Truncated Hilbert Space Research Articles

Many-body density matrices for free fermions

Building upon an analytical technique introduced by Chung and Peschel [Phys. Rev. B 64, 064412 (2001)], we calculated the many-body density matrix ${\ensuremath{\rho}}_{B}$ of a finite block of B sites within an infinite system of free spinless fermions in arbitrary dimensions. In terms of the block Green function matrix G (whose elements are ${G}_{\ifmmode \bar{\imath}\else \={\i}\fi{}j}=〈{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}〉,$ where ${c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${c}_{j}$ are fermion creation and annihilation operators acting on sites i and j within the block, respectively), the density matrix can be written as ${\ensuremath{\rho}}_{B}=\mathrm{det}(1\ensuremath{-}G)\mathrm{exp}({\ensuremath{\sum}}_{\mathrm{ij}}[\mathrm{ln}G(1\ensuremath{-}{G)}^{\ensuremath{-}1}{]}_{\mathrm{ij}}{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}).$ Our results suggest that Hilbert space truncation schemes should retain the states created by a subset of the ${c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}$'s (in any combination), rather than selecting eigenvectors of ${\ensuremath{\rho}}_{B}$ independently based on the eigenvalue.

Localization properties of two interacting electrons in a disordered quasi one-dimensional potential

We study the transport properties of two electrons in a quasi one-dimensional disordered wire. The electrons are subject to both, a disorder potential and a short range two-body interaction. Using the approach developed by Iida et al. [ Ann. Phys. (N.Y.) 200 (1990) 219 ], the supersymmetry technique, and a suitable truncation of Hilbert space, we work out the two-point correlation function in the framework of a non-linear sigma model. We study the loop corrections to arbitrary order. We obtain a remarkably simple and physically transparent expression for the change of the localization length caused by the two-body interaction.

Path-Integral Renormalization Group Method for Numerical Study on Ground States of Strongly Correlated Electronic Systems

A new efficient numerical algorithm for interacting fermion systems is proposed and examined in detail. The ground state is expressed approximately by a linear combination of numerically chosen basis states in a truncated Hilbert space. Two procedures lead to a better approximation. The first is a numerical renormalization, which optimizes the chosen basis and projects onto the ground state within the fixed dimension, $L$, of the Hilbert space. The second is an increase of the dimension of the truncated Hilbert space, which enables the linear combination to converge to a better approximation. The extrapolation $L\to\infty$ after the convergence removes the approximation error systematically. This algorithm does not suffer from the negative sign problem and can be applied to systems in any spatial dimension and arbitrary lattice structure. The efficiency is tested and the implementation explained for two-dimensional Hubbard models where Slater determinants are employed as chosen basis. Our results with less than 400 chosen basis indicate good accuracy within the errorbar of the best available results as those of the quantum Monte Carlo for energy and other physical quantities.

Path-Integral Renormalization Group Method for Numerical Study of Strongly Correlated Electron Systems

A numerical algorithm for studying strongly correlated electron systems is proposed. The groundstate wavefunction is projected out after a numerical renormalization procedure in the path integral formalism. The wavefunction is expressed from the optimized linear combination of retained states in the truncated Hilbert space with a numerically chosen basis. This algorithm does not suffer from the negative sign problem and can be applied to any type of Hamiltonian in any dimension. The efficiency is tested in examples of the Hubbard model where the basis of Slater determinants is numerically optimized. We show results on fast convergence and accuracy achieved with a small number of retained states.

Nobel Lecture: Electronic structure of matter—wave functions and density functionals

In the intervening more than six decades enormous progress has been made in finding approximate solutions of Schrodinger's wave equation for systems with several electrons, decisively aided by modern electronic com- puters. The outstanding contributions of my Nobel Prize co-winner John Pople are in this area. The main objec- tive of the present account is to explicate DFT, which is an alternative approach to the theory of electronic struc- ture, in which the electron density distribution n(r), rather than the many-electron wave function, plays a central role. I felt that it would be useful to do this in a comparative context; hence the wording ''Wave Func- tions and Density Functionals'' in the title. In my view DFT makes two kinds of contribution to the science of multiparticle quantum systems, including problems of electronic structure of molecules and of condensed matter. The first is in the area of fundamental understanding. Theoretical chemists and physicists, following the path of the Schrodinger equation, have become accustomed to think in terms of a truncated Hilbert space of single-

Peierls instability and optical response in the one-dimensional half-filled Holstein model of spinless fermions

The effects of quantum lattice fluctuations on the Peierls transition are studied within the one--dimensional Holstein molecular crystal model by means of exact diagonalization methods. Applying a very efficient variational Lanczos technique, the ground--state phase diagram is obtained in excellent agreement with predictions of recent density matrix renormalization group calculations. The transition to the charge--density--wave regime is signaled by a strong increase in the charge structure factor. In the metallic regime, the non--universal Luttinger liquid parameters (charge velocity and coupling constant) are deduced from a finite--size scaling analysis. The variational results are supported by a complete numerical solution of the quantum phonon Holstein model on small clusters, which is based on a well--controlled phonon Hilbert space truncation procedure. The metallic and charge--density--wave phases are characterized by significant differences in the calculated optical absorption spectra.

Optical absorption and single-particle excitations in the two-dimensional Holsteint−Jmodel

To discuss the interplay of electronic and lattice degrees of freedom in systems with strong Coulomb correlations we have performed an extensive numerical study of the two-dimensional Holstein t-J model. The model describes the interaction of holes, doped in a quantum antiferromagnet, with a dispersionsless optical phonon mode. We apply finite-lattice Lanczos diagonalization, combined with a well-controlled phonon Hilbert space truncation, to the Hamiltonian. The focus is on the dynamical properties. In particular we have evaluated the single-particle spectral function and the optical conductivity for characteristic hole-phonon couplings, spin exchange interactions and phonon frequencies. The results are used to analyze the formation of hole polarons in great detail. Links with experiments on layered perovskites are made. Supplementary we compare the Chebyshev recursion and maximum entropy algorithms, used for calculating spectral functions, with standard Lanczos methods.

Theory of superconducting Tc of doped fullerenes.

We develop the nonadiabatic polaron theory of superconductivity of ${\mathit{M}}_{\mathit{x}}$${\mathrm{C}}_{60}$ taking into account the polaron band narrowing and realistic electron-phonon and Coulomb interactions. We argue that the crossover from the BCS weak-coupling superconductivity to the strong-coupling polaronic and bipolaronic superconductivity occurs at the BCS coupling constant \ensuremath{\lambda}\ensuremath{\sim}1 independent of the adiabatic ratio, and there is nothing ``beyond'' Migdal's theorem except small polarons for any realistic electron-phonon interaction. By the use of the polaronic-type function and the ``exact'' diagonalization in the truncated Hilbert space of vibrons (``phonons'') we calculate the ground-state energy and the electron spectral density of the ${\mathrm{C}}_{60}^{\mathrm{\ensuremath{-}}}$ molecule. This allows us to describe the photoemission spectrum of ${\mathrm{C}}_{60}^{\mathrm{\ensuremath{-}}}$ in a wide energy region and determine the electron-phonon interaction. The strongest coupling is found with the high-frequency pinch ${\mathit{A}}_{\mathit{g}}$(2) mode and with the Frenkel exciton. We clarify the crucial role of high-frequency bosonic excitations in doped fullerenes which reduce the bare bandwidth and the Coulomb repulsion allowing the intermediate- and low-frequency phonons couple two small polarons in a Cooper pair. The Eliashberg-type equations are solved for low-frequency phonons. The value of the superconducting ${\mathit{T}}_{\mathit{c}}$, its pressure dependence and the isotope effect are found in remarkable agreement with the available experimental data. \textcopyright{} 1996 The American Physical Society.

Cluster diagonalization in systematically expanded Hilbert spaces: Application to models of correlated electrons.

A method of cluster diagonalization in a systematically expanded Hilbert space is described. We discuss some applications of this procedure to models of high-T_c superconductors, like the t - J and one and three bands Hubbard models in two dimensions. The results obtained with this method are compared against results obtained with other techniques dealing with truncated Hilbert spaces. The relation between this method of diagonalization in a reduced Hilbert space, and perturbation theory and variational techniques is also discussed.

Critical and off-critical conformal analysis of the Ising quantum chain in an imaginary field

The critical curve with central charge c=-22/5 of the Ising quantum chain in an imaginary longitudinal field is calculated by finite-size extrapolation of the Yang-Lee square root branch point position. Various power laws are found. The field dependence of the off-criticality mass scale and the four lowest off-criticality scaling functions are calculated. The latter turn out to be universal along the critical line and agree very well with the results of the truncated Hilbert space method for the c=-22/5 continuum theory. The author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic Bethe ansatz result. The scattering phaseshift calculated by Luscher's method (1988) reproduces the Cardy-Mussardo minimal solution (1989) for the S-matrix. If a boundary defect is introduced, the only non-trivial primary field decouples, so that the spectrum consists only of the tower of the unit operator. In the antiferromagnetic regime with a non-staggered imaginary field the author finds boundary-dependent lines where the gap vanishes.

Dynamical thresholds for the existence of excited electronic states of fast ions in solids.

We calculate eigenenergies for bound states of electrons in the dynamical screening potential of a fast charged particle traversing a solid by diagonalization of Hamiltonian matrices in a truncated Hilbert space of 600 hydrogenic basis states. Application of classical scaling invariances yields a universal energy-level diagram. A simple approximate formula for projectile-centered one-electron bound states is obtained. We find velocity thresholds for the existence of excited states which are in surprisingly good agreement with recent data by Chevallier et al. [Phys. Rev. A 41, 1738 (1990)].

The scaling region of the tricritical Ising model in two dimensions

We study the scaling region spanned by all four relevant perturbations of the tricritical Ising model in two dimensions. We analyze the spectrum of the (1 + 1)-dimensional off-critical hamiltonian on a truncated Hilbert space, a method recently proposed by Yurov and AL Zamolodchikov. I the phase coexistence regions the massive excitations are kink states. On the temperature-driven two-phase coexistence line, they form bound states, which we analyze for periodic as well as for twisted boundary conditions. We find a new asymmetric two-phase region driven by the subleading magnetic field. There are some indications of massless states along the crossover line to the Ising model. The effects of off-critical integrability on the spectra are also observed and discussed.

Current conservation and the transverse electric multipole field.

Different prescriptions for imposing the constraints of current conservation on the Fourier transform of the electromagnetic current operator generate alternative forms for the electric multipole fields. These forms lead to quite different predictions for electromagnetic form factors in nuclear calculations that violate current conservation either by adopting an approximate current operator or by diagonalizing the nuclear Hamiltonian in a truncated Hilbert space. For typical shell model calculations of electron scattering form factors, it appears that no existing prescription for imposing the constraints of current conservation on electric multipole operators is entirely satisfactory. Numerical results are given for electric transitions in $^{12}\mathrm{C}$.

Accurate, smooth, local, energy-dependent optical potentials for electron scattering

We present a new method for obtaining local, energy-dependent, single-channel or multichannel optical potentials. The optical potential is obtained by calculating and smoothing a potential that yields the correct scattering matrix for a sequence of truncated interaction-potential matrices. The new method is illustrated by applications to electron-hydrogen-atom scattering in a truncated Hilbert space. The resulting optical potentials accurately reproduce the original close-coupling calculations in the same Hilbert space, and they show clearly the effects of charge polarization and flux absorption. The coordinate dependence and energy dependence of the smoothed optical potentials are compared to models in the literature.

Coupled reaction channels compared with coupled channels array

It is argued that the coupled reaction channels approach to rearrangement reactions consistently solves the Schrödinger equation for the system in a truncated Hilbert space. A formulation of the coupled channels array approach in the same severely truncated space bears no consistent relation to the Schrödinger equation.

Unitarity and off-shell effects in the impulse approximation

For elastic scattering of a particle from a finite many-body target, the procedure of eliminating the two-body potential in favor of a two-body scattering matrix is examined. It is shown that the theory of Kerman, McManus, and Thaler leads to unitary relations which are not of a particularly convenient form. A simple procedure which identifies the appropriate off-shell scattering matrix to be used in the impulse approximation is presented. The projection operator technique is employed to define an optical potential and then, in a truncated Hilbert space, a relation similar to a two-body Low equation is used to identify the appropriate two-body $T$ matrix. With this latter procedure the unitarity relations are maintained in a form which allows them to be used as a guide to the validity of the impulse approximation.

Weak-Coupling Model forPb212andPb204

The weak-coupling model is applied to the study of the low-lying states of $^{212}\mathrm{Pb}$ and $^{204}\mathrm{Pb}$. We first obtain the phonon states of $^{210}\mathrm{Pb}$ and $^{206}\mathrm{Pb}$ by shell-model calculations. The states of $^{212}\mathrm{Pb}$ ($^{204}\mathrm{Pb}$) are then obtained by solving the eigenvalue problem in a truncated Hilbert space consisting of a small number of basis vectors, each of which is a product of two phonon states of $^{210}\mathrm{Pb}$ ($^{206}\mathrm{Pb}$). A canonical transformation is used to orthonormalize these two-phonon basis vectors. The energies as well as the wave functions of many low-lying states of $^{212}\mathrm{Pb}$ ($^{204}\mathrm{Pb}$) given by the exact shell-model calculations are reproduced surprisingly well by the weakcoupling model where only three or four low-lying $^{210}\mathrm{Pb}$ ($^{206}\mathrm{Pb}$) phonons are included.[NUCLEAR STRUCTURE $^{212}\mathrm{Pb}$, $^{204}\mathrm{Pb}$; energies and wave functions calculated in terms of the $^{210}\mathrm{Pb}$ and $^{206}\mathrm{Pb}$ phonons.]

Use of a Truncated Hilbert Space in Nuclear Reaction Calculations

This formulation of nuclear reaction calculations stays as close as possible to the usual bound-state shell-model calculations. The Hilbert space used in the bound-state calculations is enlarged by adding the scattering states only at that energy for which the reaction is calculated. This Hilbert space is finite-dimensional. The eliminated Hilbert space is partly accounted for by an effective interaction. A variational principle is used to derive the equations for the $K$ and the $S$ matrix, which are then solved by algebraical methods. The numerical application of this method to the reaction ${\mathrm{N}}^{15}(n,{n}^{\ensuremath{'}}){\mathrm{N}}^{15}$ shows good agreement with the results of previous calculations.