization due to interactions in the prequench state gives rise to ``ultrarelativistic'' density waves that travel at the maximum band velocity. The system is initially prepared in the ground state of the chain within the gapless $XY$ phase, which admits a Luttinger liquid (LL) description at low energies and long wavelengths. The Hamiltonian is then suddenly quenched to a band insulator, after which the chain evolves unitarily. Through the gapped dispersion of the insulator spectrum, the postquench dynamics serve as a ``velocity microscope,'' revealing initial-state particle correlations via space-time density propagation. We show that the ultrarelativistic wave production is tied to the particular way in which fractionalization evades Pauli blocking in the zero-temperature initial LL state.

arXiv:1105.1786•Physical Review B•Aug 31, 2011•29 citations
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 a XXZ spin-chain in a gapless Tomonaga-Luttinger liquid (TLL) phase with time dependent anisotropy of spin exchange interactions. To begin we focus on a linear ramp of JzJ_zJz​, starting at XX point and slowly increasing towards the anti-ferromagnetic Heisenberg point. Although the problem of a linear ramp in the TLL has been recently under intense scrutiny in a perturbative \emph{g-ology} framework, an aspect that has been overlooked so far is the role of the Galilean invariance breaking. We find that, although the differential equation that needs to be solved to find time evolution of the system is substantially more complicated, in some cases exact analytic solutions can be given. We obtain them for the linear ramp in the limit of small JzJ_zJz​ as well as Jz→1J_z\rightarrow 1Jz​→1, and for such protocols that are tailored to keep the Galilean invariance breaking term constant for every JzJ_zJz​. We point out the features of dynamics during the quench that stays unaltered, and those that need to be taken with care when Galilean invariance breaking is present. We are able to deduce that it is the shape of the propagating front that is affected in the most pronounced way.
We consider an interaction quench in the critical $\text{spin}\ensuremath{-}\frac{1}{2}$ Heisenberg XXZ chain. We numerically compute the time evolution of the two-point correlation functions of spin operators in the thermodynamic limit and compare the results to predictions obtained in the framework of the Luttinger liquid approximation. We find that the transverse correlation function $\ensuremath{\langle}{S}_{j}^{x}{S}_{j+\ensuremath{\ell}}^{x}\ensuremath{\rangle}$ agrees with the Luttinger model prediction to a surprising level of accuracy. The agreement for the longitudinal two-point function $\ensuremath{\langle}{S}_{j}^{z}{S}_{j+\ensuremath{\ell}}^{z}\ensuremath{\rangle}$ is found to be much poorer. We speculate that this difference between transverse and longitudinal correlations has its origin in the locality properties of the respective spin operator with respect to the underlying fermionic modes.
Recent experimental achievements in controlling ultracold gases in optical lattices open a new perspective on quantum many-body physics. In these experimental setups, it is possible to study coherent time evolution of isolated quantum systems. These dynamics reveal new physics beyond the low-energy properties that are usually relevant in solid-state many-body systems. In this paper, we study the time evolution of antiferromagnetic order in the Heisenberg chain after a sudden change of the anisotropy parameter, using various numerical and analytical methods. As a generic result, we find that the order parameter, which can show oscillatory or non-oscillatory dynamics, decays exponentially except for the effectively non-interacting case of the XX limit. For weakly ordered initial states, we also find evidence for an algebraic correction to the exponential law. The study is based on numerical simulations using a numerical matrix product method for infinite system sizes (iMPS), for which we provide a detailed description and an error analysis. Additionally, we investigate in detail the exactly solvable XX limit. These results are compared to approximative analytical approaches including an effective description by the XZ model as well as by mean-field, Luttinger-liquid and sine-Gordon theories. The comparison reveals which aspects of non-equilibrium dynamics can, as in equilibrium, be described by low-energy theories and which are the novel phenomena specific to quantum quench dynamics. The relevance of the energetically high part of the spectrum is illustrated by means of a full numerical diagonalization of the Hamiltonian.
It is common in physics to treat 1D system as a toy model in order to test a physical theory because it is easier to work with low dimensional systems and find analytical solutions. One naively will say that one-dimensional models serve as toy models and they do not apply in the real world. But, one-dimensional systems, e.g quantum wires, exists in nature and have quite different physics from higher dimensional systems. It is remarkable that, in 1D, an interacting fermionic problem can be turned to a free bosonic theory. An interesting question to ask is how the system evolves after a change in the systems internal parameters. This is called a quantum quench. In this thesis we investigate quantum quenches in the Luttinger liquid model, which is a 1D model and serves as the analog of the Fermi liquid theory in one dimension. More specifically, we study the kinetic energy, when we periodically drive the internal parameters of the system, and the two point correlation function, after a periodic quench of finite duration.
We study the time evolution of the two-point correlation functions in the XXZ Heisenberg chain after a finite-time quantum quench in the anisotropy. We compare results from numerical simulations to ones obtained in the Luttinger model and find good agreement. We analyze the spreading of the correlations and the associated light-cone features. We observe a delay in the appearance of the light cone as compared to the sudden-quench setup, and link this delay to the properties of the quench protocol.
We compute the dynamics of localized excitations produced by a quantum quench in the spin-1/2 XXZ chain. Using numerics combining the density-matrix renormalization group and exact time evolution, as well as analytical arguments, we show that fractionalization due to interactions in the prequench state gives rise to ``ultrarelativistic'' density waves that travel at the maximum band velocity. The system is initially prepared in the ground state of the chain within the gapless XYXYXY phase, which admits a Luttinger liquid (LL) description at low energies and long wavelengths. The Hamiltonian is then suddenly quenched to a band insulator, after which the chain evolves unitarily. Through the gapped dispersion of the insulator spectrum, the postquench dynamics serve as a ``velocity microscope,'' revealing initial-state particle correlations via space-time density propagation. We show that the ultrarelativistic wave production is tied to the particular way in which fractionalization evades Pauli blocking in the zero-temperature initial LL state.