We consider a random matrix model of interaction between a small $n$ -level system, $S$ , and its environment, a $N$ -level heat reservoir, $R$ . The interaction between $S$ and $R$ is modeled by a tensor product of a fixed $% n\times n$ matrix and a $N\times N$ hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on $R$ , the reduced density matrix of the system $ρ_{S}=\mathrm{Tr}_{R}ρ_{S\cup R}^{(eq)} $, is given by $ρ_{S}^{(c)}\sim \exp {\{-βH_{S}\}}$ , where $H_{S}$ is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using $% ρ_{S}^{(c)}$ to describe some nano-systems, like biopolymers, in equilibrium with their environment \cite{Se:12}. Our results extend those obtained previously in \cite{Le-Pa:03,Le-Co:07} for a special two-level system.We consider a random matrix model of interaction between a small $n$-level system, $S$, and its environment, a $N$-level heat reservoir, $R$. The interaction between $S$ and $R$ is modeled by a tensor product of a fixed $% n\times n$ matrix and a $N\times N$ hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on $R$, the reduced density matrix of the system $ρ_{S}=\mathrm{Tr}_{R}ρ_{S\cup R}^{(eq)} $, is given by $ρ_{S}^{(c)}\sim \exp {\{-βH_{S}\}}$, where $H_{S}$ is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using $% ρ_{S}^{(c)}$ to describe some nano-systems, like biopolymers, in equilibrium with their environment \cite{Se:12}. Our results extend those obtained previously in \cite{Le-Pa:03,Le-Co:07} for a special two-level system.

arXiv:1502.05004Journal of Physics A Mathematical and TheoreticalJun 8, 20158 citations

Two-Level Systems with a Time-Dependent Interaction

WORLD SCIENTIFIC eBooksNov 1, 20180 citations

Randomly interacting bosons on two spin levels

Declan Mulhall

The problem of random interactions leading to regular spectra in shell model type simulations is described. The key results are reviewed alnog with a selection of the explanations. A model system of N particles on 2 spin levels having random 2-body collisions that conserve angular momentum is examined. Preliminary results are described, including the ground state spin distributions peaked at extreme values of angular momentum, signatures of rotational bands, and smooth parabolic yrast lines. A simple random matrix theory analysis shows signatures of quantum chaos in the level spacing distribution and the Δ3 statistic.The problem of random interactions leading to regular spectra in shell model type simulations is described. The key results are reviewed alnog with a selection of the explanations. A model system of N particles on 2 spin levels having random 2-body collisions that conserve angular momentum is examined. Preliminary results are described, including the ground state spin distributions peaked at extreme values of angular momentum, signatures of rotational bands, and smooth parabolic yrast lines. A simple random matrix theory analysis shows signatures of quantum chaos in the level spacing distribution and the Δ3 statistic.

AIP conference proceedingsJan 1, 20170 citations

Level statistics of the disordered Haldane-Shastry model with $1/r^α$ interactionLevel statistics of the disordered Haldane-Shastry model with $1/r^α$ interaction

Vengatesan Ganapathy, Pranay Patil, Ajit C. Balram

Understanding how the interaction range and various types of disorder affect the level statistics of many-body quantum systems, leading to the emergence of many-body localization (MBL), is a challenging open frontier. We study the level statistics of a variant of the spin- $1/2$ Haldane-Shastry model with $1/r^α$ interactions, where $α{\geq}0$ parametrizes the range of the interactions, in the presence of position disorder and/or random magnetic fields. We find that neither position disorder nor random magnetic fields alone yields pristine Poisson statistics in this system; however, Poisson statistics emerge in their combined presence, suggesting the emergence of MBL when both types of disorder coexist. Interestingly, once random magnetic fields break the $SU(2)$ symmetry, the strength of the position disorder, $δ$ , appears to play an important role, as evidenced by an approximate scaling collapse of the disorder-averaged gap ratios that is parametrized in terms of a single parameter, $αδ$ .Understanding how the interaction range and various types of disorder affect the level statistics of many-body quantum systems, leading to the emergence of many-body localization (MBL), is a challenging open frontier. We study the level statistics of a variant of the spin-$1/2$ Haldane-Shastry model with $1/r^α$ interactions, where $α{\geq}0$ parametrizes the range of the interactions, in the presence of position disorder and/or random magnetic fields. We find that neither position disorder nor random magnetic fields alone yields pristine Poisson statistics in this system; however, Poisson statistics emerge in their combined presence, suggesting the emergence of MBL when both types of disorder coexist. Interestingly, once random magnetic fields break the $SU(2)$ symmetry, the strength of the position disorder, $δ$, appears to play an important role, as evidenced by an approximate scaling collapse of the disorder-averaged gap ratios that is parametrized in terms of a single parameter, $αδ$.

cond-mat.str-elCross-listedarXiv:2604.14695Physical review. B./Physical review. BApr 17, 20260 citations

Level statistics of XXZ spin chains with a random magnetic field

Kazue Kudo, Tetsuo Deguchi

The level-spacing distribution of a spin- $\frac{1}{2}$ $\mathrm{XXZ}$ chain is numerically studied under random magnetic field. We show explicitly how the level statistics depends on the lattice size L, the anisotropy parameter $\ensuremath{\Delta},$ and the mean amplitude of the random magnetic field h. In the energy spectrum, quantum integrability competes with nonintegrability derived from the randomness, where the $\mathrm{XXZ}$ interaction is modified by the parameter $\ensuremath{\Delta}.$ When $\ensuremath{\Delta}\ensuremath{\ne}0,$ the level-spacing distribution mostly shows Wigner-like behavior, while when $\ensuremath{\Delta}=0,$ Poisson-like behavior appears although the system is nonintegrable due to randomness. Poisson-like behavior also appears for $\ensuremath{\Delta}\ensuremath{\ne}0$ in the large h limit. Furthermore, the level-spacing distribution depends on the lattice size L, particularly when the random field is weak.The level-spacing distribution of a spin-$\frac{1}{2}$ $\mathrm{XXZ}$ chain is numerically studied under random magnetic field. We show explicitly how the level statistics depends on the lattice size L, the anisotropy parameter $\ensuremath{\Delta},$ and the mean amplitude of the random magnetic field h. In the energy spectrum, quantum integrability competes with nonintegrability derived from the randomness, where the $\mathrm{XXZ}$ interaction is modified by the parameter $\ensuremath{\Delta}.$ When $\ensuremath{\Delta}\ensuremath{\ne}0,$ the level-spacing distribution mostly shows Wigner-like behavior, while when $\ensuremath{\Delta}=0,$ Poisson-like behavior appears although the system is nonintegrable due to randomness. Poisson-like behavior also appears for $\ensuremath{\Delta}\ensuremath{\ne}0$ in the large h limit. Furthermore, the level-spacing distribution depends on the lattice size L, particularly when the random field is weak.

arXiv:cond-mat/0310752Physical Review BApr 8, 200434 citations

We consider a random matrix model of interaction between a small

n

-level system,

S

, and its environment, a

N

-level heat reservoir,

R

. The interaction between

S

and

R

is modeled by a tensor product of a fixed

% n\times n

matrix and a

N\times N

hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on

R

, the reduced density matrix of the system $ρ_{S}=\mathrm{Tr}_{R}ρ_{S\cup R}^{(eq)} $, is given by

ρ_{S}^{(c)}\sim \exp {\{-βH_{S}\}}

, where

H_{S}

is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using

% ρ_{S}^{(c)}

to describe some nano-systems, like biopolymers, in equilibrium with their environment \cite{Se:12}. Our results extend those obtained previously in \cite{Le-Pa:03,Le-Co:07} for a special two-level system.

Level statistics of the disordered Haldane-Shastry model with

1/r^α

interaction

1/2

Haldane-Shastry model with

1/r^α

interactions, where

α{\geq}0

parametrizes the range of the interactions, in the presence of position disorder and/or random magnetic fields. We find that neither position disorder nor random magnetic fields alone yields pristine Poisson statistics in this system; however, Poisson statistics emerge in their combined presence, suggesting the emergence of MBL when both types of disorder coexist. Interestingly, once random magnetic fields break the

SU(2)

symmetry, the strength of the position disorder,

δ

, appears to play an important role, as evidenced by an approximate scaling collapse of the disorder-averaged gap ratios that is parametrized in terms of a single parameter,

αδ

The level-spacing distribution of a spin-

\frac{1}{2}

\mathrm{XXZ}

chain is numerically studied under random magnetic field. We show explicitly how the level statistics depends on the lattice size L, the anisotropy parameter $\ensuremath{\Delta},$ and the mean amplitude of the random magnetic field h. In the energy spectrum, quantum integrability competes with nonintegrability derived from the randomness, where the

\mathrm{XXZ}

interaction is modified by the parameter $\ensuremath{\Delta}.$ When $\ensuremath{\Delta}\ensuremath{\ne}0,$ the level-spacing distribution mostly shows Wigner-like behavior, while when $\ensuremath{\Delta}=0,$ Poisson-like behavior appears although the system is nonintegrable due to randomness. Poisson-like behavior also appears for $\ensuremath{\Delta}\ensuremath{\ne}0$ in the large h limit. Furthermore, the level-spacing distribution depends on the lattice size L, particularly when the random field is weak.