One-particle Density Matrix Research Articles

Quasicondensation and off-diagonal long-range order of hard-core bosons during a free expansion

Abstract Quasicondensation in one dimension is known to occur for equilibrium systems of hard-core bosons (HCBs) at zero temperature. This phenomenon arises due to the off-diagonal long-range order in the ground state, characterized by a power-law decay of the one-particle density matrix $g_1(x,y)\sim |x-y|^{-1/2}$~--~a well-known outcome of Luttinger liquid theory. 
Remarkably, HCBs, when allowed to freely expand from an initial product state (i.e., characterized by initial zero correlation), exhibit quasicondensation and demonstrate the emergence of off-diagonal long-range order during nonequilibrium dynamics. 
This phenomenon has been substantiated by numerical and experimental investigations in the early 2000s.
In this work, we revisit the dynamical quasicondensation of HCBs, providing a fully analytical treatment of the issue. 
In particular, we derive an exact asymptotic formula for the equal-time one-particle density matrix by borrowing ideas from the framework of quantum Generalized Hydrodynamics. 
Our findings elucidate the phenomenology of quasicondensation and of dynamical fermionization occurring at different stages of the time evolution, as well as the crossover between the two. 

Entanglement in Bound States of Two Particles with a Localized Center-of-Mass Wave-Function

Abstract We explore some entanglement-related features of systems consisting of two particles coupled through a central potential. We consider two types of systems, constituted by particles in one spatial dimension interacting through an attractive Dirac Delta potential, or through a harmonic quadratic potential. The degree of freedom corresponding to the system’s center of mass is either regarded as confined within a one dimensional box, or described by a localized Gaussian wave-packet. As a quantitative indicator of the degree of entanglement between the particles, we use the linear entropy of the one-particle marginal density matrix. In general, this quantity cannot be calculated analytically (except in some special cases related to the Harmonic interaction). Consequently, the linear entropy has to be evaluated numerically. Since the concomitant numerical problem consist in evaluating a multi-dimensional integral in four dimensions, the Monte Carlo integration method is an appropriate one for this task. We explore numerically how the system’s entanglement depends on the different interaction potentials, on the different types of confinement for the system’s center of mass, and on the associated parameters describing the size and geometry of the system.

Interacting local topological markers: A one-particle density matrix approach for characterizing the topology of interacting and disordered states

While topology is a property of a quantum state itself, most existing methods for characterizing the topology of interacting phases of matter require direct knowledge of the underlying Hamiltonian. We offer an alternative by utilizing the one-particle density matrix formalism to extend the concept of the Chern, chiral, and Chern-Simons markers to include interactions. The one-particle density matrix of a free-fermion state is a projector onto the occupied bands, defining a Brillouin zone bundle of the given topological class. This is no longer the case in the interacting limit, but as long as the one-particle density matrix is gapped, its spectrum can be adiabatically flattened, connecting it to a topologically equivalent projector. The corresponding topological markers thus characterize the topology of the interacting phase. Importantly, the one-particle density matrix is defined in terms of a given state alone, making the local markers numerically favorable, and providing a valuable tool for characterizing topology of interacting systems when only the state itself is available. To demonstrate the practical use of the markers we use the chiral marker to identify the topology of midspectrum eigenstates of the Ising-Majorana chain across the transition between the ergodic and many-body localized phases. We also apply the chiral marker to random states with a known topology, and compare it with the entanglement spectrum degeneracy. Published by the American Physical Society 2024

Predicting the One-Particle Density Matrix with Machine Learning.

Two of the most widely used electronic-structure theory methods, namely, Hartree-Fock and Kohn-Sham density functional theory, require the iterative solution of a set of Schrödinger-like equations. The speed of convergence of such a process depends on the complexity of the system under investigation, the self-consistent-field algorithm employed, and the initial guess for the density matrix. An initial density matrix close to the ground-state matrix will effectively allow one to cut out many of the self-consistent steps necessary to achieve convergence. Here, we predict the density matrix of Kohn-Sham density functional theory by constructing a neural network that uses only the atomic positions as information. Such a neural network provides an initial guess for the density matrix far superior to that of any other recipes available. Furthermore, the quality of such a neural-network density matrix is good enough for the evaluation of interatomic forces. This allows us to run accelerated ab initio molecular dynamics with little to no self-consistent steps.

The diagonal behaviour of the one-particle Coulombic density matrix

The diagonal behaviour of the one-particle Coulombic density matrix

Exact one-particle density matrix for SU(N) fermionic matter-waves in the strong repulsive limit

We consider a gas of repulsive NN-component fermions confined in a ring-shaped potential, subjected to an effective magnetic field. For large repulsion strengths, we work out a Bethe ansatz scheme to compute the two-point correlation matrix and then the one-particle density matrix. Our results hold in the mesoscopic regime of finite but sufficiently large number of particles and system size that are not accessible by numerics. We access the momentum distribution of the system and analyse its specific dependence of interaction, magnetic field and number of components NN. In the context of cold atoms, the exact computation of the correlation matrix to determine the interference patterns that are produced by releasing cold atoms from ring traps is carried out.

Dynamics of mean-field bosons at positive temperature

We study the time evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in Deuchert–Seiringer (2021) that the one-particle density matrix of Gibbs states of the interacting trapped gas is given, to leading order in N , as N \rightarrow \infty , by that of the ideal gas, with the condensate wave function replaced by the minimizer of the Hartree energy functional. We show that this structure is stable with respect to the many-body evolution in the following sense: the dynamics can be approximated in terms of the time-dependent Hartree equation for the condensate wave function and in terms of the free evolution for the thermally excited particles. The main technical novelty of our work is the use of the Hartree–Fock–Bogoliubov equations to define a fluctuation dynamics.

A non-self-consistent tight-binding electronic structure potential in a polarized double-ζ basis set for all spd-block elements up to Z = 86.

Existing semiempirical molecular orbital methods suffer from the usually minimal atomic-orbital (AO) basis set used to simplify the calculations. Here, a completely new and consistently parameterized tight-binding electronic structure Hamiltonian evaluated in a deeply contracted, properly polarized valence double-zeta basis set (vDZP) is described. The inner-shell electrons are accounted for by standard, large-core effective potentials and approximations to them. The primary target of this so-called density matrix tight-binding method is to reproduce the one-particle density matrix P of a molecular ωB97X-V range-separated hybrid density functional theory (DFT) calculation in exactly the same basis set. Additional properties considered are orbital energies, dipole polarizabilities and dipole moments, and dipole polarizability derivatives. The key features of the method are as follows: (a) it is non-self-consistent with an overall fixed number of only three required matrix diagonalizations; (b) only AO overlap integrals are needed to construct the effective Hamiltonian matrix; (c) new P-dependent terms emulating non-local exchange are included; and (d) only element-specific empirical parameters (about 50 per element) need to be determined. The method globally achieves a high accuracy for the target properties at a speedup compared to the ωB97X-V/vDZP reference of about 3-4 orders of magnitude. It performs robustly for difficult transition metal complexes, for highly charged or zwitterionic systems, and for chemically unusual bonding situations, indicating a generally robust approximation of the (self-consistent) Kohn-Sham potential. As an example application, the vibrational Raman spectrum of an entire protein with 327 atoms with respect to the DFT reference calculation is shown. This method may be used out-of-the-box to generate molecular/atomic features for machine learning applications or as the basis for accurate high-speed DFT methods.

Effective Dynamics of Extended Fermi Gases in the High-Density Regime

We study the quantum evolution of many-body Fermi gases in three dimensions, in arbitrarily large domains. We consider both particles with non-relativistic and with relativistic dispersion. We focus on the high-density regime, in the semiclassical scaling, and we consider a class of initial data describing zero-temperature states. In the non-relativistic case we prove that, as the density goes to infinity, the many-body evolution of the reduced one-particle density matrix converges to the solution of the time-dependent Hartree equation, for short macroscopic times. In the case of relativistic dispersion, we show convergence of the many-body evolution to the relativistic Hartree equation for all macroscopic times. With respect to previous work, the rate of convergence does not depend on the total number of particles, but only on the density: in particular, our result allows us to study the quantum dynamics of extensive many-body Fermi gases.

Coherent and incoherent structures in fuzzy dark matter haloes

ABSTRACTWe show that fuzzy dark matter haloes exhibit spatial differentiation in the coherence of the field configuration, ranging from completely coherent in the central solitonic core to incoherent outside it, with a significant drop of the dimensionless phase-space density with increasing radius. The core is a pure condensate, overlapping perfectly with the Penrose-Onsager mode corresponding to the largest eigenvalue of the one-particle density matrix. The virialized outer halo exhibits no clear coherence as a whole upon radial and temporal averaging but can be described as a collection of local, short-lived quasi-condensate lumps, exhibiting suppressed fluctuations, which can be identified with the structures commonly referred to as granules. These localized regions are separated by vortices that form a dynamical web, inhibiting phase coherence across the entire halo. We further examine the core oscillations, finding that they are accurately described by two time-dependent parameters characterizing the size of the core, rc(t), and a crossover region, rt(t). For the haloes in our merger simulations, this feature is reflected in the (anti-)correlated oscillation of the peak value of the density power-spectrum. The halo’s turbulent vortex tangle appears to reach a quasi-equilibrium state over probed time-scales, with the incompressible component of the kinetic energy exhibiting a characteristic k−3 tail in its spectrum, indicative of a ρ ∼ r2 density profile around the quantum vortex cores. Comparison of the peak wavenumbers in the corresponding power-spectra shows the inter-vortex spacing and the granule length scale in the outer halo to be very similar and slightly above the core size.

Eigenvalue asymptotics for the one-particle density matrix

The one-particle density matrix γ(x,y) for a bound state of an atom or molecule is one of the key objects in the quantum-mechanical approximation schemes. We prove the asymptotic formula λk∼(Ak)−8∕3, A≥0, as k→∞, for the eigenvalues λk of the self-adjoint operator Γ≥0 with kernel γ(x,y).

Superconductivity, generalized random phase approximation and linear scaling methods

The superfluid weight is an important observable of superconducting materials since it is related to the London penetration depth of the Meissner effect. It can be computed from the change in the grand potential (or free energy) in response to twisted boundary conditions in a torus geometry. Here we review the Bardeen–Cooper–Schrieffer mean-field theory emphasizing its origin as a variational approximation for the grand potential. The variational parameters are the effective fields that enter in the mean-field Hamiltonian, namely the Hartree–Fock potential and the pairing potential. The superfluid weight is usually computed by ignoring the dependence of the effective fields on the twisted boundary conditions. However, it has been pointed out in recent works that this can lead to unphysical results, particularly in the case of lattice models with flat bands. As a first result, we show that taking into account the dependence of the effective fields on the twisted boundary conditions leads in fact to the generalized random phase approximation. Our second result is providing the mean-field grand potential as an explicit function of the one-particle density matrix. This allows us to derive the expression for the superfluid weight within the generalized random phase approximation in a transparent manner. Moreover, reformulating mean-field theory as a well-posed minimization problem in terms of the one-particle density matrix is a first step towards the application to superconducting systems of the linear scaling methods developed in the context of electronic structure theory.

Eigenvalue asymptotics for the one-particle kinetic energy density operator

The kinetic energy of a multi-particle system is described by the one-particle kinetic energy density matrix τ(x,y). Alongside the one-particle density matrix γ(x,y) it is one of the key objects in the quantum-mechanical approximation schemes. We prove the asymptotic formula λk∼(Bk)−2, B⩾0, as k→∞, for the eigenvalues λk of the self-adjoint operator T⩾0 with kernel τ(x,y).

Making optical excitations visible – An exciton wavefunction extension to the time-dependent configuration interaction method

We report the animation and analysis of laser-driven many-electron dynamics by means of time-dependent local densities and quantities derived from the one-particle transition density matrix, such as time-dependent natural transition orbital densities, particle and hole positions and exciton size. The time-dependent configuration interaction method was used to revisit studies on hydrogen molecule and lithium cyanide by Saalfrank and coworkers and our study shines new light on the optical transitions in these small molecules to benchmark the implemented time-dependent exciton properties. One focus of our simulations is the comparison of the local densities to the quantities of the two-body exciton wavefunction for resonant excitations leading either to state-to-state transitions or to the creation of a wave packet.

Deformation of localized states and state transitions in systems of randomly hopping interacting fermions

We numerically study the random-hopping fermions (the Cruetz ladder) with repulsion and investigate how the interactions deform localized eigenstates by means of the one particle-density matrix (OPDM). The ground state exhibits resurgence of localization from the compact localized state to strong-repulsion-induced localization. On the other hand, excited states in the middle of the spectrum tend to extend by the repulsion. The transition property obtained by numerical calculations of the OPDM is deeply understood by studying a solvable model in which local integrals of motion (LIOMs) are obtained explicitly. The present work clarifies the utility of the OPDM and also how compact-support LIOMs in non-interacting limit are deformed by the repulsion.

Constructing Hubbard models for the hydrogen chain using sliced-basis density matrix renormalization group

Sliced-basis DMRG(sb-DMRG) is used to simulate a chain of hydrogen atoms and to construct low-energy effective Hubbard-like models. The downfolding procedure first involves a change of basis to a set of atom-centered Wannier functions constructed from the natural orbitals of the exact DMRG one-particle density matrix. The Wannier function model is then reduced to a fewer-parameter Hubbard-like model, whose parameters are determined by minimizing the expectation value of the Wannier Hamiltonian in the ground state of the Hubbard Hamiltonian. This indirect variational procedure not only yields compact and simple models for the hydrogen chain, but also allows us to explore the importance of constraints in the effective Hamiltonian, such as the restricting the range of the single-particle hopping and two-particle interactions, and to assess the reliability of more conventional downfolding. The entanglement entropy for a model's ground state, cut in the middle, is an important property determining the ability of DMRG and tensor networks to simulate the model, and we study its variation with the range of the interactions. Counterintuitively, we find that shorter ranged interactions often have larger entanglement.

Longitudinal and transversal resonant tunneling of interacting bosons in a two-dimensional Josephson junction

We unravel the out-of-equilibrium quantum dynamics of a few interacting bosonic clouds in a two-dimensional asymmetric double-well potential at the resonant tunneling scenario. At the single-particle level of resonant tunneling, particles tunnel under the barrier from, typically, the ground-state in the left well to an excited state in the right well, i.e., states of different shapes and properties are coupled when their one-particle energies coincide. In two spatial dimensions, two types of resonant tunneling processes are possible, to which we refer to as longitudinal and transversal resonant tunneling. Longitudinal resonant tunneling implies that the state in the right well is longitudinally-excited with respect to the state in the left well, whereas transversal resonant tunneling implies that the former is transversely-excited with respect to the latter. We show that interaction between bosons makes resonant tunneling phenomena in two spatial dimensions profoundly rich, and analyze these phenomena in terms of the loss of coherence of the junction and development of fragmentation, and coupling between transverse and longitudinal degrees-of-freedom and excitations. To this end, a detailed analysis of the tunneling dynamics is performed by exploring the time evolution of a few physical quantities, namely, the survival probability, occupation numbers of the reduced one-particle density matrix, and the many-particle position, momentum, and angular-momentum variances. To accurately calculate these physical quantities from the time-dependent many-boson wavefunction, we apply a well-established many-body method, the multiconfigurational time-dependent Hartree for bosons (MCTDHB), which incorporates quantum correlations exhaustively. By comparing the survival probabilities and variances at the mean-field and many-body levels of theory and investigating the development of fragmentation, we identify the detailed mechanisms of many-body longitudinal and transversal resonant tunneling in two dimensional asymmetric double-wells. In particular, we find that the position and momentum variances along the transversal direction are almost negligible at the longitudinal resonant tunneling, whereas they are substantial at the transversal resonant tunneling which is caused by the combination of the density and breathing mode oscillations. We show that the width of the interparticle interaction potential does not affect the qualitative physics of resonant tunneling dynamics, both at the mean-field and many-body levels. In general, we characterize the impact of the transversal and longitudinal degrees-of-freedom in the many-boson tunneling dynamics at the resonant tunneling scenarios.

Analyticity of the One-Particle Density Matrix

It is proved that the one-particle density matrix gamma (x, y) for multi-particle Coulombic systems is real analytic away from the nuclei and from the diagonal x = y.

Scaling properties of a spatial one-particle density-matrix entropy in many-body localized systems

We investigate a spatial subsystem entropy extracted from the one-particle density matrix (OPDM) in one-dimensional disordered interacting fermions that host a many-body localized (MBL) phase. Deep in the putative MBL regime, this OPDM entropy exhibits the salient features of localization, despite not being a proper entanglement measure. We numerically show that the OPDM entropy of the eigenstates obeys an area law. Similar to the von-Neumann entropy, the OPDM entropy grows logarithmically with time after a quantum quench, albeit with a different prefactor. Both these features survive at moderately large interactions and well towards the transition into the ergodic phase. The computational cost to calculate the OPDM entropy scales only polynomially with the system size, suggesting that the OPDM provides a promising starting point for developing diagnostic tools for MBL in simulations and experiments.

Multifractality and Fock-space localization in many-body localized states: One-particle density matrix perspective

Many-body localization (MBL) is well characterized in Fock space. To quantify the degree of this Fock-space localization, the multifractal dimension ${D}_{q}$ is employed; it has been claimed that ${D}_{q}$ shows a jump from the delocalized value ${D}_{q}=1$ in the eigenstate thermalization hypothesis (ETH) phase to a smaller value $0<{D}_{q}<1$ at the ETH-MBL transition, yet exhibiting a conspicuous discrepancy from the fully localized value ${D}_{q}=0$, which indicates that multifractality remains inside the MBL phase. Here, to better quantify the situation, we employ, instead of the commonly used computational basis, the one-particle density matrix (OPDM) and use its eigenstates (natural orbitals) as a Fock state basis for representing many-body eigenstates $|\ensuremath{\psi}\ensuremath{\rangle}$ of the system. Using this basis, we compute ${D}_{q}$ and other indices quantifying the Fock-space localization, such as the local purity $S$, which is derived from the occupation spectrum ${{n}_{\ensuremath{\alpha}}}$ (eigenvalues of the OPDM). We highlight the statistical distribution of Hamming distance ${x}_{\ensuremath{\mu}\ensuremath{\nu}}$ occurring in the pairwise coefficients $|{a}_{\ensuremath{\mu}}{|}^{2}{|{a}_{\ensuremath{\nu}}|}^{2}$ in $S$, and compare this with a related quantity considered in the literature.