arXiv:2512.17333 · Physical review. B./Physical review. B

Quantum quenches across continuous and first-order quantum transitions in one-dimensional quantum Ising models

Andrea Pelissetto, Davide Rossini, Ettore Vicari

cond-mat.stat-mechquant-phSubmitted Dec 19, 2025• Announced Dec 22, 2025• 1 citation• Revised Jan 29, 2026 (v2)

Abstract

We investigate the quantum dynamics generated by quantum quenches (QQs) of the Hamiltonian parameters in many-body systems, focusing on protocols that cross first-order and continuous quantum transitions, both in finite-size systems and in the thermodynamic limit. As a paradigmatic example, we consider the quantum Ising chain in the presence of homogeneous transverse ($g$) and longitudinal ($h$) magnetic fields. This model exhibits a continuous quantum transition (CQT) at $g=g_c$ and $h=0$, and first-order quantum transitions (FOQTs) driven by $h$ along the line $h=0$ ($g<g_c$). In the integrable limit $h=0$, the system can be mapped onto a quadratic fermionic theory; however, any nonvanishing longitudinal field breaks integrability and the spectrum of the resulting Hamiltonian is generally expected to enter a chaotic regime. We analyze QQs in which the longitudinal field is suddenly changed from a negative value $h_i < 0$ to a positive value $h_f>0$. We focus on values of $h_f$ such that the spectrum of the post-QQ Hamiltonian $H(g,h_f)$ lies in the chaotic regime, where thermalization may emerge at asymptotically long times. We study the out-of-equilibrium dynamics for different values of $g$, finding qualitatively distinct behaviors for $g > g_c$ (where the chain is in the disordered phase), for $g = g_c$ (QQ across the CQT), and for $g<g_c$ (QQ across the FOQT line).

Journal reference: Phys. Rev. B 113, 144316 (2026)
DOI: 10.1103/plrd-q8vq
Comments: 25 pages, some comments and refs added

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arXiv:2512.17333 · Physical review. B./Physical review. B

Quantum quenches across continuous and first-order quantum transitions in one-dimensional quantum Ising models

Andrea Pelissetto, Davide Rossini, Ettore Vicari

cond-mat.stat-mechquant-phSubmitted Dec 19, 2025• Announced Dec 22, 2025• 1 citation• Revised Jan 29, 2026 (v2)

arXiv HTML PDF

Abstract

g

) and longitudinal (

h

) magnetic fields. This model exhibits a continuous quantum transition (CQT) at

g=g_c

and

h=0

, and first-order quantum transitions (FOQTs) driven by

h

along the line

h=0

(

g<g_c

). In the integrable limit

h=0

, the system can be mapped onto a quadratic fermionic theory; however, any nonvanishing longitudinal field breaks integrability and the spectrum of the resulting Hamiltonian is generally expected to enter a chaotic regime. We analyze QQs in which the longitudinal field is suddenly changed from a negative value

h_i < 0

to a positive value

h_f>0

. We focus on values of

h_f

such that the spectrum of the post-QQ Hamiltonian

H(g,h_f)

lies in the chaotic regime, where thermalization may emerge at asymptotically long times. We study the out-of-equilibrium dynamics for different values of

g

, finding qualitatively distinct behaviors for

g > g_c

(where the chain is in the disordered phase), for

g = g_c

(QQ across the CQT), and for

g<g_c

(QQ across the FOQT line).

Journal reference: Phys. Rev. B 113, 144316 (2026)
DOI: 10.1103/plrd-q8vq
Comments: 25 pages, some comments and refs added

Related papers

Dynamics after quenches in one-dimensional quantum Ising-like systems

Davide Rossini, Ettore Vicari

We study the out-of-equilibrium dynamics of one-dimensional quantum Ising-like systems, arising from sudden quenches of the Hamiltonian parameter $g$ driving quantum transitions between disordered and ordered phases. In particular, we consider quenches to values of $g$ around the critical value ${g}_{c}$ , and mainly address the question whether, and how, the quantum transition leaves traces in the evolution of the transverse and longitudinal magnetizations during such a deep out-of-equilibrium dynamics. We shed light on the emergence of singularities in the thermodynamic infinite-size limit, likely related to the integrability of the model. Finite systems in periodic and open boundary conditions develop peculiar power-law finite-size scaling laws related to revival phenomena, but apparently unrelated to the quantum transition, because their main features are generally observed in quenches to generic values of $g$ . We also investigate the effects of dissipative interactions with an environment, modeled by a Lindblad equation with local decay and pumping dissipation operators within the quadratic fermionic model obtainable by a Jordan-Wigner mapping. Dissipation tends to suppress the main features of the unitary dynamics of closed systems. We finally address the effects of integrability breaking, due to further lattice interactions, such as in anisotropic next-to-nearest-neighbor Ising (ANNNI) models. We show that some qualitative features of the post-quench dynamics persist, in particular, the different behaviors when quenching to quantum ferromagnetic and paramagnetic phases, and the revival phenomena due to the finite size of the system.We study the out-of-equilibrium dynamics of one-dimensional quantum Ising-like systems, arising from sudden quenches of the Hamiltonian parameter $g$ driving quantum transitions between disordered and ordered phases. In particular, we consider quenches to values of $g$ around the critical value ${g}_{c}$, and mainly address the question whether, and how, the quantum transition leaves traces in the evolution of the transverse and longitudinal magnetizations during such a deep out-of-equilibrium dynamics. We shed light on the emergence of singularities in the thermodynamic infinite-size limit, likely related to the integrability of the model. Finite systems in periodic and open boundary conditions develop peculiar power-law finite-size scaling laws related to revival phenomena, but apparently unrelated to the quantum transition, because their main features are generally observed in quenches to generic values of $g$. We also investigate the effects of dissipative interactions with an environment, modeled by a Lindblad equation with local decay and pumping dissipation operators within the quadratic fermionic model obtainable by a Jordan-Wigner mapping. Dissipation tends to suppress the main features of the unitary dynamics of closed systems. We finally address the effects of integrability breaking, due to further lattice interactions, such as in anisotropic next-to-nearest-neighbor Ising (ANNNI) models. We show that some qualitative features of the post-quench dynamics persist, in particular, the different behaviors when quenching to quantum ferromagnetic and paramagnetic phases, and the revival phenomena due to the finite size of the system.

arXiv:2007.09041Physical review. B./Physical review. BAug 31, 202021 citations

Scaling properties of the dynamics at first-order quantum transitions when boundary conditions favor one of the two phases

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We address the out-of-equilibrium dynamics of a many-body system when one of its Hamiltonian parameters is driven across a first-order quantum transition (FOQT). In particular, we consider systems subject to boundary conditions favoring one of the two phases separated by the FOQT. These issues are investigated within the paradigmatic one-dimensional quantum Ising model, at the FOQTs driven by the longitudinal magnetic field h, with boundary conditions that favor the same magnetized phase (EFBC) or opposite magnetized phases (OFBC). We study the dynamic behavior for an instantaneous quench and for a protocol in which h is slowly varied across the FOQT. We develop a dynamic finite-size scaling theory for both EFBC and OFBC, which displays some remarkable differences with respect to the case of neutral boundary conditions. The corresponding relevant timescale shows a qualitative different size dependence in the two cases: it increases exponentially with the size in the case of EFBC, and as a power of the size in the case of OFBC.We address the out-of-equilibrium dynamics of a many-body system when one of its Hamiltonian parameters is driven across a first-order quantum transition (FOQT). In particular, we consider systems subject to boundary conditions favoring one of the two phases separated by the FOQT. These issues are investigated within the paradigmatic one-dimensional quantum Ising model, at the FOQTs driven by the longitudinal magnetic field h, with boundary conditions that favor the same magnetized phase (EFBC) or opposite magnetized phases (OFBC). We study the dynamic behavior for an instantaneous quench and for a protocol in which h is slowly varied across the FOQT. We develop a dynamic finite-size scaling theory for both EFBC and OFBC, which displays some remarkable differences with respect to the case of neutral boundary conditions. The corresponding relevant timescale shows a qualitative different size dependence in the two cases: it increases exponentially with the size in the case of EFBC, and as a power of the size in the case of OFBC.

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We study the out-of-equilibrium dynamics of one-dimensional quantum Ising models in a transverse field $g$ , driven by a time-dependent longitudinal field $h$ across their {\em magnetic} first-order quantum transition at $h=0$ , for sufficiently small values of $|g|$ . We consider nearest-neighbor Ising chains of size $L$ with periodic boundary conditions. We focus on the out-of-equilibrium behavior arising from Kibble-Zurek protocols, in which $h$ is varied linearly in time with time scale $t_s$ , i.e., $h(t)=t/t_s$ . The system starts from the ground state at $h_i\equiv h(t_i)<0$ , where the longitudinal magnetization $M$ is negative. Then it evolves unitarily up to positive values of $h(t)$ , where $M(t)$ becomes eventually positive. We identify several scaling regimes characterized by a nontrivial interplay between the size $L$ and the time scale $t_s$ , which can be observed when the system is close to one of the many avoided level crossings that occur for $h\ge 0$ . In the $L\to\infty$ limit, all these crossings approach $h=0^+$ , making the study of the thermodynamic limit, defined as the limit $L\to\infty$ keeping $t$ and $t_s$ constant, problematic. We study such limit numerically, by first determining the large- $L$ quantum evolution at fixed $t_s$ , and then analyzing its behavior with increasing $t_s$ . Our analysis shows that the system switches from the initial state with $M<0$ to a positively magnetized state at $h = h_*(t_s)>0$ , where $h_*(t_s)$ decreases with increasing $t_s$ , apparently as $h_*\sim 1/\ln t_s$ . This suggests the existence of a scaling behavior in terms of the rescaled time $Ω= t \ln t_s/t_s$ . The numerical results also show that the system converges to a nontrivial stationary state in the large- $t$ limit, characterized by an energy significantly larger than that of the corresponding homogeneously magnetized ground state.We study the out-of-equilibrium dynamics of one-dimensional quantum Ising models in a transverse field $g$, driven by a time-dependent longitudinal field $h$ across their {\em magnetic} first-order quantum transition at $h=0$, for sufficiently small values of $|g|$. We consider nearest-neighbor Ising chains of size $L$ with periodic boundary conditions. We focus on the out-of-equilibrium behavior arising from Kibble-Zurek protocols, in which $h$ is varied linearly in time with time scale $t_s$, i.e., $h(t)=t/t_s$. The system starts from the ground state at $h_i\equiv h(t_i)<0$, where the longitudinal magnetization $M$ is negative. Then it evolves unitarily up to positive values of $h(t)$, where $M(t)$ becomes eventually positive. We identify several scaling regimes characterized by a nontrivial interplay between the size $L$ and the time scale $t_s$, which can be observed when the system is close to one of the many avoided level crossings that occur for $h\ge 0$. In the $L\to\infty$ limit, all these crossings approach $h=0^+$, making the study of the thermodynamic limit, defined as the limit $L\to\infty$ keeping $t$ and $t_s$ constant, problematic. We study such limit numerically, by first determining the large-$L$ quantum evolution at fixed $t_s$, and then analyzing its behavior with increasing $t_s$. Our analysis shows that the system switches from the initial state with $M<0$ to a positively magnetized state at $h = h_*(t_s)>0$, where $h_*(t_s)$ decreases with increasing $t_s$, apparently as $h_*\sim 1/\ln t_s$. This suggests the existence of a scaling behavior in terms of the rescaled time $Ω= t \ln t_s/t_s$. The numerical results also show that the system converges to a nontrivial stationary state in the large-$t$ limit, characterized by an energy significantly larger than that of the corresponding homogeneously magnetized ground state.

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Andrea Pelissetto C, Davide Rossini, Ettore Vicari

We study the out-of-equilibrium Kibble-Zurek (KZ) dynamics in quantum Ising chains in a transverse field, driven by a time-dependent longitudinal field $h(t)=t/t_s$ ( $t_s$ is the time scale of the protocol), across their first-order quantum transitions (FOQTs) at $h=0$ . The KZ protocol starts at time $t_i<0$ from the negatively magnetized ground state for $h_i = t_i/t_s<0$ . Then, the system evolves unitarily up to a time $t_f > 0$ , such that the magnetization of the state at time $t_f$ is positive. In finite-size systems, the KZ dynamics develops out-of-equilibrium finite-size scaling (OFSS) behaviors. Their scaling variables depend either exponentially or with a power law on the size, depending on the boundary conditions (BC). The OFSS functions can be computed in effective models restricted to appropriate low-energy (magnetized and/or kink) states. The KZ scaling behavior drastically changes in the thermodynamic limit (TL), defined as the infinite-size limit keeping $t$ and $t_s$ fixed, which appears substantially unrelated with the OFSS regime, because it involves higher-energy multi-kink states, which are irrelevant in the OFSS limit. The numerical analyses of the KZ dynamics in the TL show the emergence of a quantum spinodal-like scaling behavior at the FOQTs for all considered BC, which is independent of the BC. The longitudinal magnetization changes sign at $h(t)=h*>0$ , where $h*$ decreases with increasing $t_s$ , as $h*\sim 1/\ln t_s$ . Moreover, in the large- $t_s$ limit, the time-dependence of the magnetization is described by a universal function of $Ω= t/τ_s$ , with $τ_s = t_s/\ln t_s$ .We study the out-of-equilibrium Kibble-Zurek (KZ) dynamics in quantum Ising chains in a transverse field, driven by a time-dependent longitudinal field $h(t)=t/t_s$ ($t_s$ is the time scale of the protocol), across their first-order quantum transitions (FOQTs) at $h=0$. The KZ protocol starts at time $t_i<0$ from the negatively magnetized ground state for $h_i = t_i/t_s<0$. Then, the system evolves unitarily up to a time $t_f > 0$, such that the magnetization of the state at time $t_f$ is positive. In finite-size systems, the KZ dynamics develops out-of-equilibrium finite-size scaling (OFSS) behaviors. Their scaling variables depend either exponentially or with a power law on the size, depending on the boundary conditions (BC). The OFSS functions can be computed in effective models restricted to appropriate low-energy (magnetized and/or kink) states. The KZ scaling behavior drastically changes in the thermodynamic limit (TL), defined as the infinite-size limit keeping $t$ and $t_s$ fixed, which appears substantially unrelated with the OFSS regime, because it involves higher-energy multi-kink states, which are irrelevant in the OFSS limit. The numerical analyses of the KZ dynamics in the TL show the emergence of a quantum spinodal-like scaling behavior at the FOQTs for all considered BC, which is independent of the BC. The longitudinal magnetization changes sign at $h(t)=h*>0$, where $h*$ decreases with increasing $t_s$, as $h*\sim 1/\ln t_s$. Moreover, in the large-$t_s$ limit, the time-dependence of the magnetization is described by a universal function of $Ω= t/τ_s$, with $τ_s = t_s/\ln t_s$.

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We address the quantum dynamics of a system composed of a qubit globally coupled to a many-body system characterized by short-range interactions. We employ a dynamic finite-size scaling framework to investigate the out-of-equilibrium dynamics arising from the sudden variation (turning on) of the interaction between the qubit and the many-body system, in particular when the latter is in proximity to a quantum first-order or continuous-phase transition. Although the approach is quite general, we consider $d$ -dimensional quantum Ising spin models in the presence of transverse and longitudinal fields as paradigmatic quantum many-body systems. To characterize the out-of-equilibrium dynamics, we focus on a number of quantum-information-oriented properties of the model. Namely, we concentrate on the decoherence features of the qubit, the energy interchange between the qubit and the many-body system during the out-of-equilibrium dynamics, and the work distribution associated with the quench. The scaling behaviors predicted by the dynamic finite-size scaling theory are verified through extensive numerical computations for the one-dimensional Ising model, which reveal a fast convergence to the expected asymptotic behavior with increasing system size.We address the quantum dynamics of a system composed of a qubit globally coupled to a many-body system characterized by short-range interactions. We employ a dynamic finite-size scaling framework to investigate the out-of-equilibrium dynamics arising from the sudden variation (turning on) of the interaction between the qubit and the many-body system, in particular when the latter is in proximity to a quantum first-order or continuous-phase transition. Although the approach is quite general, we consider $d$-dimensional quantum Ising spin models in the presence of transverse and longitudinal fields as paradigmatic quantum many-body systems. To characterize the out-of-equilibrium dynamics, we focus on a number of quantum-information-oriented properties of the model. Namely, we concentrate on the decoherence features of the qubit, the energy interchange between the qubit and the many-body system during the out-of-equilibrium dynamics, and the work distribution associated with the quench. The scaling behaviors predicted by the dynamic finite-size scaling theory are verified through extensive numerical computations for the one-dimensional Ising model, which reveal a fast convergence to the expected asymptotic behavior with increasing system size.

arXiv:1903.02019Physical Review AMay 16, 201915 citations

Non equilibrium dynamics of quantum Ising chains in the presence of transverse and longitudinal magnetic fields

Zahra Mokhtari

There has been much recent interest in the study of out of equilibrium quantum systems due to experimental advances in trapped cold atomic gases that allow systematic studies of these effects.In this work we study a quantum Ising model in the presence of transverse and longitudinal magnetic fields.The model lacks an analytical solution in the most general case and has to be treated numerically.We first investigate the static properties of the model, including the ground state energy and the zero temperature phase diagram, using the Density Matrix Renormalization Group (DMRG) method.We identify ordered and disordered phases separated by a quantum phase transition.We then consider the time evolution of the system following an abrupt quench of the fields.We study the response of the order parameter to the quenches within the ordered and disordered phases as well as quenches between the phases using the time-dependent DMRG method.The results are in qualitative agreement with experiments and the implemented numerical method in this work can be extended to allow for more general interactions.iii I would like to thank my supervisor, Dr. Malcolm Kennett, whose constant support and hope in the most disappointing situations, his helpful guidance, and endless energy were all new and very invaluable to me.I like to thank myThere has been much recent interest in the study of out of equilibrium quantum systems due to experimental advances in trapped cold atomic gases that allow systematic studies of these effects.In this work we study a quantum Ising model in the presence of transverse and longitudinal magnetic fields.The model lacks an analytical solution in the most general case and has to be treated numerically.We first investigate the static properties of the model, including the ground state energy and the zero temperature phase diagram, using the Density Matrix Renormalization Group (DMRG) method.We identify ordered and disordered phases separated by a quantum phase transition.We then consider the time evolution of the system following an abrupt quench of the fields.We study the response of the order parameter to the quenches within the ordered and disordered phases as well as quenches between the phases using the time-dependent DMRG method.The results are in qualitative agreement with experiments and the implemented numerical method in this work can be extended to allow for more general interactions.iii I would like to thank my supervisor, Dr. Malcolm Kennett, whose constant support and hope in the most disappointing situations, his helpful guidance, and endless energy were all new and very invaluable to me.I like to thank my

Aug 26, 20130 citations

g

) and longitudinal (

h

) magnetic fields. This model exhibits a continuous quantum transition (CQT) at

g=g_c

and

h=0

, and first-order quantum transitions (FOQTs) driven by

h

along the line

h=0

(

g<g_c

). In the integrable limit

h=0

h_i < 0

to a positive value

h_f>0

. We focus on values of

h_f

such that the spectrum of the post-QQ Hamiltonian

H(g,h_f)

lies in the chaotic regime, where thermalization may emerge at asymptotically long times. We study the out-of-equilibrium dynamics for different values of

g

, finding qualitatively distinct behaviors for

g > g_c

(where the chain is in the disordered phase), for

g = g_c

(QQ across the CQT), and for

g<g_c

(QQ across the FOQT line).

We study the out-of-equilibrium dynamics of one-dimensional quantum Ising-like systems, arising from sudden quenches of the Hamiltonian parameter

g

driving quantum transitions between disordered and ordered phases. In particular, we consider quenches to values of

g

around the critical value

{g}_{c}

, and mainly address the question whether, and how, the quantum transition leaves traces in the evolution of the transverse and longitudinal magnetizations during such a deep out-of-equilibrium dynamics. We shed light on the emergence of singularities in the thermodynamic infinite-size limit, likely related to the integrability of the model. Finite systems in periodic and open boundary conditions develop peculiar power-law finite-size scaling laws related to revival phenomena, but apparently unrelated to the quantum transition, because their main features are generally observed in quenches to generic values of

g

. We also investigate the effects of dissipative interactions with an environment, modeled by a Lindblad equation with local decay and pumping dissipation operators within the quadratic fermionic model obtainable by a Jordan-Wigner mapping. Dissipation tends to suppress the main features of the unitary dynamics of closed systems. We finally address the effects of integrability breaking, due to further lattice interactions, such as in anisotropic next-to-nearest-neighbor Ising (ANNNI) models. We show that some qualitative features of the post-quench dynamics persist, in particular, the different behaviors when quenching to quantum ferromagnetic and paramagnetic phases, and the revival phenomena due to the finite size of the system.

We study the out-of-equilibrium dynamics of one-dimensional quantum Ising models in a transverse field

g

, driven by a time-dependent longitudinal field

h

across their {\em magnetic} first-order quantum transition at

h=0

, for sufficiently small values of

|g|

. We consider nearest-neighbor Ising chains of size

L

with periodic boundary conditions. We focus on the out-of-equilibrium behavior arising from Kibble-Zurek protocols, in which

h

is varied linearly in time with time scale

t_s

, i.e.,

h(t)=t/t_s

. The system starts from the ground state at

h_i\equiv h(t_i)<0

, where the longitudinal magnetization

M

is negative. Then it evolves unitarily up to positive values of

h(t)

, where

M(t)

becomes eventually positive. We identify several scaling regimes characterized by a nontrivial interplay between the size

L

and the time scale

t_s

, which can be observed when the system is close to one of the many avoided level crossings that occur for

h\ge 0

. In the

L\to\infty

limit, all these crossings approach

h=0^+

, making the study of the thermodynamic limit, defined as the limit

L\to\infty

keeping

t

and

t_s

constant, problematic. We study such limit numerically, by first determining the large-

L

quantum evolution at fixed

t_s

, and then analyzing its behavior with increasing

t_s

. Our analysis shows that the system switches from the initial state with

M<0

to a positively magnetized state at

h = h_*(t_s)>0

, where

h_*(t_s)

decreases with increasing

t_s

, apparently as

h_*\sim 1/\ln t_s

. This suggests the existence of a scaling behavior in terms of the rescaled time

Ω= t \ln t_s/t_s

. The numerical results also show that the system converges to a nontrivial stationary state in the large-

t

limit, characterized by an energy significantly larger than that of the corresponding homogeneously magnetized ground state.

We study the out-of-equilibrium Kibble-Zurek (KZ) dynamics in quantum Ising chains in a transverse field, driven by a time-dependent longitudinal field

h(t)=t/t_s

(

t_s

is the time scale of the protocol), across their first-order quantum transitions (FOQTs) at

h=0

. The KZ protocol starts at time

t_i<0

from the negatively magnetized ground state for

h_i = t_i/t_s<0

. Then, the system evolves unitarily up to a time

t_f > 0

, such that the magnetization of the state at time

t_f

is positive. In finite-size systems, the KZ dynamics develops out-of-equilibrium finite-size scaling (OFSS) behaviors. Their scaling variables depend either exponentially or with a power law on the size, depending on the boundary conditions (BC). The OFSS functions can be computed in effective models restricted to appropriate low-energy (magnetized and/or kink) states. The KZ scaling behavior drastically changes in the thermodynamic limit (TL), defined as the infinite-size limit keeping

t

and

t_s

fixed, which appears substantially unrelated with the OFSS regime, because it involves higher-energy multi-kink states, which are irrelevant in the OFSS limit. The numerical analyses of the KZ dynamics in the TL show the emergence of a quantum spinodal-like scaling behavior at the FOQTs for all considered BC, which is independent of the BC. The longitudinal magnetization changes sign at

h(t)=h*>0

, where

h*

decreases with increasing

t_s

, as

h*\sim 1/\ln t_s

. Moreover, in the large-

t_s

limit, the time-dependence of the magnetization is described by a universal function of

Ω= t/τ_s

, with

τ_s = t_s/\ln t_s

d

-dimensional quantum Ising spin models in the presence of transverse and longitudinal fields as paradigmatic quantum many-body systems. To characterize the out-of-equilibrium dynamics, we focus on a number of quantum-information-oriented properties of the model. Namely, we concentrate on the decoherence features of the qubit, the energy interchange between the qubit and the many-body system during the out-of-equilibrium dynamics, and the work distribution associated with the quench. The scaling behaviors predicted by the dynamic finite-size scaling theory are verified through extensive numerical computations for the one-dimensional Ising model, which reveal a fast convergence to the expected asymptotic behavior with increasing system size.