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Physical Review A

Inverse problem for the time-dependent density matrix

H. L. Frisch, Pál Ruján

Published Apr 1, 1982• 3 citations
Publisher

Abstract

The solution of the inverse problem for the time-dependent density matrix means finding the time-dependent Hamiltonian operator $\stackrel{^}{H}$ corresponding to a given time-dependent density matrix $\stackrel{^}{\ensuremath{\rho}}$. We point out that the standard requirements for the time-dependent density operator imply a number of compatibility relations. Accordingly, a given $N\ifmmode\times\else\texttimes\fi{}N$ density matrix might result from a whole family of Hamiltonians, characterized by $N$ free, real parameters. The cases of a 2\ifmmode\times\else\texttimes\fi{}2 and 3\ifmmode\times\else\texttimes\fi{}3 density matrix and of the free-fermion chain are worked out explicitly.

DOI
10.1103/physreva.25.1815

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The solution of the inverse problem for the time-dependent density matrix means finding the time-dependent Hamiltonian operator \stackrel{^}{H} corresponding to a given time-dependent density matrix \stackrel{^}{\ensuremath{\rho}}. We point out that the standard requirements for the time-dependent density operator imply a number of compatibility relations. Accordingly, a given N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN density matrix might result from a whole family of Hamiltonians, characterized by NNN free, real parameters. The cases of a 2\ifmmode\times\else\texttimes\fi{}2 and 3\ifmmode\times\else\texttimes\fi{}3 density matrix and of the free-fermion chain are worked out explicitly.
We formulate the inverse Kohn–Sham equations based on the density equation theory. The present formulation relies on neither action functionals nor force balance equation, but is constructed by comparing the first-order density equation for the electron interacting real system and that for the electron non-interacting Kohn–Sham system. The derived inverse Kohn–Sham integral equation requires information of an exact ground-state electron density to calculate an exact Kohn–Sham potential for a ground state and it requires an exact time-dependent electron density and a time-derivative of the exact time-dependent first-order density matrix to calculate an exact time-dependent Kohn–Sham potential. A possible algorithm to construct the Kohn–Sham potential for a ground state is suggested as well as a solution algorithm for the time-dependent inverse Kohn–Sham problem without resorting to any approximations. These two solution algorithms, in combination, suggest a way to perform an exact time-dependent density functional calculation as a predictive method.
A new family of free fermionic quantum spin chains with multispin interactions was recently introduced. Here we show that it is possible to build standard quantum Ising chains---but with inhomogeneous couplings---which have the same spectra as the novel spin chains with multispin interactions. The Ising models are obtained by associating an antisymmetric tridiagonal matrix to the polynomials that characterize the quasienergies of the system via a modified Euclidean algorithm. For the simplest nontrivial case, corresponding to the Fendley model, the phase diagram of the inhomogeneous Ising model is investigated numerically. It is characterized by gapped phases separated by critical lines with order-disorder transitions depending on the parity of the total number of energy density operators in the Hamiltonian.
The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time- dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schrodinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time- independent equation by replacing the energy E with ih @ @t . The perturbation theory formulas for the density equation have also been given for both the time- dependent and time-independent cases.
Abstract We present a simple construction for a tridiagonal matrix T that commutes with the hopping matrix for the entanglement Hamiltonian of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since T is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of , and therefore the entanglement entropy.
We present a general study of a large family of exactly integrable quantum chains with multispin interactions. The exact integrability follows from the algebraic properties of the energy density operators defining the quantum chains. The Hamiltonians are characterized by a parameter $p=1,2,\ensuremath{\cdots}$ related to the number of interacting spins in the multispin interaction. In the general case, the quantum spins are of infinite dimension. In special cases, characterized by the parameter N=2,3,...N=2,3,...N=2,3,..., the quantum chains describe the dynamics of Z(N)Z(N)Z(N) quantum spin chains with open boundary conditions. The simplest case p=1p=1p=1 corresponds to the free fermionic quantum Ising chain (N=2N=2N=2) or the Z(N)Z(N)Z(N) free parafermionic quantum chain. The eigenenergies of the quantum chains are given in terms of the roots of special polynomials, and for general values of ppp the quantum chains are characterized by a free fermionic (N=2N=2N=2) or free parafermionic (N>2N>2N>2) eigenspectrum. The models have a special critical point when all coupling constants are equal. At this point, the ground-state energy is exactly calculated in the bulk limit, and our analytical and numerical analyses indicate that the models belong to universality classes of critical behavior with dynamical critical exponent z=(p+1)/Nz=(p+1)/Nz=(p+1)/N and specific-heat exponent $\ensuremath{\alpha}=max{0,1\ensuremath{-}(p+1)/N}$.