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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Berry’s phase in a one-dimensional quantum many-body system

Gunter Markus Schutz

Published Mar 1, 1994• 6 citations
Publisher

Abstract

We study an interacting one-dimensional quantum lattice gas of massive fermions on a ring with L lattice sites. The ring is threaded by a magnetic flux corresponding to a twist in boundary conditions. We compute the periodicity of the ground state under an adiabatically increasing flux and the associated Berry's phase occurring in this process. The model has a second-order phase transition line which coincides with a line where the Berry phase changes nonanalytically.

DOI
10.1103/physreve.49.2461

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We study an interacting one-dimensional quantum lattice gas of massive fermions on a ring with L lattice sites. The ring is threaded by a magnetic flux corresponding to a twist in boundary conditions. We compute the periodicity of the ground state under an adiabatically increasing flux and the associated Berry's phase occurring in this process. The model has a second-order phase transition line which coincides with a line where the Berry phase changes nonanalytically.
The one-dimensional attractive lattice fermion gas equivalent to the Heisenberg-Ising spin-1/2 chain is studied for a ring geometry threaded by magnetic flux. We find that for charged fermions having interaction strength \ensuremath{\Delta}=cos(\ensuremath{\pi}/p) with p noninteger, the adiabatic ground state is periodic in the magnetic flux threading the chain, with period two flux quanta, as found by Shastry and Sutherland for the repulsive case. We find that, at particular values of the threading field, a sequence of initially zero-energy bound states form at the Fermi surface during the adiabatic process, the largest containing [p] (the integer part of p) fermions. This largest bound state moves rapidly around to the other Fermi point where it sequentially unbinds. We find Berry's phase for the whole process to be [p]\ensuremath{\pi}. For p integer, as \ensuremath{\Phi} increases, eventually all the particles in the system go into bound states of size [p]. The period in this case is of order the size of the system. The charge-carrying mass of the fermions is calculated by finding the energy of the adiabatic ground state with twisted boundary conditions.
We study a system of strongly interacting one-dimensional (1D) bosons on a ring pierced by a synthetic magnetic flux tube. By the Fermi-Bose mapping, this system is related to the system of spin-polarized noninteracting electrons confined on a ring and pierced by a solenoid (magnetic flux tube). On the ring there is an external localized $\ensuremath{\delta}$-function potential barrier, $V(\ensuremath{\phi})=g\ensuremath{\delta}(\ensuremath{\phi}\ensuremath{-}{\ensuremath{\phi}}_{0})$. We study the Berry phase associated with the adiabatic motion of the $\ensuremath{\delta}$-function barrier around the ring as a function of the strength of the potential ggg and the number of particles NNN. The behavior of the Berry phase can be explained via quantum mechanical reflection and tunneling through the moving barrier which pushes the particles around the ring. The barrier produces a cusp in the density to which one can associate a missing charge $\mathrm{\ensuremath{\Delta}}q$ (missing density) for the case of electrons (bosons, respectively). We show that the Berry phase (i.e., the Aharonov-Bohm phase) cannot be identified with the quantity $\mathrm{\ensuremath{\Delta}}q/\ensuremath{\hbar}\ensuremath{\oint}\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}d\mathbf{l}$. This means that the missing charge cannot be identified as a (quasi)hole. We point out the connection of this result and recent studies of synthetic anyons in noninteracting systems. In addition, for bosons we study the weakly interacting regime, which is related to the strongly interacting electrons via Fermi-Bose duality in 1D systems.
We consider a system of charged spinless fermions in a ring governed by the antiferromagnetic Heisenberg-Ising Hamiltonian and in the presence of a magnetic flux. We find that the two broken-symmetry ground states evolve adiabatically with increasing flux with a period corresponding to two flux quanta, while the period of the total spectrum is one flux quantum. This behavior, already observed for this system in the gapless regime [B. Sutherland and B. S. Shastry, Phys. Rev. Lett. 65, 1833 (1990)], is shown to be a natural consequence of the relevant degrees of freedom involved in the low-energy physics of this system: antiferromagnetic domain walls or solitons, with half the charge of the original particles. A space-time approach is introduced to describe the dynamics of these objects, affording a complete topological classification of space-time histories of the system. This allows a physically complete understanding of the ground-state-subspace evolution with increasing flux in the antiferromagnetic broken-symmetry regime. In this case, the flux period doubling can be explained in terms of the Berry's phase gained by the two degenerate broken-symmetry ground states upon adiabatic switching of the flux.
I generalize the concept of Berry's geometrical phase for quasicyclic Hamiltonians to the case in which the ground state evolves adiabatically to an excited state after one cycle, but returns to the ground state after an integer number of cycles. This allows to extend the charge Berry phase γ c related to the macroscopic polarization, to many-body systems with fractional number of particles per site. Under certain conditions, γ c and the spin Berry phase γ s jump in π at the boundary of superconducting phases. In the extended Hubbard chain with on-site attraction U and nearest-neighbor interaction V at quarter filling, the transitions detected agree very well with exact results in two limits solved by the Bethe ansatz, and with previous numerical studies. In chains with spin SU (2) symmetry, γ s jumps when a spin gap opens.