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Low Temperature Physics

Spinor order parameter and equilibrium states of spin s = 1 Bose systems

Anton V. Hlushchenko, Mikhail Yurievich Kovalevsky

Published Sep 1, 2017• 3 citations
Publisher

Abstract

The problem of classification of degenerate equilibrium states in spin s = 1 systems with the phase and magnetic symmetries simultaneously broken is considered. An assumption about the residual symmetry of degenerate equilibrium states and the transformation properties of the spinor order-parameter operator under transformations generated by additive integrals of motion yields equations that classify its equilibrium values. The anisotropic structure of the equilibrium values of the order parameter is presented in terms of the parameters of the residual symmetry generator. Based on the model with a separated condensate, the structure of the densities of the magnetic additive integrals of motion in the equilibrium state is revealed. A comparison is made with the results of existing studies on superfluid spin s = 1 systems.

DOI
10.1063/1.5004448

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The problem of classification of degenerate equilibrium states in spin s = 1 systems with the phase and magnetic symmetries simultaneously broken is considered. An assumption about the residual symmetry of degenerate equilibrium states and the transformation properties of the spinor order-parameter operator under transformations generated by additive integrals of motion yields equations that classify its equilibrium values. The anisotropic structure of the equilibrium values of the order parameter is presented in terms of the parameters of the residual symmetry generator. Based on the model with a separated condensate, the structure of the densities of the magnetic additive integrals of motion in the equilibrium state is revealed. A comparison is made with the results of existing studies on superfluid spin s = 1 systems.
We propose a method for systematically finding ground states of spinor Bose-Einstein condensates by utilizing the symmetry properties of the system. By this method, we can find not only an inert state, whose symmetry is maximal in the manifold under consideration, but also a noninert state, which has lower symmetry and depends on the parameters in the Hamiltonian. We establish the symmetry-classification method for the spin-1, 2, and 3 cases at zero magnetic field, and find an additional phase in the last case. The properties of the vortices in the spin-3 system are also discussed.
The classification of equilibrium states of superfluid with scalar, vector and tensor order parameters is carried out on the basis of the quasiaveragues concept. The generalization of a requirement of the residual symmetry for nonuniform equilibrium states is given. The admissible requirements of a spatial symmetry in the terms of integrals of motion are found. The connection of these requirements with helicoidal structure of vectors of a spin and spatial anisotropy is established. At some restrictions is shown, that the equilibrium structure of an order parameter can be represented as product of a nonuniform part, depending on spatial coordinates, and homogeneous part of an order parameter.
The s=1s=1s=1 spinor Bose condensate at zero temperature supports ferromagnetic and polar phases that combine magnetic and superfluid ordering. We investigate the formation of magnetic domains at finite temperature and magnetic field in two dimensions in an optical trap. We study the general ground state phase diagram of a spin-1 system and focus on a phase that has a magnetic Ising order parameter and numerically determine the nature of the finite-temperature superfluid and magnetic phase transitions. We then study three different dynamical models: model A, which has no conserved quantities, model F, which has a conserved second-sound mode, and the Gross-Pitaevskii (GP) equation, which has a conserved density and magnetization. We find the dynamic critical exponent to be the same for models A and F (z=2)(z=2)(z=2) but different for GP $(z\ensuremath{\approx}3)$. Externally imposed magnetization conservation in models A and F yields the value $z\ensuremath{\approx}3$, which demonstrates that the only conserved density relevant to domain formation is the magnetization density.
A classification of the equilibrium states of a quantum liquid with d pairing is given on the basis of the concept of quasi-averages. The order-parameter operator is represented in terms of Fermi operators. It is shown that the set of such equilibrium states can be classified in terms of the quantum number corresponding to the projection of the orbital angular momentum of a Cooper pair on the anisotropy direction. The explicit form of the three admissible generators of the unbroken symmetry is found and the corresponding equilibrium values of the order parameter are obtained. A generalization to possible nonuniform equilibrium structures is carried out, and the corresponding form of the order parameter of such structures is found.
Degenerate spinor Bose gases with repulsive density-density interaction and anti-ferromagnetic spin-spin coupling in one spatial dimension are shown to be described by a quantum integrable matrix extension of the nonlinear Schrödinger model, whose fundamental fields are described by an m × nm\,\times\,nm×n matrix of bosonic field operators. The eigenstates of this model are constructed for arbitrarily sized matrix field operators by means of algebraic Bethe-ansatz techniques, and the corresponding Bethe equations governing the spectra of conserved quantities are derived. The approach thus generalizes previously chosen techniques to account for arbitrary spin multiplets and their spin-spin interaction. Focusing on the specific case of the 2×22\times22×2 model, which is shown to correspond to a spin-111 Bose gas, a set of integral equations is derived, which describe its equilibrium thermodynamic properties. From these, the ground state phase diagram is computed both, numerically and analytically in the parameter plane spanned by the chemical potential and an external magnetic field. Furthermore, we establish a sufficient condition for interacting bosons in one dimension to obey the Pauli exclusion principle. In particular, we show that in the presence of paired bound states, no two quasiparticle rapidities can coincide, provided that the Lieb parameter satisfies γ>4/3γ>4/3γ>4/3.
Spinor Bose gases form a family of quantum fluids manifesting both magnetic order and superfluidity. This article reviews experimental and theoretical progress in understanding the static and dynamic properties of these fluids. The connection between system properties and the rotational symmetry properties of the atomic states and their interactions are investigated. Following a review of the experimental techniques used for characterizing spinor gases, their mean-field and many-body ground states, both in isolation and under the application of symmetry-breaking external fields, are discussed. These states serve as the starting point for understanding low-energy dynamics, spin textures and topological defects, effects of magnetic dipole interactions, and various non-equilibrium collective spin-mixing phenomena. The paper aims to form connections and establish coherence among the vast range of works on spinor Bose gases, so as to point to open questions and future research opportunities.