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arXiv:0812.0535 · Physical Review B

U(1) symmetry breaking and violated axial symmetry in TlCuCl3 and other insulating spin systems

Raffaele Dell’Amore, A. Schilling, Karl W. Krämer

Published Jan 29, 2009• 28 citations
arXivHTMLPDF

Abstract

We describe the Bose-Einstein condensate of magnetic bosonic quasiparticles in insulating spin systems using a phenomenological standard functional method for $T=0$. We show that results that are already known from advanced computational techniques immediately follow. The inclusion of a perturbative anisotropy term that violates the axial symmetry allows us to remarkably well explain a number of experimental features of the dimerized spin-1/2 system ${\text{TlCuCl}}_{3}$. Based on an energetic argument we predict a general intrinsic instability of an axially symmetric magnetic condensate toward a violation of this symmetry, which leads to the spontaneous formation of an anisotropy gap in the energy spectrum above the critical field. We, therefore, expect that a true Goldstone mode in insulating spin systems, i.e., a strictly linear energy-dispersion relation down to arbitrarily small excitations energies, cannot be observed in any real material.

DOI
10.1103/physrevb.79.014438

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We describe the Bose-Einstein condensate of magnetic bosonic quasiparticles in insulating spin systems using a phenomenological standard functional method for T=0T=0T=0. We show that results that are already known from advanced computational techniques immediately follow. The inclusion of a perturbative anisotropy term that violates the axial symmetry allows us to remarkably well explain a number of experimental features of the dimerized spin-1/2 system TlCuCl3{\text{TlCuCl}}_{3}TlCuCl3​. Based on an energetic argument we predict a general intrinsic instability of an axially symmetric magnetic condensate toward a violation of this symmetry, which leads to the spontaneous formation of an anisotropy gap in the energy spectrum above the critical field. We, therefore, expect that a true Goldstone mode in insulating spin systems, i.e., a strictly linear energy-dispersion relation down to arbitrarily small excitations energies, cannot be observed in any real material.
Starting from a phenomenological standard energy functional to describe the condensation of a dilute gas of bosonic quasiparticles in magnetic insulators, we find that the inclusion of a perturbation term that explicitly violates the axial symmetry significantly modifies the details of the resulting Bose Einstein condensation. Systems with an originally axial symmetry must show an intrinsic tendency to spontaneously violate this symmetry as soon as the condensation sets in, and the phase transition at the respective critical field may even become of first order. We can explain a number of features in the experimental data of various insulating spin systems, such as a slightly nonlinear magnetization near the critical field as well as hysteresis effects and peculiarities in the energy-level scheme of TlCuCl3. We also offer a consistent explanation for certain anomalies in the magnetocaloric effect and in the magnetization of BaCuSi2O6 by assuming a spontaneous violation of axial symmetry at the magnetic phase transition on an energy scale of about 1 micro-eV. The resulting anisotropy gap in the magnetic excitation spectrum, that inevitably forms at the critical field of all such systems, lifts the linear Goldstone mode and is therefore seriously limiting the lifetime of magnetic condensates to a few nanoseconds at most.
The low-temperature properties of certain quantum magnets can be described in terms of a Bose–Einstein condensation (BEC) of magnetic quasiparticles (triplons). Some mean-field approaches (MFA) to describe these systems, based on the standard grand canonical ensemble, do not take the anomalous density into account and leads to an internal inconsistency, as it has been shown by Hohenberg and Martin, and may therefore produce unphysical results. Moreover, an explicit breaking of the U(1) symmetry as observed, for example, in TlCuCl 3 makes the application of MFA more complicated. In the present work, we develop a self-consistent MFA approach, similar to the Hartree–Fock–Bogolyubov approximation in the notion of representative statistical ensembles, including the effect of a weakly broken U(1) symmetry. We apply our results on experimental data of the quantum magnet TlCuCl 3 and show that magnetization curves and the energy dispersion can be well described within this approximation assuming that the BEC scenario is still valid. We predict that the shift of the critical temperature T c due to a finite exchange anisotropy is rather substantial even when the anisotropy parameter γ is small, e.g., of T c in H = 6 T and for .
States with a coherent precession of magnetization such as a homogeneously precessing domain or a persistently precessing domain created in the superfluid 3{}^{3}3He-BBB represent the macroscopic examples of the Bose-Einstein condensates of magnons. Once the magnons form one of these states, this many-magnon coherent quantum state is described by a ``single-magnon wave function'' (or an order parameter). A suitable external perturbation may cause the condensate to oscillate around the state of coherent precession, which demonstrates a collective rigidity of the condensate against scattering a single magnon out of it. The states corresponding to a free coherent precession of magnetization are degenerate in the phase of precession, so there exist oscillations around such states with a gapless dispersion relation, known as the Goldstone modes. Here, we present both experimental and theoretical results of the study of the spin density oscillations superposed on a homogeneously precessing domain in superfluid 3{}^{3}3He-BBB in the presence of a high-frequency excitation field Brf{B}_{\text{rf}}Brf​. We show that the presence of this field lifts the degeneracy of the precessing state with respect to the phase of precession, that is, it violates the symmetry of the magnon condensate, and former Goldstone modes become non-Goldstone ones, as they acquire the energy gap in their spectrum.
We investigate spontaneous U(1) symmetry breaking and the associated phase transitions in rotating interacting Bose gases. Using a theoretical framework that combines mean-field analysis with rotational dynamics, we analyze how rigid rotation modifies the condensate structure and critical behavior. The study identifies the emergence of Goldstone modes and clarifies their role in the low-energy excitation spectrum. The results provide insight into the interplay between symmetry, rotation, and many-body interactions, contributing to a deeper theoretical understanding of phase structures in Bose systems.
For most kinds of already known Bose-Einstein condensates experimental evidence for the existence of a macroscopic coherent quantum state has been provided, e.g., from the observation of interference phenomena, the formation vortices, the detection of Josephson effects, or even by the manifestation of superfluid behaviour. However, none of these hallmarks for a true, macroscopic phase-coherent state has ever been reported for any insulating spin system in a solid in which magnetic bosonic quasiparticles are thought to condense close to a quantum critical point. We describe an attempt to detect superfluid behaviour in TlCuCl3, and discuss in more detail a proposal for an experiment in which the a.c. Josephson effect can be probed by using a device composed of two weakly coupled magnetic insulators with different critical fields. We conclude that the detection of the a.c. Josephson effect is feasible for a proper choice of compounds with realistic material parameters.
The ground states of Bose-Einstein condensates (BECs) of spin-2 bosons are classified into three distinct phases (ferromagnetic, ``antiferromagnetic,'' and cyclic) depending on the s-wave scattering lengths of binary collisions for total-spin 0, 2, and 4 channels. Many-body spin correlations and magnetic response of the condensate in each of these phases are studied in a mesoscopic regime, while low-lying excitation spectra are investigated in the thermodynamic regime. In the mesoscopic regime, where the system is so tightly confined that the spatial degrees of freedom are frozen, the exact, many-body ground state for each phase is found to be expressed in terms of the creation operators of pair or trio bosons having spin correlations. These pairwise and trio-wise units are shown to bring about some unique features of spin-2 BECs such as a huge jump in magnetization from minimum to maximum possible values and the robustness of the minimum-magnetization state against an applied magnetic field. In the thermodynamic regime, where the system is spatially uniform, low-lying excitation spectra in the presence of magnetic field are obtained analytically using the Bogoliubov approximation. In the ferromagnetic phase, the excitation spectrum consists of one Goldstone mode and four single-particle modes. In the antiferromagnetic phase, where spin-singlet ``pairs'' undergo Bose-Einstein condensation, the spectrum consists of two Goldstone modes and three massive ones, all of which become massless when magnetic field vanishes. In the cyclic phase, where boson ``trios'' condense into a spin-singlet state, the spectrum is characterized by two Goldstone modes, one single-particle mode having a magnetic-field-independent energy gap, and a gapless single-particle mode that becomes massless in the absence of magnetic field.