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arXiv:1108.5068 · Physical Review B

Quantum quenches, dynamical transitions, and off-equilibrium quantum criticality

Bruno Sciolla, Giulio Biroli

Published Nov 25, 2013• 135 citations
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Abstract

Several mean-field computations have revealed the existence of an out-of-equilibrium dynamical transition induced by quantum-quenching an isolated system starting from its symmetry-broken phase. In this work we focus on the quantum ${\ensuremath{\phi}}^{4}$ $N$-component field theory. By taking into account dynamical fluctuations at the Hartree-Fock level, corresponding to the leading order of the $1/N$ expansion, we derive the critical properties of the dynamical transition beyond mean-field theory (including at finite temperature). We find diverging time and length scales, dynamic scaling, and aging. Finally, we unveil a relationship with critical coarsening, an off-equilibrium regime that can be induced by quenching from the symmetric toward the symmetry-broken phase.

DOI
10.1103/physrevb.88.201110

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Several mean-field computations have revealed the existence of an out-of-equilibrium dynamical transition induced by quantum-quenching an isolated system starting from its symmetry-broken phase. In this work we focus on the quantum \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 NNN-component field theory. By taking into account dynamical fluctuations at the Hartree-Fock level, corresponding to the leading order of the 1/N1/N1/N expansion, we derive the critical properties of the dynamical transition beyond mean-field theory (including at finite temperature). We find diverging time and length scales, dynamic scaling, and aging. Finally, we unveil a relationship with critical coarsening, an off-equilibrium regime that can be induced by quenching from the symmetric toward the symmetry-broken phase.
The nonequilibrium dynamics of a system that is located in the vicinity of a quantum critical point is affected by the critical slowing down of order-parameter correlations with the potential for novel out-of-equilibrium universality. After a quantum quench, i.e., a sudden change of a parameter in the Hamiltonian, such a system is expected to almost instantly fall out of equilibrium and undergo aging dynamics, i.e., dynamics that depends on the time passed since the quench. Investigating the quantum dynamics of an NNN-component ${\ensuremath{\varphi}}^{4}$ model coupled to an external bath, we determine this universal aging and demonstrate that the system undergoes a coarsening, governed by a critical exponent that is unrelated to the equilibrium exponents of the system. We analyze this behavior in the large-NNN limit, which is complementary to our earlier renormalization-group analysis, allowing in particular the direct investigation of the order-parameter dynamics in the symmetry-broken phase and at the upper critical dimension. By connecting the long-time limit of fluctuations and response, we introduce a distribution function that shows that the system remains nonthermal and exhibits quantum coherence even on long time scales.
The nonequilibrium dynamics of an isolated quantum system after a sudden quench to a dynamical critical point is expected to be characterized by scaling and universal exponents due to the absence of time scales. We explore these features for a quench of the parameters of a Hamiltonian with O(N) symmetry, starting from a ground state in the disordered phase. In the limit of infinite N, the exponents and scaling forms of the relevant two-time correlation functions can be calculated exactly. Our analytical predictions are confirmed by the numerical solution of the corresponding equations. Moreover, we find that the same scaling functions, yet with different exponents, also describe the coarsening dynamics for quenches below the dynamical critical point.
A quenched second order phase transition is modeled by an effective ${\ensuremath{\Phi}}^{4}$ theory with a time-dependent Hamiltonian $\ifmmode \hat{H}\else \^{H}\fi{}(t),$ whose symmetry is broken spontaneously in time. The quantum field evolves out of equilibrium (nonequilibrium) during the phase transition as the density operator significantly deviates from $\stackrel{^}{\ensuremath{\rho}}{(t)=e}^{\ensuremath{-}\ensuremath{\beta}\ifmmode \hat{H}\else \^{H}\fi{}(t)}{/Z}_{H}.$ The recently developed Liouville--von Neumann method provides various quantum states for the phase transition in terms of a complex solution to the mean-field equation, which is equivalent to the Gaussian effective potential in the static case and to the time-dependent Hartree-Fock equation in the nonequilibrium case. Using multiple-scale perturbation theory we solve the mean-field equation analytically to the first order of the coupling constant and find the quantum states during the quenched second order phase transition. We propose a renormalization scheme during the process of phase transition to regularize the divergences, which originate from the mode coupling between hard and hard modes or between soft and hard modes. The effect of mode coupling is discussed.
This work is devoted to the study of relaxation--dissipation processes in systems described by Quantum Field Theory. In the first part, I focus on the phi^4 scalar quantum field theory in finite volume in the large N limit. I find that the zero-mode quantum fluctuations cannot grow macroscopically large starting from microscopic initial conditions, while the long-wavelength fluctuations do scale with the linear size of the system, signalling dynamical infrared properties quite different from the equilibrium ones. I also introduce a non-gaussian Hartree-Fock approximation (tdHF) and I derive the mean-field coupled time-dependent Schroedinger equations for the modes of the scalar field. The dynamical evolution of a uniform condensate is studied in a further controlled gaussian approximation, for N=1. Results similar to the large N case show an internal inconsistency of this approximation. I also study the dynamics of the system in infinite volume in the broken symmetry phase. The data show that the fixed points of the evolution cover at most the classically metastable part of the static effective potential. I finally study the dynamical evolution for the O(N+1) nonlinear sigma model in 1+1 dimensions in the large N limit. I first of all verify that the large coupling limit of the O(N+1) phi^4 model, which renders the model non linear, commutes with the large N limit, so that the O(infty) nonlinear sigma model is uniquely defined. I then study the evolution of several observables, like the spectrum of produced particles, finding no evidence for parametric resonance. I conclude with some remarks on the ``state of art'' in gauge theories and some comments about the open issues in the subject.
The λϕ4λϕ^4λϕ4 model in a finite volume is studied in the infinite NNN limit and within a non-gaussian Hartree-Fock approximation both at equilibrium and out of equilibrium, with particular attention to certain fundamental features of the broken symmetry phase. The numerical solution of the dynamical evolution equations show that the zero-mode quantum fluctuations cannot grow macroscopically large starting from microscopic initial conditions. Thus we conclude that there is no evidence for a dynamical Bose-Einstein condensation. On the other hand, out of equilibrium the long-wavelength fluctuations do scale with the linear size of the system, signalling dynamical infrared properties quite different from the equilibrium ones characteristic of the same approximation schemes. This result suggests the cause, and the possible remedy, of some unlikely features of the application to out--of--equilibrium dynamics of the standard HF factorization scheme, which coincides with the gaussian restriction of our Hartree--Fock approximation.
We present a simple theory accounting for two central observations in a recent experiment on quantum coarsening and collective dynamics on a programmable quantum simulator [Manovitz et al., Nature (London) 638, 86 (2025).NATUAS0028-083610.1038/s41586-024-08353-5]: an apparent speeding up of the coarsening process as the phase transition is approached and persistent oscillations of the order parameter after quenches within the ordered phase. Our theory, based on the Hamiltonian structure of the equations of motion in the classical limit of the quantum model, finds a speeding up already deep within the ordered phase, with subsequent slowing down as the domain wall tension vanishes upon approaching the critical line. Further, the oscillations are captured within a mean-field treatment of the order parameter field. For quenches within the ordered phase, small spatially varying fluctuations in the initial mean field lead to a remarkable long-time effect, wherein the system dynamically destroys its long-range order and has to coarsen to re-establish it. We term this phenomenon symmetry rebreaking, as the resulting late-time magnetization can have a sign opposite to the initial magnetization.