Elsevier

Annals of Physics

Volume 424, January 2021, 168361
Annals of Physics

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

https://doi.org/10.1016/j.aop.2020.168361Get rights and content

Highlights

  • The phase of the wave function of the whole system is arbitrary, as expected.

  • Phase angle of condensate wave function of BEC should be discrete.

  • Effects of exchange and Dzyaloshinsky-Moriya anisotropies lead to crossover.

  • Predictions for Kibble-Zurek mechanism in magnets may be checked experimentally.

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 Ω of a triplon gas taking into account the strength of DM interaction up to second order. The minimization of Ω with respect to self-energies Σn and Σan yields the equation for X1,2=Σn±Σanμ, which define the dispersion of quasiparticles Ek=ϵk+X1ϵk+X2 where ϵk is the bare dispersion of triplons. The minimization of Ω with respect to the magnitude ρ0 and the phase Θ of triplon condensate leads to coupled equations for ρ0 and Θ. We discuss the restrictions on ρ0 and Θ imposed by these equations for systems with and without anisotropy. The requirement of dynamical stability conditions (X1>0,X2>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 Θ=πn (n=0, ±1,±2...) while that of BEC with even a tiny DM interaction results in Θ=π2+2πn. 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. In such quantum magnets the emergence of a triplon condensate leads to a finite staggered magnetization M, whose direction in the xy-plane is related to the condensate phase Θ. We also discuss the possible Kibble–Zurek mechanism in dimerized magnets and its influence on M.

Introduction

Triplons are bosonic quasi-particles introduced in bond operator formalism [1] to describe the singlet–triplet excitations in spin-gapped magnetic materials. Measurement of the magnetization of such antiferromagnetic compounds have shown [2] an interesting dependence of magnetization M(T) at low temperatures: M(T) decreases with decreasing temperature and unexpectedly starts to increase when temperature becomes lower than a critical temperature, Tc. Moreover, it is observed that such behavior of M(T) can take place only when the external magnetic field exceeds a critical value Hc, i.e., (HHc). Further measurements based on inelastic neutron scattering [3] have revealed that in some compounds such as KCuCl3 or TlCuCl3 two Cu2++ ions are antiferromagnetically coupled to form a dimer in crystalline network. There is a gap Δst0.6  meV between the ground singlet (S = 0) and excited triplet (S = 1) states, which can be closed owing to the Zeeman effect for HHc with Δst=gμBHc, where Landé electron g-factor is g2 and Bohr magneton is μB=0.672  K/T. Subsequently, the list of spin-gapped magnets has been extended as reviewed in Ref. [4].

In the ideal case, the triplons have axial symmetry O(3) which can be spontaneously broken leading to a Bose–Einstein condensation (BEC), similar to a BEC of atoms arising from the spontaneous breaking of U(1) symmetry. Now, relating the uniform magnetization M to the total number of triplons as M=gμBN, and the staggered magnetization to the condensate fraction as M=gμBN02, one may describe experimental data on magnetization [4]. As to the data on energy dispersion of collective excitations, Rüegg et al. [5] showed that at low temperatures the spectrum becomes gapless and they may be naturally explained by the existence of a Goldstone mode in a BEC.

Further neutron scattering experiments [6] have shown that the staggered magnetization of dimerized spin gap system of TlCuCl3 remains finite even at T>Tc. This contradicts a pure BEC model based on spontaneous symmetry breaking, when, by definition, N0(T>Tc)=0. Therefore, to improve the situation one may suppose that the rotational (axial) symmetry is weakly explicitly broken in this sample. In fact, in real systems, there may always be weak anisotropy, breaking O(3) symmetry, such as crystalline anisotropies, spin–orbit coupling and dipole interactions. Even when they are weak, the anisotropies may become important at low temperatures and modify the physical properties [7].

In general, the exchange interaction between two moments has the form ΣijSriTijSr+eνj [8] where T=(13)Tr(T)I+Tas+TSM. Here the first term leads to the usual isotropic exchange coupling, Tas is an antisymmetric tensor that describes the Dzyaloshinsky–Moriya (DM) interaction D[SrSr+eν], where D is the DM vector. The last term contains the so-called symmetric exchange anisotropy (EA) and has contributions from the classical dipole–dipole interaction between magnetic moments. Clearly, such interactions break axial symmetry not spontaneously but explicitly.

Spontaneous symmetry breaking (SSB) was originally developed to explain spontaneous magnetization in ferromagnetic systems. Spontaneous coherence in all its forms can be viewed as another type of symmetry breaking. In SSB the Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all particles [9]. Therefore, when the system of triplons is invariant under rotational symmetry, one deals with spontaneous symmetry breaking and hence with pure BEC is termed as isotropic. On the other hand, when DM or EA interactions exist, one has to deal with an explicitly broken axial symmetry where strictly speaking no BEC can take place [10]. Nevertheless, due to the weakness of observed anisotropy we shall apply BEC model calling it the anisotropic case, and study its consequences for physical observables.

A similar study has been performed for the first time by Sirker et al. [11]. They attempted to describe the experimental data [2], [6] on TlCuCl3 and came to an important conclusion: the inclusion of DM interaction into the standard Hamiltonian with a contact interaction smears out the phase transition into a crossover. In other words, in the presence of DM anisotropy a critical temperature Tc (defined as (dNdT)T=Tc=0) may still exist, but the condensate fraction N0(T) subsides asymptotically with increasing temperature. The effect of the presence of only EA interaction has been also studied within BEC concept [12], [13]. Delamore et al. [13] based on an energetic argument on classical level predicted a general intrinsic instability of triplon condensate due to EA. It is interesting to know if this prediction remains true when quantum fluctuations are taken into account. As to the works by Sirker et al.[11], [14] there are two main points we wish to improve. (i) The anomalous density, σ has been neglected. This corresponds to the mean field approach (MFA) called in the literature as Hartree–Fock–Popov (HFP) approximation2 and predicts an unexpected cusp in magnetization near Tc in the pure BEC case for compounds whose anisotropies are negligibly small [15]. Further, it was shown that [16], [17] application of Hartree–Fock–Bogoliubov (HFB) approximation, which includes anomalous density may improve the theoretical description. (ii) The DM interaction with intensity γ, has been taken into account up to the first order in γ as a perturbation. However in reality, it is well known that among the magnetic anisotropies, DM interaction is the strongest and dominate over e.g., the EA interaction.

The main goal of the present work is to study the low temperature properties of spin gapped magnets with DM and exchange anisotropies by taking into account their contribution more systematically than in Refs. [12], [13], [14] including anomalous density. For this purpose, we shall represent DM and EA interactions by a linear (in fields) and quadratic terms, respectively, and use δ-expansion method developed in quantum field theory [18], [19], [20]. We will show that the present approach gives a better description for the proper choice of the phase of the condensate.

The phase of the condensate plays an important role in interference experiments, where two BECs are released and resulting interference patterns are measured [21]. However, in the experiments performed, it is unknown whether the condensate has a well-defined phase before the measurement was made or only afterward. We show that a BEC of a noninteracting system has an arbitrary, random phase, while the presence of interparticle interactions make BEC to acquire a preferred phase which guaranties its stability. Then, we study the consequences of this conclusion for some physical phenomena such as Josephson junctions and Kibble–Zurek mechanism (KZM). In the present work we suggest a method for observing KZM in dimerized magnets. Namely, we propose experimentally to make a rapid quench in a spin gapped quantum magnet below T<Tc and measure the staggered magnetization M. We predict that for a material with even a tiny DM anisotropy, M is not sensitive to the time of quenching, while that of a pure magnet without DM anisotropy vanishes in “rapid quenching” and remains finite in “slow quenching”.

The rest of this article is organized as follows. In Section 2, we present the thermodynamic potential Ω, whose detailed derivation is moved to Appendix. In Section 3, we discuss the phase of the condensate and properties of our equations with respect to self-energies; in Section 4, we discuss interference and Josephson junction of two Bose systems and then we study the KZM in spin gapped magnets in Section 5. The Section 6 summarizes our main conclusions.

Section snippets

The thermodynamic potential including EA and DM interactions

In bond operator formalism for magnetic fields greater than the critical field, HHc, the Hamiltonian of a triplon gas with exchange and DM anisotropies can be presented as the sum of “isotropic” and “anisotropic” terms3 H=Hiso+Haniso,Hiso=drψ+(r)(Kˆμ)ψ(r)+U2(ψ+(r)ψ(r))2,Haniso=HEA+HDM,HEA=γ2drψ+(r)ψ+(r)+ψ(r)ψ(r),HDM=iγdrψ(r)ψ+(r), where ψ(r) is the bosonic field operator of a quasiparticle — triplon, U,γ,γ are the interaction strengths (U0,γ0,γ0)

The condensate fraction and its phase

The condensate fraction ρ0 and its phase ξ=exp(iΘ) may be found from Eqs. (5) and (15) as Ωρ0=cos2Θ(Uσ+γ)+U(ρ0+2ρ1)μγsinΘρ0=0,2Ωρ02=U+γsinΘ2ρ0320,ΩΘ=2cosΘ2ρ0(Uσ+γ)sinΘ+γρ0=0,2ΩΘ2=2ρ0cos2Θ(Uσ+γ)+sinΘγρ00.

The first couple of the equations is convenient to determine ρ0, while the other two are used to fix the phase angle. In particular, Eq. (27c) has two branches of solutions for the phase angle which we call mode-1 and mode-2 as Θ=arcsin(s̃)+2πn,ξ=1s̃2is̃:mode-1π2+πn,ξ=±i:

Interference of two condensates and Josephson effect

The best way of studying the phase of matter experimentally is through measurements with interference patterns or Josephson junctions. The former is usually performed with atomic Bose condensates and the latter with superconductors or, possibly, quantum magnets. Below we discuss the consequences of our results summarized in the previous section, to each of these effects.

Kibble–Zurek mechanism in staggered magnetization

The Kibble–Zurek mechanism (KZM) predicts the spontaneous formation of topological defects in systems that cross a second-order phase transition with SSB at a finite rate. The mechanism was first proposed by Kibble [50] in the context of cosmology to explain how the rapid cooling below a critical temperature induced a cosmological phase transition resulting in the creation domain structures and e.g., baryons from quark–quark plasma. Further, Zurek has extended this paradigm to the condensed

Conclusion

We have derived an explicit expression for the grand canonical thermodynamic potential Ω for a triplon system in dimerized spin-gapped magnets, taking into account both the EA and weak DM anisotropies. The thermodynamic potential embodies all information about the equilibrium homogenous Bose gas at low temperatures. Particularly, minimization of Ω by self-energies (X1,X2) yields two coupled equations, which define the spectrum of quasiparticles, Ek=(εk+X1)(εk+X2) and the densities of triplons.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

AR acknowledges support by TUBITAK-BIDEB, Turkey (2221), LR acknowledges support by TUBITAK-ARDEB, Turkey (1001), AK is supported by the Ministry of Innovative Development of the Republic of Uzbekistan, BT is supported by TUBITAK and TUBA, Turkey 9 .

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