Restricted Hilbert Space Research Articles

Twisted bilayer graphene. VI. An exact diagonalization study at nonzero integer filling

Using exact diagonalization, we study the projected Hamiltonian with Coulomb interaction in the 8 flat bands of first magic angle twisted bilayer graphene. Employing the U(4) (U(4)$\times$U(4)) symmetries in the nonchiral (chiral) flat band limit, we reduced the Hilbert space to an extent which allows for study around $\nu=\pm 3,\pm2,\pm1$ fillings. In the first chiral limit $w_0/w_1=0$ where $w_0$ ($w_1$) is the $AA$ ($AB$) stacking hopping, we find that the ground-states at these fillings are extremely well-described by Slater determinants in a so-called Chern basis, and the exactly solvable charge $\pm1$ excitations found in [arXiv:2009.14200] are the lowest charge excitations up to system sizes $8\times8$ (for restricted Hilbert space) in the chiral-flat limit. We also find that the Flat Metric Condition (FMC) used in [arXiv:2009.11301,2009.11872,2009.12376,2009.13530,2009.14200] for obtaining a series of exact ground-states and excitations holds in a large parameter space. For $\nu=-3$, the ground state is the spin and valley polarized Chern insulator with $\nu_C=\pm1$ at $w_0/w_1\lesssim0.9$ (0.3) with (without) FMC. At $\nu=-2$, we can only numerically access the valley polarized sector, and we find a spin ferromagnetic phase when $w_0/w_1\gtrsim0.5t$ where $t\in[0,1]$ is the factor of rescaling of the actual TBG bandwidth, and a spin singlet phase otherwise, confirming the perturbative calculation [arXiv:2009.13530]. The analytic FMC ground state is, however, predicted in the intervalley coherent sector which we cannot access [arXiv:2009.13530]. For $\nu=-3$ with/without FMC, when $w_0/w_1$ is large, the finite-size gap $\Delta$ to the neutral excitations vanishes, leading to phase transitions. Further analysis of the ground state momentum sectors at $\nu=-3$ suggests a competition among (nematic) metal, momentum $M_M$ ($\pi$) stripe and $K_M$-CDW orders at large $w_0/w_1$.

Type II superstring field theory with cyclic $L_\infty$ structure

Abstract We construct a complete type II superstring field theory that includes all the NS–NS, R–NS, NS–R, and R–R sectors. As in the open and heterotic superstring cases, the R–NS, NS–R, and R–R string fields are constrained by using the picture-changing operators. In particular, we use a non-local inverse picture-changing operator for the constraint on the R–R string field, which seems to be inevitable due to the compatibility of the extra constraint with the closed string constraints. The natural symplectic form in the restricted Hilbert space gives a non-local kinetic action for the R–R sector, but it correctly provides the propagator expected from the first-quantized formulation. Extending the prescription previously obtained for the heterotic string field theory, we give a construction of general type II superstring products, which realizes a cyclic $L_\infty$ structure, and thus provides a gauge-invariant action based on the homotopy algebraic formulation. Three typical four-string amplitudes derived from the constructed string field theory are demonstrated to agree with those in the first-quantized formulation. We also give the half-Wess–Zumino–Witten action defined in the medium Hilbert space whose left-moving sector is still restricted to the small Hilbert space.

Charge dynamics of correlated electrons: Variational description with inclusion of composite fermions

We propose a method to calculate the charge dynamical structure factors for the ground states of correlated electron systems based on the variational Monte Carlo method. Our benchmarks for the one- and two-dimensional Hubbard models show that inclusion of composite-fermion excitations in the basis set greatly improves the accuracy, in reference to the exact charge dynamical structure factors for clusters. Together with examination for larger systems beyond tractable sizes by the exact diagonalization, our results indicate that the variational Monte Carlo method is a promising way for studies on the nature of charge dynamics in correlated materials such as the copper oxide superconductors, if the composite-fermion excitations are properly included in the restricted Hilbert space of intermediate states in the linear response theory. Our results are consistent with the particle-hole excitations inferred from the single-particle spectral function $A(\mathbf{k}, \omega)$ in the literature. We also discuss the importance of incorporating nonlocal composite fermion for more accurate description. Future issues for further improvements are also discussed.

Multimodeling Approach to Ferromagnetic Spin-Wave Excitations in the High-Spin Cluster Mn18Sr Observed by Inelastic Neutron Scattering.

The magnetism of the mixed-valence high-spin cluster [Mn18SrO8(N3)7Cl(MedhmpH)12(MeCN)6]Cl2 (1) exhibiting intramolecular ferromagnetic interactions was studied using inelastic neutron scattering (INS), and reliable values for the exchange coupling constants were determined based on the quality of simultaneous fits to the INS and magnetic data. The challenge of the huge size of the Hilbert space (3 375 000) and many exchange coupling constants (7 assuming a C3 symmetry) generally encountered in large spin clusters was resolved as follows: (a) The results of the restricted Hilbert space ferromagnetic cluster spin wave theory were compared to the experimental spectroscopic data. The observed INS transitions were thus assigned to spin wave excitations in a bounded ferromagnetic spin cluster and moreover could be visualized in a straightforward way based on this theory. (b) Simultaneously, Quantum Monte Carlo (QMC) calculations of the temperature-dependent magnetic susceptibility with the same parameter set were compared to the experimental data. Application of state-of-the-art QMC algorithms, as available in the open source ALPS package, in ferromagnetic clusters avoids the full Hamiltonian diagonalization without sacrificing calculation accuracy of the magnetic susceptibility down to the lowest temperatures, which was crucial for the successful analysis. The combined fits revealed two exchange-coupling models with equally good overall agreement to the data. Our preferred model was inspired by magnetostructural correlations and is consistent with them. The model involves three different exchange interactions, one describing the interaction between the core MnIII spins Ja = 14.3(1.0) K and two interactions linking the core and the peripheral MnII spins: Jb = 8.3(4) K and J6 = 3.6(4) K. The use of open-source QMC software and our systematic approach to fitting multiple sets of data obtained by different experimental techniques are described in detail and are generally applicable for understanding large ferromagnetically coupled clusters.

Density–driven superfluid transition of the constrained bosons on a lattice

We formulate the Landau theory and study its consequences for lattice bosons, which are stabilized by a three-body on-site constraint. This is relevant for experiments with cold atoms in optical lattices, where the presence of three-body dissipative loss processes renders the occupation number n for bosons on short timescales effectively constrained to n≤2. We elaborate on the Landau free energy expansion resulting from the Hamiltonian of bosons in the restricted Hilbert space, and present the phase diagram of the constrained Bose lattice gas.

Gutzwiller projection for exclusion of holes: Application to strongly correlated ionic Hubbard model and binary alloys

We consider strongly correlated limit of variants of the Hubbard model (HM) in which on parts of the system it is energetically favourable to project out doublons from the low energy Hilbert space while on other sites of the system it is favourable to project out holes while still allowing for doublons. As an effect the low energy Hilbert space itself varies with sites of the system. Though the formalism is well developed for the case of doublon projection in the literature, case of hole projection has not been explored in detail so far. We derive basic framework by defining creation and annihilation operators for electrons in a restricted Hilbert space where holes are projected out but which still allows for doublons. We generalise the idea of Gutzwiller approximation for case of hole projection which has been done in literature for the case of doublon projection. To be specific, we provide detailed analysis of strongly correlated limit of the ionic Hubbard model (IHM) which has a staggered potential $\Delta$ on two sublattices of a bipartite lattice and the correlated binary alloys which have binary disorder $\pm V/2$ randomly distributed on sites of the lattice. In both the cases, for $\Delta \sim U \gg t$ and for $V\sim U \gg t$, where $U$ is the Hubbard energy cost for having a doublon at a site, there are sites on which doublons are allowed while holes are the maximum energy states. We do a systematic generalization of similarity transformation for both these cases and obtain the effective low energy Hamiltonian. We further derive Gutzwiller approximation factors which provide renormalization of various terms in the effective low energy Hamiltonian due to the Gutzwiller projection operators, excluding holes on some sites and doublons on the remaining sites.

Spin-charge transformation of lattice fermion models: duality approach for diagrammatic simulation of strongly correlated systems

I derive a dual description of lattice fermions, specifically focusing on the t-J and Hubbard models, that allow diagrammatic techniques to be employed efficiently in the strongly correlated regime, as well as for systems with a restricted Hilbert space. These constructions are based on spin-charge transformation, where the lattice fermions of the original model are mapped onto spins and spin-less fermions. This mapping can then be combined with Popov-Fedotov fermionisation, where the spins are mapped onto lattice fermions with imaginary chemical potential. The resulting models do not contain any large expansion parameters, even for strongly correlated systems. Also, they exhibit dramatically smaller corrections to the density matrix from nonlinear terms in the Hamiltonian. The combination of these two properties means that they can be addressed with diagrammatic methods, including simulation techniques based on stochastic sampling of diagrammatic expansions.

Open superstring field theory on the restricted Hilbert space

Recently an action for open superstring field theory was proposed where the Neveu-Schwarz sector is formulated in the large Hilbert space while the Ramond sector lives in a restriction of the small Hilbert space. The purpose of this note is to clarify the relation of the restricted Hilbert space with other approaches and to formulate open superstring field theory entirely in the small Hilbert space.

Incommensurability effects in odd lengthJ1-J2quantum spin chains: On-site magnetization and entanglement

For the antiferromagnetic ${J}_{1}$-${J}_{2}$ quantum spin chain with an even number of sites, the point ${J}_{2}^{d}={J}_{1}/2$ is a disorder point. It marks the onset of incommensurate real space correlations for ${J}_{2}>{J}_{2}^{d}$. At a distinct larger value of ${J}_{2}^{L}=0.520\phantom{\rule{0.16em}{0ex}}36(6){J}_{1}$, the Lifshitz point, the peak in the static structure factor begins to move away from $k=\ensuremath{\pi}$. Here, we focus on chains with an odd number of sites. In this case, the disorder point is also at ${J}_{2}^{d}={J}_{1}/2$ but the behavior close to the Lifshitz point, ${J}_{2}^{L}\ensuremath{\simeq}0.538{J}_{1}$, is quite different: starting at ${J}_{2}^{L}$, the ground state goes through a sequence of level crossings as its momentum changes away from $k=\ensuremath{\pi}/2$. An even length chain, on the other hand, is gapped for any ${J}_{2}>0.24{J}_{1}$ and has the ground-state momentum $k=0$. This gradual change in the ground-state wave function for chains with an odd number of sites is reflected in a dramatic manner directly in the ground-state on-site magnetization as well as in the bipartite von Neumann entanglement entropy. Our results are based on DMRG calculations and variational calculations performed in a restricted Hilbert space defined in the valence bond picture. In the vicinity of the point ${J}_{2}={J}_{1}/2$, we expect the variational results to be very precise.

Diffusion Monte Carlo for fermions with replica reduction

We propose an improved framework of diffusion Monte Carlo method for quantum many-fermion systems. This approach enables us to find the ground state of the system without any a priori information on the nodal structure of the wave function. The main idea in this approach is to use sign-carrying replicas which are the basis vectors in a restricted Hilbert space. For efficiency, we propose to reduce the number of replicas in a proper manner. We test our method on a spinless fermion system, and compare the results with the exact values. We also discuss this method in relation to exact diagonalization.

Incoherence induced by phonon excitations in domain-growth dynamics: Role of the domain center-of-gravity motion

An ultrafast growth dynamics of a quantum domain after visible or near-infrared photoexcitation is discussed in a one-dimensional system that resides on the first-order phase boundary. Special emphasis is placed on the effects of phonon excitations that are expected to occur in the course of the dynamics. To treat such effects as transparently as possible, we incorporate phonon degrees of freedom into a domain model that consists of two fundamental domain degrees of freedom, namely the spatial extension and the center of gravity. An exact time-evolution calculation within a restricted Hilbert space that almost converges the results indicates that the phonon excitations provide appreciable incoherence to the domain growth especially via overall distributions of the domain center-of-gravity momentum over its nonzero values.

Brunnian and Efimov N-Body States

The Efimov effect was originally formulated for three particles. The underlying principle of model independence is extended in this article in several directions. We present our definitions of the concepts of universality and scale independence. In this context we review briefly the scaling relations established for two- and three-body structures of nuclear halos. We emphasize the difference between the two extremes of weak binding named Efimov and Brunnian states. They arise respectively for two-body interactions at threshold of binding either two or N particles. We restrict the Hilbert space to include no more than two-body correlations, and discuss the derived excited N-body Efimov states both for zero- and finite-range two-body interactions. Then we investigate the relation between radius and binding energy extremely close to threshold of binding the Brunnian N-body system. Radii of both ground and first excited states for N = 4, 5, 6 remain finite as the binding energy vanishes, and the distances between pairs of particles are substantially larger than the range of the two-body potential. The radii decrease with N and increase with excitation energy. The computed radii are larger for the complete than for the restricted Hilbert space. Model independence at the Brunnian threshold is strongly indicated.

Excitation spectrum of Mott shells in optical lattices

We theoretically study the excitation spectrum of confined macroscopic optical lattices in the Mott-insulating limit. For large systems, a fast numerical method is proposed to calculate the ground-state filling and excitation energies. We introduce many-particle on-site energies capturing multi-band effects and discuss tunnelling on a perturbative level using an effectively restricted Hilbert space. Results for small one-dimensional lattices obtained by this method are in good agreement with the exact multi-band diagonalization of the Hamiltonian. Spectral properties associated with the formation of regions with constant filling, so-called Mott shells, are investigated and interfaces between the shells with strong particle fluctuations are characterized by gapless local excitations.

Effective Hamiltonian of a three-orbital Hubbard model on the pyrochlore lattice: Application toLiV2O4

To investigate heavy fermion behavior in the vanadium spinel ${\text{LiV}}_{2}{\text{O}}_{4}$, we start from a three-orbital Hubbard model on the pyrochlore lattice and derive its low-energy effective Hamiltonian by an approach of real-space renormalization-group type. We first derive the effective Hamiltonian numerically and then succeed in representing the results into an analytic form with physical operators for low-energy degrees of freedom in tetrahedron unit. The effective Hamiltonian is defined on the coarse-grained lattice, i.e., face-centered-cubic (fcc) lattice, and it operates in a restricted Hilbert space defined in terms of a specific molecular orbital ${T}_{2}$ in the unit. One important tetrahedron configuration has a threefold orbital degeneracy and spin $S=1$, and correspondingly, the effective Hamiltonian has spin and orbital exchange interactions of Kugel-Khomskii type as well as correlated electron hoppings. The coupling constants in the effective Hamiltonian are determined from the numerically obtained renormalized Hamiltonian and also by means of perturbation. We calculate and analyze low-energy states of the effective Hamiltonian for the unit of four coupled tetrahedra both analytically and numerically. Effective hopping elements in the effective Hamiltonian are renormalized to about 1/10 of the original hopping integral. It is important that different virtual processes make opposite contributions to the exchange term, and consequently the coupling constant is given by a remaining small value. This is particularly prominent in the spin-spin channel, where ferromagnetic double-exchange processes compete with antiferromagnetic superexchange processes. Another important point is that various spin and orbital exchange processes are competing to each other. Together with geometrical frustration of the effective fcc lattice, these two features result in nearly degenerate three lowest-energy states of different types in the four coupled tetrahedra, and each of the three has a finite degeneracy in spin and/or orbital. We also calculate spatial correlations of spin and orbital and found that short-range spin-spin correlations are strongly entangled with orbital configurations. This indicates that large remaining entropy at low temperature is related to slow coupled fluctuations of spin and orbital. These results suggest the absence of phase transition in spin and orbital spaces down to very low temperatures and their large fluctuations in the low-energy sector, which are key issues for understanding the heavy fermion behavior in ${\text{LiV}}_{2}{\text{O}}_{4}$.

ELECTRONIC STRUCTURE OF KINETIC ENERGY DRIVEN CUPRATE SUPERCONDUCTORS

In this paper, we review the low energy electronic structure of the kinetic energy driven d-wave cuprate superconductors. We give a general description of the charge-spin separation fermion-spin theory, where the constrained electron is decoupled as the gauge invariant dressed holon and spin. In particular, we show that under the decoupling scheme, the charge-spin separation fermion-spin representation is a natural representation of the constrained electron defined in a restricted Hilbert space without double electron occupancy. Based on the charge-spin separation fermion-spin theory, we have developed the kinetic energy driven superconducting mechanism, where the superconducting state is controlled by both superconducting gap parameter and quasiparticle coherence. Within this kinetic energy driven superconductivity, we have discussed the low energy electronic structure of the single layer and bilayer cuprate superconductors in both superconducting and normal states, and qualitatively reproduced all main features of the angle-resolved photoemission spectroscopy measurements on the single layer and bilayer cuprate superconductors. We show that the superconducting state in cuprate superconductors is the conventional Bardeen-Cooper-Schrieffer like with the d-wave symmetry, so that the basic Bardeen-Cooper-Schrieffer formalism with the d-wave gap function is still valid in discussions of the low energy electronic structure of cuprate superconductors, although the pairing mechanism is driven by the kinetic energy by exchanging spin excitations. We also show that the well-pronounced peak-dip-hump structure of the bilayer cuprate superconductors in the superconducting state and double-peak structure in the normal state are mainly caused by the bilayer splitting.

Dissociative electron attachment to theH2Omolecule I. Complex-valued potential-energy surfaces for theB12,A12, andB22metastable states of the water anion

We present the results of calculations defining global, three-dimensional representations of the complex-valued potential-energy surfaces of the $^{2}B_{1}$, $^{2}A_{1}$, and $^{2}B_{2}$ metastable states of the water anion that underlie the physical process of dissociative electron attachment to water. The real part of the resonance energies is obtained from configuration-interaction calculations performed in a restricted Hilbert space, while the imaginary part of the energies (the widths) is derived from complex Kohn scattering calculations. A diabatization is performed on the $^{2}A_{1}$ and $^{2}B_{2}$ surfaces, due to the presence of a conical intersection between them. We discuss the implications that the shapes of the constructed potential-energy surfaces will have on the nuclear dynamics of dissociative electron attachment to ${\mathrm{H}}_{2}\mathrm{O}$.

Construction of Wannier functions from localized atomiclike orbitals

The problem of construction of the Wannier functions (WFs) in a restricted Hilbert space of eigenstates of the one-electron Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ (forming the so-called low-energy part of the spectrum) can be formulated in several different ways. One possibility is to use the projector-operator techniques, which pick up a set of trial atomic orbitals and project them onto the given Hilbert space. Another possibility is to employ the downfolding method, which eliminates the high-energy part of the spectrum and incorporates all related to it properties into the energy-dependence of an effective Hamiltonian. We show that by modifying the high-energy part of the spectrum of the original Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$, which is rather irrelevant to the construction of the WFs in the low-energy part of the spectrum, these two methods can be formulated in an absolutely exact and identical form, so that the main difference between them is reduced to the choice of the trial orbitals. Concerning the latter part of the problem, we argue that an optimal choice for the trial orbitals can be based on the maximization of the site-diagonal part of the density matrix. This idea is illustrated on a simple toy model consisting of only two bands, as well as on a more realistic example of ${t}_{2g}$ bands in ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$. Using the model analysis, we explicitly show that a bad choice of the trial orbitals can be linked to the discontinuity of phase of the Bloch waves in the reciprocal space, which leads to the delocalization of the WFs in the real space. Nevertheless, such a discontinuity does not necessary contribute to the matrix elements of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ in the Wannier basis. Similar tendencies are seen in more realistic calculations for the ${t}_{2g}$ bands in ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$, though with some variations caused by the multi-orbital effects. An analogy with the search of the ground state of a many-electron system is also discussed.

Variational DVR Calculations

In the discrete variable representation (DVR) method the potential energy matrix \({\cal V}\) has a particular simple form. It is diagonal with values of the interaction potential at the discrete points. However, this simple form is obtained by making approximations in the calculation of the matrix elements of \({\cal V}\). As a consequence the results cannot be considered as variational estimates. We will show how to recover the variational character of the method using the discrete variable representation eigenvectors as trial functions and performing a variational calculation in a restricted Hilbert space.

Charge and spin ordering and charge transport properties in a two-dimensional inhomogeneoust−Jmodel

We study a two-dimensional t-J model close to the Ising limit in which charge inhomogeneity is stabilized by an on-site potential e_s, by using diagonalization in a restricted Hilbert space and finite temperature Quantum Monte Carlo. Both site and bond centered stripes are considered and their similitudes and differences are analyzed. The amplitude of charge inhomogeneity is studied as e_s -> 0. Moreover, we show that the anti-phase domain ordering occurs at a much lower temperature than the formation of charge inhomogeneities and charge localization. Hole-hole correlations indicate a metallic behavior of the stripes with no signs of hole attraction. Kinetic energies and current susceptibilities are computed and indications of charge localization are discussed. The study of the doping dependence in the range 0.083 < x < 0.167 suggests that these features are characteristic of the whole underdoped region.

Siegert-pseudostate representation of quantal time evolution: A harmonic oscillator kicked by periodic pulses

Finiteness of the computational resources is a hindrance to representing the time evolution of an infinitely extended system. Several numerical techniques are available for mimicking the nonboundedness of the system despite the restricted Hilbert space of the employed expansion basis set. We present a formulation based on the outgoing-wave Siegert pseudostates. A harmonic oscillator exposed to a periodic train of impulsive pulses (``kicks'') demonstrate the efficiency of the Siegert method.