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An investigation of equilibration in small quantum systems

J. M. Luck

Published Oct 21, 2015• 0 citations
OpenAlex

Abstract

We investigate the equilibration of a small isolated quantum system by means of its matrix of asymptotic transition probabilities in a preferential basis. The trace of this matrix is put forward as a quantitative measure of the degree of equilibration in the system, from the standpoint of the chosen basis. This approach is illustrated on the example of a tight-binding particle in one dimension. In the regime of free ballistic propagation, the above trace saturates to a finite limit, testifying good equilibration. In the presence of a random potential, the trace grows linearly with the system size, testifying poor equilibration in the insulating regime induced by Anderson localization. In the weak-disorder situation of most interest, a universal finite-size scaling law describes the crossover between the ballistic and localized regimes. The associated crossover exponent 2/3 is dictated by the anomalous band-edge scaling characterizing the most localized energy eigenstates.

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An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

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Abstract We investigate the equilibration of a small isolated quantum system by means of its matrix of asymptotic transition probabilities in a preferential basis. The trace of this matrix is shown to measure the degree of equilibration of the system launched from a typical state, from the standpoint of the chosen basis. This approach is substantiated by an in-depth study of the example of a tight-binding particle in one dimension. In the regime of free ballistic propagation, the above trace saturates to a finite limit, testifying good equilibration. In the presence of a random potential, the trace grows linearly with the system size, testifying poor equilibration in the insulating regime induced by Anderson localization. In the weak-disorder situation of most interest, a universal finite-size scaling law describes the crossover between the ballistic and localized regimes. The associated crossover exponent 2/3 is dictated by the anomalous band-edge scaling characterizing the most localized energy eigenstates.

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Abstract It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace T of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension N of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of T can be investigated by analytical means. We first study the statistics of T when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U ( N ) or O ( N ) . We then investigate a quantum spin S in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter—such as the tilt angle or the electric field—through some scaling regime induces a continuous crossover from good to bad equilibration properties.

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Equilibration properties of small quantum systems: further examples

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It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace $T$ of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension $N$ of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of $T$ can be investigated by analytical means. We first study the statistics of $T$ when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U$(N)$ or O$(N)$. We then investigate a quantum spin $S$ in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter -- such as the tilt angle or the electric field -- through some scaling regime induces a continuous transition from good to bad equilibration properties.

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A disordered quantum system can be driven, e.g., by increasing disorder, through the Anderson transition between a delocalized and localized phase. Recently, Anderson localization on treelike graphs has attracted much attention in view of its connections with problems of many-body localization. One of the central questions in this context is that of ergodicity of eigenstates. The authors of this paper perform a numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder. The problem can be described as a tight-binding model on a lattice with $N$ sites that is locally a tree with constant connectivity. Focusing on the delocalized side of the transition, they show that the data can be interpreted in terms of the finite-size crossover from a small ($N\ensuremath{\ll}{N}_{c}$) to a large ($N\ensuremath{\gg}{N}_{c}$) system, where ${N}_{c}$ is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. The results support an analytical prediction that states in the delocalized phase (and at $N\ensuremath{\gg}{N}_{c}$) are ergodic.

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We investigate the equilibration of a small isolated quantum system by means of its matrix of asymptotic transition probabilities in a preferential basis. The trace of this matrix is put forward as a quantitative measure of the degree of equilibration in the system, from the standpoint of the chosen basis. This approach is illustrated on the example of a tight-binding particle in one dimension. In the regime of free ballistic propagation, the above trace saturates to a finite limit, testifying good equilibration. In the presence of a random potential, the trace grows linearly with the system size, testifying poor equilibration in the insulating regime induced by Anderson localization. In the weak-disorder situation of most interest, a universal finite-size scaling law describes the crossover between the ballistic and localized regimes. The associated crossover exponent 2/3 is dictated by the anomalous band-edge scaling characterizing the most localized energy eigenstates.
Abstract We investigate the equilibration of a small isolated quantum system by means of its matrix of asymptotic transition probabilities in a preferential basis. The trace of this matrix is shown to measure the degree of equilibration of the system launched from a typical state, from the standpoint of the chosen basis. This approach is substantiated by an in-depth study of the example of a tight-binding particle in one dimension. In the regime of free ballistic propagation, the above trace saturates to a finite limit, testifying good equilibration. In the presence of a random potential, the trace grows linearly with the system size, testifying poor equilibration in the insulating regime induced by Anderson localization. In the weak-disorder situation of most interest, a universal finite-size scaling law describes the crossover between the ballistic and localized regimes. The associated crossover exponent 2/3 is dictated by the anomalous band-edge scaling characterizing the most localized energy eigenstates.
Abstract It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace T of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension N of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of T can be investigated by analytical means. We first study the statistics of T when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U ( N ) or O ( N ) . We then investigate a quantum spin S in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter—such as the tilt angle or the electric field—through some scaling regime induces a continuous crossover from good to bad equilibration properties.
It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace TTT of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension NNN of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of TTT can be investigated by analytical means. We first study the statistics of TTT when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U(N)(N)(N) or O(N)(N)(N). We then investigate a quantum spin SSS in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter -- such as the tilt angle or the electric field -- through some scaling regime induces a continuous transition from good to bad equilibration properties.
In this thesis, we have explored the commonalities and connections between different classes of quantum systems that do not thermalize. Specifically, we have (1) shown that localized systems possess conservation laws like integrable systems, which can be constructed in a systematic way and used to detect localization-delocalization transitions , (2) studied the phenomenon of many-body localization in a model with a single particle mobility edge, (3) shown that interesting finite-size scaling emerges, with universal exponents, when athermal quantum systems are forced to thermalize through the application of perturbations and (4) shown that these scaling laws also arise when a perturbation causes a crossover between quantum systems described by different random matrix ensembles. We conclude with a brief summary of each chapter. In Chapter 2, we have investigated the effects of finite size on the crossover between quantum integrable systems and non-integrable systems. Using exact diagonalization of finite-sized systems, we have studied this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L → ∞, non-integrability sets in for an arbitrarily small integrabilitybreaking perturbation. The crossover value of the perturbation scales as a power law ∼ L−3 when the integrable system is gapless and the scaling appears to be robust to microscopic details and the precise form of the perturbation. In Chapter 3, we have studied the crossover among different random matrix ensembles CHAPTER 6. CONCLUSION 127 [Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE)] realized in different microscopic models. We have found that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We have also found that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. Finally,we have conjectured that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system. In Chapter 4, we have outlined a procedure to construct conservation laws for Anderson localized systems. These conservation laws are found as power series in the hopping parameters. We have also obtained the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended depending on the strength of a coupling constant. We have formulated a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using…
Quantum phase transitions are usually observed in ground states of correlated systems. Remarkably, eigenstate phase transitions can also occur at finite energy density in disordered, isolated quantum systems. Such transitions fall outside the framework of statistical mechanics as they involve the breakdown of ergodicity. Here, we consider what general constraints can be imposed on the nature of eigenstate transitions due to the presence of disorder. We derive Harris-type bounds on the finite-size scaling exponents of the mean entanglement entropy and level statistics at the many-body localization phase transition using several different arguments. Our results are at odds with recent small-size numerics, for which we estimate the crossover scales beyond which the Harris bound must hold.
A disordered quantum system can be driven, e.g., by increasing disorder, through the Anderson transition between a delocalized and localized phase. Recently, Anderson localization on treelike graphs has attracted much attention in view of its connections with problems of many-body localization. One of the central questions in this context is that of ergodicity of eigenstates. The authors of this paper perform a numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder. The problem can be described as a tight-binding model on a lattice with NNN sites that is locally a tree with constant connectivity. Focusing on the delocalized side of the transition, they show that the data can be interpreted in terms of the finite-size crossover from a small ($N\ensuremath{\ll}{N}_{c}$) to a large ($N\ensuremath{\gg}{N}_{c}$) system, where Nc{N}_{c}Nc​ is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. The results support an analytical prediction that states in the delocalized phase (and at $N\ensuremath{\gg}{N}_{c}$) are ergodic.