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Physical Review Letters

Exact Time-Dependent Analysis for the One-Dimensional XY Model

Eytan Barouch, Max Dresden

Published Jul 21, 1969• 21 citations
Publisher

Abstract

An exact solution of the Liouville equation for the $\mathrm{XY}$ model with time-dependent external magnetic fields is obtained. Standard transformations allow a decomposition of the space of states into a direct product of four-dimensional subspaces. The Liouville equation is reduced to a set of coupled differential equations in each subspace. These equations have been explicitly solved for several cases.

DOI
10.1103/physrevlett.23.114

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An exact solution of the Liouville equation for the XY\mathrm{XY}XY model with time-dependent external magnetic fields is obtained. Standard transformations allow a decomposition of the space of states into a direct product of four-dimensional subspaces. The Liouville equation is reduced to a set of coupled differential equations in each subspace. These equations have been explicitly solved for several cases.
The Liouville equation for the XY\mathrm{XY}XY model is solved exactly, and the magnetization is computed explicitly. Nonergodic behavior of the magnetization is found for a general class of time-dependent magnetic fields.
The Liouville-Bloch equation for a general spin jjj in time-dependent external fields is solved as a unitary product of exponential operators. Operator noncommutativity is explicitly handled so as to leave behind a set of first-order differential equations for classical functions of time. Of these, only one is nontrivial and, once solved, the remaining yield to simple quadrature. Exact solutions or perturbative expansions of the classical equations can be used for analytical or numerical integration of the quantal equations.
The one-dimensional Schrödinger equation associated with a time-dependent Coulomb potential is studied. The invariant operator method (Lewis and Riesenfeld) and unitary transformation approach are employed to derive quantum solutions of the system. We obtain an ordinary second-order differential equation whose analytical exact solution has been unknown. It is confirmed that the form of this equation is similar to the radial Schrödinger equation for the hydrogen atom in a (arbitrary) strong magnetic field. The qualitative properties for the eigenstates spectrum are described separately for the different values of the parameter ω 0 appearing in the x 2 term, x being the position, i.e., ω 0 > 0, ω 0 < 0 and ω 0 = 0. For the ω 0 = 0 case, the eigenvalue equation of invariant operator reduces to a solvable form and, consequently, we have provided exact eigenstates of the time-dependent Hamiltonian system.
We present an exact analytical solution for the quantum dynamics of a charged particle subjected to both a time-dependent electric field and a static magnetic field aligned along the zzz -direction. Using a systematic approach based on successive unitary transformations, we reduce the original three-dimensional problem to a two-dimensional system of decoupled, time-dependent harmonic oscillators. This technique produces free parameters that allow us to impose constrains to derive the exact solution of Schrödinger equation with the time-dependent Hamiltonian through the explicit derivation of the quantum propagator and shows the equivalence of this approach to established path integral methods for such systems. The developed framework provides new insights into quantum systems with time-dependent electromagnetic fields and offers analytical solutions.