Numerical Diagonalization Research Articles

Majorana fermions in finite-size strips with in-plane magnetic fields

We study the Majorana bound states arising in quasi-one-dimensional systems with Rashba spin-orbit coupling in the presence of an in-plane Zeeman magnetic field. Using two different methods, first, the numerical diagonalization of the tight-binding Hamiltonian, and second, finding the singular points of the Hamiltonian (see Refs. [1-4]), we obtain the topological phase diagram for these systems as a function of the chemical potential and the magnetic field, and we demonstrate the consistency of these two methods. By introducing disorder into these systems we confirm that the states with even number of Majorana pairs are not topologically protected. Finally, we show that a formal calculation of the $\mathbb{Z}_2$ topological invariants recovers correctly the parity of the number of Majorana bound states pairs, and it is thus fully consistent with the phase diagrams of the disordered systems.

Thermal inclusions: how one spin can destroy a many-body localized phase.

Many-body localized (MBL) systems lie outside the framework of statistical mechanics, as they fail to equilibrate under their own quantum dynamics. Even basic features of MBL systems, such as their stability to thermal inclusions and the nature of the dynamical transition to thermalizing behaviour, remain poorly understood. We study a simple central spin model to address these questions: a two-level system interacting with strength J with N≫1 localized bits subject to random fields. On increasing J, the system transitions from an MBL to a delocalized phase on the vanishing scale Jc(N)∼1/N, up to logarithmic corrections. In the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either 'on' (strongly entangled) or 'off' (weakly entangled) in eigenstates. The clusters of 'on' bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogeneous discontinuous transition out of the localized phase in single-site observables. We obtain these results by perturbative mapping to bond percolation on the hypercube at small J and by numerical exact diagonalization of the full many-body system. Our results support the arguments that the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions d that are high but finite.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

Numerical Study of Quantum Hall Bilayers at Total Filling ν_{T}=1: A New Phase at Intermediate Layer Distances.

We study the phase diagram of quantum Hall bilayer systems with total filing ν_{T}=1/2+1/2 of the lowest Landau level as a function of layer distances d. Based on numerical exact diagonalization calculations, we obtain three distinct phases, including an exciton superfluid phase with spontaneous interlayer coherence at small d, a composite Fermi liquid at large d, and an intermediate phase for 1.1<d/l_{B}<1.8 (l_{B} is the magnetic length). The transition from the exciton superfluid to the intermediate phase is identified by (i)a dramatic change in the Berry curvature of the ground state under twisted boundary conditions on the two layers and (ii)an energy level crossing of the first excited state. The transition from the intermediate phase to the composite Fermi liquid is identified by the vanishing of the exciton superfluid stiffness. Furthermore, from our finite-size study, the energy cost of transferring one electron between the layers shows an even-odd effect and possibly extrapolates to a finite value in the thermodynamic limit, indicating the enhanced intralayer correlation. Our identification of an intermediate phase and its distinctive features shed new light on the theoretical understanding of the quantum Hall bilayer system at total filling ν_{T}=1.

Diagonalizing the Hamiltonian of [formula omitted]4 theory in 2 space–time dimensions

We propose a new non-perturbative technique for calculating the scattering amplitudes of field-theory directly from the eigenstates of the Hamiltonian. Our method involves a discretized momentum space and a momentum cutoff, thereby truncating the Hilbert space and making numerical diagonalization of the Hamiltonian achievable. We show how to do this in the context of a simplified λϕ4 theory in two space–time dimensions. We present the results of our diagonalization, its dependence on time, its dependence on the parameters of the theory and its renormalization.

Hard-wall confinement of a fractional quantum Hall liquid

We make use of numerical exact diagonalization calculations to explore the physics of $\nu = 1/2$ bosonic fractional quantum Hall (FQH) droplets in the presence of experimentally realistic cylindrically symmetric hard-wall potentials. This kind of confinement is found to produce very different many-body spectra compared to a harmonic trap or the so-called extremely steep limit. For a relatively weak confinement, the degeneracies are lifted and the low-lying excited states organize themselves in energy branches that can be explained in terms of their Jack polynomial representation. For a strong confinement, a strong spatial deformation of the droplet is found, with an unexpected depletion of its central density.

Reprint of : Gapless quantum spin liquid of the kagome-lattice antiferromagnet

The S=1/2 kagome-lattice antiferromagnet is studied using the numerical diagonalization of finite clusters and a finite-size scaling. The susceptibility analysis at the zero magnetization indicates that the system has a gapless spin excitation. It is consistent with our previous finite-size scaling analysis of the spin gap.

Gapless spin excitations in the [formula omitted] Kagome- and triangular-lattice Heisenberg antiferromagnets

The S=1/2 kagome- and triangular-lattice Heisenberg antiferromagnets are investigated using the numerical exact diagonalization and the finite-size scaling analysis. The behaviour of the field derivative at zero magnetization is examined for both systems. The present result indicates that the spin excitation is gapless for each system.

Quantum signature of breathers in 1D ultracold bosons in optical lattices involving next-nearest neighbor interactions

The dynamics and the energy spectrum of an ultracold gas of bosonic atoms in an optical lattice can be described by a Bose–Hubbard model for which the system parameters can be controlled by laser light. We study by means of the perturbation theory in addition to the numerical diagonalization, the energy spectrum and the related features of the band structures of the ultracold bosons in optical lattices containing a few number of quanta interacting with next-nearest neighbor interactions (NNNI) modeled by the Bose–Hubbard Hamiltonian. The energy spectra of such system display the bound states signature, which are analyzed in the first Brillouin zone for different wave numbers. The finding, i.e., quantum breathers, shows that their probabilities’ weight depends on the wave vector. The influence of NNNI on both the probabilities’ amplitude and the correlation function is also realized in case of a system with a small number of sites, respectively.

Exciton-phonon dynamics on complex networks: Comparison between a perturbative approach and exact calculations.

A method combining perturbation theory with a simplifying ansatz is used to describe the exciton-phonon dynamics in complex networks. This method, called PT^{*}, is compared to exact calculations based on the numerical diagonalization of the exciton-phonon Hamiltonian for eight small-sized networks. It is shown that the accuracy of PT^{*} depends on the nature of the network, and three different situations were identified. For most graphs, PT^{*} yields a very accurate description of the dynamics. By contrast, for the Wheel graph and the Apollonian network, PT^{*} reproduces the dynamics only when the exciton occupies a specific initial state. Finally, for the complete graph, PT^{*} breaks down. These different behaviors originate in the interplay between the degenerate nature of the excitonic energy spectrum and the strength of the exciton-phonon interaction so that a criterion is established to determine whether or not PT^{*} is relevant. When it succeeds, our study shows the undeniable advantage of PT^{*} in that it allows us to perform very fast simulations when compared to exact calculations that are restricted to small-sized networks.

Finite-temperature dynamics and thermal intraband magnon scattering in Haldane spin-one chains

The antiferromagnetic spin-one chain is considerably one of the most fundamental quantum many-body systems, with symmetry protected topological order in the ground state. Here, we present results for its dynamical spin structure factor at finite temperatures, based on a combination of exact numerical diagonalization, matrix-product-state calculations and quantum Monte Carlo simulations. Open finite chains exhibit a sub-gap band in the thermal spectral functions, indicative of localized edge-states. Moreover, we observe the thermal activation of a distinct low-energy continuum contribution to the spin spectral function with an enhanced spectral weight at low momenta and its upper threshold. This emerging thermal spectral feature of the Haldane spin-one chain is shown to result from intra-band magnon scattering due to the thermal population of the single-magnon branch, which features a large bandwidth-to-gap ratio. These findings are discussed with respect to possible future studies on spin-one chain compounds based on inelastic neutron scattering.

Evidence of a fractional quantum Hall nematic phase in a microscopic model

The fractional quantum Hall (FQH) nematic state is a conjectured state of matter in which FQH topological order coexists with spontaneously broken rotation symmetry. Recent experiments on GaAs quantum wells suggest this state, as well as the concomitant isotropic-nematic phase transition, may be realized in a two-dimensional electron gas at certain rational filling factors. Although effective field theories of the FQH nematic state have been constructed, there has been so far no explicit realization of this state as the ground state of a microscopic model of interacting electrons. The authors show via a variational approach and numerical exact diagonalization that tuning Haldane pseudopotentials in a model of interacting electrons in the lowest Landau level induces a quantum phase transition out of the isotropic Laughlin FQH liquid and into the FQH nematic state.

Phase Transitions in Spin-1/2 Falicov–Kimball Model on a Two-dimensional Triangular Lattice

Phase transitions from low-temperature (ordered) phases to high-temperature (disordered/homogeneous) phases for different fillings are studied on a triangular lattice using the spin-dependent Falicov–Kimball model. Numerical diagonalization and Monte Carlo simulation methods are used to study thermodynamic properties of the system. It has been observed that low-temperature ordered phases persist up to a finite temperature and after reaching a critical temperature ( $$T_c$$ ), homogeneous phases are observed for all parameter space. We have also calculated the temperature dependence of specific heat and observed a sharp jump at $$T_c$$ indicating the phase transition, and this $$T_c$$ increases with increase in on-site Coulomb correlation U and electron fillings.

Excitonic complexes in strain-free and highly symmetric GaAs quantum dots fabricated by filling of self-assembled nanoholes

We perform a theoretical study of the optical transitions for different excitonic complexes in highly symmetric strain-free GaAs quantum dots (QDs) fabricated by epitaxially filling nanoholes (NHs) in an AlGaAs surface. NHs are formed by local droplet etching. As a first step, we propose a QD shape modeling consistent with atomic force microscopy (AFM) profiles and an experimental growth procedure. We investigate the QD height dependence of s- and p- shell exciton recombination energies in the framework of the effective mass approximation with an exact numerical diagonalization method. A comparison between theoretical results and available spectroscopic data is carried out. Systematic evolution of the binding energies of neutral (X), charged excitons (X–, X+) and the biexciton (XX), with QD height, is interpreted in terms of a balance between the Coulomb interactions and charge carrier correlation effects. Our calculations demonstrate the important role of the correlation energies in elucidating the bound character of all few-particle states especially the biexciton.

Hilbert space and ground-state structure of bilayer quantum Hall systems at ν=2/λ

We analyze the Hilbert space and ground state structure of bilayer quantum Hall (BLQH) systems at fractional filling factors $\nu=2/\lambda$ ($\lambda$ odd) and we also study the large $SU(4)$ isospin-$\lambda$ limit. The model Hamiltonian is an adaptation of the $\nu=2$ case [Z.F. Ezawa {\it et al.}, Phys. Rev. {B71} (2005) 125318] to the many-body situation (arbitrary $\lambda$ flux quanta per electron). The semiclassical regime and quantum phase diagram (in terms of layer distance, Zeeeman, tunneling, etc, control parameters) is obtained by using previously introduced Grassmannian $\mathbb{G}^4_{2}=U(4)/[U(2)\times U(2)]$ coherent states as variational states. The existence of three quantum phases (spin, canted and ppin) is common to any $\lambda$, but the phase transition points depend on $\lambda$, and the instance $\lambda=1$ is recovered as a particular case. We also analyze the quantum case through a numerical diagonalization of the Hamiltonian and compare with the mean-field results, which give a good approximation in the spin and ppin phases but not in the canted phase, where we detect exactly $\lambda$ energy level crossings between the ground and first excited state for given values of the tunneling gap. An energy band structure at low and high interlayer tunneling (spin and ppin phases, respectively) also appears depending on angular momentum and layer population imbalance quantum numbers.

Quantum Spin Liquid in the Kagome-Lattice Antiferromagnet and Related Systems

The S=1/2 kagome-lattice antiferromagnet is investigated using the numerical diagonalization up to the 42-spin cluster. The analysis of the field derivative of the magnetization at the zero magnetization indicates that the magnetic excitation ot the system is gapless. It is consistent with our previous finite-scaling analysis of the spin gap.

Anisotropy-driven transition from the Moore-Read state to quantum Hall stripes

We investigate the nature of the quantum Hall liquid in a half-filled second Landau level ($n=1$) as a function of band mass anisotropy using numerical exact diagonalization (ED) and density matrix renormalization group (DMRG) methods. We find increasing the mass anisotropy induces a quantum phase transition from the Moore-Read state to a charge density wave state. By analyzing the energy spectrum, guiding center structure factors and by adding weak pinning potentials, we show that this charge density wave is a uni-directional quantum Hall stripe, which has a periodicity of a few magnetic lengths and survives in the thermodynamic limit. We find smooth profiles for the guiding center occupation function that reveal the strong coupling nature of the array of chiral Luttinger liquids residing at the stripe edges.

Quantum phase transition in the twisted three-leg spin tube

We investigate the spin-1/2 twisted three-leg antiferromagnetic quantum spin tube in the presence of the easy-plane anisotropy, using the numerical diagonalization of finite-size clusters. We show cases of the existence and the absence of magnetization plateau at 1/3 height of saturation magnetization when the anisotropy is tuned. The phenomenological-renormalization group analysis indicates a quantum phase transition between the 1/3 magnetization plateau phase and the plateauless one. The phase diagram is also presented.

Permutation-symmetric three-particle hyper-spherical harmonics based on the S3 ⊗ SO(3)rot ⊂ O(2)⊗SO(3)rot ⊂ U(3)⋊S2 ⊂ O(6) subgroup chain

We construct the three-body permutation symmetric hyperspherical harmonics to be used in the non-relativistic three-body Schrödinger equation in three spatial dimensions (3D). We label the state vectors according to the S3⊗SO(3)rot⊂O(2)⊗SO(3)rot⊂U(3)⋊S2⊂O(6) subgroup chain, where S3 is the three-body permutation group and S2 is its two element subgroup containing transposition of first two particles, O(2) is the “democracy transformation”, or “kinematic rotation” group for three particles; SO(3)rot is the 3D rotation group, and U(3),O(6) are the usual Lie groups. We discuss the good quantum numbers implied by the above chain of algebras, as well as their relation to the S3 permutation properties of the harmonics, particularly in view of the SO(3)rot⊂SU(3) degeneracy. We provide a definite, practically implementable algorithm for the calculation of harmonics with arbitrary finite integer values of the hyper angular momentum K, and show an explicit example of this construction in a specific case with degeneracy, as well as tables of K≤6 harmonics. All harmonics are expressed as homogeneous polynomials in the Jacobi vectors (λ,ρ) with coefficients given as algebraic numbers unless the “operator method” is chosen for the lifting of the SO(3)rot⊂SU(3) multiplicity and the dimension of the degenerate subspace is greater than four – in which case one must resort to numerical diagonalization; the latter condition is not met by any K≤15 harmonic, or by any L≤7 harmonic with arbitrary K. We also calculate a certain type of matrix elements (the Gaunt integrals of products of three harmonics) in two ways: 1) by explicit evaluation of integrals and 2) by reduction to known SU(3) Clebsch–Gordan coefficients. In this way we complete the calculation of the ingredients sufficient for the solution to the quantum-mechanical three-body bound state problem.

AWESoMe 1.1: A code for the calculation of phase and group velocities of acoustic waves in homogeneous solids

AWESoMe 1.1: A code for the calculation of phase and group velocities of acoustic waves in homogeneous solids

Persistence of non-local correlations and quantum information theoretic measures in the thermal state of frustrated molecular wheels

ABSTRACTGenuine quantumness present in quantum systems is the resource for implementing quantum information and computation protocols which can outperform the classical counterparts. These quantumness measures encompass non-local ones known as quantum entanglement (QE) and quantum information theoretic ones, e.g. quantum discord (QD). In this paper, some well-known measures of QE and QD have been studied in two wheel-like frustrated molecular magnetic systems, with special emphasis on the survival of the measures in the thermal states. A molecular magnetic system similar to the first one (with half-integer spins ) has already been synthesised using coordination chemistry, and the other is hypothetical, where the dominant interaction is the spin–spin exchange interaction. Exact numerical diagonalisation methods have been used. Some non-trivial features, like nonmonotonicity of threshold temperatures with external magnetic field, persistence of multipartite entanglement and QD over bipartite entanglement, have been found. The measures, being operational ones, can be used to realise the resource of quantumness in experiments.