Exact Diagonalization Results Research Articles

Extraction of Spectra in the Shell Model MonteCarlo Method Using Imaginary-Time Correlation Matrices.

Nuclear energy levels are usually calculated using conventional diagonalization methods in the framework of the configuration-interaction (CI) shell model but these methods are prohibited in very large model spaces. The shell model MonteCarlo (SMMC) method is a powerful technique for calculating thermal and ground-state observables of nuclei in very large model spaces, but it is challenging to extract nuclear spectra in this approach. We present a novel method to extract within SMMC low-lying energy levels for given values of a set of good quantum numbers such as spin and parity. The method is based on imaginary-time correlation matrices (ITCMs) of one-body densities that satisfy asymptotically a generalized eigenvalue problem. We validate the method in a light nucleus that allows comparison with exact diagonalization results of the CI shell-model Hamiltonian. The method is broadly applicable to quantum many-body systems in other disciplines.

Time‐Resolved Optical Response of the Dicke's Model via the Nonequilibrium Green's Function Approach

Due to their conceptual appeal and computational convenience, two‐level systems (TLSs) and generalizations are used to investigate nonlinear behavior in quantum optics and assess the applicability of theoretical methods. Herein, the focus is on second‐harmonic generation (SHG) and, as system of interest, on the Dicke model, which consists of several TLSs inside an optical cavity. The main aspect addressed is the scope of nonequilibrium Green's function (NEGF) to describe the effect of inhomogeneities and electron–electron (e–e) interactions on the SHG signal. For benchmarking purposes, exact diagonalization (ED) results are also presented and discussed. SHG spectra obtained with NEGF and ED are found to be in very good mutual agreement in most situations. Furthermore, inhomogeneity in the TLS and e–e interactions reduce the SHG signal, and the reduction is stronger with inhomogeneity than with interactions. This trend is consistently noted across different (small to large) system sizes. Finally, a modified NEGF approach is proposed to account for cavity leakage, where the quantum photon fields are coupled to a bath of classical oscillators. As expected, within this mixed quantum‐classical scheme, a decrease in the intensity of the fluorescent spectra takes place depending on the entity of leakage.

Neural network quantum states analysis of the Shastry-Sutherland model

We utilize neural network quantum states (NQS) to investigate the ground state properties of the Heisenberg model on a Shastry-Sutherland lattice using the variational Monte Carlo method. We show that already relatively simple NQSs can be used to approximate the ground state of this model in its different phases and regimes. We first compare several types of NQSs with each other on small lattices and benchmark their variational energies against the exact diagonalization results. We argue that when precision, generality, and computational costs are taken into account, a good choice for addressing larger systems is a shallow restricted Boltzmann machine NQS. We then show that such NQS can describe the main phases of the model in zero magnetic field. Moreover, NQS based on a restricted Boltzmann machine correctly describes the intriguing plateaus forming in magnetization of the model as a function of increasing magnetic field.

Magnetic phases of spatially modulated spin-1 chains in Rydberg excitons: Classical and quantum simulations.

In this work, we study the magnetic phases of a spatially modulated chain of spin-1 Rydberg excitons. Using the Density Matrix Renormalization Group (DMRG) technique, we study various magnetic and topologically nontrivial phases using both single-particle properties, such as local magnetization and quantum entropy, and many-body ones, such as pair-wise Néel and long-range string correlations. In particular, we investigate the emergence and robustness of the Haldane phase, a topological phase of anti-ferromagnetic spin-1 chains. Furthermore, we devise a hybrid quantum algorithm employing restricted Boltzmann machine to simulate the ground state of such a system that shows very good agreement with the results of exact diagonalization and DMRG.

Non-Abelian chiral spin liquid on a spin-1 kagome lattice: Truncation of an exact Hamiltonian and numerical optimization

We search for short-range Hamiltonians of finite spin-1 kagome systems, maximizing the overlaps with lattice Moore-Read states. Our starting point is an exact, long-range parent Hamiltonian for such a state on a finite plane, obtained from conformal field theory. A truncation procedure is applied to it, which retains only short-range terms and makes it easy to define the Hamiltonian on a torus. Finally, the remaining coefficients are optimized, to yield maximized overlaps between exact diagonalization results and model ground states. In the best cases, these overlaps exceed 0.9 and 0.8 for the three lowest states of 12- and 18-site systems, respectively, suggesting that the obtained Hamiltonians are good parent Hamiltonians for a non-Abelian topological order.

Spectral Functions of the Holstein Polaron: Exact and Approximate Solutions.

It is generally accepted that the dynamical mean field theory gives a good solution of the Holstein model, but only in dimensions greater than two. Here, we show that this theory, which becomes exact in the weak coupling and in the atomic limit, provides an excellent, numerically cheap, approximate solution for the spectral function of the Holstein model in the whole range of parameters, even in one dimension. To establish this, we make a detailed comparison with the spectral functions that we obtain using the newly developed momentum-space numerically exact hierarchical equations of motion method, which yields electronic correlation functions directly in real time. We crosscheck these conclusions with our path integral quantum MonteCarlo and exact diagonalization results, as well as with the available numerically exact results from the literature.

Time-dependent variational Monte Carlo study of the dynamic response of bosons in an optical lattice

We study the dynamics of a one-dimensional Bose gas at unit filling in both shallow and deep optical lattices and obtain the dynamic structure factor \bm{S(k,\omega)}𝐒(𝐤,𝛚) by monitoring the linear response to a weak probe pulse. We introduce a new procedure, based on the time-dependent variational Monte Carlo method (tVMC), which allows to evolve the system in real time, using as a variational model a Jastrow-Feenberg wave function that includes pair correlations. Comparison with exact diagonalization results of \bm{S(k,\omega)}𝐒(𝐤,𝛚) obtained on a lattice in the Bose-Hubbard limit shows good agreement of the dispersion relation for sufficiently deep optical lattices, while for shallow lattices we observe the influence of higher Bloch bands. We also investigate non-linear response to strong pulses. From the power spectrum of the density fluctuations we obtain the excitation spectrum, albeit broadened, by higher harmonic generation after a strong pulse with a single low wave number. As a remarkable feature of our simulations we furthermore demonstrate that the full excitation spectrum can be retrieved from the power spectrum of the density fluctuations due to the stochastic noise inherent in any Monte Carlo method, without applying an actual perturbation.

Signature of a randomness-driven spin-liquid state in a frustrated magnet

Collective behaviour of electrons, frustration induced quantum fluctuations and entanglement in quantum materials underlie some of the emergent quantum phenomena with exotic quasi-particle excitations that are highly relevant for technological applications. Herein, we present our thermodynamic and muon spin relaxation measurements, complemented by ab initio density functional theory and exact diagonalization results, on the recently synthesized frustrated antiferromagnet Li4CuTeO6, in which Cu2+ ions (S = 1/2) constitute disordered spin chains and ladders along the crystallographic [101] direction with weak random inter-chain couplings. Our thermodynamic experiments detect neither long-range magnetic ordering nor spin freezing down to 45 mK despite the presence of strong antiferromagnetic interaction between Cu2+ moments leading to a large effective Curie-Weiss temperature of − 154 K. Muon spin relaxation results are consistent with thermodynamic results. The temperature and magnetic field scaling of magnetization and specific heat reveal a data collapse pointing towards the presence of random-singlets within a disorder-driven correlated and dynamic ground-state in this frustrated antiferromagnet.

Ground-state and thermodynamic properties of the spin-$\frac{1}{2}$ Heisenberg model on the anisotropic triangular lattice

The spin-\frac1212 triangular lattice antiferromagnetic Heisenberg model has been for a long time the prototypical model of magnetic frustration. However, only very recently this model has been proposed to be realized in the Ba_88CoNb_66O_{24}24 compound. The ground-state and thermodynamic properties are evaluated from a high-temperature series expansion interpolation method, called ``entropy method’’, and compared to experiments. We find a ground-state energy e_0 = -0.5445(2)e0=−0.5445(2) in perfect agreement with exact diagonalization results. We also calculate the specific heat and entropy at all temperatures, finding a good agreement with the latest experiments, and evaluate which further interactions could improve the comparison. We explore the spatially anisotropic triangular lattice and provide evidence that supports the existence of a gapped spin liquid between the square and triangular lattices.

Real-space density kernel method for Kohn–Sham density functional theory calculations at high temperature

Kohn-Sham density functional theory calculations using conventional diagonalization based methods become increasingly expensive as temperature increases due to the need to compute increasing numbers of partially occupied states. We present a density matrix based method for Kohn-Sham calculations at high temperatures that eliminates the need for diagonalization entirely, thus reducing the cost of such calculations significantly. Specifically, we develop real-space expressions for the electron density, electronic free energy, Hellmann-Feynman forces, and Hellmann-Feynman stress tensor in terms of an orthonormal auxiliary orbital basis and its density kernel transform, the density kernel being the matrix representation of the density operator in the auxiliary basis. Using Chebyshev filtering to generate the auxiliary basis, we next develop an approach akin to Clenshaw-Curtis spectral quadrature to calculate the individual columns of the density kernel based on the Fermi operator expansion in Chebyshev polynomials and employ a similar approach to evaluate band structure and entropic energy components. We implement the proposed formulation in the SPARC electronic structure code, using which we show systematic convergence of the aforementioned quantities to exact diagonalization results, and obtain significant speedups relative to conventional diagonalization based methods. Finally, we employ the new method to compute the self-diffusion coefficient and viscosity of aluminum at 116 045K from Kohn-Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.

Exact and many-body perturbation solutions of the Hubbard model applied to linear chains

This study reports on the accuracy of the GW approximation for the treatment of the Hubbard model compared to exact diagonalization (ED) results. We consider not only global quantities, such as the total energy and the density of states, but also the spatial and spin symmetry of wavefunctions via the analysis of the local density of states. GW is part of the more general Green’s function approach used to develop many-body approximations. We show that, for small linear chains, the GW approximation corrects the mean-field (MF) approach by reducing the total energy and the magnetization obtained from the MF approximation. The GW energy gap is in better agreement with ED, especially in systems of an even number of atoms where, in contrast to the MF approximation, no plateau is observed below the artificial predicted phase transition. In terms of density of states, the GW approximation induces quasi-particles and side satellite peaks via a splitting process of MF peaks. At the same time, GW slightly modifies the localization (e.g., edges or center) of states. We also use the GW approximation results in the context of Löwdin’s symmetry dilemma and show that GW predicts an artificial paramagnetic–antiferromagnetic phase transition at a higher Hubbard parameter than MF does.

Tail states and unusual localization transition in low-dimensional Anderson model with power-law hopping

We study deterministic power-law quantum hopping model with an amplitude J(r)∝−r−β and local Gaussian disorder in low dimensions d=1,2 under the condition d<β<3d/2. We demonstrate unusual combination of exponentially decreasing density of the ”tail states” and localization–delocalization transition (as function of disorder strength W) pertinent to a small (vanishing in thermodynamic limit) fraction of eigenstates. In a broad range of parameters density of states ν(E) decays into the tail region E<0 as simple exponential, ν(E)=ν0eE/E0, while characteristic energy E0 varies smoothly across edge localization transition. We develop simple analytic theory which describes E0 dependence on power-law exponent β, dimensionality d and W, and compare its predictions with exact diagonalization results. At low energies within the bare ”conduction band”, all eigenstates are localized due to strong quantum interference at d=1,2; however localization length grows fast with energy decrease, contrary to the case of usual Schrodinger equation with local disorder.

Real-space formulation of the stress tensor for O(N) density functional theory: Application to high temperature calculations.

We present an accurate and efficient real-space formulation of the Hellmann-Feynman stress tensor for O(N) Kohn-Sham density functional theory (DFT). While applicable at any temperature, the formulation is most efficient at high temperature where the Fermi-Dirac distribution becomes smoother and the density matrix becomes correspondingly more localized. We first rewrite the orbital-dependent stress tensor for real-space DFT in terms of the density matrix, thereby making it amenable to O(N) methods. We then describe its evaluation within the O(N) infinite-cell Clenshaw-Curtis Spectral Quadrature (SQ) method, a technique that is applicable to metallic and insulating systems, is highly parallelizable, becomes increasingly efficient with increasing temperature, and provides results corresponding to the infinite crystal without the need of Brillouin zone integration. We demonstrate systematic convergence of the resulting formulation with respect to SQ parameters to exact diagonalization results and show convergence with respect to mesh size to the established plane wave results. We employ the new formulation to compute the viscosity of hydrogen at 106 K from Kohn-Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.

Single-hole wave function in two dimensions: A case study of the doped Mott insulator

We study a ground-state ansatz for the single-hole doped $t$-$J$ model in two dimensions via a variational Monte Carlo (VMC) method. Such a single-hole wave function possesses finite angular momenta generated by hidden spin currents, which give rise to a novel ground state degeneracy in agreement with recent exact diagonalization (ED) and density matrix renormalization group (DMGR) results. We further show that the wave function can be decomposed into a quasiparticle component and an incoherent momentum distribution in excellent agreement with the DMRG results up to an $8\times 8 $ lattice. Such a two-component structure indicates the breakdown of Landau's one-to-one correspondence principle, and in particular, the quasiparticle spectral weight vanishes by a power law in the large sample-size limit. By contrast, turning off the phase string induced by the hole hopping in the so-called $\sigma\cdot t\text{-}J$ model, a conventional Bloch-wave wave function with a finite quasiparticle spectral weight can be recovered, also in agreement with the ED and DMRG results. The present study shows that a singular effect already takes place in the single-hole-doped Mott insulator, by which the bare hole is turned into a non-Landau quasiparticle with translational symmetry breaking. Generalizations to pairing and finite doping are briefly discussed.

Large magnetic thermal conductivity induced by frustration in low-dimensional quantum magnets

We study the magnetic field-dependence of the thermal conductivity due to magnetic excitations in frustrated spin-1/2 Heisenberg chains. Near the saturation field, the system is described by a dilute gas of weakly-interacting fermions (free-fermion fixed point). We show that in this regime the thermal conductivity exhibits a non-monotonic behavior as a function of the ratio $\alpha= J_2/J_1$ between second and first nearest-neighbor antiferromagnetic exchange interactions. This result is a direct consequence of the splitting of the single-particle dispersion minimum into two minima that takes place at the Lifshitz point $\alpha=1/4$. Upon increasing $\alpha$ from zero, the inverse mass vanishes at $\alpha=1/4$ and it increases monotonically from zero for $\alpha \geq 1/4$. By deriving an effective low-energy theory of the dilute gas of fermions, we demonstrate that the Drude weight $K_{\rm th}$ of the thermal conductivity exhibits a similar dependence on $\alpha$ near the saturation field. Moreover, this theory predicts a transition between a two-component Tomonaga-Luttinger liquid and a vector-chiral phase at a critical value $\alpha=\alpha_c$ that agrees very well with previous density matrix renormalization group results. We also show that the resulting curve $K_{\rm th}(\alpha)$ is in excellent agreement with exact diagonalization (ED) results. Our ED results also show that $K_{\rm th}(\alpha)$ has a pronounced minimum at $\alpha\simeq 0.7$ and it decreases for sufficiently large $\alpha$ at lower magnetic field values. We also demonstrate that the thermal conductivity is significantly affected by the presence of magnetothermal coupling.

Fractional Chern insulators in singular geometries

The fractional quantum anomalous Hall (FQAH) states or fractional Chern insulator (FCI) states have been studied on two-dimensional (2D) flat lattices with different boundary conditions. Here, we propose the geometry-dependent FCI/FQAH states that interacting particles are bounded on 2D singular lattices with arbitrary $n$-fold rotational symmetry. Based on the generalized Pauli principle, we construct trial wave functions for the singular-lattice FCI/FQAH states with the aid of an effective projection approach, and compare them with the exact diagonalization results. High wave-function overlaps show that the singular-lattice FCI/FQAH states are certainly related to the geometric factor $\beta$. More interestingly, we observe some exotic degeneracy sequences of edge excitations in these singular-lattice FCI/FQAH states, and provide an explanation that two branches of edge excitations mix together.

Statistics of eigenstates near the localization transition on random regular graphs

Dynamical and spatial correlations of eigenfunctions as well as energy level correlations in the Anderson model on random regular graphs (RRG) are studied. We consider the critical point of the Anderson transition and the delocalized phase. In the delocalized phase near the transition point, the observables show a broad critical regime for system sizes $N$ below the correlation volume $N_{\xi}$ and then cross over to the ergodic behavior. Eigenstate correlations allow us to visualize the correlation length $\xi \sim \log N_{\xi}$ that controls the finite-size scaling near the transition. The critical-to-ergodic crossover is very peculiar, since the critical point is similar to the localized phase, whereas the ergodic regime is characterized by very fast "diffusion" which is similar to the ballistic transport. In particular, the return probability crosses over from a logarithmically slow variation with time in the critical regime to an exponentially fast decay in the ergodic regime. Spectral correlations in the delocalized phase near the transition are characterized by level number variance $\Sigma_2(\omega)$ crossing over, with increasing freqyency $\omega$, from ergodic behavior $\Sigma_2=\left(2/\pi^2\right)\ln\omega/\Delta$ to $\Sigma_2\propto \omega^2$ at $\omega_c\sim (N N_{\xi})^{-1/2}$ and finally to Poissonian behavior $\Sigma_2 = \omega/\Delta$ at $\omega_{\xi}\sim N_{\xi}^{-1}$. We find a perfect agreement between results of exact diagonalization and those resulting from the solution of the self-consistency equation obtained within the saddle-point analysis of the effective supersymmetric action. We show that the RRG model can be viewed as an intricate $d\to\infty$ limit of the Anderson model in $d$ spatial dimensions.

Variational study of the interacting, spinless Su–Schrieffer–Heeger model

We study the phase diagram and the total polarization distribution of the Su–Schrieffer–Heeger model with nearest neighbor interaction in one dimension at half-filling. To obtain the ground state wave-function, we extend the Baeriswyl variational wave function to account for alternating hopping parameters. The ground state energies of the variational wave functions compare well to exact diagonalization results. For the case of uniform hopping for all bonds, where it is known that an ideal conductor to insulator transition takes place at finite interaction, we also find a transition at an interaction strength somewhat lower than the known value. The ideal conductor phase is a Fermi sea. The phase diagram in the whole parameter range shows a resemblance to the phase diagram of the Kane–Mele–Hubbard model. We also calculate the gauge invariant cumulants corresponding to the polarization (Zak phase) and use these to reconstruct the distribution of the polarization. We calculate the reconstructed polarization distribution along a path in parameter space which connects two points with opposite polarization in two ways. In one case we cross the metallic phase line, in the other, we go through only insulating states. In the former case, the average polarization changes discontinuously after passing through the metallic phase line, while in the latter the distribution ‘walks across’ smoothly from one polarization to its opposite. This state of affairs suggests that the correlation acts to break the chiral symmetry of the Su–Schrieffer–Heeger model, in the same way as it happens when a Rice–Mele onsite potential is turned on.

Phase dilemma in natural orbital functional theory from the N-representability perspective

Any rigorous approach to first-order reduced density (1RDM) matrix functional theory faces the phase dilemma, that is, having to deal with a large number of possible combinations of signs in terms of the electron-electron interaction energy. This problem was discovered by reducing a ground-state energy generated from an approximate N-particle wavefunction into a functional of the 1RDM, known as the top-down method. Here, we show that the phase dilemma also appears in the bottom-up method, in which the functional E[1RDM] is generated by progressive inclusion of N-representability conditions on the reconstructed two-particle reduced density matrix. It is shown that an adequate choice of signs is essential to accurately describe model systems with strong non-dynamic (static) electron correlation, specifically, the one-dimensional Hubbard model with periodic boundary conditions and hydrogen rings. For the latter, the Piris natural orbital functional 7 (PNOF7), with phases equal to -1 for the inter-pair energy terms containing the exchange-time-inversion integrals, agrees with exact diagonalization results.

Fractional quantum Hall states of bosons on cones

Motivated by a recent experiment which synthesizes Landau levels for photons on cones [Schine {\em et al.}, Nature 534, 671 (2016)], and more generally the interest in understanding gravitational responses of quantum Hall states, we study fractional quantum Hall states of bosons on cones. A variety of trial wave functions for conical systems are constructed and compared with exact diagonalization results. The tip of a cone is a localized geometrical defect with singular curvature which can modify the density profiles of quantum Hall states. The density profiles on cones can be used to extract some universal information about quantum Hall states. The values of certain quantities are computed numerically using the density profiles of some quantum Hall states and they agree with analytical predictions.