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arXiv:1710.06710 · Nuclear Physics B

Macroscopic length correlations in non-equilibrium systems and their possible realizations

Zohar Nussinov

Published Feb 6, 2020• 13 citations
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Abstract

We consider general systems that start from and/or end in thermodynamic equilibrium while experiencing a finite rate of change of their energy density or other intensive quantities q at intermediate times. We demonstrate that at these times, during which q varies at a finite rate, the associated covariance, the connected pair correlator Gij=〈qiqj〉−〈qi〉〈qj〉, between any two (far separated) sites i and j in a macroscopic system may, on average, become finite. Once the global mean q no longer changes, the average of Gij over all site pairs i and j may tend to zero. However, when the equilibration times are significant (e.g., as in a glass that is not in true thermodynamic equilibrium yet in which the energy density (or temperature) reaches a final steady state value), these long range correlations may persist also long after q ceases to change. We explore viable experimental implications of our findings and speculate on their potential realization in glasses (where a prediction of a theory based on the effect that we describe here suggests a universal collapse of the viscosity that agrees with all published viscosity measurements over sixteen decades) and non-Fermi liquids. We discuss effective equilibrium in driven systems and derive uncertainty relation based inequalities that connect the heat capacity to the dynamics in general open thermal systems. These rigorous thermalization inequalities suggest the shortest possible fluctuation times scales in open equilibrated systems at a temperature T are typically “Planckian” (i.e., O(ħ/(kBT))). We briefly comment on parallels between quantum measurements, unitary quantum evolution, and thermalization and on how Gaussian distributions may generically emerge.

DOI
10.1016/j.nuclphysb.2020.114948

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Infinite Range Correlations in Non-Equilibrium Quantum Systems and Their Possible Experimental Realizations

Zohar Nussinov

We consider systems that start from and/or end in thermodynamic equilibrium while experiencing a finite rate of change of their energy density or of other intensive quantities $q$ at intermediate times. We demonstrate that at these times, during which the global intensive quantities $q$ vary, the size of the associated covariance, the connected pair correlator $|G_{ij}| = |\langle q_{i} q_{j} \rangle - \langle q_{i} \rangle \langle q_{j} \rangle|$, between any two (arbitrarily far separated) sites $i$ and $j$ is, on average, finite. This non-vanishing character of the connected correlations for asymptotically distant sites also applies to theories with purely local interactions. In simple models, these correlations may be traced to the generic volume law entanglement of finite temperature states. Once the global mean of $q$ no longer changes, the average of $|G_{ij}|$ over all spatial separations $|i-j|$ may tend to zero. However, when the equilibration times are significant (e.g., as in a glass that is not in true thermodynamic equilibrium yet in which the energy density (or temperature) reaches a final steady state value), these long range correlations may persist also long after $q$ ceases to change. We briefly discuss possible experimental implications of our findings and speculate on their potential realization in glasses (where a prediction of a theory based on the effect that we describe here suggests a universal collapse of the viscosity that agrees with all published viscosity measurements) and non-Fermi liquids.

•Oct 18, 2017•1 citation

Dynamic correlations, fluctuation-dissipation relations, and effective temperatures after a quantum quench of the transverse field Ising chain

Laura Foini, Leticia F. Cugliandolo, Andrea Gambassi

Abstract Fluctuation-dissipation relations, i.e., the relation between two-time correlation and linear response functions, were successfully used to search for signs of equilibration and to identify effective temperatures in the non-equilibrium behavior of a number of macroscopic classical and quantum systems in contact with thermal baths. Among the most relevant cases in which the effective temperatures thus defined were shown to have a thermodynamic meaning one finds the stationary dynamics of driven super-cooled liquids and vortex glasses, and the relaxation of glasses. Whether and under which conditions an effective thermal behavior can be found in quantum isolated many-body systems after a global quench is a question of current interest. We propose to study the possible emergence of thermal behavior long after the quench by studying fluctuation-dissipation relations in which (possibly time- or frequency-dependent) parameters replace the equilibrium temperature. If thermalization within the Gibbs ensemble eventually occurs these parameters should be constant and equal for all pairs of observables in ‘partial’ or ‘mutual’ equilibrium. We analyze these relations in the paradigmatic quantum system, i.e., the quantum Ising chain, in the stationary regime after a quench of the transverse field. The lack of thermalization to a Gibbs ensemble becomes apparent within this approach.

arXiv:1207.1650•Journal of Statistical Mechanics Theory and Experiment•Sep 1, 2012•75 citations

Thermalization and Out-of-Equilibrium Dynamics in Open Quantum Many-Body Systems

Michael Buchhold

Thermalization, the evolution of an interacting many-body system towards a thermal Gibbs ensemble after initialization in an arbitrary non-equilibrium state, is currently a phenomenon of great interest, both in theory and experiment. As the time evolution of a quantum system is unitary, the proposed mechanism of thermalization in quantum many-body systems corresponds to the so-called eigenstate thermalization hypothesis (ETH) and the typicality of eigenstates. Although this formally solves the contradiction of thermalizing but unitary dynamics in a closed quantum many-body system, it does neither make any statement on the dynamical process of thermalization itself nor in which way the coupling of the system to an environment can hinder or modify the relaxation dynamics. 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From a theoretical point of view, one-dimensional quantum fluids are particular interesting, as Luttinger Liquids are integrable and therefore, due to an infinite number of constants of motion, do not thermalize. The leading order correction to the quadratic theory is irrelevant in the sense of the renormalization group and does therefore not modify static correlation functions, however, it breaks integrability and will therefore, even if irrelevant, induce a completely different non-equilibrium dynamics as the quadratic Luttinger theory alone. In this thesis, we derive for the first time a kinetic equation for interacting Luttinger Liquids, which describes the time evolution of the excitation densities for arbitrary initial states. The resonant character of the interaction makes a straightforward derivation of the kinetic equation, using Fermi\'s golden rule, impossible and we have to develop non-perturbative techniques in the Keldysh framework. We derive a closed expression for the time evolution of the excitation densities in terms of self-energies and vertex corrections. Close to equilibrium, the kinetic equation describes the exponential decay of excitations, with a decay rate $\\sigma^R=\\mbox{Im}\\Sigma^R$, determined by the self-energy at equilibrium. However, for long times $\\tau$, it also reveals the presence of dynamical slow modes, which are the consequence of exactly energy conserving dynamics and lead to an algebraic decay $\\sim\\tau^{-\\eta\\sub{D}}$ with $\\eta_D=0.58$. The presence of these dynamical slow modes is not contained in the equilibrium Matsubara formalism, while they emerge naturally in the non-equilibrium formalism developed in this thesis. In order to initialize a one-dimensional quantum fluid out of equilibrium, we consider an interaction quench in a model of interacting, dispersive fermions in Chap.~\\ref{sec:Quench}. 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•Jun 29, 2015•1 citation

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Zohar Nussinov, Saurish Chakrabarty

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In this Thesis we discussed several issues related to the dynamics of isolated quantum many-body systems. The brief overview of Chapter 1 highlighted a series of open fundamental theoretical questions which are crucial for the investigation of nonequilibrium phenomena. Among them, we mainly focused on the emergence of thermal behaviors and on the characterization of dynamical phase transitions and of the associated universal dynamics. To this end, we analyzed some paradigmatic models, such as the Fermi-Hubbard model, the quantum Ising chain, and the O(N) model. -Chapter 2: Absence of thermalization in a Fermi liquid. In this Chapter we report two results that question the common expectation of thermalization in a Landau-Fermi liquid after an interaction quench. We first consider the perturbative expansion in the interaction strength of the long-wavelength structure factor S(q) and show that it does not satisfy the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) has an analytical component ∼ const. + O(q^2) for q → 0, compatible with thermalization, but it retains also a nonanalytic term ∼ |q| characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity turns in a power-law decay of the density-density correlations, in contrast with the exponential decay associated to thermalization. We next analyze the low-density case, where one can obtain results valid at any order in interaction but at leading in the density, and find that in the steady state the momentum distribution jump at the Fermi surface is strictly finite, though smaller than at equilibrium. -Chapter 3: Nonadiabatic stationary behavior in a driven Ising chain. In this Chapter we discuss the emergence of a nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional system subject to a linear ramp of one of its parameters within a gapped phase, which should be the most favorable situation for an adiabatic evolution. We study in details the dynamics of the spontaneous magnetization m_x(t) in a quantum Ising chain after a linear variation in time of the transverse field within the ordered phase. In particular, focusing on the value of the order parameter at the end of the ramp m_x(τ), we show that the smaller the switching rate of the transverse field is the closer m_x(τ) gets to its ground state value. Nonetheless, this apparently small correction to adiabaticity eventually leads to the disruption of the order exponentially fast in the subsequent time evolution, no matter how slowly the ramp is performed. -Chapter 4: Aging and coarsening in isolated quantum systems after a quench. This Chapter analyzes the nonequilibrium dynamics of an isolated quantum system after a sudden quench to the dynamical critical point, where the emergence of scaling and universal exponents is expected, due to the absence of time and length scales. We explore these features for a bosonic interacting scalar field theory with O(N) symmetry in the large-N limit, where the model is exactly solvable and the exponents and scaling forms of the relevant two-time correlation functions can be analytically calculated. Moreover, we provide evidence of the emergence of a dynamical scaling behavior also for quenches below the dynamical critical point, associated with coarsening. We find that the latter case is characterized by the same scaling functions as those describing the critical case, yet with different exponents. -Chapter 5: Dynamical transitions and statistics of excitations. In this Chapter we study the dynamics of the O(N) model (introduced in the previous Chapter) resulting from a different protocol: a linear ramp of one of its parameters. We find that the presence of a dynamical phase transition, as well as its critical properties, are robust against the change of the protocol. We show that a characterization based on the critical dimensions and exponents would suggest that the dynamical phase transition is analogous to the equilibrium thermal one. However, its nonequilibrium nature becomes evident analyzing the statistics of excitations produced in the ramp process. In particular, the critical properties are encoded in the fluctuations in the number of excitations, which display qualitatively different behaviors depending on the ramp being performed above, at, or below the dynamical critical point. These behaviors bear no dependence on the duration of the protocol.

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We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.

•Jan 1, 2013•0 citations
We consider general systems that start from and/or end in thermodynamic equilibrium while experiencing a finite rate of change of their energy density or other intensive quantities q at intermediate times. We demonstrate that at these times, during which q varies at a finite rate, the associated covariance, the connected pair correlator Gij=〈qiqj〉−〈qi〉〈qj〉, between any two (far separated) sites i and j in a macroscopic system may, on average, become finite. Once the global mean q no longer changes, the average of Gij over all site pairs i and j may tend to zero. However, when the equilibration times are significant (e.g., as in a glass that is not in true thermodynamic equilibrium yet in which the energy density (or temperature) reaches a final steady state value), these long range correlations may persist also long after q ceases to change. We explore viable experimental implications of our findings and speculate on their potential realization in glasses (where a prediction of a theory based on the effect that we describe here suggests a universal collapse of the viscosity that agrees with all published viscosity measurements over sixteen decades) and non-Fermi liquids. We discuss effective equilibrium in driven systems and derive uncertainty relation based inequalities that connect the heat capacity to the dynamics in general open thermal systems. These rigorous thermalization inequalities suggest the shortest possible fluctuation times scales in open equilibrated systems at a temperature T are typically “Planckian” (i.e., O(ħ/(kBT))). We briefly comment on parallels between quantum measurements, unitary quantum evolution, and thermalization and on how Gaussian distributions may generically emerge.
We consider systems that start from and/or end in thermodynamic equilibrium while experiencing a finite rate of change of their energy density or of other intensive quantities qqq at intermediate times. We demonstrate that at these times, during which the global intensive quantities qqq vary, the size of the associated covariance, the connected pair correlator ∣Gij∣=∣⟨qiqj⟩−⟨qi⟩⟨qj⟩∣|G_{ij}| = |\langle q_{i} q_{j} \rangle - \langle q_{i} \rangle \langle q_{j} \rangle|∣Gij​∣=∣⟨qi​qj​⟩−⟨qi​⟩⟨qj​⟩∣, between any two (arbitrarily far separated) sites iii and jjj is, on average, finite. This non-vanishing character of the connected correlations for asymptotically distant sites also applies to theories with purely local interactions. In simple models, these correlations may be traced to the generic volume law entanglement of finite temperature states. Once the global mean of qqq no longer changes, the average of ∣Gij∣|G_{ij}|∣Gij​∣ over all spatial separations ∣i−j∣|i-j|∣i−j∣ may tend to zero. However, when the equilibration times are significant (e.g., as in a glass that is not in true thermodynamic equilibrium yet in which the energy density (or temperature) reaches a final steady state value), these long range correlations may persist also long after qqq ceases to change. We briefly discuss possible experimental implications of our findings and speculate on their potential realization in glasses (where a prediction of a theory based on the effect that we describe here suggests a universal collapse of the viscosity that agrees with all published viscosity measurements) and non-Fermi liquids.
Abstract Fluctuation-dissipation relations, i.e., the relation between two-time correlation and linear response functions, were successfully used to search for signs of equilibration and to identify effective temperatures in the non-equilibrium behavior of a number of macroscopic classical and quantum systems in contact with thermal baths. Among the most relevant cases in which the effective temperatures thus defined were shown to have a thermodynamic meaning one finds the stationary dynamics of driven super-cooled liquids and vortex glasses, and the relaxation of glasses. Whether and under which conditions an effective thermal behavior can be found in quantum isolated many-body systems after a global quench is a question of current interest. We propose to study the possible emergence of thermal behavior long after the quench by studying fluctuation-dissipation relations in which (possibly time- or frequency-dependent) parameters replace the equilibrium temperature. If thermalization within the Gibbs ensemble eventually occurs these parameters should be constant and equal for all pairs of observables in ‘partial’ or ‘mutual’ equilibrium. We analyze these relations in the paradigmatic quantum system, i.e., the quantum Ising chain, in the stationary regime after a quench of the transverse field. The lack of thermalization to a Gibbs ensemble becomes apparent within this approach.
Thermalization, the evolution of an interacting many-body system towards a thermal Gibbs ensemble after initialization in an arbitrary non-equilibrium state, is currently a phenomenon of great interest, both in theory and experiment. As the time evolution of a quantum system is unitary, the proposed mechanism of thermalization in quantum many-body systems corresponds to the so-called eigenstate thermalization hypothesis (ETH) and the typicality of eigenstates. Although this formally solves the contradiction of thermalizing but unitary dynamics in a closed quantum many-body system, it does neither make any statement on the dynamical process of thermalization itself nor in which way the coupling of the system to an environment can hinder or modify the relaxation dynamics. In this thesis, we address both the question whether or not a quantum system driven away from equilibrium is able to relax to a thermal state, which fulfills detailed balance, and if one can identify universal behavior in the non-equilibrium relaxation dynamics. As a first realization of driven quantum systems out of equilibrium, we investigate a system of Ising spins, interacting with the quantized radiation field in an optical cavity. For multiple cavity modes, this system forms a highly entangled and frustrated state with infinite correlation times, known as a quantum spin glass. In the presence of drive and dissipation, introduced by coupling the intra-cavity radiation field to the photon vacuum outside the cavity via lossy mirrors, the quantum glass state is modified in a universal manner. For frequencies below the photon loss rate, the dissipation takes over and the system shows the universal behavior of a dissipative spin glass, with a characteristic spectral density mathcalA(omega)simsqrtomega\\mathcal{A}(\\omega)\\sim\\sqrt{\\omega}mathcalA(omega)simsqrtomega. On the other hand, for frequencies above the loss rate, the system retains the universal behavior of a zero temperature, quantum spin glass. Remarkably, at the glass transition, the two subsystems of spins and photons thermalize to a joint effective temperature, even in the presence of photon loss. This thermalization is a consequence of the strong spin-photon interactions, which favor detailed balance in the system and detain photons from escaping the cavity. In the thermalized system, the features of the spin glass are mirrored onto the photon degrees of freedom, leading to an emergent photon glass phase. Exploiting the inherent photon loss of the cavity, we make predictions of possible measurements on the escaping photons, which contain detailed information of the state inside the cavity and allow for a precise, non-destructive measurement of the glass state. As a further set of non-equilibrium systems, we consider one-dimensional quantum fluids driven out of equilibrium, whose universal low energy theory is formed by the so-called Luttinger Liquid description, which, due to its large degree of universality, is of intense theoretical and experimental interest. A set of recent experiments in research groups in Vienna, Innsbruck and Munich have probed the non-equilibrium time-evolution of one-dimensional quantum fluids for different experimental realizations and are pushing into a time regime, where thermalization is expected. From a theoretical point of view, one-dimensional quantum fluids are particular interesting, as Luttinger Liquids are integrable and therefore, due to an infinite number of constants of motion, do not thermalize. The leading order correction to the quadratic theory is irrelevant in the sense of the renormalization group and does therefore not modify static correlation functions, however, it breaks integrability and will therefore, even if irrelevant, induce a completely different non-equilibrium dynamics as the quadratic Luttinger theory alone. In this thesis, we derive for the first time a kinetic equation for interacting Luttinger Liquids, which describes the time evolution of the excitation densities for arbitrary initial states. The resonant character of the interaction makes a straightforward derivation of the kinetic equation, using Fermi\'s golden rule, impossible and we have to develop non-perturbative techniques in the Keldysh framework. We derive a closed expression for the time evolution of the excitation densities in terms of self-energies and vertex corrections. Close to equilibrium, the kinetic equation describes the exponential decay of excitations, with a decay rate sigmaR=mboxImSigmaR\\sigma^R=\\mbox{Im}\\Sigma^RsigmaR=mboxImSigmaR, determined by the self-energy at equilibrium. However, for long times tau\\tautau, it also reveals the presence of dynamical slow modes, which are the consequence of exactly energy conserving dynamics and lead to an algebraic decay simtau−etasubD\\sim\\tau^{-\\eta\\sub{D}}simtau−etasubD with etaD=0.58\\eta_D=0.58etaD​=0.58. The presence of these dynamical slow modes is not contained in the equilibrium Matsubara formalism, while they emerge naturally in the non-equilibrium formalism developed in this thesis. In order to initialize a one-dimensional quantum fluid out of equilibrium, we consider an interaction quench in a model of interacting, dispersive fermions in Chap.~\\ref{sec:Quench}. In this scenario, the fermionic interaction is suddenly changed at time t=0t=0t=0, such that for t>0t>0t>0 the system is not in an eigenstate and therefore undergoes a non-trivial time evolution. For the quadratic theory, the stationary state in the limit trightarrowinftyt\\rightarrow\\inftytrightarrowinfty is a non-thermal, or prethermal, state, described by a generalized Gibbs ensemble (GGE). The GGE takes into account for the conservation of all integrals of motion, formed by the eigenmodes of the Hamiltonian. On the other hand, in the presence of non-linearities, the final state for trightarrowinftyt\\rightarrow\\inftytrightarrowinfty is a thermal state with a finite temperature T>0T>0T>0. . The spatio-temporal, dynamical thermalization process can be decomposed into three regimes: A prequench regime on the largest distances, which is determined by the initial state, a prethermal plateau for intermediate distances, which is determined by the metastable fixed point of the quadratic theory and a thermal region on the shortest distances. The latter spreads sub-ballistically simtalpha\\sim t^{\\alpha}simtalpha in space with 0<alpha<10<\\alpha<10<alpha<1 depending on the quench. Until complete thermalization (i.e. for times t<inftyt<\\inftyt<infty), the thermal region contains more energy than the prethermal and prequench region, which is expressed in a larger temperature Tt>TinftyT_{t}>T_{\\infty}Tt​>Tinfty​, decreasing towards its final value TinftyT_{\\infty}Tinfty​. As the system has achieved local detailed balance in the thermalized region, energy transport to the non-thermal region can only be performed by the macroscopic dynamical slow modes and the decay of the temperature Tt−Tinftysimt−etasubDT_{t}-T_{\\infty}\\sim t^{-\\eta\\sub{D}}Tt​−Tinfty​simt−etasubD again witnesses the presence of these slow modes. The very slow spreading of thermalization is consistent with recent experiments performed in Vienna, which observe a metastable, prethermal state after a quench and only observe the onset of thermalization on much larger time scales. As an immediate indication of thermalization, we determine the time evolution of the fermionic momentum distribution after a quench from non-interacting to interacting fermions. For this quench scenario, the step in the Fermi distribution at the Fermi momentum ksubFk\\sub{F}ksubF decays to zero algebraically in the absence of a non-linearity but as a stretched exponential (the exponent being proportional to the non-linearity) in the presence of a finite non-linearity. This can serve as a proof for the presence or absence of the non-linearity even on time-scales for which thermalization can not yet be observed. Finally, we consider a bosonic quantum fluid, which is driven away from equilibrium by permanent heating. The origin of the heating is atomic spontaneous emission of laser photons, which are used to create a coherent lattice potential in optical lattice experiments. This process preserves the system\'s U(1)U(1)U(1)-invariance, i.e. conserves the global particle number, and the corresponding long-wavelength description is a heated, interacting Luttinger Liquid, for which phonon modes are continuously populated with a momentum dependent rate partialtnqsimgamma∣q∣\\partial_tn_q\\sim\\gamma |q|partialt​nq​simgamma∣q∣. In the dynamics, we identify a quasi-thermal regime for large momenta, featuring an increasing time-dependent effective temperature. In this regime, due to fast phonon-phonon scattering, detailed balance has been achieved and is expressed by a time-local, increasing temperature. The thermal region emerges locally and spreads in space sub-ballistically according to xtsimt4/5x_t\\sim t^{4/5}xt​simt4/5. For larger distances, the system is described by an non-equilibrium phonon distribution nqsim∣q∣n_q\\sim |q|nq​sim∣q∣, which leads to a new, non-equilibrium behavior of large distance observables. For instance, the phonon decay rate scales universally as gammaqsim∣q∣5/3\\gamma_q\\sim |q|^{5/3}gammaq​sim∣q∣5/3, with a new non-equilibrium exponent eta=5/3\\eta=5/3eta=5/3, which differs from equilibrium. This new, universal behavior is guaranteed by the U(1)U(1)U(1) invariant dynamics of the system and is insensitive to further subleading perturbations. The non-equilibrium long-distance behavior can be determined experimentally by measuring the static and dynamic structure factor, both of which clearly indicate the exponents for phonon decay, eta=5/3\\eta=5/3eta=5/3 and for the spreading of thermalization etaT=4/5\\eta_T=4/5etaT​=4/5. Remarkably, even in the presence of this strong external drive, the interactions and their aim to achieve detailed balance are strong enough to establish a locally emerging and spatially spreading thermal region. The physical setups in this thesis do not only reveal interesting and new dynamical features in the out-of-equilibrium time evolution of interacting systems, but they also strongly underline the high degree of universality of thermalization for the classes of models studied here. May it be a system of coupled spins and photons, where the photons are pulled away from a thermal state by Markovian photon d
Using local uncertainty relations in thermal many body systems, we illustrate that there exist universal speed limit (regardless of interaction range), bounds on acceleration or force/stress, acceleration or material stress rates, diffusion constant, viscosity, other transport coefficients, electromagnetic or other gauge field strengths, correlation functions of arbitrary spatio-temporal derivatives, Lyapunov exponents, thermalization times, and transition times between orthogonal states in non-relativistic quantum thermal systems, and derive analogs of the Ioffe-Regel limit. In the ℏ→0\hbar \to 0ℏ→0 limit, all of our bounds either diverge (e.g., our speed and acceleration limit) or vanish (as in, e.g., our viscosity and diffusion constant bounds). Our inequalities hold at all temperatures and, as corollaries, imply general power law bounds on response functions at both asymptotically high and low temperatures. Our results shed light on measurements and on how apparent nearly instantaneous effective collapse to an eigenstate may arise in macroscopic interacting many body quantum systems.
In this Thesis we discussed several issues related to the dynamics of isolated quantum many-body systems. The brief overview of Chapter 1 highlighted a series of open fundamental theoretical questions which are crucial for the investigation of nonequilibrium phenomena. Among them, we mainly focused on the emergence of thermal behaviors and on the characterization of dynamical phase transitions and of the associated universal dynamics. To this end, we analyzed some paradigmatic models, such as the Fermi-Hubbard model, the quantum Ising chain, and the O(N) model. -Chapter 2: Absence of thermalization in a Fermi liquid. In this Chapter we report two results that question the common expectation of thermalization in a Landau-Fermi liquid after an interaction quench. We first consider the perturbative expansion in the interaction strength of the long-wavelength structure factor S(q) and show that it does not satisfy the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) has an analytical component ∼ const. + O(q^2) for q → 0, compatible with thermalization, but it retains also a nonanalytic term ∼ |q| characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity turns in a power-law decay of the density-density correlations, in contrast with the exponential decay associated to thermalization. We next analyze the low-density case, where one can obtain results valid at any order in interaction but at leading in the density, and find that in the steady state the momentum distribution jump at the Fermi surface is strictly finite, though smaller than at equilibrium. -Chapter 3: Nonadiabatic stationary behavior in a driven Ising chain. In this Chapter we discuss the emergence of a nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional system subject to a linear ramp of one of its parameters within a gapped phase, which should be the most favorable situation for an adiabatic evolution. We study in details the dynamics of the spontaneous magnetization m_x(t) in a quantum Ising chain after a linear variation in time of the transverse field within the ordered phase. In particular, focusing on the value of the order parameter at the end of the ramp m_x(τ), we show that the smaller the switching rate of the transverse field is the closer m_x(τ) gets to its ground state value. Nonetheless, this apparently small correction to adiabaticity eventually leads to the disruption of the order exponentially fast in the subsequent time evolution, no matter how slowly the ramp is performed. -Chapter 4: Aging and coarsening in isolated quantum systems after a quench. This Chapter analyzes the nonequilibrium dynamics of an isolated quantum system after a sudden quench to the dynamical critical point, where the emergence of scaling and universal exponents is expected, due to the absence of time and length scales. We explore these features for a bosonic interacting scalar field theory with O(N) symmetry in the large-N limit, where the model is exactly solvable and the exponents and scaling forms of the relevant two-time correlation functions can be analytically calculated. Moreover, we provide evidence of the emergence of a dynamical scaling behavior also for quenches below the dynamical critical point, associated with coarsening. We find that the latter case is characterized by the same scaling functions as those describing the critical case, yet with different exponents. -Chapter 5: Dynamical transitions and statistics of excitations. In this Chapter we study the dynamics of the O(N) model (introduced in the previous Chapter) resulting from a different protocol: a linear ramp of one of its parameters. We find that the presence of a dynamical phase transition, as well as its critical properties, are robust against the change of the protocol. We show that a characterization based on the critical dimensions and exponents would suggest that the dynamical phase transition is analogous to the equilibrium thermal one. However, its nonequilibrium nature becomes evident analyzing the statistics of excitations produced in the ramp process. In particular, the critical properties are encoded in the fluctuations in the number of excitations, which display qualitatively different behaviors depending on the ramp being performed above, at, or below the dynamical critical point. These behaviors bear no dependence on the duration of the protocol.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.