arXiv subjects

Juan P. Garrahan

Publications and source records attributed to Juan P. Garrahan.

At least 37 records · Page 2Linked to original sources

arXiv2024

Non-thermal eigenstates and slow relaxation in quantum Fredkin spin chains

We study the dynamics and thermalization of the Fredkin spin chain, a system with local three-body interactions, particle conservation and explicit kinetic constraints. We consider deformations away from its stochastic point in order to tune between regimes where kinetic energy dominates and those where potential energy does. By means of exact diagonalisation, perturbation theory and variational matrix product states, we show there is a sudden change of behaviour in the dynamics that occurs, from one of fast thermalization to one of slow metastable (prethermal) dynamics near the stochastic point. This change in relaxation is connected to the emergence of additional kinetic constraints which lead to the fragmentation of Hilbert space in the limit of a large potential energy. We also show that this change can lead to thermalization being evaded for special initial conditions due to non-thermal eigenstates (akin to quantum many-body scars). We provide clear evidence for the existence of these non-thermal states for large system sizes even when far from the large-potential-energy limit, and explain their connection to the emergent kinetic constraints.

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arXiv2024

Exact pre-transition effects in kinetically constrained circuits: dynamical fluctuations in the Floquet-East model

We study the dynamics of a classical circuit corresponding to a discrete-time kinetically constrained East model. We show that this classical "Floquet-East" model displays pre-transition behaviour which is a dynamical equivalent of the hydrophobic effect in water. For the deterministic version of the model we prove exactly: (i) a change in scaling with size in the probability of inactive space-time regions (akin to the "energy-entropy" crossover of the solvation free energy in water), (ii) a first-order phase transition in the dynamical large deviations, (iii) the existence of the optimal geometry for local phase separation to accommodate space-time solutes, and (iv) a dynamical analog of "hydrophobic collapse".

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arXiv2024

Dynamical heterogeneity and large deviations in the open quantum East glass model from tensor networks

We study the non-equilibrium dynamics of the dissipative quantum East model via numerical tensor networks. We use matrix product states to represent evolution under quantum-jump unravellings for sizes beyond those accessible to exact diagonalisation. This allows us to demonstrate that dynamical heterogeneity accompanies slow relaxation, in analogy with what is seen in classical glassy systems. Furthermore, using variational matrix product operators we: (i) compute the spectral gap of the Lindbladian, and show that glassiness is enhanced in the presence of weak quantum fluctuations compared to the pure classical case, and (ii) obtain the dynamical large deviations by calculating the leading eigenvector of the tilted Lindbladian, and find clear evidence for a first-order active-inactive dynamical phase transition. We also show how to directly sample the rare quantum trajectories associated to the large deviations.

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arXiv2024

Optimal sampling of dynamical large deviations in two dimensions via tensor networks

We use projected entangled-pair states (PEPS) to calculate the large deviations (LD) statistics of the dynamical activity of the two dimensional East model, and the two dimensional symmetric simple exclusion process (SSEP) with open boundaries, in lattices of up to 40x40 sites. We show that at long-times both models have phase transitions between active and inactive dynamical phases. For the 2D East model we find that this trajectory transition is of the first-order, while for the SSEP we find indications of a second order transition. We then show how the PEPS can be used to implement a trajectory sampling scheme capable of directly accessing rare trajectories. We also discuss how the methods described here can be extended to study rare events at finite times.

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arXiv2024

Rejection-free quantum Monte Carlo in continuous time from transition path sampling

Continuous-time quantum Monte Carlo refers to a class of algorithms designed to sample the thermal distribution of a quantum Hamiltonian through exact expansions of the Boltzmann exponential in terms of stochastic trajectories which are periodic in imaginary time. Here, we show that for (sign-problem-free) quantum many-body systems with discrete degrees of freedom -- such as spins on a lattice -- this sampling can be done in a rejection-free manner using transition path sampling (TPS). The key idea is to converge the trajectory ensemble through updates where one individual degree of freedom is modified across all time while the remaining unaltered ones provide a time-dependent background. The ensuing single-body dynamics provides a way to generate trajectory updates exactly, allowing one to obtain the target ensemble efficiently via rejection-free TPS. We demonstrate our method on the transverse field Ising model in one and two dimensions, and on the quantum triangular plaquette (or Newman-Moore) model. We show that despite large autocorrelation times, our method is able to efficiently recover the respective quantum phase transition of each model. We also discuss the connection to rare event sampling in continuous-time Markov dynamics.

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arXiv2024

Discrete generative diffusion models without stochastic differential equations: a tensor network approach

Diffusion models (DMs) are a class of generative machine learning methods that sample a target distribution by transforming samples of a trivial (often Gaussian) distribution using a learned stochastic differential equation. In standard DMs, this is done by learning a ``score function'' that reverses the effect of adding diffusive noise to the distribution of interest. Here we consider the generalisation of DMs to lattice systems with discrete degrees of freedom, and where noise is added via Markov chain jump dynamics. We show how to use tensor networks (TNs) to efficiently define and sample such ``discrete diffusion models'' (DDMs) without explicitly having to solve a stochastic differential equation. We show the following: (i) by parametrising the data and evolution operators as TNs, the denoising dynamics can be represented exactly; (ii) the auto-regressive nature of TNs allows to generate samples efficiently and without bias; (iii) for sampling Boltzmann-like distributions, TNs allow to construct an efficient learning scheme that integrates well with Monte Carlo. We illustrate this approach to study the equilibrium of two models with non-trivial thermodynamics, the $d=1$ constrained Fredkin chain and the $d=2$ Ising model.

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arXiv2024

Exact results on the dynamics of the stochastic Floquet-East model

We introduce a stochastic generalisation of the classical deterministic Floquet-East model, a discrete circuit with the same kinetic constraint as the East model of glasses. We prove exactly that, in the limit of long time and large size, this model has a large deviation phase transition between active and inactive dynamical phases. We also compute the finite time and size scaling of general space-time fluctuations, which for the case of inactive regions gives rise to dynamical hydrophobicity. We also discuss how, through the Trotter limit, these exact results also hold for the continuous-time East model, thus proving long-standing observations in kinetically constrained models. Our results here illustrate the applicability of exact tensor network methods for solving problems in many-body stochastic systems.

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arXiv2024

Bounds on Fluctuations of First Passage Times for Counting Observables in Classical and Quantum Markov Processes

We study the statistics of first passage times (FPTs) of trajectory observables in both classical and quantum Markov processes. We consider specifically the FPTs of counting observables, that is, the times to reach a certain threshold of a trajectory quantity which takes values in the positive integers and is non-decreasing in time. For classical continuous-time Markov chains we rigorously prove: (i) a large deviation principle (LDP) for FPTs, whose corollary is a strong law of large numbers; (ii) a concentration inequality for the FPT of the dynamical activity, which provides an upper bound to the probability of its fluctuations to all orders; and (iii) an upper bound to the probability of the tails for the FPT of an arbitrary counting observable. For quantum Markov processes we rigorously prove: (iv) the quantum version of the LDP, and subsequent strong law of large numbers, for the FPTs of generic counts of quantum jumps; (v) a concentration bound for the the FPT of total number of quantum jumps, which provides an upper bound to the probability of its fluctuations to all orders, together with a similar bound for the sub-class of quantum reset processes which requires less strict irreducibility conditions; and (vi) a tail bound for the FPT of arbitrary counts. Our results allow to extend to FPTs the so-called "inverse thermodynamic uncertainty relations" that upper bound the size of fluctuations in time-integrated quantities. We illustrate our results with simple examples.

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arXiv2024

Slow dynamics and large deviations in classical stochastic Fredkin chains

The Fredkin spin chain serves as an interesting theoretical example of a quantum Hamiltonian whose ground state exhibits a phase transition between three distinct phases, one of which violates the area law. Here we consider a classical stochastic version of the Fredkin model, which can be thought of as a simple exclusion process subject to additional kinetic constraints, and study its classical stochastic dynamics. The ground state phase transition of the quantum chain implies an equilibrium phase transition in the stochastic problem, whose properties we quantify in terms of numerical matrix product states (MPS). The stochastic model displays slow dynamics, including power law decaying autocorrelation functions and hierarchical relaxation processes due to exponential localization. Like in other kinetically constrained models, the Fredkin chain has a rich structure in its dynamical large deviations - which we compute accurately via numerical MPS - including an active-inactive phase transition, and a hierarchy of trajectory phases connected to particular equilibrium states of the model. We also propose, via its height field representation, a generalization of the Fredkin model to two dimensions in terms of constrained dimer coverings of the honeycomb lattice.

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arXiv2024

Exact quench dynamics of the Floquet quantum East model at the deterministic point

We study the nonequilibrium dynamics of the Floquet quantum East model (a Trotterized version of the kinetically constrained quantum East spin chain) at its "deterministic point", where evolution is defined in terms of CNOT permutation gates. We solve exactly the thermalization dynamics for a broad class of initial product states by means of "space evolution". We prove: (i) the entanglement of a block of spins grows at most at one-half the maximal speed allowed by locality (i.e., half the speed of dual-unitary circuits); (ii) if the block of spins is initially prepared in a classical configuration, speed of entanglement is a quarter of the maximum; (iii) thermalization to the infinite temperature state is reached exactly in a time that scales with the size of the block.

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arXiv2024

Combining Reinforcement Learning and Tensor Networks, with an Application to Dynamical Large Deviations

We present a framework to integrate tensor network (TN) methods with reinforcement learning (RL) for solving dynamical optimisation tasks. We consider the RL actor-critic method, a model-free approach for solving RL problems, and introduce TNs as the approximators for its policy and value functions. Our "actor-critic with tensor networks" (ACTeN) method is especially well suited to problems with large and factorisable state and action spaces. As an illustration of the applicability of ACTeN we solve the exponentially hard task of sampling rare trajectories in two paradigmatic stochastic models, the East model of glasses and the asymmetric simple exclusion process (ASEP), the latter being particularly challenging to other methods due to the absence of detailed balance. With substantial potential for further integration with the vast array of existing RL methods, the approach introduced here is promising both for applications in physics and to multi-agent RL problems more generally.

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arXiv2023

Quantum reaction-limited reaction-diffusion dynamics of annihilation processes

We investigate the quantum reaction-diffusion dynamics of fermionic particles which coherently hop in a one-dimensional lattice and undergo annihilation reactions. The latter are modelled as dissipative processes which involve losses of pairs $2A \to \emptyset$, triplets $3A \to \emptyset$, and quadruplets $4A \to \emptyset$ of neighbouring particles. When considering classical particles, the corresponding decay of their density in time follows an asymptotic power-law behavior. The associated exponent in one dimension is different from the mean-field prediction whenever diffusive mixing is not too strong and spatial correlations are relevant. This specifically applies to $2A\to \emptyset$, while the mean-field power-law prediction just acquires a logarithmic correction for $3A \to \emptyset$ and is exact for $4A \to \emptyset$. A mean-field approach is also valid, for all the three processes, when the diffusive mixing is strong, i.e., in the so-called reaction-limited regime. Here, we show that the picture is different for quantum systems. We consider the quantum reaction-limited regime and we show that for all the three processes power-law behavior beyond mean field is present as a consequence of quantum coherences, which are not related to space dimensionality. The decay in $3A\to \emptyset$ is further, highly intricate, since the power-law behavior therein only appears within an intermediate time window, while at long times the density decay is not power-law. Our results show that emergent critical behavior in quantum dynamics has a markedly different origin, based on quantum coherences, to that applying to classical critical phenomena, which is, instead, solely determined by the relevance of spatial correlations.

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arXiv2023

Universal and nonuniversal probability laws in Markovian open quantum dynamics subject to generalized reset processes

We consider quantum jump trajectories of Markovian open quantum systems subject to stochastic in time resets of their state to an initial configuration. The reset events provide a partitioning of quantum trajectories into consecutive time intervals, defining sequences of random variables from the values of a trajectory observable within each of the intervals. For observables related to functions of the quantum state, we show that the probability of certain orderings in the sequences obeys a universal law. This law does not depend on the chosen observable and, in case of Poissonian reset processes, not even on the details of the dynamics. When considering (discrete) observables associated with the counting of quantum jumps, the probabilities in general lose their universal character. Universality is only recovered in cases when the probability of observing equal outcomes in a same sequence is vanishingly small, which we can achieve in a weak reset rate limit. Our results extend previous findings on classical stochastic processes [N.~R.~Smith et al., EPL {\bf 142}, 51002 (2023)] to the quantum domain and to state-dependent reset processes, shedding light on relevant aspects for the emergence of universal probability laws.

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arXiv2023

Minibatch training of neural network ensembles via trajectory sampling

Most iterative neural network training methods use estimates of the loss function over small random subsets (or minibatches) of the data to update the parameters, which aid in decoupling the training time from the (often very large) size of the training datasets. Here, we show that a minibatch approach can also be used to train neural network ensembles (NNEs) via trajectory methods in a highly efficient manner. We illustrate this approach by training NNEs to classify images in the MNIST datasets. This method gives an improvement to the training times, allowing it to scale as the ratio of the size of the dataset to that of the average minibatch size which, in the case of MNIST, gives a computational improvement typically of two orders of magnitude. We highlight the advantage of using longer trajectories to represent NNEs, both for improved accuracy in inference and reduced update cost in terms of the samples needed in minibatch updates.

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arXiv2023

Reaction-limited quantum reaction-diffusion dynamics

We consider the quantum nonequilibrium dynamics of systems where fermionic particles coherently hop on a one-dimensional lattice and are subject to dissipative processes analogous to those of classical reaction-diffusion models. Particles can either annihilate in pairs, $A+A \to \emptyset$, coagulate upon contact, $A+A \to A$, and possibly also branch, $A \to A+A$. In classical settings, the interplay between these processes and particle diffusion leads to critical dynamics as well as to absorbing-state phase transitions. Here, we analyze the impact of coherent hopping and of quantum superposition, focusing on the so-called reaction-limited regime. Here, spatial density fluctuations are quickly smoothed out due to fast hopping, which for classical systems is described by a mean-field approach. By exploiting the time-dependent generalized Gibbs ensemble method, we demonstrate that quantum coherence and destructive interference play a crucial role in these systems and are responsible for the emergence of locally protected dark states and collective behavior beyond mean-field. This can manifest both at stationarity and during the relaxation dynamics. Our results highlight fundamental differences between classical nonequilibrium dynamics and their quantum counterpart and show that quantum effects indeed change collective universal behavior.

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arXiv2023

Training neural network ensembles via trajectory sampling

In machine learning, there is renewed interest in neural network ensembles (NNEs), whereby predictions are obtained as an aggregate from a diverse set of smaller models, rather than from a single larger model. Here, we show how to define and train a NNE using techniques from the study of rare trajectories in stochastic systems. We define an NNE in terms of the trajectory of the model parameters under a simple, and discrete in time, diffusive dynamics, and train the NNE by biasing these trajectories towards a small time-integrated loss, as controlled by appropriate counting fields which act as hyperparameters. We demonstrate the viability of this technique on a range of simple supervised learning tasks. We discuss potential advantages of our trajectory sampling approach compared with more conventional gradient based methods.

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arXiv2023

Stochastic strong zero modes and their dynamical manifestations

Strong zero modes (SZMs) are conserved operators localised at the edges of certain quantum spin chains, which give rise to long coherence times of edge spins. Here we define and analyse analogous operators in one-dimensional classical stochastic systems. For concreteness, we focus on chains with single occupancy and nearest-neighbour transitions, in particular particle hopping and pair creation and annihilation. For integrable choices of parameters we find the exact form of the SZM operators. Being in general non-diagonal in the classical basis, the dynamical consequences of stochastic SZMs are very different from those of their quantum counterparts. We show that the presence of a stochastic SZM is manifested through a class of exact relations between time-correlation functions, absent in the same system with periodic boundaries.

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arXiv2023

Slow heterogeneous relaxation due to constraints in dual XXZ models

With the aim to understand the role of the constraints in the thermalisation of quantum systems, we study the dynamics of a family of kinetically constrained models arising through duality from the XXZ spin chain. We find that integrable and nonintegrable deformations around the stochastic point give rise to ground state phase transitions between localised and delocalised phases, which in turn determine the nature of the relaxation dynamics at finite energy densities. While in the delocalised phase thermalisation is fast and homogeneous, in the localised phase relaxation is slow, temporal autocorrelations exhibit plateaus indicative of metastability, and the growth of entanglement is heterogeneous in space. Furthermore, by considering relaxation from initial product states, we demostrate that this slow thermalisation can be rationalised directly from the presence of constraints in the dynamics.

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