arXiv:2405.04364 · Physical review. B./Physical review. B

Interplay between short-range and critical long-range fluctuations in the out-of-equilibrium behavior of the particle density at quantum transitions

Davide Rossini, Ettore Vicari

Published Jul 8, 2024• 4 citations

Abstract

We address the equilibrium and out-of-equilibrium behavior of the particle density in many-body systems undergoing quantum transitions driven by the chemical potential $\ensuremath{\mu}$. They originate from a nontrivial interplay between noncritical short-range and critical long-range quantum fluctuations. As a paradigmatic model, we consider the one-dimensional fermionic Kitaev model, for which very accurate numerical studies can be performed, up to $O({10}^{4})$ chain sites. The search for dynamic scaling behaviors of the particle density is complicated by the fact that its equilibrium (ground-state) behavior is dominated by short-range fluctuations, giving rise to regular background terms and peculiar logarithmic terms from resonances between renormalization-group perturbations associated with the energy and identity operator families within the conformal field theory. To study these issues, we focus on two dynamic protocols, either instantaneous quenches or quasiadiabatic changes of $\ensuremath{\mu}$, to the critical value ${\ensuremath{\mu}}_{c}$, unveiling out-of-equilibrium scaling behaviors of the particle density, which arise from the critical modes, within a dynamic finite-size scaling framework.

DOI: 10.1103/physrevb.110.035126

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arXiv:2405.04364 · Physical review. B./Physical review. B

Interplay between short-range and critical long-range fluctuations in the out-of-equilibrium behavior of the particle density at quantum transitions

Davide Rossini, Ettore Vicari

Published Jul 8, 2024• 4 citations

arXiv HTML PDF

Abstract

We address the equilibrium and out-of-equilibrium behavior of the particle density in many-body systems undergoing quantum transitions driven by the chemical potential

\ensuremath{\mu}

. They originate from a nontrivial interplay between noncritical short-range and critical long-range quantum fluctuations. As a paradigmatic model, we consider the one-dimensional fermionic Kitaev model, for which very accurate numerical studies can be performed, up to

O({10}^{4})

chain sites. The search for dynamic scaling behaviors of the particle density is complicated by the fact that its equilibrium (ground-state) behavior is dominated by short-range fluctuations, giving rise to regular background terms and peculiar logarithmic terms from resonances between renormalization-group perturbations associated with the energy and identity operator families within the conformal field theory. To study these issues, we focus on two dynamic protocols, either instantaneous quenches or quasiadiabatic changes of

\ensuremath{\mu}

, to the critical value

{\ensuremath{\mu}}_{c}

, unveiling out-of-equilibrium scaling behaviors of the particle density, which arise from the critical modes, within a dynamic finite-size scaling framework.

DOI: 10.1103/physrevb.110.035126

Related papers

Out-of-equilibrium scaling of the particle density in quantum fermionic wires after a critical quenching of the chemical potential

H. Panagopoulos, Ettore Vicari

We study the out-of-equilibrium scaling behavior of the particle density in quantum fermionic Kitaev wires, after instantaneous quantum quenches (QQs) of the chemical potential within their quantum critical region. The critical scaling of the ground-state particle density is known to be subleading at its Ising-like quantum transition, hidden by regular and logarithmic terms arising from peculiar mixings with the identity operator. This situation changes along the out-of-equilibrium dynamics arising from QQs of the chemical potential to the critical point, starting from the ground state for Hamiltonian parameters within the critical region. We analytically show that the difference between the post-QQ particle density and its critical value develops an out-of-equilibrium scaling behavior, in terms of the dynamic scaling variable $θ\sim t/ξ^z$ (where $t>0$ is the post-QQ time, $ξ$ is the length scale of the initial state, and $z$ is the dynamic critical exponent) associated with the post-QQ time evolution. The scaling function turns out to have a peculiar singular behavior in the $θ\to 0$ limit, apparently related to the anomalous equilibrium scaling behavior of the particle density at the starting point of the QQ protocol. This provides analytical evidence of earlier conjectures on the general emergence of post-QQ dynamic scaling behaviors of the subtracted particle density (supported by numerical finite-size scaling analyses), unlike their equilibrium counterpart which turns out to be dominated by nonuniversal contributions.We study the out-of-equilibrium scaling behavior of the particle density in quantum fermionic Kitaev wires, after instantaneous quantum quenches (QQs) of the chemical potential within their quantum critical region. The critical scaling of the ground-state particle density is known to be subleading at its Ising-like quantum transition, hidden by regular and logarithmic terms arising from peculiar mixings with the identity operator. This situation changes along the out-of-equilibrium dynamics arising from QQs of the chemical potential to the critical point, starting from the ground state for Hamiltonian parameters within the critical region. We analytically show that the difference between the post-QQ particle density and its critical value develops an out-of-equilibrium scaling behavior, in terms of the dynamic scaling variable $θ\sim t/ξ^z$ (where $t>0$ is the post-QQ time, $ξ$ is the length scale of the initial state, and $z$ is the dynamic critical exponent) associated with the post-QQ time evolution. The scaling function turns out to have a peculiar singular behavior in the $θ\to 0$ limit, apparently related to the anomalous equilibrium scaling behavior of the particle density at the starting point of the QQ protocol. This provides analytical evidence of earlier conjectures on the general emergence of post-QQ dynamic scaling behaviors of the subtracted particle density (supported by numerical finite-size scaling analyses), unlike their equilibrium counterpart which turns out to be dominated by nonuniversal contributions.

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Out-of-equilibrium dynamics arising from slow round-trip variations of Hamiltonian parameters across quantum and classical critical points

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We address the out-of-equilibrium dynamics of many-body systems subject to time-dependent round-trip protocols across quantum and classical (thermal) phase transitions. They are realized by slowly changing one relevant parameter $w$ across its critical point ${w}_{c}=0$ , linearly in time with a large timescale ${t}_{s}$ , from ${w}_{i}<0$ to ${w}_{f}>0$ and then back to ${w}_{i}<0$ , thus entailing multiple passages through the critical point. Analogously to the one-way Kibble-Zurek protocols across a critical point, round-trip protocols develop dynamic scaling behaviors at both classical and quantum transitions put forward within renormalization-group frameworks. The scaling scenario is analyzed within some paradigmatic models undergoing quantum and classical transitions belonging to the two-dimensional Ising universality class, such as one-dimensional quantum Ising models and fermionic wires, and two-dimensional classical Ising models (supplemented with a purely relaxational dynamics). While the dynamic scaling frameworks are similar for classical and quantum systems, substantial differences emerge due to the different nature of their dynamics, which is purely relaxational for classical systems (implying thermalization in the large-time limit at fixed model parameters), and unitary in the case of quantum systems. In particular, when the critical point separates two gapped (short-ranged) phases and the extreme value ${w}_{f}>0$ is kept fixed in the large- ${t}_{s}$ limit of the round-trip protocol, we observe hysteresislike scenarios in classical systems, while quantum systems do not apparently develop a sufficiently robust scaling limit along the return way, due to the presence of rapidly oscillating relative phases among the relevant quantum states.We address the out-of-equilibrium dynamics of many-body systems subject to time-dependent round-trip protocols across quantum and classical (thermal) phase transitions. They are realized by slowly changing one relevant parameter $w$ across its critical point ${w}_{c}=0$, linearly in time with a large timescale ${t}_{s}$, from ${w}_{i}<0$ to ${w}_{f}>0$ and then back to ${w}_{i}<0$, thus entailing multiple passages through the critical point. Analogously to the one-way Kibble-Zurek protocols across a critical point, round-trip protocols develop dynamic scaling behaviors at both classical and quantum transitions put forward within renormalization-group frameworks. The scaling scenario is analyzed within some paradigmatic models undergoing quantum and classical transitions belonging to the two-dimensional Ising universality class, such as one-dimensional quantum Ising models and fermionic wires, and two-dimensional classical Ising models (supplemented with a purely relaxational dynamics). While the dynamic scaling frameworks are similar for classical and quantum systems, substantial differences emerge due to the different nature of their dynamics, which is purely relaxational for classical systems (implying thermalization in the large-time limit at fixed model parameters), and unitary in the case of quantum systems. In particular, when the critical point separates two gapped (short-ranged) phases and the extreme value ${w}_{f}>0$ is kept fixed in the large-${t}_{s}$ limit of the round-trip protocol, we observe hysteresislike scenarios in classical systems, while quantum systems do not apparently develop a sufficiently robust scaling limit along the return way, due to the presence of rapidly oscillating relative phases among the relevant quantum states.

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Out-of-equilibrium finite-size scaling in generalized Kibble-Zurek protocols crossing quantum phase transitions in the presence of symmetry-breaking perturbations

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Equilibrium and nonequilibrium scaling phenomena in strongly correlated systems

D. Mesterházy

In this thesis we examine universal scaling properties of strongly-correlated systems near and far from equilibrium. We discuss quantum phase transitions at vanishing temperature, multicritical and dynamic critical behavior near thermal equilibrium, and scaling properties of nonequilibrium steady states. We employ nonperturbative methods including the functional renormalization group as well as Monte Carlo simulations. A general outline of the functional renormalization group is given in the introductory chapters. \n \nIn the first part of this thesis, we investigate spinless fermions on the honeycomb lattice interacting via short-range repulsive interactions. Such a system can be seen as a simple model for suspended graphene. The short-range interactions control the ground state properties of the system that may lead to a chiral phase transition from the semimetal to the charge density wave (CDW)/Kekulé ordered state. We determine the universal scaling properties at the chiral transition, and establish the presence of large anomalous dimensions indicating the importance of strong fluctuations. \n \nThe competition of two nonvanishing order parameters and their corresponding multicritical behavior are investigated in the subsequent chapter. We characterize the bicritical and tetracritical behavior in the purely bosonic O(N_1) + O(N_2) symmetric model and comment on possible applications to condensed-matter and high-energy physics. \n \nIn the following chapter we discuss the long-time relaxational behavior at criticality of an order parameter with O(N) symmetry coupled to an additional conserved density. We find an anomalous diffusion phase with new dynamic scaling properties. Using the functional renormalization group we determine the complete dynamic critical behavior of the model in 2 < d < 4 dimensions and compare our results to experiments. \n \nFinally, we investigate the scaling properties of stationary states far from equilibrium. At the example of the one-dimensional Burgers’ equation we develop a novel approach to hydrodynamic turbulence using lattice Monte Carlo methods. We apply these techniques to determine the statistical properties of small-scale fluctuations in this model and identify the anomalous scaling behavior. \nIn this thesis we examine universal scaling properties of strongly-correlated systems near and far from equilibrium. We discuss quantum phase transitions at vanishing temperature, multicritical and dynamic critical behavior near thermal equilibrium, and scaling properties of nonequilibrium steady states. We employ nonperturbative methods including the functional renormalization group as well as Monte Carlo simulations. A general outline of the functional renormalization group is given in the introductory chapters. \n \nIn the first part of this thesis, we investigate spinless fermions on the honeycomb lattice interacting via short-range repulsive interactions. Such a system can be seen as a simple model for suspended graphene. The short-range interactions control the ground state properties of the system that may lead to a chiral phase transition from the semimetal to the charge density wave (CDW)/Kekulé ordered state. We determine the universal scaling properties at the chiral transition, and establish the presence of large anomalous dimensions indicating the importance of strong fluctuations. \n \nThe competition of two nonvanishing order parameters and their corresponding multicritical behavior are investigated in the subsequent chapter. We characterize the bicritical and tetracritical behavior in the purely bosonic O(N_1) + O(N_2) symmetric model and comment on possible applications to condensed-matter and high-energy physics. \n \nIn the following chapter we discuss the long-time relaxational behavior at criticality of an order parameter with O(N) symmetry coupled to an additional conserved density. We find an anomalous diffusion phase with new dynamic scaling properties. Using the functional renormalization group we determine the complete dynamic critical behavior of the model in 2 < d < 4 dimensions and compare our results to experiments. \n \nFinally, we investigate the scaling properties of stationary states far from equilibrium. At the example of the one-dimensional Burgers’ equation we develop a novel approach to hydrodynamic turbulence using lattice Monte Carlo methods. We apply these techniques to determine the statistical properties of small-scale fluctuations in this model and identify the anomalous scaling behavior. \n

Jan 1, 20130 citations

We address the equilibrium and out-of-equilibrium behavior of the particle density in many-body systems undergoing quantum transitions driven by the chemical potential

\ensuremath{\mu}

O({10}^{4})

\ensuremath{\mu}

, to the critical value

{\ensuremath{\mu}}_{c}

, unveiling out-of-equilibrium scaling behaviors of the particle density, which arise from the critical modes, within a dynamic finite-size scaling framework.

θ\sim t/ξ^z

(where

t>0

is the post-QQ time,

ξ

is the length scale of the initial state, and

z

is the dynamic critical exponent) associated with the post-QQ time evolution. The scaling function turns out to have a peculiar singular behavior in the

θ\to 0

limit, apparently related to the anomalous equilibrium scaling behavior of the particle density at the starting point of the QQ protocol. This provides analytical evidence of earlier conjectures on the general emergence of post-QQ dynamic scaling behaviors of the subtracted particle density (supported by numerical finite-size scaling analyses), unlike their equilibrium counterpart which turns out to be dominated by nonuniversal contributions.

w

across its critical point

{w}_{c}=0

, linearly in time with a large timescale

{t}_{s}

, from

{w}_{i}<0

{w}_{f}>0

and then back to

{w}_{i}<0

, thus entailing multiple passages through the critical point. Analogously to the one-way Kibble-Zurek protocols across a critical point, round-trip protocols develop dynamic scaling behaviors at both classical and quantum transitions put forward within renormalization-group frameworks. The scaling scenario is analyzed within some paradigmatic models undergoing quantum and classical transitions belonging to the two-dimensional Ising universality class, such as one-dimensional quantum Ising models and fermionic wires, and two-dimensional classical Ising models (supplemented with a purely relaxational dynamics). While the dynamic scaling frameworks are similar for classical and quantum systems, substantial differences emerge due to the different nature of their dynamics, which is purely relaxational for classical systems (implying thermalization in the large-time limit at fixed model parameters), and unitary in the case of quantum systems. In particular, when the critical point separates two gapped (short-ranged) phases and the extreme value

{w}_{f}>0

is kept fixed in the large-

{t}_{s}

limit of the round-trip protocol, we observe hysteresislike scenarios in classical systems, while quantum systems do not apparently develop a sufficiently robust scaling limit along the return way, due to the presence of rapidly oscillating relative phases among the relevant quantum states.

{t}_{s}

of the parameter variations in the KZ protocol, the size

L

of the system, and the strength

h

of the symmetry-breaking perturbation, in the limit of large

{t}_{s}

and

L

. Moreover, scaling arguments based on the first-order adiabatic approximation of slow variations in quantum systems allow us to characterize the approach to the adiabatic regimes for some limits of the model parameters (for example, when we take ${t}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ before $L\ensuremath{\rightarrow}\ensuremath{\infty}$), predicting asymptotic power-law suppressions of the nonadiabatic behaviors in the adiabatic limits. This out-of-equilibrium FSS is supported by numerical analyses for the paradigmatic quantum Ising chain along generalized KZ protocols, with a time-dependent transverse field crossing its CQT, in the presence of a static longitudinal field breaking the

{\mathbb{Z}}_{2}

symmetry.

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.