arXiv subjects

Wouter Buijsman

Publications and source records attributed to Wouter Buijsman.

17 recordsLinked to original sources

arXiv2025

Average weighted ratio of consecutive level spacings for infinite-dimensional orthogonal random matrices

The onset of quantum ergodicity is often quantified by the average ratio of consecutive level spacings. The reference values for ergodic quantum systems have been obtained numerically from the spectra of large but finite-dimensional random matrices. This work introduces a weighted ratio of consecutive level spacings, having the propery that the average can be computed numerically for random matrices of infinite dimension. A Painlev\'e differential equation is solved numerically in order to determine this average for infinite-dimensional orthogonal random matrices, thereby providing a reference value for ergodic quantum systems obeying time-reversal symmetry (provided that the time-reversal operator squares to the identity matrix). A Wigner surmise-inspired analytical calculation is found to yield a qualitatively accurate picture for the statistics of high-dimensional random matrices from each of the symmetry classes. For Poissonian level statistics, a significantly different average is found, indicating that the average weighted ratio of consecutive level spacings can be used as a probe for quantum ergodicity.

cond-mat.stat-mech↗

arXiv2025

Power-law banded random matrix ensemble as a model for quantum many-body Hamiltonians

We explore interpretations of the power-law banded random matrix (PLBRM) ensemble as Hamiltonians of one-dimensional quantum many-body systems. We introduce and compare a number of labeling schemes for assigning random matrix basis indices to many-body basis vectors. We compare the physical properties of the resulting Hamiltonians, focusing on the half-system eigenstate bipartite entanglement entropy. We show and quantify how the different PLBRM phases (ergodic, weakly ergodic, localized), known from the single-particle interpretation, can be interpreted as entanglement transitions in the quantum many-body interpretation. For the weakly ergodic phase, where spectral edge and bulk eigenstates show distinct behavior, we perform a detailed scaling analysis to provide a quantitative picture of the boundaries between different types of entanglement scaling behaviors. In particular, we identify and characterize an intermediate set of eigenstates whose entanglement entropy have volume law scaling but nonvanishing deviation from the Page value expected for maximally ergodic states.

cond-mat.dis-nn↗

arXiv2024

Periodic revivals from supersymmetry in a fermionic kinetically constrained model

Supersymmetry provides a natural playground for the construction of dynamically constrained lattice fermion models. We here illustrate how supersymmetry can be used to construct a fermionic equivalent of the PXP model with an adjustable chemical potential. This model is closely related to the $\mathcal{N} = 2$ supersymmetric $M_1$ model, inheriting its integrability. The supersymmetric algebra additionally implies that the dynamics exhibit periodic revivals for specific initial states, including the $\mathbb{Z}_2$-ordered (every second site occupied) product state. These dynamics are reminiscent to those of the PXP model, a paradigmatic effective model in the field of quantum many-body scars. We draw a further parallel by uncovering eigenstates obeying sub-thermal entanglement scaling at energies given by (plus or minus) square roots of integers and relate these to special eigenstates of the $M_1$ model. While we focus on a concrete model, our proposed approach is applicable to more general supersymmetric algebras, where it is expected to lead to non-ergodic dynamics.

quant-ph↗

arXiv2024

Efficient circular Dyson Brownian motion algorithm

Circular Dyson Brownian motion describes the Brownian dynamics of particles on a circle (periodic boundary conditions), interacting through a logarithmic, long-range two-body potential. Within the log-gas picture of random matrix theory, it describes the level dynamics of unitary ("circular") matrices. A common scenario is that one wants to know about an initial configuration evolved over a certain interval of time, without being interested in the intermediate dynamics. Numerical evaluation of this is computationally expensive as the time-evolution algorithm is accurate only on short time intervals because of an underlying perturbative approximation. This work proposes an efficient and easy-to-implement improved circular Dyson Brownian motion algorithm for the unitary class (Dyson index $β= 2$, physically corresponding to broken time-reversal symmetry). The algorithm allows one to study time evolution over arbitrarily large intervals of time at a fixed computational cost, with no approximations being involved.

cond-mat.stat-mech↗

arXiv2024

Long-range spectral statistics of the Rosenzweig-Porter model

The Rosenzweig-Porter model is a single-parameter random matrix ensemble that supports an ergodic, fractal, and localized phase. The names of these phases refer to the properties of the (midspectrum) eigenstates. This work focuses on the long-range spectral statistics of the recently introduced unitary equivalent of this model. By numerically studying the Thouless time obtained from the spectral form factor, it is argued that long-range spectral statistics can be used to probe the transition between the ergodic and the fractal phases. The scaling of the Thouless time as a function of the model parameters is found to be similar to the scaling of the spreading width of the eigenstates. Provided that the transition between the fractal and the localized phases can be probed through short-range level statistics, such as the average ratio of consecutive level spacings, this work establishes that spectral statistics are sufficient to probe both transitions present in the phase diagram.

cond-mat.dis-nn↗

arXiv2023

Gaussian state approximation of quantum many-body scars

Quantum many-body scars are atypical, highly nonthermal eigenstates embedded in a sea of thermal eigenstates that have been observed in, for example, kinetically constrained quantum many-body models. These special eigenstates are characterized by a bipartite entanglement entropy that scales as most logarithmically with the subsystem size. We use numerical optimization techniques to investigate if quantum many-body scars of the experimentally relevant PXP model can be well approximated by Gaussian states. Gaussian states are described by a number of parameters that scales quadratically with system size, thereby having a much lower complexity than generic quantum many-body states, for which this number scales exponentially. We find that while quantum many-body scars can typically be well approximated by (symmetrized) Gaussian states, this is not the case for ergodic (thermal) eigenstates. This observation suggests that the non-ergodic part of the PXP Hamiltonian is related to certain quadratic parent Hamiltonians, thereby hinting on the origin of the quantum many-body scars.

cond-mat.str-el↗

arXiv2023

Solvable random matrix ensemble with a logarithmic weakly confining potential

This work identifies a solvable (in the sense that spectral correlation functions can be expressed in terms of orthogonal polynomials), rotationally invariant random matrix ensemble with a logarithmic weakly confining potential. The ensemble, which can be interpreted as a transformed Jacobi ensemble, is in the thermodynamic limit characterized by a Lorentzian eigenvalue density. It is shown that spectral correlation functions can be expressed in terms of the nonclassical Gegenbauer polynomials $C_n^{(-1/2)}(x)$ with $n \ge 2$, which have been proven to form a complete orthogonal set with respect to the proper weight function. A procedure to sample matrices from the ensemble is outlined and used to provide a numerical verification for some of the analytical results. This ensemble is pointed out to potentially have applications in quantum many-body physics.

cond-mat.stat-mech↗

arXiv2022

Many-body localization of ${\mathbb Z}_3$ Fock parafermions

We study the effects of a random magnetic field on a one-dimensional (1D) spin-1 chain with {\it correlated} nearest-neighbor $XY$ interaction. We show that this spin model can be exactly mapped onto the 1D disordered tight-binding model of ${\mathbb Z}_3$ Fock parafermions (FPFs), exotic anyonic quasiparticles that generalize usual spinless fermions. Thus, we have a peculiar case of a disordered Hamiltonian that, despite being bilinear in the creation and annihilation operators, exhibits a many-body localization (MBL) transition owing to the nontrivial statistics of FPFs. This is in sharp contrast to conventional bosonic and fermionic quadratic disordered Hamiltonians that show single-particle (Anderson) localization. We perform finite-size exact diagonalization calculations of level-spacing statistics, fractal dimensions, and entanglement entropy, and provide convincing evidence for the MBL transition at finite disorder strength.

cond-mat.dis-nn↗

arXiv2022

Ground-state energy distribution of disordered many-body quantum systems

Extreme-value distributions are studied in the context of a broad range of problems, from the equilibrium properties of low-temperature disordered systems to the occurrence of natural disasters. Our focus here is on the ground-state energy distribution of disordered many-body quantum systems. We derive an analytical expression that, upon tuning a parameter, reproduces with high accuracy the ground-state energy distribution of the systems that we consider. For some models, it agrees with the Tracy-Widom distribution obtained from Gaussian random matrices. They include transverse Ising models, the Sachdev-Ye model, and a randomized version of the PXP model. For other systems, such as Bose-Hubbard models with random couplings and the disordered spin-1/2 Heisenberg chain used to investigate many-body localization, the shapes are at odds with the Tracy-Widom distribution. Our analytical expression captures all of these distributions, thus playing a role to the lowest energy level similar to that played by the Brody distribution to the bulk of the spectrum.

cond-mat.dis-nn↗

arXiv2022

Number of zero-energy eigenstates in the PXP model

The PXP model is paradigmatic in the field of quantum many-body scars. This model has a number of zero-energy eigenstates that is exponentially large in system size. Lower bounds on the number of zero-energy eigenstates are obtained for both open and periodic (zero and $π$-momentum sectors) boundary conditions. These bounds are found to be tight up to system sizes accessible by numerical exact diagonalization, and can be expected to be tight in general. In addition to previous results, separate lower bounds are obtained for the spatial inversion-symmetric and inversion-antisymmetric symmetry sectors. Furthermore, the derivations improve on previous ones as these are free of assumptions.

quant-ph↗

arXiv2021

Circular Rosenzweig-Porter random matrix ensemble

The Rosenzweig-Porter random matrix ensemble serves as a qualitative phenomenological model for the level statistics and fractality of eigenstates across the many-body localization transition in static systems. We propose a unitary (circular) analogue of this ensemble, which similarly captures the phenomenology of many-body localization in periodically driven (Floquet) systems. We define this ensemble as the outcome of a Dyson Brownian motion process. We show numerical evidence that this ensemble shares some key statistical properties with the Rosenzweig-Porter ensemble for both the eigenvalues and the eigenstates.

cond-mat.dis-nn↗

arXiv2020

Sensitivity of the spectral form factor to short-range level statistics

The spectral form factor is a dynamical probe for level statistics of quantum systems. The early-time behaviour is commonly interpreted as a characterization of two-point correlations at large separation. We argue that this interpretation can be too restrictive by indicating that the self-correlation imposes a constraint on the spectral form factor integrated over time. More generally, we indicate that each expansion coefficient of the two-point correlation function imposes a constraint on the properly weighted time-integrated spectral form factor. We discuss how these constraints can affect the interpretation of the spectral form factor as a probe for ergodicity. We propose a new probe, which eliminates the effect of the constraint imposed by the self-correlation. The use of this probe is demonstrated for a model of randomly incomplete spectra and a Floquet model supporting many-body localization.

cond-mat.dis-nn↗

arXiv2019

Gumbel statistics for entanglement spectra of many-body localized eigenstates

An entanglement spectrum encodes statistics beyond the entanglement entropy, of which several have been studied in the context of many-body localization. We numerically study the extreme value statistics of entanglement spectra of many-body localized eigenstates. The physical information encoded in these spectra is almost fully carried by the few smallest elements, suggesting the extreme value statistics to have physical significance. We report the surprising observation of Gumbel statistics. Our result provides an analytical, parameter-free characterization of many-body localized eigenstates.

cond-mat.dis-nn↗

arXiv2018

Random Matrix Ensemble for the Level Statistics of Many-Body Localization

We numerically study the level statistics of the Gaussian $β$ ensemble. These statistics generalize Wigner-Dyson level statistics from the discrete set of Dyson indices $β= 1,2,4$ to the continuous range $0 < β< \infty$. The Gaussian $β$ ensemble covers Poissonian level statistics for $β\to 0$, and provides a smooth interpolation between Poissonian and Wigner-Dyson level statistics. We establish the physical relevance of the level statistics of the Gaussian $β$ ensemble by showing near-perfect agreement with the level statistics of a paradigmatic model in studies on many-body localization over the entire crossover range from the thermal to the many-body localized phase. In addition, we show similar agreement for a related Hamiltonian with broken time-reversal symmetry.

cond-mat.dis-nn↗

arXiv2018

Many-body localization in the Fock space of natural orbitals

We study the eigenstates of a paradigmatic model of many-body localization in the Fock basis constructed out of the natural orbitals. By numerically studying the participation ratio, we identify a sharp crossover between different phases at a disorder strength close to the disorder strength at which subdiffusive behaviour sets in, significantly below the many-body localization transition. We repeat the analysis in the conventionally used computational basis, and show that many-body localized eigenstates are much stronger localized in the Fock basis constructed out of the natural orbitals than in the computational basis.

cond-mat.dis-nn↗

arXiv2017

Non-ergodicity in the Anisotropic Dicke model

We study the ergodic -- non-ergodic transition in a generalized Dicke model with independent co- and counter rotating light-matter coupling terms. By studying level statistics, the average ratio of consecutive level spacings, and the quantum butterfly effect (out-of-time correlation) as a dynamical probe, we show that the ergodic -- non-ergodic transition in the Dicke model is a consequence of the proximity to the integrable limit of the model when one of the couplings is set to zero. This can be interpreted as a hint for the existence of a quantum analogue of the classical Kolmogorov-Arnold-Moser theorem. Besides, we show that there is no intrinsic relation between the ergodic -- non-ergodic transition and the precursors of the normal -- superradiant quantum phase transition.

quant-ph↗

arXiv2014

Efficient fold-change detection based on protein-protein interactions

Various biological sensory systems exhibit a response to a relative change of the stimulus, often referred to as fold-change detection. In the last few years fold-change detecting mechanisms, based on transcriptional networks, have been proposed. Here we present fold-change detecting mechanism, based on protein-protein interactions, consisting of two interacting proteins. This mechanism, in contrast to previously proposed mechanisms, does not consume chemical energy and is not subject to transcriptional and translational noise. We show by analytical and numerical calculations, that the mechanism can have a fast, precise and efficient response for parameters that are relevant to eukaryotic cells.

physics.bio-ph↗