Exact Numerical Diagonalization Research Articles

Stoner Ferromagnetism in a Momentum-Confined Interacting 2D Electron Gas.

In this work we investigate the ground state of a momentum-confined interacting 2D electron gas, a momentum-space analog of an infinite quantum well. The study is performed by combining analytical results with a numerical exact diagonalization procedure. We find a ferromagnetic ground state near a particular electron density and for a range of effective electron (or hole) masses. We argue that this observation may be relevant to the generalized Stoner ferromagnetism recently observed in multilayer graphene systems. The collective magnon excitations exhibit a linear dispersion, which originates from a diverging spin stiffness.

Thermodynamics of resonating-valence-bond states toward the understanding of quantum spin liquid phenomena.

A quantum spin liquid (QSL) is a state of matter in which spins do not exhibit magnetic order. In contrast to paramagnets, the spins in a QSL interact strongly, similar to conventional ordered magnets; however the thermodynamic stability of QSLs is rarely studied. Here, the thermodynamic properties of centered hexagon nanoclusters were investigated using the Hubbard model, which was solved using exact numerical diagonalization. The total spin, spin-spin correlation functions, local magnetic moments, charge and spin gaps, and magnetocaloric effect were analyzed for a half-filled band as a function of the ratio between the on-site Coulomb repulsion and electronic hopping (U/t). The centered hexagon nanocluster exhibited an antiferromagnetic (AFM) behavior with exotic magnetic ordering. Resonating-valence-bond (RVB) states were observed for intermediate values of U/t, in which short-range spin-spin correlation functions were suppressed to minimize spin frustration. The AFM order was examined in terms of the Néel-like temperature derived from the temperature dependence of the magnetic susceptibility. An interesting result is that the systems under external magnetic fields exhibited an inverse magnetocaloric effect, which was remarkable for intermediate values of U/t, where the RVB state was observed. Owing to the novel discovery of exotic magnetic ordering in triangular moiré patterns in twisted bilayer graphene (TBLG) systems, these results provide insights into the onset of magnetism and the possible spin liquid states in these graphene moiré materials.

Z3 and (×Z3)3 symmetry protected topological paramagnets

We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by (×Z3)3 ≡ Z3 × Z3 × Z3 symmetry and smaller Z3 symmetry. We derive microscopic models for the gapless edge, uncover their symmetries and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary field. We argue, that the low energy limit of our edge modes defined by the SUk(3)/SUk(2) coset conformal field theory with the level k = 2. The discussed two-dimensional models realize a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.

Transition to chaos in extended systems and their quantum impurity models

Chaos sets a fundamental limit to quantum-information processing schemes. We study the onset of chaos in spatially extended quantum many-body systems that are relevant to quantum optical devices. We consider an extended version of the Tavis–Cummings model on a finite chain. By studying level-spacing statistics, adjacent gap ratios, and spectral form factors, we observe the transition from integrability to chaos as the hopping between the Tavis–Cummings sites is increased above a finite value. The results are obtained by means of exact numerical diagonalization which becomes notoriously hard for extended lattice geometries. In an attempt to circumvent these difficulties, we identify a minimal single-site quantum impurity model that successfully captures the spectral properties of the lattice model. This approach is intended to be adaptable to other lattice models with large local Hilbert spaces.

Fragility of the Schrödinger Cat in thermal environments

We describe the decoherence instability of Schrödinger Cat states in the two-site Bose-Hubbard model with an attractive on-site interaction between particles. For N particles with onsite attractive energy U and hopping amplitude between sites t, Cat states exist for uequiv frac{UN}{2t}<-1 at zero temperature. However, they are increasingly unstable to small thermal fluctuations as the Cat itself is increasingly well-defined and its components become well-separated. For any given u<-1, the decoherence temperature becomes smaller for large N. The loss of off-diagonal coherence peaks in the equilibrium density matrix is dominated by the thermal admixture of the first excited state of the many-body system with its ground state. Particle number fluctuations, described in the grand canonical ensemble also reduce coherence, but to a lesser degree than thermal fluctuations. The full density matrix of the Schrödinger Cat is obtained by exact numerical diagonalization of the many-body Hamiltonian and a narrow regime in the parameter space of the particle number, temperature, and U/t is identified where small Cat states may survive decoherence in a physical environment.

Entropy and temperature in finite isolated quantum systems.

We investigate how the temperature calculated from the microcanonical entropy compares with the canonical temperature for finite isolated quantum systems. We concentrate on systems with sizes that make them accessible to numerical exact diagonalization. We thus characterize the deviations from ensemble equivalence at finite sizes. We describe multiple ways to compute the microcanonical entropy and present numerical results for the entropy and temperature computed in these various ways. We show that using an energy window whose width has a particular energy dependence results in a temperature with minimal deviations from the canonical temperature.

Supergravity model of the Haldane-Rezayi fractional quantum Hall state

Supersymmetry and supergravity were invented in the 1970s to solve fundamental problems in high-energy physics. Even though neither of these ideas has yet been confirmed in high-energy and cosmology experiments, they have been beneficial in constructing numerous theoretical models including superstring theory. Despite the absence of supersymmetry in particle physics, it can potentially emerge in exotic phases of strongly correlated condensed matter systems. In this paper, we propose a supergravity model that describes the low-energy physics of the Haldane-Rezayi state, a gapless quantum Hall state that occurs in a half-filled Landau level. We show that the corresponding edge modes of the Haldane-Rezayi state and the Girvin-MacDonald-Platzman algebra appear naturally in the supergravity model. Finally, we substantiate our theoretical findings with numerical exact diagonalization calculations that support the appearance of the emergent graviton and gravitino excitations in the Haldane-Rezayi state.

Excited-eigenstate entanglement properties of XX spin chains with random long-range interactions

Quantum information theoretical measures are useful tools for characterizing quantum dynamical phases. However, employing them to study excited states of random spin systems is a challenging problem. Here, we report results for the entanglement entropy (EE) scaling of excited eigenstates of random XX antiferromagnetic spin chains with long-range (LR) interactions decaying as a power law with distance with exponent $\alpha$. To this end, we extend the real-space renormalization group technique for excited states (RSRG-X) to solve this problem with LR interaction. For comparison, we perform numerical exact diagonalization (ED) calculations. From the distribution of energy level spacings, as obtained by ED for up to $N\sim 18$ spins, we find indications of a delocalization transition at $\alpha_c \approx 1$ in the middle of the energy spectrum. With RSRG-X and ED, we show that for $\alpha>\alpha^*$ the entanglement entropy (EE) of excited eigenstates retains a logarithmic divergence similar to the one observed for the ground state of the same model, while for $\alpha<\alpha^*$ EE displays an algebraic growth with the subsystem size $l$, $S_l\sim l^{\beta}$, with $0<\beta<1$. We find that $\alpha^* \approx 1$ coincides with the delocalization transition $\alpha_c$ in the middle of the many-body spectrum. An interpretation of these results based on the structure of the RG rules is proposed, which is due to {\it rainbow} proliferation for very long-range interactions $\alpha\ll 1$. We also investigate the effective temperature dependence of the EE allowing us to study the half-chain entanglement entropy of eigenstates at different energy densities, where we find that the crossover in EE occurs at $\alpha^* < 1$.

Localized states in local isomorphism classes of pentagonal quasicrystals

A family of pentagonal quasicrystals can be defined by projecting a section of the five-dimensional cubic lattice to two dimensions. A single parameter, the sum of intercepts $\Gamma=\sum_j \gamma_j$, describes this family. Each value of $0\le \Gamma \le \frac{1}{2}$ defines a unique local isomorphism class for these quasicrystals, with $\Gamma=0$ giving the Penrose lattice. Except for a few special values of $\Gamma$, these lattices lack simple inflation-deflation rules making it hard to count how frequently a given local configuration is repeated. We consider the vertex-tight-binding model on these quasicrystals and investigate the strictly localized states (LS) for all values of $\Gamma$. We count the frequency of localized states both by numerical exact diagonalization on lattices of $~10^5$ sites and by identifying localized state types and calculating their perpendicular space images. While the imbalance between the number of sites of the sublattices of the bipartite quasicrystal grows monotonically with $\Gamma$, we find that the localized state fraction first decreases and then increases as the distance from the Penrose lattice grows. The highest LS fraction of $~10.17\%$ is attained at $\Gamma=0.5$ while the minimum is $~4.5\%$ at $\Gamma \simeq 0.12$. The LS on the even sublattice are generally concentrated near sites with high symmetry, while the LS on the odd sublattice are more uniformly distributed. The odd sublattice has a higher LS fraction, having almost three times the LS frequency of the even sublattice at $\Gamma=0.5$. We identify 20 LS types on the even sublattice, and their total frequency agrees well with the numerical exact diagonalization result for all values of $\Gamma$. For the odd sublattice, we identify 45 LS types. However, their total frequency remains significantly below the numerical calculation, indicating the possibility of more independent LS types.

Number of zero-energy eigenstates in the PXP model

The PXP model is paradigmatic in the field of quantum many-body scars. This model has a number of zero-energy eigenstates that is exponentially large in system size. Lower bounds on the number of zero-energy eigenstates are obtained for both open and periodic (zero and $\pi$-momentum sectors) boundary conditions. These bounds are found to be tight up to system sizes accessible by numerical exact diagonalization, and can be expected to be tight in general. In addition to previous results, separate lower bounds are obtained for the spatial inversion-symmetric and inversion-antisymmetric symmetry sectors. Furthermore, the derivations improve on previous ones as these are free of assumptions.

Vibration‐induced symmetry breaking in hybrid light‐matter dimer states

AbstractWe investigate the influence of vibronic coupling on a molecular dimer strongly coupled to a single cavity mode. In the framework of the Holstein‐Tavis‐Cummings model, the energy structure of the molecular dimer is analyzed by numerical exact diagonalization and perturbation theory. Under numerical exact diagonalization, we find that the degeneracy of lower polaritons vanishes in the presence of vibronic coupling. Under the second‐order degenerate perturbation theory, the degeneracy breaking of lower polaritons can be associated with asymmetric indirect interactions mediated by the upper polaritons and the dark states. The consistency of the two approaches confirms the robustness of our simulations, indicating that the vibration‐induced symmetry breaking should be experimentally observed.

Two-Hole Ground State: Dichotomy in Pairing Symmetry

A single-hole ground state Ansatz for the two-dimensional t-J model has been recently studied by the variational Monte Carlo (VMC) method. Such a doped hole behaves like a "twisted" non-Landau quasiparticle characterized by an emergent quantum number in agreement with exact numerics. In this work, we further investigate the ground state of two holes by VMC. It is found that the two holes strongly attract each other to form a pairing state with a new quantum number the same as obtained by the numerical exact diagonalization and density matrix renormalization group (DMRG) calculations. A unique feature of this pairing state is a dichotomy in the pairing symmetry, i.e., a d-wave in terms of the electron c operators and an s-wave in terms of the new quasiparticles, as explicitly illustrated in the ground state wave function. A similar VMC study of a two-hole wave function for the t-J two-leg ladder also yields a good agreement with the DMRG result. We demonstrate that the pairing mechanism responsible for the strong binding here is not due to the long-range antiferromagnetic order nor the resonating-valence-bound pairing in the spin background but is the consequence of the quantum phase-strings created by the hopping of holes. The resulting spin-current pattern mediating the pairing force is explicitly illustrated in the VMC calculation. Physical implications to superconductivity at finite doping are also discussed.

The quantum vortex states in extended Bose–Hubbard model: effects of lattice geometries, inter-particle interactions and spatial inhomogeneity

We study quantum vortex states of strongly interacting bosons in a two-dimensional rotating optical lattice. The system is modeled by Bose-Hubbard Hamiltonian with rotation. We consider lattices of different geometries, such as square, rectangular and triangular. Using numerical exact diagonalization method we show how the rotation introduces vortex states of different ground-state symmetries and the transition between these states at discrete rotation frequencies. We show how the geometry of the lattice plays crucial role in determining the maximum number of vortex states as well as the general characteristics of these states such as, the average angular momentum $<L_z>$, the current at the perimeter of the lattice, phase winding, the relation between the maximum phase difference, the maximum current and also the saturation of the current between the two neighboring lattice points. The effect of the two- and three-body interactions between the particles, both attractive and repulsive, also depends on the geometry of the lattice as the current flow or the lattice current depends on the interactions. We also consider the effect of the spatial inhomogeneity introduced by the presence of an additional confining harmonic trap potential. It is shown that the curvature of the trap potential and the position of the minimum of the trap potential with respect to the axis of rotation or the center of the lattice have a significant effect on the general characteristics these vortex states.

Josephson and Persistent Currents in a Quantum Ring between Topological Superconductors

AbstractIn this work, the spectra in an Aharonov–Bohm quantum‐ring interferometer forming a Josephson junction between two topological superconductor (TSC) nanowires are investigated. The TSCs host Majorana bound states at their edges, and both the magnetic flux and the superconducting phase difference between the TSCs are used as control parameters. A tight‐binding approach is used to model the quantum ring coupled to both TSCs, described by the Kitaev effective Hamiltonian. The problem is solved by means of exact numerical diagonalization of the Bogoliubov‐de Gennes Hamiltonian and obtain the spectra for two sizes of the quantum ring as a function of the magnetic flux and the phase difference between the TSCs. Depending on the size of the quantum ring and the coupling, the spectra display several patterns. Those are denoted as line, point, and undulated nodes, together with flat bands, which are topologically protected. The first three patterns can be possibly detected by means of persistent and Josephson currents. Hence, the results could be useful to understand the spectra and their relation with the behavior of the current signals.

Dimerization of Many-Body Subradiant States in Waveguide Quantum Electrodynamics.

We theoretically study subradiant states in an array of atoms coupled to photons propagating in a one-dimensional waveguide focusing on the strongly interacting many-body regime with large excitation fill factor f. We introduce a generalized many-body entropy of entanglement based on exact numerical diagonalization followed by a high-order singular value decomposition. This approach has allowed us to visualize and understand the structure of a many-body quantum state. We reveal the breakdown of fermionized subradiant states with increase of f with the emergence of short-ranged dimerized antiferromagnetic correlations at the critical point f=1/2 and the complete disappearance of subradiant states at f>1/2.

Localization dynamics from static and mobile impurities

We study the superfluid response and localization dynamics from static and mobile impurities. The superfluidity is formed in the rung-Mott phase of a bosonic ladder model producing spin-Meissner currents induced by a $\mathbb{U}(1)$ gauge field or a uniform magnetic field. Impurities are described through two-state systems which act as a two-peak random potential. An impurity sits either at the top or at the bottom of the ladder on each rung equally, producing a telegraph signal. The impurities-matter coupling gives rise to a classical Ising symmetry for static and mobile impurities associated to the inversion symmetry of the two legs of the ladder. From the decoupled rungs limit, we also identify a local $\mathbb{Z}_2$ gauge theory for mobile impurities. The properties of the system are studied from an effective quantum spin model including the possibility of four-body coupling in the limit of a strong interaction between bosons and impurities. Through analytical approaches and numerical exact diagonalization, we study the superfluid currents both in the weakly-coupled and strongly-coupled rungs limits for the bosons. In the weakly-coupled rungs situation, we find a smooth power-law localization whereas the strongly-coupled rungs limit produces a steep localization or insulating phase for various configurations of the two-peak random potential. In the strongly disordered situation, through entanglement and bipartite fluctuation measures, we also identify a many-body localization regime in time after a quench of the system when prepared in a N\' eel state.

Ab initio based ligand field approach to determine electronic multiplet properties

A method is developed to calculate the ligand field (LF) parameters and the multiplet spectra of local magnetic centers with open $d$ and $f$ shells in solids in a parameter-free way. This method proceeds from density functional theory and employs Wannier projections of nonmagnetic band structures onto local $d$ or $f$ orbitals. Energies of multiplets and optical, as well as x-ray spectra are determined by exact numerical diagonalization of a local Hamiltonian describing Coulomb, LF, and spin-orbit interactions. The method is tested for several $3d$ and $5f$ compounds for which the LF parameters and multiplet spectra are experimentally well known. In this way, we obtain good agreement with the experiments for ${\mathrm{La}}_{2}{\mathrm{NiO}}_{4}, {\mathrm{CaCuO}}_{2}, {\mathrm{Li}}_{2}{\mathrm{CuO}}_{2}$, ZnO:Co, and ${\mathrm{UO}}_{2}$.

Ground-state properties of a quantum frustrated spin chain with side spins and arbitrary spin s

The properties of the spin-s frustrated antiferromagnetic Heisenberg chain with side spins are studied by means of the coupled cluster method and the numerical exact diagonalization method. The results of the two methods both manifest that the ground state quantum phase diagram of the model consists of the Néel phase and the canted phase characterized by two pitch angles, irrespective of the value of the spin quantum number s. Although the phenomenon of quantum melting of magnetic long-range order is absent in the model, quantum fluctuation exerts an important influence on the property of the model. In addition to the reduction of magnetization, the nature of the phase transition from the Néel state to the canted state transforms from a classical second-order transition to a quantum first-order transition at any finite value of s. It is observed that the ways one of the two pitch angles evolves with frustration in quantum and classical cases, are qualitatively different in the large frustration parameter region. The location of the critical point and the physical quantities, including the ground state energy per spin and the magnetic order parameter, move toward their classical values in the form of power series in s.

Exact closed-form analytic wave functions in two dimensions: Contact-interacting fermionic spinful ultracold atoms in a rapidly rotating trap

Exact two-dimensional analytic wave functions for an arbitrary number $N$ of contact-interacting lowest-Landau-level (LLL) spinful fermions are derived with the use of combined numerical and symbolic computational approaches via analysis of exact Hamiltonian numerical diagonalization data. Closed-form analytic expressions are presented for two families of zero-interaction-energy states at given total angular momentum and total spin $0 \leq S \leq N/2$ in the neighborhood of the $\nu=1$ filling, covering the range from the maximum density droplet to the first quasihole. Our theoretical predictions for higher-order spatial and momentum correlations reveal intrinsic polygonal, multi-ring crystalline-type structures, which can be tested with ultracold-atom experiments in rapidly rotating traps, simulating quantum Hall physics (including quantum LLL skyrmions).

Emergent symmetries and slow quantum dynamics in a Rydberg-atom chain with confinement

Rydberg atoms in optical tweezer arrays provide a playground for nonequilibrium quantum many-body physics. The PXP model describes the dynamics of such systems in the strongly interacting Rydberg blockade regime and notably exhibits weakly nonergodic dynamics due to quantum many-body scars. Here, we study the PXP model in a strong staggered external field, which has been proposed to manifest quasiparticle confinement in light of a mapping to a lattice gauge theory. We characterize this confining regime using both numerical exact diagonalization and perturbation theory around the strong-field limit. In addition to the expected emergent symmetry generated by the staggered field, we find a second emergent symmetry that is special to the PXP model. The interplay between these emergent symmetries and the Rydberg blockade constraint dramatically slows down the system's dynamics beyond naive expectations. We devise a nested Schrieffer-Wolff perturbation theory to properly account for the new emergent symmetry, and we show that this treatment is essential to understand the numerically observed relaxation time scales. We also discuss connections to Hilbert space fragmentation, and we trace the origin of the new emergent symmetry to a ``nearly-$\text{SU}(2)$'' algebra discovered in the context of many-body scarring.