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

Some years ago, Tan derived a set of exact relations that link the short distance large-momentum correlations to the bulk thermodynamic properties of a fermionic system with short-range inter-particle interactions Tan1 ; Tan2 ; Tan3 . They are connected by a single coefficient $C$, referred to as the integrated contact intensity or “contact”. Further, Tan’s ideas were developed Br8 ; Br9 and extended to Bose systems also Br10 ; Lang11 ; Werner1 ; Combescot13 . It has been established that Tan’s contact measures the density of pairs at short distances and determines the exact large-momentum or high frequency behavior of various physical observables. It serves as an important quantity to characterize strongly interacting many-body systems Pitbook14 .

Experimentally, Tan’s relations have been confirmed both for fermions (especially for quasi-1D systems) kuhle2011 ; sagi2012 ; stewart2010 and Bose gases chang2016 ; Fletcher2016 ; makotyu2014 ; wild2012 ; zou2021 .

Although, there are several theoretical studies of contact of low dimensional Fermi systems even at finite temperatures Hoffman ; yuPRA80 ; Matveeva ; Br8 ; brateen2013 ; vigndofermi ; Patu , the temperature dependence of bosonic contact in three dimensions is mostly unknown.

Studying the temperature dependence of contact parameter gives an opportunity to know its critical behavior near phase transitions. For example, Chen et al. chencrit have shown that contact and its derivatives are uniquely determined by the universality class of the phase transition for fermions in the critical region. In a system of bosons, critical behavior of thermodynamic quantities have some specifics due to Bose-Einstein condensation (BEC) at low temperatures. For instance, the specific heat is discontinuous at the critical temperature $T_{c}$, and the Grüneisen parameter changes its sign garst ; ourMCE .

The main goal of the present work is to study the temperature dependence of Tan’s contact for bosons, as well as its derivatives, to draw conclusions about their critical behavior near $T_{c}$ and close to the quantum critical point (QCP) of the quantum phase transition (QPT).

It is well known that physical systems at equilibrium are studied in statistical mechanics through statistical ensembles. For example, a system that the canonical ensemble exchanges only energy with the surroundings is called $NVT$, where $N$ is the number of atoms of a gas in the volume $V$ at temperature $T$. On the other hand, a grand canonical ensemble also makes it possible for particles to exchange, with the parameter $\mu$, which may be interpreted as the chemical potential, and is called $\mu VT$. In the former case, $N$ or $\rho=N/V$ is fixed and the chemical potential is determined by the equation of state, while in the latter case $\mu$ is fixed as an input parameter which defines number of particles through a thermodynamic equation Marzolino . Tan’s relations including the contact have been thoroughly studied for $NVT$ ensembles, however, the investigations of contact for a $\mu VT$ ensemble is still missing. In this work, we choose a system of bosonic quasiparticles introduced in bond operator formalism Sachdev ; OurAnn2 to describe the singlet-triplet excitations in spin gapped magnetic materials Zapf . It is well established that low temperature properties of such a class of quantum magnets could be described within the paradigm of Bose-Einstein condensation of these quasiparticles called triplons. Therefore, one may conclude that, at low temperature, thermodynamic properties of such materials are mainly determined (but not only) by the condensation (and depletion) of triplons Zapf ; ourMCE ; ourJT ; ourCharac . Triplon concept has found application in other spin-gapped models as well kumar .

There is one more reason of our choice of spin gapped quantum magnets to study Tan’s contact. In such systems, the number of particles $N$ and the condensate fraction $N_{0}/N$ can be directly evaluated by measuring the uniform magnetization $M$ and staggered magnetization $M_{\perp}$ respectively, as $N=M/g\mu_{B}$, $N_{0}^{2}=2M_{\perp}^{2}/(g\mu_{B})^{2}$ delamorre , where $g$ is the electron Landé factor and $\mu_{B}$ is the Bohr magneton. Evidently, the final analytic expression for $C$ will include $M$ and $M_{\perp}$, making the contact parameter to be easily measured.

In practice, Tan’s contact for bosons with zero range interaction may be evaluated by any of following Tan’s relations Pitbook14 . For example,

1. 1.

   By the asymptotic behavior of momentum distribution $n_{k}$,

   $$C=\lim_{k\rightarrow\infty}k^{4}n_{k}\,.$$

   (1.1)

2. 2.

   By Tan’s sweep theorem as

   $$C=\frac{8\pi ma^{2}}{V}\ \begin{cases}\left(\displaystyle\frac{\partial F}{\partial a}\right)_{N,T}\ \ \ &\text{for $NVT$ ensembles}\,,\\ \left(\displaystyle\frac{\partial\Omega}{\partial a}\right)_{\mu,T}\ \ \ &\text{for ${\mu}VT$ensembles}\,,\end{cases}$$

   (1.2)

   where $F$ and $\Omega$ are the free energy and the grand canonical potential, respectively bouchul21 ; $a$ is the scattering length and $m$ is the effective mass of the particle.

3. 3.

   By using Hellman-Feynman theorem Br8

   $$C=\frac{16\pi^{2}a^{2}}{V}\int d\vec{r}\langle\psi^{\dagger}(\vec{r})\psi^{\dagger}(\vec{r})\psi(\vec{r})\psi(\vec{r})\rangle\,.$$

   (1.3)

In our previous paper ourTan1 , it has been shown that if one uses mean field theory (MFT), evaluation of $C$ from Eq. (1.2) will be the most convenient and reliable. Consequently, we study $C$ of a triplon gas at finite temperature in the framework of MFT, namely in the Hartree-Fock-Bogoliubov (HFB) approximation ourTan13 . We stress here that below we shall discuss $C$ only for homogeneous systems. For inhomogeneous systems, especially at very low temperature, one may also use the Gross-Pitaevskii equation to study the dynamics of the condensate wangpra81 ; wangpra84 as well as that of the Tan’s contact.

For convenience, we adopt units such that $\hbar=1$, $k_{B}=1$ and $V=1$ in the following text. In these units $C$ is dimensionless. In natural units $C$ may be obtained further by dividing it by ${\bar{a}}^{4}$, getting it in (length)^-4 where ${\bar{a}}$ is the average lattice parameter, ${\bar{a}}=V^{1/3}$, and $V$ is the unit cell volume of the crystal.

This article is organized as follows. In Section II, we outline main equations for a triplon gas at finite temperature in the HFB approximation, which will be used to evaluate the Tan’s contact and its derivatives in Section III. Our discussions and conclusions will be presented in Section IV.

We start with the effective Hamiltonian of triplons as a non-ideal Bose gas with contact repulsive interaction

where $\psi(\vec{r})$ is the bosonic field operator, $U=4\pi a/m$ is the interaction strength, and $\hat{K}$ is the kinetic energy operator which defines the bare triplon dispersion $\varepsilon_{\vec{k}}$ in momentum space. Since the triplon BEC occurs in solids, the integration is performed over the unit cell of the crystal with the corresponding momenta defined in the first Brillouin zone ouraniz1 . The parameter $\mu$ characterizes an additional direct contribution to the triplon energy due to the external field

and it can be interpreted as the chemical potential of the $S_{z}=-1$ triplons Giamarchi . In Eq. (2.2), $\Delta=g\mu_{B}H_{c}$, is the spin gap separating the ground-state from the lowest energy triplet excitation and $H_{c}$ is the critical external magnetic field grundmann . The effective Hamiltonian ${\hat{H}}$ has the following free parameters characterizing a given material: $g$, $H_{c}$, $U$ and possibly, an effective triplon mass $m$ for the case of a simple bare dispersion given as $\varepsilon_{\vec{k}}={\vec{k}}^{2}/2m$.¹¹1 In general, a realistic $\varepsilon_{\vec{k}}$ includes several other parameters misguich .

In general the Hamiltonian in Eq. (2.1) is invariant under global $U(1)$ gauge transformation $\psi(\vec{r})\rightarrow e^{i\alpha}\psi(\vec{r})$ with a real number $\alpha$. However, this symmetry is broken in the condensate phase, where $T<T_{c}$, and it is restored in the normal phase, $T\geqslant T_{c}$.

This transition temperature $T_{c}$, which corresponds to the vanishing of the condensate density $\rho_{0}(T_{c})=0$ may be calculated from the following equation

Note that for existing quantum magnets with a spin gap $T_{c}$ is rather large, $T_{c}\approx 2$ K, in contrast to the critical temperature of BEC in atomic gases, where the number of particles is fixed and the critical temperature is of the order of nanokelvins. However, in the $\mu VT$ ensemble under discussion, the particle density depends on the external magnetic field through the chemical potential $\mu$ as

in the normal phase. In the BEC phase, an explicit expression for $\rho(T<T_{c})$ strongly depends on the chosen version of an approximation. For example, in the HFB approximation, which is employed here, the particle density and condensate fraction are given by the following set of equations ourAnn

where $\rho_{1}$ and $\sigma$ correspond to the density of non-condensed particles and anomalous density, respectively. The hyperbolic function in Eq. (2.5) can be represented also as $\coth(\beta x/2)=1/2+f_{B}(x)$, where $f_{B}(x)=1/(\exp(\beta x)-1)$ is the Bose distribution function.

Note that neglecting $\sigma$ leads to an unexpected cusp in magnetization ourPRB81 . In Eq. (2.5), $E_{k}$ is the dispersion of quasiparticles (bogolons) given as $E_{k}=\sqrt{\varepsilon_{k}}\sqrt{\varepsilon_{k}+2\mu_{eff}}$ with the speed of sound $c=\sqrt{\mu_{eff}/m}$, where $m$ has the meaning of the triplon effective mass, corresponding to the limit of small momenta $\varepsilon_{k}\approx\vec{k}^{2}/2m$. The typical value of $m$, used in the literature, Zapf is rather small, $m\approx 0.02$ K.

As it is pointed out in the Introduction, for the $\mu VT$ ensemble it is convenient to calculate $C$ using

where the grand thermodynamic potential $\Omega$ has the following total derivative Pitbook14 ; ourJT ; ourCharac

in which $S$ is the entropy and $P$ is the pressure. Note that the relation between the magnetization $M$ and the triplon density may be directly obtained from Eq. (3.2) as $M=g\mu_{B}\rho$. Moreover, using Eq. (3.2) one may find useful expressions for the derivatives of $C$ as

It is well known that from BEC to a normal phase transition is a second order phase transition, in which the entropy $S(T,H,a)$ is continuous across the critical temperature i.e.,

Therefore, using Eqs. (3.2)-(3.4) leads to an extended Ehrenfest relation:

where $C_{H}=T(\partial S/\partial T)$ is the heat capacity and $\Delta[f]\equiv f(T=T_{c}^{-})-f(T=T_{c}^{+})$ is the jump in the function $f$ at $T=T_{c}$.²²2A similar relation for $NVT$ ensembles will be $T_{c}^{-1}\Delta[C_{P}]=-\Delta\left[\left(\displaystyle\frac{\partial P}{\partial T}\right)_{V,U}\right]\left(\displaystyle\frac{dV}{dT}\right)+\displaystyle\frac{1}{2U^{2}m^{2}}\Delta\left[\left(\displaystyle\frac{\partial C}{\partial T}\right)_{V,U}\right]\left(\displaystyle\frac{dU}{dT}\right)$ when the entropy is considered as a function of temperature, volume and the strength of the contact interaction.

In the following we discuss the normal and BEC phases, separately.

We have studied the Tan’s contact for spin-gapped quantum magnets, assuming that their low temperature properties are related to that of a triplon gas. To the best of our knowledge, this is the first time the parameter $C$ and its dependence on temperature is investigated for a solid. Naturally, it will be interesting to compare our results with existing ones in the literature. However, there is neither a theoretical nor an experimental study of Tan’s contact for $\mu VT$ ensembles at finite temperatures and most of the literature on $C(T)$ concern 1D gases in $NVT$ ensembles.

The temperature dependence of $C$ of harmonically trapped 1D Leib-Liniger Bose gas has been studied by Yao et al. YaoPRE121 . They have found that contact increases at low temperatures, reaches a maximum at $T=T^{*}$ and then starts to decrease, i.e., $\left.(\partial C/\partial T)\right|_{T<T^{*}}>0$, $\left.(\partial C/\partial T)\right|_{T=T^{*}}=0$ and $\left.(\partial C/\partial T)\right|_{T>T^{*}}<0$. From Figs. 1 and 2, one may note that for a 3D system of bosons the situation is quite the opposite; $\left.(\partial C/\partial T)\right|_{T<T_{c}}<0$ and $\left.(\partial C/\partial T)\right|_{T>T_{c}}>0$. But at the critical temperature $T_{c}$, in both cases $C(T)$ exhibits an extremum as a function of temperature, maximum for 1D bosons and minimum for triplons, i.e., in both cases $(\partial C/\partial T)$, changes its sign near the critical temperature. Bearing in mind the relation $(\partial C/\partial T)\sim(\partial S/\partial U)$, one may conclude that at $T=T_{c}$ the interaction dependence of the entropy displays a maximum. Note that in the case of 1D bosons this maximum provides a signature of the crossover to the fermionized regime, while in the case of quantum magnets it corresponds just to the point of finite temperature phase transition, where accumulation of entropy occurs garst .

In the present work the temperature dependence of $C(T)$ at large temperatures $T>T_{c}$, is found to be almost linear. This is to be compared with the results by Vignolo and Minguzzi Vignolo who obtained the large temperature behavior as $C\sim\sqrt{T}$ for a 1D Bose gas in the Tonks-Girardeau limit. It would be interesting to study the effect of dimensionality on $C(T)$ in $\mu VT$ ensembles also kosterlic .

We have studied the critical behavior of $C$ at the quantum phase transition and found that $C$ and $(\partial C/\partial\mu)$ are regular at QCP which is in good agreement with predictions made by Chen et al. chencrit .

In conclusion it should be underlined that our analytical expressions for Tan’s contact of spin gapped compounds are expressed in terms of magnetizations which are directly observable. For example, at very low temperatures, $T\ll T_{c}$, one evaluates $C$ simply from the expression $C=m^{2}\mu^{2}/{\bar{a}}^{4}$, where the only parameter, effective mass should be estimated. Note that, in this region contact of a $\mu VT$ ensemble is almost not sensitive to the strength of the boson-boson interaction $U$, while the dependence on $U$ of the contact of an $NVT$ ensemble is rather strong even in the classical approximation, when $C$ is given by $C(NVT)\approx 16\pi^{2}a^{2}\rho^{2}=U^{2}m^{2}\rho^{2}$ ourTan1 ; Pitbook14 .

Finally, we venture to suggest a way to measure $C$ in quantum magnets, which can be performed in a similar way as it was done in Ref. [wild2012, ] (see Appendix B for details).

We hope our work will stimulate more studies, especially experimental ones exploring the universality of $\mu VT$ ensembles in quantum magnets. Results presented here will deepen our understanding of the connection between short-ranged two-body correlations and magnetic phase transitions in antiferromagnetic materials, which have the potential to be used in developing the next generation of spintronic devices.

In this appendix we provide explicit expressions for the integrals introduced in Eqs. (LABEL:drodSUBg), (3.15) and (3.16):

where $f_{B}(x)=1/(e^{\beta x}-1)$, $W_{k}=(1/2+f_{B}(E_{k}))$, $\omega_{k}=\varepsilon_{k}-\mu+2U\rho$ and $E_{k}^{2}=\varepsilon_{k}(\varepsilon_{k}+2\mu_{eff})$.

Equation (3.15) includes $(\partial\rho_{1}/\partial\mu_{eff})$, $(\partial\sigma/\partial\mu_{eff})$ and $(\partial\mu_{eff}/\partial U)$. The derivatives with respect to $\mu_{eff}$ can be found by differentiating Eqs. (2.5) to give

where $W_{k}^{\prime}=\beta(1-4W_{k}^{2})$.

As to $(\partial\mu_{\rm eff}/\partial U)$, it can be obtained by differentiation of both sides of the equation $\mu_{\rm eff}(U)=\mu+2U\left[\sigma(\mu_{eff}(U))-\rho_{1}(\mu_{eff}(U))\right]$ with respect to $U$ and then solving it for $(\partial\mu_{eff}/\partial U)$. This leads to

Now, we proceed to evaluate $(\partial C/\partial H)=-2m^{2}U^{2}g\mu_{B}(\partial\rho/\partial U)$ and $(\partial C/\partial T)=-2m^{2}U^{2}(\partial S/\partial U)$ near the critical temperature $T_{c}$.

From the explicit expression

where $\omega_{k}=\varepsilon_{k}-\mu+2U\rho(U)$, one obtains

where $\omega^{\prime}_{k}=(\partial\omega_{k}/\partial U)=2\rho+2U(\partial\rho/\partial U)$ and $(\partial\rho/\partial U)$ is given by Eq. (LABEL:drodSUBg). It is straightforward to show that the integral $I_{2}$ in Eq. (LABEL:drodSUBg) is divergent at $T=T_{c}$, where $\omega_{k}=\varepsilon_{k}\sim k^{2}/2m$, and hence $\left.(\partial\rho/\partial U)\right|_{T=T_{c}}=-\rho/U$. This means that $\omega^{\prime}_{k}(T=T_{c}^{+})=0$ and $\left.(\partial S/\partial U)\right|_{T=T_{c}^{+}}=0$. Therefore $\left.(\partial C/\partial T)\right|_{T=T_{c}^{+}}=0$ and $\left.(\partial\rho/\partial U)\right|_{T=T_{c}^{+}}=-\rho/U$.