Exact Numerical Diagonalization Research Articles

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.

Static properties of two linearly coupled discrete circuits

Bosonic two-ring ladders constitute an important class of atomtronic circuits, where coherent current flows not only can offer a new insight into many-body physics, but also can play the role of actual degrees of freedom, and hence allow for a viable implementation of cold-atom based devices and qubit systems. In this work, we exhaustively investigate the ground state properties and the low-lying energy spectrum of two linearly coupled Bose–Hubbard rings. We show that the competition among interactions, intra- and inter-ring hopping processes gives place to a rather rich physical scenario, where Mott-like states and (different kinds of) superfluid-like states emerge. The latter ones depend also on the (in)commensurate filling of the atoms. Our analysis, carried out within a simple analytical framework and by means of the exact numerical diagonalization of the system Hamiltonian, provides one with a rather complete characterization of the static properties of the two-ring ladder, including, but not limited to, coherence, fragmentation, correlations, and entanglement. We complement our investigation by studying how these indicators depend on the commensurability of the total number of bosons with respect to the total number of sites and show that the two stacked rings are always entangled for an odd number of atoms.

Interaction-resistant metals in multicomponent Fermi systems

We analyze two different fermionic systems that defy Mott localization showing a metallic ground state at integer filling and very large Coulomb repulsion. The first is a multiorbital Hubbard model with a Hund's coupling (this physics has been widely studied, and the new metallic state is called a Hund's metal), and the second is a SU(3) Hubbard model with a patterned single-particle potential designed to display a similar interaction-resistant metal in a setup which can be implemented with $\mathrm{SU}(N)$ ultracold atoms. With simple analytical arguments and exact numerical diagonalization of the Hamiltonians for a minimal three-site system, we demonstrate that the interaction-resistant metal emerges in both cases as a compromise between two different insulating solutions which are stabilized by different terms of the models. This provides strong evidence that the Hund's metal is a specific realization of a more general phenomenon which can be realized in various strongly correlated systems.

Many-body localization: Transitions in spin models

We study the transitions between ergodic and many-body localized phases in spin systems, subject to quenched disorder, including the Heisenberg chain and the central spin model. In both cases systems with common spin lengths $1/2$ and $1$ are investigated via exact numerical diagonalization and random matrix techniques. Particular attention is paid to the sample-to-sample variance $(\Delta_sr)^2$ of the averaged consecutive-gap ratio $\langle r\rangle$ for different disorder realizations. For both types of systems and spin lengths we find a maximum in $\Delta_sr$ as a function of disorder strength, accompanied by an inflection point of $\langle r\rangle$, signaling the transition from ergodicity to many-body localization. The critical disorder strength is found to be somewhat smaller than the values reported in the recent literature. Further information about the transitions can be gained from the probability distribution of expectation values within a given disorder realization.

Quantum geometric tensor and quantum phase transitions in the Lipkin-Meshkov-Glick model

We study the quantum metric tensor and its scalar curvature for a particular version of the Lipkin-Meshkov-Glick model. We build the classical Hamiltonian using Bloch coherent states and find its stationary points. They exhibit the presence of a ground state quantum phase transition, where a bifurcation occurs, showing a change of stability associated with an excited state quantum phase transition. Symmetrically, for a sign change in one Hamiltonian parameter, the same phenomenon is observed in the highest energy state. Employing the Holstein-Primakoff approximation, we derive analytic expressions for the quantum metric tensor and compute the scalar and Berry curvatures. We contrast the analytic results with their finite-size counterparts obtained through exact numerical diagonalization and find an excellent agreement between them for large sizes of the system in a wide region of the parameter space, except in points near the phase transition where the Holstein-Primakoff approximation ceases to be valid.

Flux mobility delocalization in the Kitaev spin ladder

We study the Kitaev spin-$1/2$ ladder, a model which exhibits self-localization due to fractionalization caused by exchange frustration. When a weak magnetic field is applied, the model is described by an effective fermionic Hamiltonian, with an additional time reversal symmetry breaking term. We show that this term alone is not capable of delocalizing the system but flux mobility is a prerequisite. For magnetic fields larger but comparable to the flux gap, fluxes become mobile and drive the system into a delocalized regime, featuring finite dc transport coefficients. Our findings are based on numerical techniques, exact diagonalization and dynamical quantum typicality, from which, we present results for the specific heat, the dynamical energy current correlation function, as well as the inverse participation ratio, contrasting the spin against the fermion representation. Implications of our results for two-dimensional extensions of the model will be speculated on.

Itinerant Topological Magnons in SU(2) Symmetric Topological Hubbard Models with Nearly Flat Electronic BandsSupported by the National Natural Science Foundation of China (Grant No. 11774152), and National Key R&D Program of China (Grant No. 2016YFA0300401).

We show that a suitable combination of flat-band ferromagnetism, geometry and nontrivial electronic band topology can give rise to itinerant topological magnons. An SU(2) symmetric topological Hubbard model with nearly flat electronic bands, on a Kagome lattice, is considered as the prototype. This model exhibits ferromagnetic order when the lowest electronic band is half-filled. Using the numerical exact diagonalization method with a projection onto this nearly flat band, we can obtain the magnonic spectra. In the flat-band limit, the spectra exhibit distinct dispersions with Dirac points, similar to those of free electrons with isotropic hoppings, or a local spin magnet with pure ferromagnetic Heisenberg exchanges on the same geometry. Significantly, the non-flatness of the electronic band may induce a topological gap at the Dirac points, leading to a magnonic band with a nonzero Chern number. More intriguingly, this magnonic Chern number changes its sign when the topological index of the electronic band is reversed, suggesting that the nontrivial topology of the magnonic band is related to its underlying electronic band. Our work suggests interesting directions for the further exploration of, and searches for, itinerant topological magnons.

Topology and many-body localization

We discuss the problem of localization in two dimensional electron systems in the quantum Hall (single Landau level) regime. After briefly summarizing the well-studied problem of Anderson localization in the non-interacting case, we concentrate on the problem of disorder induced many-body localization (MBL) in the presence of electron–electron interactions using numerical exact diagonalization and eigenvalue spacing statistics as a function of system size. We provide evidence showing that MBL is not attainable in a single Landau level with short range (white noise) disorder in the thermodynamic limit. We then study the interplay of topology and localization, by contrasting the behavior of topological and nontopological subbands arising from a single Landau level in two models — (i) a pair of extremely flat Hofstadter bands with an optimally chosen periodic potential, and (ii) a Landau level with a split-off nontopological impurity band. Both models provide convincing evidence for the strong effect of topology on the feasibility of many-body localization as well as slow dynamics starting from a nonequilibrium state with charge imbalance.

Unconventional filling factor of 4/11: A closed-form ground-state wave function

The ground state at 4/11 filling factor is very well understood, Mukherjee et al. [Phys. Rev. Lett. 112, 016801 (2014)] in terms of the 1/3-filled second effective Landau level of the composite fermions whose correlations resemble that of electrons in the ground state of a two-body Haldane pseudo-potential of relative angular momentum 3, ${V}_{3}$. We here propose a closed-form ground-state wave function for ${V}_{3}$ at 1/3 filling factor. We successfully compare it with the exact wave function for systems with a few electrons by calculating their mutual overlap, pair-correlation function, and entanglement spectra. By numerical exact diagonalization for a few electron systems, we find a window of nonzero ${V}_{3}$ is essential together with ${V}_{1}$ for being 4/11 state incompressible. The constructed wave function for the 4/11 state using this proposed wave function has satisfactorily high overlap with previously studied composite-fermion-diagonalized ground-state wave function.

Moiré versus Mott: Incommensuration and Interaction in One-Dimensional Bichromatic Lattices.

Inspired by the rich physics of twisted 2D bilayer moiré systems, we study Coulomb interacting systems subjected to two overlapping finite 1D lattice potentials of unequal periods through exact numerical diagonalization. Unmatching underlying lattice periods lead to a 1D bichromatic "moiré" superlattice with a large unit cell and consequently a strongly flattened band, exponentially enhancing the effective dimensionless electron-electron interaction strength and manifesting clear signatures of enhanced Mott gaps at discrete fillings. An important nonperturbative finding is a remarkable fine-tuning effect of the precise lattice commensuration, where slight variations in the relative lattice periods may lead to a suppression of the correlated insulating phase, in qualitative agreement with the observed fragility of the correlated insulating phase in twisted bilayer graphene. Our predictions, which should be directly verifiable in bichromatic optical lattices, establish that the competition between interaction and incommensuration is a key element of the physics of moiré superlattices.

Nucleation of Ergodicity by a Single Mobile Impurity in Supercooled Insulators.

We consider a disordered Hubbard model and show that, at sufficiently weak disorder, a single spin-down mobile impurity can thermalize an extensive initially localized system of spin-up particles. Thermalization is enabled by resonant processes that involve correlated hops of the impurity and localized particles. This effect indicates that Anderson localized insulators behave as "supercooled" systems, with mobile impurities acting as ergodic seeds. We provide analytical estimates, supported by numerical exact diagonalization, showing how the critical disorder strength for such mechanism depends on the particle density of the localized system. In the U→∞ limit, doublons are stable excitations, and they can thermalize mesoscopic systems by a similar mechanism. The emergence of an additional conservation law leads to an eventual localization of doublons. Our predictions apply to fermionic and bosonic systems and are readily accessible in ongoing experiments simulating synthetic quantum lattices with tunable disorder.

Fractionalized spin excitations in the ferromagnetic edge state of graphene: Signature of the ferromagnetic Luttinger liquid

The elementary excitations from the conventional magnetic ordered states, such as ferromagnets and antiferromagnets, are magnons. Here, we elaborate a case where the well-defined magnons are absent completely and the spin excitation spectra exhibit an entire continuum in the itinerant edge ferromagnetic state of graphene arising from the flatband edge electronic states. Based on the further studies of the entanglement entropy and finite-size analysis, we show that the continuum other than the Stoner part results from the spin-1/2 spinons deconfined from magnons. The spinon continuum in a magnetically ordered state is ascribed to a ferromagnetic Luttinger liquid in this edge ferromagnet. The investigation is carried out by using the numerical exact diagonalization method with a projection of the interacting Hamiltonian onto the flat band.

Entanglement properties of disordered quantum spin chains with long-range antiferromagnetic interactions

We examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\alpha$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\alpha > 1/2$. A numerical implementation of the SDRG method yields a power law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\alpha$, corresponding to a critical behavior with an effective central charge $c = {\rm ln} 2$, independent of $\alpha$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\alpha$. For a wide range of distances $l$, the EE fits a critical behavior with a central charge close to $c=1$, which is the same as for the clean Haldane-Shastry model with a power-la-decaying interaction with $\alpha =2$. Consistent with this observation, we find using ED that the concurrence shows power law decay, albeit with smaller power exponents than obtained by SDRG.

Spin-1/2 XXZ Heisenberg cupolae: magnetization process and related enhanced magnetocaloric effect

The magnetization curves and magnetocaloric effect during the adiabatic demagnetization of antiferromagnetic spin-1/2 XXZ Heisenberg clusters with the shape of Johnson's solids (triangular cupola, square cupola and pentagonal cupola) are investigated using the exact numerical diagonalization for different values of the exchange anisotropy ranging in between the Ising and fully isotropic limit. It is demonstrated that spin-1/2 XXZ Heisenberg cupolae display, in comparison with their Ising counterparts, at least one more magnetization plateau. The novel magnetization plateaux extends over a wider range of the magnetic fields with increasing of a quantum part of the XXZ exchange coupling at the expense of the original plateaux present in the limiting Ising case. It is shown that the spin-1/2 XXZ Heisenberg triangular and pentagonal cupolae exhibit an enhanced magnetocaloric effect in the vicinity of zero magnetic field, which makes these magnetic systems promising refrigerants for cooling down to ultra-low zero temperatures.

Eigenstate thermalization hypothesis beyond standard indicators: Emergence of random-matrix behavior at small frequencies

Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with all standard indicators of the eigenstate thermalization hypothesis, the eigenvalue distribution still exhibits clear signatures of the original operator, implying correlations between matrix elements. Moreover, we demonstrate that at much smaller energy scales, the eigenvalue distribution approximately assumes the universal semicircle shape, indicating transition to the random-matrix behavior, and in particular that matrix elements become uncorrelated.

Spectral transitions and universal steady states in random Kraus maps and circuits

The study of dissipation and decoherence in generic open quantum systems recently led to the investigation of spectral and steady-state properties of random Lindbladian dynamics. A natural question is then how realistic and universal those properties are. Here, we address these issues by considering a different description of dissipative quantum systems, namely, the discrete-time Kraus map representation of completely positive quantum dynamics. Through random matrix theory (RMT) techniques and numerical exact diagonalization, we study random Kraus maps, allowing for a varying dissipation strength, and their local circuit counterpart. We find the spectrum of the random Kraus map to be either an annulus or a disk inside the unit circle in the complex plane, with a transition between the two cases taking place at a critical value of dissipation strength. The eigenvalue distribution and the spectral transition are well described by a simplified RMT model that we can solve exactly in the thermodynamic limit, by means of non-Hermitian RMT and quaternionic free probability. The steady state, on the contrary, is not affected by the spectral transition. It has, however, a perturbative crossover regime at small dissipation, inside which the steady state is characterized by uncorrelated eigenvalues. At large dissipation (or for any dissipation for a large-enough system), the steady state is well described by a random Wishart matrix. The steady-state properties thus coincide with those already observed for random Lindbladian dynamics, indicating their universality. Quite remarkably, the statistical properties of the local Kraus circuit are qualitatively the same as those of the nonlocal Kraus map, indicating that the latter, which is more tractable, already captures the realistic and universal physical properties of generic open quantum systems.

Influence of Majorana Bound States in Quantum Rings

AbstractA quantum ring coupled to a 1D topological superconductor hosting Majorana bound states (MBSs) is investigated. The MBSs effects over the spectrum and persistent current along the quantum ring are studied. The spectra of the system are obtained by an exact numerical diagonalization of the Bogoliubov‐de Gennes Hamiltonian in the Majorana representation. In addition, Green's function formalism is implemented for analytical calculations and obtained a switching condition in the MBSs fermionic parity. Three different patterns that could be obtained for the spatial separation of the MBSs, named: bowtie, diamond, and asymmetric, are reported here, which are present only in odd parity in the quantum ring, while only a single pattern (bowtie) is obtained for even parity. Those patterns are subject strictly to the switching condition for the MBSs. Besides, quantum ring with the presence of a Majorana zero mode presents gapped/gapless spectra in odd/even parity showing in the even case a subtle signature in the persistent current.