arXiv subjects

Jonathon Riddell

Publications and source records attributed to Jonathon Riddell.

15 recordsLinked to original sources

arXiv2025

Group word dynamics from local random matrix Hamiltonians and beyond

We study one dimensional quantum spin chains whose nearest neighbor interactions are random matrices that square to one. By employing free probability theory, we establish a mapping from the many-body quantum dynamics of energy density in the original chain to a single-particle hopping dynamics when the local Hilbert space dimension is large. The hopping occurs on the Cayley graph of an infinite Coxeter reflection group. Adjacency matrices on large finite clusters of this Cayley graph can be constructed numerically by leveraging the automatic structure of the group. The density of states and two-point functions of the local energy density are approximately computed and consistent with the physics of a generic local Hamiltonian: Gaussian density of states and thermalization of energy density. We then ask what happens to the physics if we modify the group on which the hopping dynamics occurs, and conjecture that adding braid relations into the group leads to integrability. Our results put into contact ideas in free probability theory, quantum mechanics of hyperbolic lattices, and the physics of both generic and integrable Hamiltonian dynamics.

cond-mat.stat-mech↗

arXiv2025

Quantum State Designs from Minimally Random Quantum Circuits

Random many-body states are both a useful tool to model certain physical systems and an important asset for quantum computation. Realising them, however, generally requires an exponential (in system size) amount of resources. Recent research has presented a way out by showing that one can generate random states, or more precisely a controlled approximation of them, by applying a quantum circuit built in terms of few-body unitary gates. Most of this research, however, has been focussed on the case of quantum circuits composed by completely random unitary gates. Here we consider what happens for circuits that, instead, involve a minimal degree of randomness. Specifically, we concentrate on two different settings: (a) brickwork quantum circuits with a single one-qudit random matrix at a boundary; (b) brickwork quantum circuits with fixed interactions but random one-qudit gates everywhere. We show that, for any given initial state, (a) and (b) produce a distribution of states approaching the Haar distribution in the limit of large circuit depth. More precisely, we show that the moments of the distribution produced by our circuits can approximate the ones of the Haar distribution in a depth proportional to the system size. Interestingly we find that in both Cases (a) and (b) the relaxation to the Haar distribution occurs in two steps - this is in contrast with what happens in fully random circuits. Moreover, we show that choosing appropriately the fixed interactions, for example taking the local gate to be a dual-unitary gate with high enough entangling power, minimally random circuits produce a Haar random distribution more rapidly than fully random circuits. In particular, dual-unitary circuits with maximal entangling power - i.e. perfect tensors - appear to provide the optimal quantum state design preparation for any design number.

quant-ph↗

arXiv2025

Energy dynamics in a class of local random matrix Hamiltonians

Random matrix theory yields valuable insights into the universal features of quantum many-body chaotic systems. Although all-to-all interactions are traditionally studied, many interesting dynamical questions, such as transport of a conserved density, require a notion of spatially local interactions. We study the transport of the energy, the most basic conserved density, in few-body and 1D chains of nearest-neighbor random matrix terms that square to one. In the few-body but large local Hilbert space dimension case, we develop a mapping for the energy dynamics to a single-particle hopping picture. This allows for the computation of the energy density autocorrelators and an out-of-time-ordered correlator of the energy density. In the 1D chain, we numerically study the energy transport for a small local Hilbert space dimension. We also discuss the density of states throughout and touch upon the relation to free probability theory.

cond-mat.stat-mech↗

arXiv2024

Rationally independent free fermions with local hopping

Rationally independent free fermions are those where sums of single-particle energies multiplied by arbitrary rational coefficients vanish only if the coefficients are all zero. This property guaranties that they have no degeneracies in the many-body spectrum and gives them relaxation properties more similar to those of generic systems. Using classic results from number theory we provide minimal examples of rationally independent free fermion models for every system size in one dimension. This is accomplished by considering a free fermion model with a chemical potential, and hopping terms corresponding to all the divisors of the number of sites, each one with an incommensurate complex amplitude. We further discuss the many-body spectral statistics for these models and show that local probes -- like the ratio of consecutive level spacings -- look very similar to what is expected for the Poisson statistics. We however demonstrate that free fermion models can never have Poisson statistics with an analysis of the moments of the spectral form factor.

quant-ph↗

arXiv2024

Structural Stability Hypothesis of Dual Unitary Quantum Chaos

Having spectral correlations that, over small enough energy scales, are described by random matrix theory is regarded as the most general defining feature of quantum chaotic systems as it applies in the many-body setting and away from any semiclassical limit. Although this property is extremely difficult to prove analytically for generic many-body systems, a rigorous proof has been achieved for dual-unitary circuits -- a special class of local quantum circuits that remain unitary upon swapping space and time. Here we consider the fate of this property when moving from dual-unitary to generic quantum circuits focussing on the \emph{spectral form factor}, i.e., the Fourier transform of the two-point correlation. We begin with a numerical survey that, in agreement with previous studies, suggests that there exists a finite region in parameter space where dual-unitary physics is stable and spectral correlations are still described by random matrix theory, although up to a maximal quasienergy scale. To explain these findings, we develop a perturbative expansion: it recovers the random matrix theory predictions, provided the terms occurring in perturbation theory obey a relatively simple set of assumptions. We then provide numerical evidence and a heuristic analytical argument supporting these assumptions.

cond-mat.stat-mech↗

arXiv2023

No-resonance conditions, random matrices, and quantum chaotic models

In this article we investigate no-resonance conditions for quantum many body chaotic systems and random matrix models. No-resonance conditions are properties of the spectrum of a model, usually employed as a theoretical tool in the analysis of late time dynamics. The first order no-resonance condition holds when a spectrum is non-degenerate, while higher order no-resonance conditions imply sums of an equal number of energies are non-degenerate outside of permutations of the indices. This resonance condition is usually assumed to hold for quantum chaotic models. In this work we use several tests from random matrix theory to demonstrate that the statistics of sums of eigenvalues, that are of interest to due to the no-resonance conditions, have Poisson statistics, and lack level repulsion. This result is produced for both a quantum chaotic Hamiltonian as well as the Gaussian Unitary Ensemble and Gaussian Orthogonal Ensemble. This implies some models may have violations of the no-resonance condition or "near" violations. We finish the paper by generalizing important bounds in quantum equilibration theory to cases where the no-resonance conditions are violated, and to the case of random matrix models.

quant-ph↗

arXiv2023

Islands of Chiral Solitons in Integer Spin Kitaev Chains

An intriguing chiral soliton phase has recently been identified in the $S$=1/2 Kitaev spin chain. Here we show that for $S$=1,2,3,4,5 an analogous phase can be identified, but contrary to the $S$=1/2 case the chiral soliton phases appear as islands within the sea of the polarized phase. In fact, a small field applied in a general direction will adiabatically connect the integer spin Kitaev chain to the polarized phase. Only at sizable intermediate fields along symmetry directions does the soliton phase appear centered around the special point $h^\star_x$=$h^\star_y$=$S$ where two exact product ground-states can be identified. The large $S$ limit can be understood from a semi-classical analysis, and variational calculations provide a detailed picture of the $S$=1 soliton phase. Under open boundary conditions, the chain has a single soliton in the ground-state which can be excited, leading to a proliferation of in-gap states. In contrast, even length periodic chains exhibit a gap above a twice degenerate ground-state. The presence of solitons leaves a distinct imprint on the low temperature specific heat.

cond-mat.str-el↗

arXiv2022

Field Induced Chiral Soliton Phase in the Kitaev Spin Chain

The bond-dependent Ising interaction present in the Kitaev model has attracted considerable attention. The appearance of an unexpected intermediate phase under a magnetic field is particularly intriguing, and one may wonder if a similar phase occurs in the Kitaev spin chain with alternating $x$- and $y$-bond Ising interactions. Previous studies have focused on a transverse field, $h_z$, and reported a direct transition to the polarized state. Here, we investigate phases with arbitrary angle of two longitudinal fields, $h_x$ and $h_y$. For a magnetic field applied along the diagonal, $h_x$=$h_y$, the chain remains gapless up to a critical field $h^{c_1}_{xy}$. Surprisingly, above $h^{c1}_{xy}$ it enters an unusual intermediate phase before reaching the polarized state at $h^{c_2}_{xy}$. This phase is characterized by a staggered vector chirality and for periodic boundary conditions, a two-fold degeneracy with a finite gap. For open boundary systems the ground-state exhibits a single soliton, lowering the energy, and gapless excitations. However, the corresponding anti-soliton raises the energy sufficiently that a gap appears for soliton and anti-soliton pairs in periodic systems. An intuitive variational picture is developed describing the soliton phase.

cond-mat.str-el↗

arXiv2022

Concentration of quantum equilibration and an estimate of the recurrence time

We show that the dynamics of generic quantum systems concentrate around their equilibrium value when measuring at arbitrary times. This means that the probability of finding them away from equilibrium is exponentially suppressed, with a decay rate given by the effective dimension. Our result allows us to place a lower bound on the recurrence time of quantum systems, since recurrences corresponds to the rare events of finding a state away from equilibrium. In many-body systems, this bound is doubly exponential in system size. We also show corresponding results for free fermions, which display a weaker concentration and earlier recurrences.

quant-ph↗

arXiv2022

Relaxation of non-integrable systems and correlation functions

We investigate early-time equilibration rates of observables in closed many-body quantum systems and compare them to those of two correlation functions, first introduced by Kubo and Srednicki. We explore whether these different rates coincide at a universal value that sets the timescales of processes at a finite energy density. We find evidence for this coincidence when the initial conditions are sufficiently generic, or typical. We quantify this with the effective dimension of the state and with a state-observable effective dimension, which estimate the number of energy levels that participate in the dynamics. Our findings are confirmed by proving that these different timescales coincide for dynamics generated by Haar-random Hamiltonians. This also allows to quantitatively understand the scope of previous theoretical results on equilibration timescales and on random matrix formalisms. We approach this problem with exact, full spectrum diagonalization. The numerics are carried out in a non-integrable Heisenberg-like Hamiltonian, and the dynamics are investigated for several pairs of observables and states.

quant-ph↗

arXiv2021

Scaling at the OTOC Wavefront: free versus chaotic models

Out of time ordered correlators (OTOCs) are useful tools for investigating foundational questions such as thermalization in closed quantum systems because they can potentially distinguish between integrable and nonintegrable dynamics. Here we discuss the properties of wavefronts of OTOCs by focusing on the region around the main wavefront at $x=v_{B}t$, where $v_{B}$ is the butterfly velocity. Using a Heisenberg spin model as an example, we find that the leading edge of a propagating Gaussian with the argument $-m(x)\left( x-v_B t \right)^2 +b(x)t$ gives an excellent fit to the region around $x=v_{B}t$ for both the free and chaotic cases. However, the scaling in these two regimes is very different: in the free case the coefficients $m(x)$ and $b(x)$ have an inverse power law dependence on $x$ whereas in the chaotic case they decay exponentially. We conjecture that this result is universal by using catastrophe theory to show that, on the one hand, the wavefront in the free case has to take the form of an Airy function and its local expansion shows that the power law scaling seen in the numerics holds rigorously, and on the other hand an exponential scaling of the OTOC wavefront must be a signature of nonintegrable dynamics. We find that the crossover between the two regimes is smooth and characterized by an S-shaped curve giving the lifting of Airy nodes as a function of a chaos parameter. This shows that the Airy form is qualitatively stable against weak chaos and consistent with the concept of a quantum Kolmogorov-Arnold-Moser theory.

cond-mat.stat-mech↗

arXiv2020

Time evolution of correlation functions in quantum many-body systems

We give rigorous analytical results on the temporal behavior of two-point correlation functions --also known as dynamical response functions or Green's functions-- in closed many-body quantum systems. We show that in a large class of translation-invariant models the correlation functions factorize at late times $\langle A(t) B\rangle_β\rightarrow \langle A \rangle_β\langle B \rangle_β$, thus proving that dissipation emerges out of the unitary dynamics of the system. We also show that for systems with a generic spectrum the fluctuations around this late-time value are bounded by the purity of the thermal ensemble, which generally decays exponentially with system size. For auto-correlation functions we provide an upper bound on the timescale at which they reach the factorized late time value. Remarkably, this bound is only a function of local expectation values, and does not increase with system size. We give numerical examples that show that this bound is a good estimate in non-integrable models, and argue that the timescale that appears can be understood in terms of an emergent fluctuation-dissipation theorem. Our study extends to further classes of two point functions such as the symmetrized ones and the Kubo function that appears in linear response theory, for which we give analogous results.

quant-ph↗

arXiv2019

Out of Time Order Correlations in the Quasi-Periodic Aubry-André model

We study out of time ordered correlators (OTOC) in a free fermionic model with a quasi-periodic potential. This model is equivalent to the Aubry-André model and features a phase transition from an extended phase to a localized phase at a non-zero value of the strength of the quasi-periodic potential. We investigate five different time-regimes of interest for out of time ordered correlators; early, wavefront, $x=v_Bt$, late time equilibration and infinite time. For the early time regime we observe a power law for all potential strengths. For the time regime preceding the wavefront we confirm a recently proposed universal form and use it to extract the characteristic velocity of the wavefront for the present model. A Gaussian waveform is observed to work well in the time regime surrounding $x=v_Bt$. Our main result is for the late time equilibration regime where we derive a finite time equilibration bound for the OTOC, bounding the correlator's distance from its late time value. The bound impose strict limits on equilibration of the OTOC in the extended regime ans is valid not only for the Aubry-André model but for any quadratic model model. Finally, momentum out of time ordered correlators for the Aubry-André model are studied where large values of the OTOC are observed at late times at the critical point.

cond-mat.stat-mech↗

arXiv2018

Out of Time Ordered Correlators and Entanglement Growth in the Random Field XX Spin Chain

We study out of time order correlations, $C(x,t)$ and entanglement growth in the random field XX model with open boundary conditions using the exact Jordan-Wigner transformation to a fermionic Hamiltonian. For any non-zero strength of the random field this model describes an Anderson insulator. Two scenarios are considered: A global quench with the initial state corresponding to a product state of the Néel form, and the behaviour in a typical thermal state at $β=1$. As a result of the presence of disorder the information spreading as described by the out of time correlations stops beyond a typical length scale, $ξ_{OTOC}$. For $|x|<ξ_{OTOC}$ information spreading occurs at the maximal velocity $v_{max}=J$ and we confirm predictions for the early time behaviour of $C(x,t)\sim t^{2|x|}$. For the case of the quench starting from the Néel product state we also study the growth of the bipartite entanglement, focusing on the late and infinite time behaviour. The approach to a bounded entanglement is observed to be slow for the disorder strengths we study.

cond-mat.stat-mech↗

arXiv2018

Generalized eigenstate typicality in translation-invariant quasifree fermionic models

We demonstrate a generalized notion of eigenstate thermalization for translation-invariant quasifree fermionic models: the vast majority of eigenstates satisfying a finite number of suitable constraints (e.g. fixed energy and particle number) have the property that their reduced density matrix on small subsystems approximates the corresponding generalized Gibbs ensemble. To this end, we generalize analytic results by Lai and Yang (Phys. Rev. B 91, 081110 (2015)) and illustrate the claim numerically by example of the Jordan-Wigner transform of the XX spin chain.

cond-mat.stat-mech↗