Spin-gapped magnets with weak anisotropies I: Constraints on the phase of the condensate wave function

TL;DR

This paper investigates the thermodynamic properties and phase constraints of spin-gapped quantum magnets with anisotropies, revealing that the phase of the condensate wave function is discretely locked by anisotropy effects, impacting interference and Josephson phenomena.

Contribution

It introduces a mean-field approach accounting for anisotropies to determine phase restrictions of the condensate wave function in spin-gapped magnets, contrasting with previous arbitrary phase assumptions.

Findings

Pure BEC phase angle restricted to multiples of π without anisotropy

Tiny Dzyaloshinsky-Moriya interaction locks phase at π/2

Discrete phase values influence interference and Josephson effects

Abstract

We study the thermodynamic properties of dimerized spin-gapped quantum magnets with and without exchange anisotropy (EA) and Dzyaloshinsky and Moriya (DM) anisotropies within the mean-field approximation (MFA). For this purpose we obtain the thermodynamic potential $\Omega$ of a triplon gas taking into account the strength of DM interaction up to second order. The minimization of $\Omega$ with respect to self-energies $\Sigma_n$ and $\Sigma_{an}$ yields the equation for $X_{1,2}=\Sigma_n\pm\Sigma_{an}-\mu$, which define the dispersion of quasiparticles $E_k=\sqrt{\epsilon_k+X_1}\sqrt{\epsilon_k+X_2}$ where $\epsilon_k$ is the bare dispersion of triplons. The minimization of $\Omega$ with respect to the magnitude $\rho_0$ and the phase $\Theta$ of triplon condensate leads to coupled equations for $\rho_0$ and $\Theta$. We discuss the restrictions on $\rho_0$ and $\Theta$ imposed by these equations for systems with and without anisotropy. The requirement of dynamical stability conditions $(X_1>0, X_2>0)$ in equilibrium, as well as the Hugenholtz-Pines theorem, particularly for isotropic Bose condensate, impose certain conditions to the physical solutions of these equations. It is shown that the phase angle of a purely homogenous Bose-Einstein condensate (BEC) without any anisotropy may only take values $\Theta=\pi n $ (n=0,$\pm 1,\pm 2$...) while that of BEC with even a tiny DM interaction results in $\Theta=\pi/2+2\pi n$. In contrast to the widely used Hartree-Fock-Popov approximation, which allows arbitrary phase angle, our approach predicts that the phase angle may have only discrete values, while the phase of the wave function of the whole system remains arbitrary as expected. The consequences of this phase locking for interference of two Bose condensates and to their possible Josephson junction is studied.