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arXiv:2104.11894 · Journal of Physics Condensed Matter

Critical behavior of Tan’s contact for bosonic systems with a fixed chemical potential

Abdulla Rakhimov, Tolibjon Abdurakhmonov, B. Tanatar

Published Aug 18, 2021• 6 citations

Abstract

Abstract The temperature dependence of Tan’s contact parameter C and its derivatives for spin gapped quantum magnets are investigated. We use the paradigm of Bose–Einstein condensation (BEC) to describe the low temperature properties of quasiparticles in the system known as triplons. Since the number of particles and the condensate fraction are not fixed we use the μVT ensemble to calculate the thermodynamic quantities. The interactions are treated at the Hartree–Fock–Bogoliubov approximation level. We obtained the temperature dependence of C and its derivative with respect to temperature and applied magnetic field both above and below T c of the phase transition from the normal phase to BEC. We have shown that C is regular, while its derivatives are discontinuous at T c in accordance with Ehrenfest’s classification of phase transitions. Moreover, we have found a sign change in ∂ C /∂ T close to the critical temperature. As to the quantum critical point, C and its derivatives are regular as a function of the control parameter r , which induces the quantum phase transition. At very low temperatures, one may evaluate C simply from the expression C = m 2 μ 2 / a ¯ 4 , where the only parameter effective mass of quasiparticles should be estimated. We propose a method for measuring of Tan’s contact for spin gapped dimerized magnets.

DOI: 10.1088/1361-648x/ac1ec6

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Abstract The temperature dependence of Tan’s contact parameter C and its derivatives for spin gapped quantum magnets are investigated. We use the paradigm of Bose–Einstein condensation (BEC) to describe the low temperature properties of quasiparticles in the system known as triplons. Since the number of particles and the condensate fraction are not fixed we use the μVT ensemble to calculate the thermodynamic quantities. The interactions are treated at the Hartree–Fock–Bogoliubov approximation level. We obtained the temperature dependence of C and its derivative with respect to temperature and applied magnetic field both above and below T c of the phase transition from the normal phase to BEC. We have shown that C is regular, while its derivatives are discontinuous at T c in accordance with Ehrenfest’s classification of phase transitions. Moreover, we have found a sign change in ∂ C /∂ T close to the critical temperature. As to the quantum critical point, C and its derivatives are regular as a function of the control parameter r , which induces the quantum phase transition. At very low temperatures, one may evaluate C simply from the expression C = m 2 μ 2 / a ¯ 4 , where the only parameter effective mass of quasiparticles should be estimated. We propose a method for measuring of Tan’s contact for spin gapped dimerized magnets.

We obtained, within a path-integral formalism (mean-field variational Gaussian approximation), analytical expressions for thermodynamic quantities such as magnetization, heat capacity, and the magnetic Gr\"uneisen parameter ${\mathrm{\ensuremath{\Gamma}}}_{H}$ of the system of triplons in spin gapped quantum magnets. Near the critical temperature ${\mathrm{\ensuremath{\Gamma}}}_{H}$ is discontinuous and changes its sign upon the Bose-Einstein condensation (BEC) of triplons. We predict that in the limit of low temperature

T

and near the critical magnetic field ${H}_{c},\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}_{H}$ diverges as ${\mathrm{\ensuremath{\Gamma}}}_{H}\ensuremath{\sim}1/{T}^{2}$, while it scales as ${\mathrm{\ensuremath{\Gamma}}}_{H}\ensuremath{\sim}1/(H\ensuremath{-}{H}_{c})$ as the magnetic field approaches

{H}_{c}

.

Exploiting the analogy between ultracold atomic gases and the system of triplons, we study magneto-thermodynamic properties of dimerized quantum magnets in the framework of Bose–Einstein condensation (BEC). Particularly, introducing the inversion (or Joule–Thomson) temperature [Formula: see text] as the point where Joule–Thomson coefficient of an isenthalpic process changes its sign, we show that for a simple paramagnet, this temperature is infinite, while for three-dimensional (3D) dimerized quantum magnets it is finite and always larger than the critical temperature [Formula: see text] of BEC. Below the inversion temperature [Formula: see text], the system of triplons may be in a liquid phase, which undergoes a transition into a superfluid phase at [Formula: see text]. The dependence of the inversion temperature on the external magnetic field [Formula: see text] has been calculated for quantum magnets of TlCuCl3 and Sr3Cr2O8.

We study properties of magnetic field-induced Bose-Einstein condensate of triplons as a function of temperature and the field within the Hartree-Fock-Bogoliubov approach including the anomalous density. We show that the magnetization is continuous across the transition, in agreement with the experiment. In sufficiently strong fields the condensate becomes unstable due to triplon-triplon repulsion. As a result, the system is characterized by two critical magnetic fields: one producing the condensate and the other destroying it. We show that nonparabolic triplon dispersion arising due to the gapped bare spectrum and the crystal structure has a strong influence on the phase diagram.

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 .

We study properties of magnetic field-induced Bose-Einstein condensate of triplons as a function of temperature and the field within the Hartree-Fock-Bogoliubov approach including the anomalous density. We show that the magnetization is continuous across the transition, in agreement with the experiment. In sufficiently strong fields the condensate becomes unstable due to triplon-triplon repulsion. As a result, the system is characterized by two critical magnetic fields: one producing the condensate and the other destroying it. We show that nonparabolic triplon dispersion arising due to the gapped bare spectrum and the crystal structure has a strong influence on the phase diagram.

The low-temperature evolution from BCS to Bose-Einstein condensation (BEC) for a two-dimensional d-wave superconductor is discussed at the saddle point (mean field) level. A systematic study of the changes of low-temperature thermodynamic properties is presented as a function of the charge carrier density and fixed interaction. It has been found that when the interaction strength is large enough, there is a critical density below which the single quasiparticle excitation spectrum develops a gap. At higher density of carriers and lower interaction strength (towards the BCS regime) the superconductor has gapless quasiparticle excitations, while at lower densities and higher interactions (towards the BEC regime) quasiparticle excitations are fully gapped. The appearence of a full gap in the quasiparticle excitation spectrum has dramatic consequences to the compressibility, specific heat, and spin susceptibility at low temperatures, as the critical density

{n}_{c}

is crossed. The change in behavior of these quantities indicates a possible quantum phase transition between a d-wave gapless phase and a d-wave fully gapped phase, as

{n}_{c}

is crossed.