Quantum quenches in isolated quantum glasses out of equilibrium

TL;DR

This paper investigates how a quantum glass system thermalizes after a quench, revealing that stronger interactions or quantum fluctuations can stabilize glassy states and lower the effective temperature.

Contribution

It introduces a novel analysis of quantum quenches in the spherical p-spin model using Schwinger-Keldysh formalism, highlighting the effects of interactions and quantum fluctuations.

Findings

Increasing interaction strength lowers effective temperature.

Quantum fluctuations stabilize glassy states.

System can equilibrate after a quench with modified properties.

Abstract

In this work, we address the question of how a closed quantum system thermalises in the presence of a random external potential. By investigating the quench dynamics of the isolated quantum spherical $p$-spin model, a paradigmatic model of a mean-field glass, we aim to shed new light on this complex problem. Employing a closed-time Schwinger-Keldysh path integral formalism, we first initialise the system in a random, infinite-temperature configuration and allow it to equilibrate in contact with a thermal bath before switching off the bath and performing a quench. We find evidence that increasing the strength of either the interactions or the quantum fluctuations can act to lower the effective temperature of the isolated system and stabilise glassy behaviour.

Equations

$$\mathcal{H}$$

$$C(t,t^{\prime })$$

$$R(t,t^{\prime })$$

$$[m{\partial }_t^2+z(t)]R(t,t^{\prime })=\delta (t-t^{\prime })+{\int }_0^{\infty }\text{d}{t}^{\prime \prime }\Sigma (t,t^{\prime \prime })R(t^{\prime \prime },t)$$

$$[m{\partial }_t^2+z(t)]C(t,t^{\prime })={\int }_0^{\infty }\text{d}{t}^{\prime \prime }\Sigma (t,t^{\prime \prime })C(t^{\prime \prime },t^{\prime })$$

$$+{\int }_0^{t^{\prime }}\text{d}{t}^{\prime \prime }D(t,t^{\prime \prime })R(t^{\prime },t^{\prime \prime })$$

$$\Sigma (t,t^{\prime })$$

$$D(t,t^{\prime })$$

$$\times \text{Re}{\left[C(t,t^{\prime })-\frac{i}{2}(\hslash (t^{\prime })R(t,t^{\prime })+\hslash (t)R(t^{\prime },t))\right]}^{p-1}$$

$$z(t)$$

$$-m{\partial }_t^2{C(t,t^{\prime })|}_{t^{\prime }\to t^{-}}$$

Full text

*Introduction - * Understanding how and why many-body systems can fail to reach thermal equilibrium is both of fundamental value, as it allows us to test the hypothesis underlying equilibrium statistical physics, and of practical interest. In fact systems which fail to equilibrate can often exhibit rich new dynamical phenomena not seen in typical thermal states Kinoshita et al. (2006); Gring et al. (2012); Fausti et al. (2011); Mitrano et al. (2016)

Two main mechanisms of ergodicity breaking in many-body quantum systems have emerged recently. On the one hand, quantum integrable systems have an extensive number of conserved charges and so do not thermalize to a state whose macroscopic properties are determined by only a few quantities (such as energy and density) Essler and Fagotti (2016). On the other hand, the interplay of disorder and interactions can given rise to a robust mechanism for ergodicity breaking, the many-body equivalent of Anderson localisation, a.k.a. Many Body Localization (MBL), which does not require fine tuning to (typically isolated) integrable points. The absence of thermalisation in MBL is related to an emergent integrability Serbyn et al. (2013); Huse et al. (2014); Ros et al. (2015); Monthus (2016); Thomson and Schiró (2018).

In between those two limits, for which thermalization fails on all time scales, there is a huge class of systems for which thermalization is possible but only on very long timescales. These are glassy systems, whose dynamics display ergodicity breaking due to metastability. In this case, the dynamical evolution is trapped by exponentially many metastable states that forbid equilibration on short timescales. In finite dimensions, such metastable states have a finite (but very long) lifetime, while in the mean field limit their lifetime diverges with the system size (or dimension) due to the divergence of the free energy barriers between them. Nevertheless those systems are never completely out of equilibrium since in the end they relax on timescales that scale exponentially in either the system size or dimension Castellani and Cavagna (2005a); Cavagna (2009).

In contrast with MBL and integrable systems, glassy systems do not depend crucially on isolation from their environment and indeed most investigations on the dynamical behavior of quantum glasses has focused on a dissipative setting, where the system is coupled to a thermal bath. Here important progress has been achieved through the solution of simplified fully connected models Cugliandolo and Lozano (1999); Cugliandolo et al. (2001, 2002); Biroli and Parcollet (2002); Cugliandolo et al. (2004a); Markland et al. (2011). An interesting question which has received far less attention concerns the dynamics of isolated quantum glasses. Recently the properties of highly excited eigenstates of paradigmatic mean field models of quantum glasses and their resulting dynamics have been investigated numerically through exact diagonalization of finite size systems Laumann et al. (2014), analytically using forward scattering approximations Baldwin et al. (2016, 2017); Baldwin and Laumann (2018), and more recently through a mapping to Rosenzweig-Porter random matrix model Faoro et al. (2019). Yet, in the thermodynamic limit the dynamical behavior of those quantum mean field models can be solved exactly using field theory techniques similar to those well developed for classical models  Castellani and Cavagna (2005a).

In this work we extend those techniques to the quantum case, by focusing on the unitary dynamics of the isolated spherical quantum $p$-spin model, a paradigmatic example of a mean-field glass, whose Hamiltonian

[TABLE]

describes a set of spins $\sigma_{i}$ all-to-all coupled by random $p$-body interactions $J_{i_{1}...i_{p}}$ drawn from a Gaussian distribution with zero mean and unit variance. To make the model more tractable but still non-trivial, we treat the spins as continuous variables Kosterlitz et al. (1976) and enforce the spherical constraint $\sum_{i}^{N}\sigma^{2}_{i}=N$ by adding a Lagrange multiplier (hereafter denoted $z$). We further add a conjugate momentum $\Pi_{i}$ where $\left[\Pi_{i},\sigma_{j}\right]=i\hbar(t)\delta_{ij}$ are canonical commutation relations, and we allow $\hbar(t)$ to be time-dependent in order to be able to change the strength of quantum fluctuations - for details, see the Supplementary Material SM . This model has been extensively studied in both its classical Derrida (1980, 1981); Gross and Mezard (1984); Kirkpatrick and Thirumalai (1987a, b); Crisanti et al. (1993); Cugliandolo and Kurchan (1993); Castellani and Cavagna (2005b) and quantum version, when coupled to a thermal bath Cugliandolo and Lozano (1998, 1999); Cugliandolo et al. (2001); Biroli and Cugliandolo (2001); Cugliandolo et al. (2002); Cugliandolo et al. (2004a, b). At low temperature it displays a dynamical glass transition $T_{d}$ due to the emergence of long-lived glassy states. Below this temperature equilibration is never reached and the system ages forever (but not on exponential timescales). The dynamical temperature is a decreasing function of the strength of quantum fluctuations, as one may expect Cugliandolo et al. (2001). Though the isolated dynamics of the quantum $p$-spin model have not previously been studied, the classical isolated dynamics was recently investigated in Cugliandolo et al. (2017).

Here we study the quantum evolution of this model: we prepare the system at some temperature $T_{0}$ in the paramagnetic phase and then we suddenly change both the strength of random couplings $\mathcal{J}(t)$ and the strength of quantum fluctuations measured by $\hbar(t)$, keeping the system isolated. The resulting non-equilibrium phase diagram, plotted in Figure 1, features a high-temperature paramagnetic phase, where the system relaxes toward equilibrium, and a low-temperature phase where aging and breakdown of time-translational invariance emerge. Surprisingly, we find that the phase boundary between the paramagnetic and aging regimes strongly depends on whether quantum fluctuations are kept constant (left panel) or suddenly changed (right panel) throughout the evolution. In the former case the aging regime shrinks with respect to its classical counterpart, as expected thermodynamically. In the latter, we find that a sudden increase of quantum fluctuations promotes rather suppresses glassy effects (right panel, top curve), in striking contrast with the expectation based on the canonical equilibrium case of a system in contact with a finite temperature bath Cugliandolo and Lozano (1999); Cugliandolo et al. (2001, 2002). Such enhancement of aging effects are due to an interplay of quantum fluctuations and non-equilibrium effects. We interpret this intriguing result in terms of an effective temperature $T_{eff}<T_{0}$ for the isolated disordered quantum system, which in the absence of an external thermal bath is able to cool itself down through quantum fluctuations, eventually crossing the glass transition.

*Dynamical Equations for Correlation and Response - * Throughout this work we will focus in particular on the dynamics of correlation and response functions, which are defined by

[TABLE]

where $[A,B]_{\pm}=AB\pm BA$. The fully connected nature of the model defined in Eq. (1) allows us to derive closed dynamical equations that describe the evolution of correlation and response functions starting from an uncorrelated infinite temperature initial state. After disorder-averaging and taking the $N\to\infty$ limit, the equations of motion for the correlation and response functions can be obtained following the method of Ref. Cugliandolo and Lozano (1998) and are given by

[TABLE]

where we have defined the self-energies $\Sigma(t,t^{\prime})$ and $D(t,t^{\prime})$ as:

[TABLE]

With respect to the classical dynamical equations Cugliandolo et al. (2017), Eqs. (6-7) have extra self-energy contributions proportional to $\hbar(t)$ which arise from purely quantum fluctuations Cugliandolo and Lozano (1999). We perform the dynamical evolution subject to a time-dependent Lagrange multiplier $z(t)$ used to enforce the global spherical constraint. We can derive the dynamical equation for this by taking the equal-time limit of Eq. 5 to obtain Cugliandolo and Lozano (1998):

[TABLE]

Equations 4,5 and 8 are the three dynamical equations whose solution will discuss in the remaining of the paper. Their causal structure allow for a simple discretisation and numerical solution- for further details, see the Supplementary Material SM .

*Finite Temperature Initial State Preparation and Double Quench - *The dynamical equations (4,5 and 8) describe the evolution of the system from an initial infinite temperature initial state uncorrelated with the disorder. Here we are instead interested in studying dynamics from an initial finite temperature state, which would in principle require a three branch Keldysh contour structure as recently discussed Cugliandolo et al. (2019). We instead perform the initial thermalisation numerically through a double-quench protocol. Specifically, we first quench from infinite temperature to some $T_{0}>T_{d}$ (where $T_{d}$ is the equilibrium dynamical temperature of the spin glass transition) and $\mathcal{J}(0<t<t_{q})\equiv\mathcal{J}_{0}=1$ and $\hbar(0<t<t_{q})\equiv\hbar_{0}$ and allow the system to thermalise in contact with a thermal bath, which we assume to be a set of harmonic oscillators in thermal equilibrium at some temperature $T_{0}$, as in Ref. Cugliandolo and Lozano, 1999. This results in modifed self-energies $\tilde{\Sigma}(t,t^{\prime})$ and $\tilde{D}(t,t^{\prime})$ in Eq. 7 due to the bath coupling, whose explicit expressions are given in SM . Then, for $t>t_{q}$ we switch off the coupling to the bath and let the system evolve unitarily with $\mathcal{J}(t\geq t_{q})\equiv\mathcal{J}_{F}$ and $\hbar(t\geq t_{q})\equiv\hbar_{F}$. All temperatures are measured in units of $\mathcal{J}_{0}$. Supporting data demonstrating that our system is well-equilibrated to the bath temperature is shown in Supplementary Material SM .

Results - For concreteness we will set $p=3$, though we expect our results to hold for any $p>2$. In Fig. 2 we plot the dynamics of correlation function $C(\tau+t_{w},t_{w})$ at fixed $\mathcal{J}_{0}=1$ for different type of quenches. We first study the dynamics keeping fixed the strength of quantum fluctuations while quenching $\mathcal{J}$ (panel a). We see that increasing $\mathcal{J}_{F}>\mathcal{J}_{0}$ results in a slow down of the dynamics and a plateau in the correlation function begins to emerge. Such a plateau is associated with a non-zero Edwards-Anderson glassy order parameter. In the classical case $\hbar_{F}=\hbar_{0}=0$ we therefore recover the results of Ref. Cugliandolo et al., 2017, while in the quantum case $\hbar_{F}=\hbar_{0}=1$ (see inset) we see that similar quenches of $\mathcal{J}$ does not lead to a well formed plateau, indicating that the quantum aging boundary shifts toward larger values of $\mathcal{J}_{F}/\mathcal{J}_{0}$. This is consistent with the naive expectation that quantum fluctuations suppress aging behavior. The resulting phase diagram is shown in Figure 1, panel a). A rather different picture emerges instead when quantum fluctuations are suddenly quenched rather than kept fixed, as we show in panel b of Figure 2. Keeping the interaction fixed, $\mathcal{J}_{F}=\mathcal{J}_{0}$, and increasing the quantum fluctuations (main panel) strongly enhances the aging behavior of the system, as shown by the formation of a plateau and a waiting time dependence. On the contrary reducing the value of $\hbar_{F}<\hbar_{0}$ leads to a rapid relaxation (inset). This surprising outcome for a quench of $\hbar$ is further highlighted in panel c, where the dynamics for different initial temperature $T_{0}$ is studied. In particular we see that for an increase of $\hbar$ (main panel) the system upon cooling crosses a dynamical glass transition, even in absence of an interaction quench ($\mathcal{J}_{F}=\mathcal{J}_{0}$), and for temperature well above the classical $T_{d}$. On the contrary, decreasing $\hbar$ always keep the system in the paramagnetic phase. Those results therefore suggest that the aging regime is increased when quantum fluctuations are suddenly switched on, as we summarize in the panel b of Figure 1.

We remark that for the timescales accessible to our current simulations, the correlation function still decays and does not display a true plateau: this is likely an effect of not being able to access sufficiently long waiting times $t_{w}$ to see the true plateau, as evidenced by the strengthening of the plateau for larger $t_{w}$. By approximating $C(\tau+t_{w},t_{w})|_{\tau\to\infty}$ by the value of the correlation function at the longest times accessible to our simulation, and identifying this value with the Edwards-Anderson order parameter $q_{EA}$, we can plot an approximate non-equilibrium phase diagram for the isolated quantum system, shown in Fig. 1. Within our simulation times, as clearly shown by Fig. 2, we cannot reach the true $t\to\infty$ value of $q_{EA}$. Instead, we can set a threshold value and approximate that all $q\leq q_{th}$ are slowly decaying paramagnetic solutions, whereas for $q>q_{th}$ the system is in a true glassy phase. The results of this are shown in the phase diagram in Fig. 1 by dashed lines, using $q_{th}\approx 0.2$, though the qualitative shape of the phase diagram does not depend strongly on this choice 111To extract the true phase boundary one would need to consider the relaxation time of the system as a function of the control parameters and try to fit its divergence, however the timescales required to perform this analysis are longer than accessible with our numerical code..

Effective Temperature and Quench-Induced Cooling - The results presented above indicate that quantum fluctuations and non-equilibrium effects can strongly enhance glassiness and increase the region of parameters where aging effects are observed. This is surprising at first, since glassiness is a low temperature property, while exciting the system with a global quantum quench injects extensive energy and should intuitively induce heating Mitra and Giamarchi (2011); Schiró and Mitra (2014). We can understand this effect in terms of an effective thermalisation of the isolated system to an effective temperature $T_{eff}$, as we show in detail by looking at the fluctuation-dissipation theorem (FDT) in the long-time regime of the dynamical equations for correlation and response SM . In Figure 3 we show that the $T_{eff}$ extracted from FDT decreases with $\hbar_{F}$ and eventually reaches the dynamical critical temperature $T_{d}$ for the glass transition, below which the system fails to thermalise. By extracting the local minimum of $T_{eff}$ from Figure 3 and identifying it with the transition in our numerical data, we can draw a phase boundary with no free parameters, shown in Fig. 1 by the solid lines. Interestingly, the same effect of cooling by quantum fluctuations emerges from basic energetic arguments: indeed the effective temperature can be also estimated by comparing the post-quench energy $E_{Q}$, which is conserved during the unitary evolution, to the equilibrium internal energy of the system at a given value of $\hbar_{F}$, i.e. $E_{Q}=U(T_{eff},\hbar_{F})$. Solving this equation for our model in the static approximation Cugliandolo et al. (2001); SM , which is valid in the high temperature phase under consideration, we obtain a thermodynamic estimate for $T_{eff}$ which almost perfectly matches the dynamical one obtained from FDT in the regime where the system thermalises (see light blue line in Fig. 3).

*Discussion - * In our specific model (1) the strength of quantum fluctuations is controlled by the magnitude of $\hbar$. A natural question concerns whether the qualitative picture we presented so far would change in more realistic situations where quantum fluctuations are controlled by the action of a transverse field $\Gamma$, such as in the Ising p-spin quantum glass Goldschmidt (1990); Dobrosavljevic and Thirumalai (1990); Obuchi et al. (2007); Jörg et al. (2008). In thermal equilibrium it is known that the spherical and the Ising p-spin share much of their physics Nieuwenhuizen and Ritort (1998); Cugliandolo et al. (2001); Cugliandolo et al. (2004a); Takahashi and Matsuda (2011), including the phase diagram which features a quantum glass to paramagnet phase transition driven by the strength of quantum fluctuations, encoded respectively in $\hbar$ or $\Gamma$. Whether this analogy remains valid also for the out of equilibrium dynamics is a priori not obvious. Using energetic arguments SM we estimate the effective temperature in the Ising p-spin after a quantum quench of the transverse field and show that, indeed, this quantity shows the same qualitative behavior in the two models. In particular we show that also in the Ising p-spin an increase of quantum fluctuation (i.e. a quench to a larger value of $\Gamma$) can lead to a decrease of the effective temperature, i.e. a cooling through quantum fluctuations that appears therefore a robust feature of isolated quantum glasses. This result is also of practical relevance, since quantum simulation of Ising $p$-spin models can be realized using arrays of superconducting qubits, which are modeled as two level systems with random Ising couplings and transverse fields, the latter tunable in real-time and therefore amenable to sudden or slow quenches. In fact, these protocols are routinely explored in the field of quantum annealing Boixo et al. (2014). Superconducting qubits also offer enough flexibility in fabrication and design such that arranging effective multi-spin interactions, such as those relevant for our p-spin with $p>2$ has indeed been already reported Schöndorf and Wilhelm (2019); Melanson et al. (2019).

*Conclusions - * In this work we have studied the quench dynamics of an isolated quantum glass. Remarkably, we have shown that suddenly increasing the strength of quantum fluctuations enhances aging behavior, in contradiction with common expectations based on the physics of quantum glasses coupled to thermal environment. The key feature of this effect relies on a ‘cooling by quantum fluctuations’ effect that we have shown to hold also for the more realistic Ising p-spin case, a model which can be quantum simulated using superconducting qubits.

Interesting future directions include starting from a low temperature glass phase at $T<T_{d}$, for which the corresponding dynamical equations are already available in Cugliandolo et al. (2019), to see how the quantum glasses respond to non equilibrium perturbations as well as to study the effect of a smooth quench protocol with finite duration, which may connect our results with investigations on quantum annealing done on related quantum glass models Jörg et al. (2008); Jorg et al. (2010). Solving the full real-time dynamics for other mean field models of isolated quantum glasses, such as the Ising $p$-spin and the quantum Random Energy Model, using similar techniques would also be an interesting direction to take.

We acknowledge helpful discussions with D. Abanin, G. Biroli, L. Cugliandolo and M. Tarzia. This work was supported by the grant “Investissements d’Avenir” from LabEx PALM (ANR-10-LABX-0039-PALM), the grant DynDisQ from DIM SIRTEQ and by the CNRS through the PICS-USA-14750. The majority of the computations were performed on the Collège de France IPH cluster computer.