We derive analytically the full distribution of the ground-state energy of K non-interacting fermions in a disordered environment, modelled by a Hamiltonian whose spectrum consists of N i.i.d. random energy levels with distribution (with ε ≥ 0), in the same spirit as the “Random Energy Model”. We show that for each fixed K , the distribution P K , N ( E 0 ) of the ground-state energy E 0 has a universal scaling form in the limit of large N . We compute this universal scaling function and show that it depends only on K and the exponent α characterizing the small ε behaviour of . We compared the analytical predictions with results from numerical simulations. For this purpose we employed a sophisticated importance-sampling algorithm that allowed us to obtain the distributions over a large range of the support down to probabilities as small as . We found asymptotically a very good agreement between analytical predictions and numerical results.
We derive analytically the full distribution of the ground-state energy of K non-interacting fermions in a disordered environment, modelled by a Hamiltonian whose spectrum consists of N i.i.d. random energy levels with distribution (with ε ≥ 0), in the same spirit as the “Random Energy Model”. We show that for each fixed K , the distribution P K , N ( E 0 ) of the ground-state energy E 0 has a universal scaling form in the limit of large N . We compute this universal scaling function and show that it depends only on K and the exponent α characterizing the small ε behaviour of . We compared the analytical predictions with results from numerical simulations. For this purpose we employed a sophisticated importance-sampling algorithm that allowed us to obtain the distributions over a large range of the support down to probabilities as small as . We found asymptotically a very good agreement between analytical predictions and numerical results.
We study the probability distribution function of the ground-state energies of the disordered one-dimensional Ising spin chain with power-law interactions using a combination of parallel tempering Monte Carlo and branch, cut, and price algorithms. By tuning the exponent of the power-law interactions we are able to scan several universality classes. Our results suggest that mean-field models have a non-Gaussian limiting distribution of the ground-state energies, whereas non-mean-field models have a Gaussian limiting distribution. We compare the results of the disordered one-dimensional Ising chain to results for a disordered two-leg ladder, for which large system sizes can be studied, and find a qualitative agreement between the disordered one-dimensional Ising chain in the short-range universality class and the disordered two-leg ladder. We show that the mean and the standard deviation of the ground-state energy distributions scale with a power of the system size. In the mean-field universality class the skewness does not follow a power-law behavior and converges to a nonzero constant value. The data for the Sherrington-Kirkpatrick model seem to be acceptably well fitted by a modified Gumbel distribution. Finally, we discuss the distribution of the internal energy of the Sherrington-Kirkpatrick model at finite temperatures and show that it behaves similar to the ground-state energy of the system if the temperature is smaller than the critical temperature.
We consider classical normal modes and noninteracting bosonic excitations in disordered systems. We emphasize generic aspects of such problems and parallels with disordered, noninteracting systems of fermions, and discuss in particular the relevance for bosonic excitations of symmetry classes known in the fermionic context. We also stress important differences between bosonic and fermionic problems. One of these follows from the fact that ground-state stability of a system requires all bosonic excitation energy levels to be positive, while stability in systems of noninteracting fermions is ensured by the exclusion principle, whatever the single-particle energies. As a consequence, simple models of uncorrelated disorder are less useful for bosonic systems than for fermionic ones, and it is generally important to study the excitation spectrum in conjunction with the problem of constructing a disorder-dependent ground state: we show how a mapping to an operator with chiral symmetry provides a useful tool for doing this. A second difference involves the distinction for bosonic systems between excitations which are Goldstone modes and those which are not. In the case of Goldstone modes we review established results illustrating the fact that disorder decouples from excitations in the low-frequency limit, above a critical dimension dc, which in different circumstances takes the values dc=2 and dc=0. For bosonic excitations which are not Goldstone modes, we argue that an excitation density varying with frequency as $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{4}$ is a universal feature in systems with ground states that depend on the disorder realization. We illustrate our conclusions with extensive analytical and some numerical calculations for a variety of models in one dimension.
Abstract The hole probability, i.e. , the probability that a region is void of particles, is a benchmark of correlations in many-body systems. We compute analytically this probability P ( R ) for a sphere of radius R in the case of N noninteracting fermions in their ground state in a d -dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large N and in the bulk of the Fermi gas, P ( R ) is described by a universal scaling function of k F R , for which we obtain an exact formula ( k F being the local Fermi wave vector). It exhibits a super-exponential tail where is a universal amplitude, in good agreement with existing numerical simulations. When R is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.
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.
We investigate numerically the nature of energy eigenstates in one-dimensional bond-disordered Anderson models with hopping amplitudes decreasing as ${H}_{\mathrm{ij}}\ensuremath{\propto}1/|i\ensuremath{-}j{|}^{\ensuremath{\alpha}}.$ The eigenstates become delocalized whenever the hopping amplitudes decay slower than 1/r. By performing an exact diagonalization scheme on finite chains, we compute the participation ratio of all energy eigenstates. Employing a finite-size scaling analysis, we report on the relevant scaling exponents characterizing this delocalization transition as well as the level-spacing distribution at the critical point $\ensuremath{\alpha}=1.$ The random hopping amplitudes are taken from both uniform and random sign distributions. We show that these models display similar critical behavior in the vicinity of $\ensuremath{\alpha}=1.$ However, the random sign model exhibits an asymptotic delocalization in the limit of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ and the universal scaling behavior in this regime is also reported.
We explicitly calculate the four leading-order terms of the formal asymptotic series for large |ε| (where ε denotes the external energy parameter) of the single-particle Green function G σ(ε), σ ϵ {−s, −s + 1, …, s}, and the three leading-order terms of that of the self-energy operator ∑σ(ε), pertaining to systems of spin-s fermions (s is a half-integer) in d-dimensional spatial space, interacting through an arbitrary two-body potential v(r-r'). These contributions, which are expressed in terms at most of a three-body static ground-state correlation function, are amenable to accurate numerical calculation through employing correlated many-particle ground-state wavefunctions such as determined within, for example, the quantum Monte Carlo framework. Such calculations will provide indisputably reliable information with regard to the spectral function of the single-particle excitations at high energies of correlated systems, as well as energy moments of this spectral function, and thus help to assess the reliability of theoretical approaches that are applied in studying such systems. We give especial attention to d = 3 and v ≡ v C, the long-range Coulomb potential, for which case we explicitly calculate the five leading-order terms of the regularized large-|ε| asymptotic series of ∑σ(ε). Our considerations reveal some interesting aspects which are very specific to the behaviour of v C(r-r') at both small and large values of ||r-r'||. In particular, we show that in these systems an inhomogeneity, even on the atomic scale, in the particle spin polarization density gives rise to a pronounced effect, directly discernible in the inverse photoemission spectra; we show this effect to be absent in models with v bounded at origin and those in which v ≡ v C but Umklapp processes are neglected, unless the ground state possesses long-range magnetic order. Our analyses shed light on the importance of the non-local part of the self-energy operator and disclose that some of the strictly non-local contributions to this operator transform into local contributions upon replacing v ≢ v C by v C, implying that a local approximation to ∑σ(ε), which for v ≢ v C proves accurate, is necessarily less accurate for v ≡ v C. These findings establish a fundamental limitation of the so-called ‘dynamical mean-field’ approximation to ∑σ(ε), which is strictly local, specifically in applications where v ≡ v C. We further explicitly establish some of the shortcomings of ∑σ(ε) as calculated within the framework of the many-body perturbation theory. In this context we demonstrate the empirically well-known inadequacy of the dynamically screened exchange self-energy operator in particular for describing the photoemission and inverse photoemission spectra of interacting systems at intermediate and large transfer energies and put forward a workable scheme that rids this self-energy of its fundamental defects. We present ample explicit analyses of our results in terms of uncorrelated many-body wavefunctions.