Large-scale Exact Diagonalization Research Articles

One-particle entanglement for one-dimensional spinless fermions after an interaction quantum quench

Particle entanglement provides information on quantum correlations in systems of indistinguishable particles. Here, we study the one particle entanglement entropy for an integrable model of spinless, interacting fermions both at equilibrium and after an interaction quantum quench. Using both large scale exact diagonalization and time dependent density matrix renormalization group calculations, we numerically compute the one body reduced density matrix for the J-V model, as well as its post-quench dynamics. We include an analysis of the fermionic momentum distribution, showcasing its time evolution after a quantum quench. Our numerical results, extrapolated to the thermodynamic limit, can be compared with field theoretic bosonization in the Tomonaga-Luttinger liquid regime. Excellent agreement is obtained using an interaction cutoff that can be determined uniquely in the ground state.

Exploring bosonic and fermionic link models on (3+1)D tubes

Quantum link models (QLMs) have attracted a lot of attention in recent times as a generalization of Wilson's lattice gauge theories (LGT), and are particularly suitable for realization on quantum simulators and computers. These models are known to host new phases of matter and act as a bridge between particle and condensed matter physics. In this article, we study the Abelian $U(1)$ lattice gauge theory in $(3+1)$-d tubes using large-scale exact diagonalization (ED). We are then able to motivate the phase diagram of the model with finite size scaling techniques (FSS), and in particular propose the existence of a Coulomb phase. Furthermore, we introduce the first models involving fermionic quantum links, which generalize the gauge degrees of freedom to be of fermionic nature. We prove that while the spectra remain identical between the bosonic and the fermionic versions of the $U(1)$-symmetric quantum link models in $(2+1)$-d, they are different in $(3+1)$-d. We discuss the prospects of realizing the magnetic field interactions as correlated hopping in quantum simulator experiments.

Quantum chaos and ensemble inequivalence of quantum long-range Ising chains

We use large-scale exact diagonalization to study the quantum Ising chain in a transverse field with long-range power-law interactions decaying with exponent $\alpha$. We numerically study various probes for quantum chaos and eigenstate thermalization {on} the level of eigenvalues and eigenstates. The level-spacing statistics yields a clear sign towards a Wigner-Dyson distribution and therefore towards quantum chaos across all values of $\alpha>0$. Yet, for $\alpha<1$ we find that the microcanonical entropy is nonconvex. This is due to the fact that the spectrum is organized in energetically separated multiplets for $\alpha<1$. While quantum chaotic behaviour develops within the individual multiplets, many multiplets don't overlap and don't mix with each other, as we analytically and numerically argue. Our findings suggest that a small fraction of the multiplets could persist at low energies for $\alpha\ll 1$ even for large $N$, giving rise to ensemble inequivalence.

Many-body localization in XY spin chains with long-range interactions: An exact-diagonalization study

We investigate the transition from the many-body localized phase to the ergodic thermalized phase at an infinite temperature in an $XY$ spin chain with $L$ spins, which experiences power-law decaying interactions in the form of $V_{ij}\propto1/\left|i-j\right|^{\alpha}$ ($i,j=1,\cdots,L$) and a random transverse field. By performing large-scale exact diagonalization for the chain size up to $L=18$, we systematically analyze the energy gap statistics, half-chain entanglement entropy, and uncertainty of the entanglement entropy of the system at different interaction exponents $\alpha$. The finite-size critical scaling allows us to determine the critical disorder strength $W_{c}$ and critical exponent $\nu$ at the many-body localization phase transition, as a function of the interaction exponent $\alpha$ in the limit $L\rightarrow\infty$. We find that both $W_{c}$ and $\nu$ diverge when $\alpha$ decreases to a critical power $\alpha_{c}\simeq1.16\pm0.17$, indicating the absence of many-body localization for $\alpha<\alpha_{c}$. Our result is useful to resolve the contradiction on the critical power found in two previous studies, $\alpha_{c}=3/2$ from scaling argument in Phys. Rev. B \textbf{92}, 104428 (2015) and $\alpha_{c}\approx1$ from quantum dynamics simulation in Phys. Rev. A \textbf{99}, 033610 (2019).

Sublattice coding algorithm and distributed memory parallelization for large-scale exact diagonalizations of quantum many-body systems

We present algorithmic improvements for fast and memory-efficient use of discrete spatial symmetries in Exact Diagonalization computations of quantum many-body systems. These techniques allow us to work flexibly in the reduced basis of symmetry-adapted wave functions. Moreover, a parallelization scheme for the Hamiltonian-vector multiplication in the Lanczos procedure for distributed memory machines avoiding load balancing problems is proposed. We demonstrate that using these methods low-energy properties of systems of up to 50 spin-1/2 particles can be successfully determined.

Large-scale exact diagonalizations reveal low-momentum scales of nuclei

Ab initio methods aim to solve the nuclear many-body problem with controlled approximations. Virtually exact numerical solutions for realistic interactions can only be obtained for certain special cases such as few-nucleon systems. Here we extend the reach of exact diagonalization methods to handle model spaces with dimension exceeding $10^{10}$ on a single compute node. This allows us to perform no-core shell model (NCSM) calculations for 6Li in model spaces up to $N_\mathrm{max} = 22$ and to reveal the 4He+d halo structure of this nucleus. Still, the use of a finite harmonic-oscillator basis implies truncations in both infrared (IR) and ultraviolet (UV) length scales. These truncations impose finite-size corrections on observables computed in this basis. We perform IR extrapolations of energies and radii computed in the NCSM and with the coupled-cluster method at several fixed UV cutoffs. It is shown that this strategy enables information gain also from data that is not fully UV converged. IR extrapolations improve the accuracy of relevant bound-state observables for a range of UV cutoffs, thus making them profitable tools. We relate the momentum scale that governs the exponential IR convergence to the threshold energy for the first open decay channel. Using large-scale NCSM calculations we numerically verify this small-momentum scale of finite nuclei.

Power-Law Entanglement Spectrum in Many-Body Localized Phases.

The entanglement spectrum of the reduced density matrix contains information beyond the von Neumann entropy and provides unique insights into exotic orders or critical behavior of quantum systems. Here, we show that strongly disordered systems in the many-body localized phase have power-law entanglement spectra, arising from the presence of extensively many local integrals of motion. The power-law entanglement spectrum distinguishes many-body localized systems from ergodic systems, as well as from ground states of gapped integrable models or free systems in the vicinity of scale-invariant critical points. We confirm our results using large-scale exact diagonalization. In addition, we develop a matrix-product state algorithm which allows us to access the eigenstates of large systems close to the localization transition, and discuss general implications of our results for variational studies of highly excited eigenstates in many-body localized systems.

Many-body localization edge in the random-field Heisenberg chain

We present a large scale exact diagonalization study of the one dimensional spin $1/2$ Heisenberg model in a random magnetic field. In order to access properties at varying energy densities across the entire spectrum for system sizes up to $L=22$ spins, we use a spectral transformation which can be applied in a massively parallel fashion. Our results allow for an energy-resolved interpretation of the many body localization transition including the existence of an extensive many-body mobility edge. The ergodic phase is well characterized by Gaussian orthogonal ensemble statistics, volume-law entanglement, and a full delocalization in the Hilbert space. Conversely, the localized regime displays Poisson statistics, area-law entanglement and non ergodicity in the Hilbert space where a true localization never occurs. We perform finite size scaling to extract the critical edge and exponent of the localization length divergence.

Absence of a gap in the Gaffnian state

We study the Gaffnian trial wavefunction proposed to describe fractional quantum Hall correlations at Bose filling factor $\nu=2/3$ and Fermi filling $\nu=2/5$. A family of Hamiltonians interpolating between a hard-core interaction for which the physics is known and a projector whose ground state is the Gaffnian is studied in detail. We give evidence for the absence of a gap by using large-scale exact diagonalizations in the spherical geometry. This is in agreement with recent arguments based on the fact that this wavefunction is constructed from a non-unitary conformal field theory. By using the cylinder geometry, we discuss in detail the nature of the underlying minimal model and we show the appearance of heterotic conformal towers in the edge energy spectrum where left and right movers are generated by distinct primary operators.

Interplay of charge and spin fluctuations of strongly interacting electrons on the kagome lattice

We study electrons hopping on a kagome lattice at third filling described by an extended Hubbard Hamiltonian with on-site and nearest-neighbor repulsions in the strongly correlated limit. As a consequence of the commensurate filling and the large interactions, each triangle has precisely two electrons in the effective low-energy description, and these electrons form chains of different lengths. The effective Hamiltonian includes the ring exchange around the hexagons as well as the nearest-neighbor Heisenberg interaction. Using large-scale exact diagonalization, we find that the effective model exhibits two phases: If the charge fluctuations are small, the magnetic fluctuations confine the charges to short loops around hexagons, yielding a gapped charge-ordered phase. When the charge fluctuations dominate, the system undergoes a quantum phase transition to a resonating plaquette phase with ordered spins and gapless spin excitations. We find that a peculiar conservation law is fulfilled: the electron in the chains can be divided into two sublattices, and this division is conserved by the ring exchange term.

Numerical study of magnetization plateaus in the spin-12kagome Heisenberg antiferromagnet

We clarify the existence of several magnetization plateaux for the kagome $S=1/2$ antiferromagnetic Heisenberg model in a magnetic field. Using approximate or exact localized magnon eigenstates, we are able to describe in a similar manner the plateau states that occur for magnetization per site $m=1/3$, $5/9$, and $7/9$ of the saturation value. These results are confirmed using large-scale Exact Diagonalization on lattices up to 63 sites.

Doping evolution of the oxygenK-edge x-ray absorption spectra of cuprate superconductors using a three-orbital Hubbard model

We study oxygen K-edge x-ray absorption spectroscopy (XAS) and investigate the validity of the Zhang-Rice singlet (ZRS) picture in overdoped cuprate superconductors. Using large-scale exact diagonalization of the three-orbital Hubbard model, we observe the effect of strong correlations manifesting in a dynamical spectral weight transfer from the upper Hubbard band to the ZRS band. The quantitative agreement between theory and experiment highlights an additional spectral weight reshuffling due to core-hole interaction. Our results confirm the important correlated nature of the cuprates and elucidate the changing orbital character of the low-energy quasi-particles, but also demonstrate the continued relevance of the ZRS even in the overdoped region.

Ground-state energy and spin gap of spin-12Kagomé-Heisenberg antiferromagnetic clusters: Large-scale exact diagonalization results

We present a comprehensive list of the ground-state energies and the spin gaps of finite Kagom\'e clusters with up to 42 spins obtained using large-scale exact diagonalization techniques. This represents the current limit of this exact approach. For a fixed number of spins $N$, we study several cluster shapes under periodic boundary conditions in both directions resulting in a toroidal geometry. The clusters are characterized by their side length and diagonal as well as the shortest ``Manhattan'' diameter of the torii. A finite-size scaling analysis of the ground-state energy as well as the spin gap is then performed in terms of the shortest toroidal diameter as well as the shortest ``Manhattan'' diameter. The structure of the spin-spin correlations further supports the importance of short loops wrapping around the torii.

Numerical studies of photon-based spectroscopies on high- [formula omitted] superconductors

A computational algorithm with efficient scaling to tens of thousands of CPUs has been applied to large scale exact diagonalization studies of small clusters. The goal is to simulate various photon-based spectra for effective single- and multi-orbital Hubbard Hamiltonians representative of the high- T c cuprate superconductors. As an example of the utility of this method, we address the optical band gap and the spectral functions calculated from these models which strongly depend on cluster geometry and size.

Emergent multipolar spin correlations in a fluctuating spiral: The frustrated ferromagnetic spin-12Heisenberg chain in a magnetic field

We present the phase diagram of the frustrated ferromagnetic $S=\frac{1}{2}$ Heisenberg ${J}_{1}\ensuremath{-}{J}_{2}$ chain in a magnetic field, obtained by large scale exact diagonalizations and density matrix renormalization group simulations. A vector chirally ordered state, metamagnetic behavior and a sequence of spin-multipolar Luttinger liquid phases up to hexadecupolar kind are found. We provide numerical evidence for a locking mechanism, which can drive spiral states toward spin-multipolar phases, such as quadrupolar or octupolar phases. Our results also shed light on previously discovered spin-multipolar phases in two-dimensional $S=\frac{1}{2}$ quantum magnets in a magnetic field.

The frustrated ferromagnetic S = 1/2 Heisenberg chain in a magnetic field – How multipolar spin correlations emerge from magnetically ordered states

We present the phase diagram of the frustrated ferromagnetic S = 1/2 Heisenberg J1-J2 chain in a magnetic field, obtained by large scale exact diagonalizations and densitymatrix-renormalization-group simulations. A vector chirally ordered state, metamagnetic behavior and a sequence of spin multipolar Luttinger liquid phases up to hexadecupolar order are found. Starting from classical considerations, we point out that various multipolar correlations are imprinted in a magnetic state and that they can survive the onset of frustration and quantum fluctuations which destroy the conventional magnetic order. Our results also shed new light on previously discovered spin multipolar phases in two-dimensional S = 1/2 quantum magnets in a magnetic field.

Competing compressible and incompressible phases in rotating atomic Bose gases at filling factorν=2

We study the groundstates of weakly interacting atomic Bose gases under conditions of rapid rotation. We present the results of large-scale exact diagonalization studies on a periodic geometry (a torus) which allows studies of compressible states with broken translational symmetry. Focusing on filling factor $\ensuremath{\nu}=2$, we show a competition between the triangular vortex lattice, a quantum smectic state, and the incompressible $k=4$ Read-Rezayi state. We discuss the corrections arising from finite size effects, and the likely behavior for large system sizes. The Read-Rezayi state is stabilized by a moderate amount of additional dipolar interactions.

Scaling of entanglement entropy in the random singlet phase

We present numerical evidences for the logarithmic scaling of the entanglement entropy in critical random spin chains. Very large scale exact diagonalizations performed at the critical XX point up to L=2000 spins 1/2 lead to a perfect agreement with recent real-space renormalization-group predictions of Refael and Moore [Phys. Rev. Lett. {\bf 93}, 260602 (2004)] for the logarithmic scaling of the entanglement entropy in the Random Singlet Phase with an effective central charge ${\tilde{c}}=c\times \ln 2$. Moreover we provide the first visual proof of the existence the Random Singlet Phase thanks to the quantum entanglement concept.