Thermal aspects of interacting quantum gases in Lorentz-violating scenarios

Abstract

In this work, we study the interaction of quantum gases in Lorentz-violating scenarios considering both boson and fermion sectors. In the latter case, we investigate the consequences of a system governed by scalar, vector, pseudovector and tensor operators. Besides, we examine the implications of $\left( \hat{k}_{a}\right) ^{\kappa }$ and $\left( \hat{k}_{c}\right) ^{\kappa \xi }$ operators for the boson case and the upper bounds are estimated. To do so, we regard the grand canonical ensemble seeking the so-called partition function, which suffices to provide analytically the calculations of interest, i.e., the mean particle number, the entropy, the mean total energy and the pressure. Furthermore, in low-temperature regime, such quantities converge until reaching a similar behavior being in contrast with what is shown in high-temperature regime, which brings out the differentiation of their effects. In addition, the particle number, the entropy and the energy exhibit an extensive characteristic even in the presence of Lorentz violation. Also, for the pseudovector and the tensor operators, we notice a remarkable feature due to the breaking process of spin degeneracy: the system turns out to have greater energy and particle number for the spin-down particles in comparison with the spin-up ones. Finally, we propose two feasible applications to corroborate our results: phosphorene and spin precession.

Appendices

Appendix A: Are they still extensive state quantities?

Here, to corroborate our results, we proceed further for the sake of verifying the validity of the derived relations in the thermodynamic limit. Indeed, we need to take $N\rightarrow \infty$, $V\rightarrow \infty$ and $N/V=$ const. Being proportional to the volume, the mean particle number, the entropy and the mean energy turn out to be extensive quantities in the ordinary case. Nevertheless, knowing whether the Lorentz violation removes such extensive property or not is an intriguing question to be checked. With this purpose, we proceed making the substitution in the following way

$$\begin{aligned} \sum _{r}\rightarrow \frac{gV}{\left( 2\pi \right) ^{3}}\int d^{3}\varvec{ p}, \end{aligned}$$

(A1)

where g is the degeneracy factor. Now, let us make some comments. Taking the thermodynamic limit is only reasonably supported when $u^{\prime }\left( \bar{n}\right)$ and $u\left( \bar{n}\right)$ do not depend upon the volume. Additionally, this also entails that the particle number density $\bar{n}=\bar{N}/V$ must not depend upon it. Now, let us verify this assumption starting with

$$\begin{aligned} \left. \frac{\partial \bar{n}}{\partial V}\right| _{\mu ,T}=\left. \frac{ \partial }{\partial V}\left( \frac{\bar{N}}{V}\right) \right| _{\mu ,T}= \frac{1}{V}\left( \left. \frac{\partial \bar{N}}{\partial V}\right| _{\mu ,T}-\frac{\bar{N}}{V}\right) , \end{aligned}$$

(A2)

which yields

$$\begin{aligned} \left. \frac{\partial \bar{N}}{\partial V}\right| _{\mu ,T}= & {} \frac{gV}{ \left( 2\pi \right) ^{3}}\int d^{3}\varvec{p}\frac{1}{\exp \left\{ \beta \left[ \epsilon _{r}\left( \varvec{p}\right) +\delta _{r}\left( \varvec{p}\right) +u^{\prime }\left( \bar{n}\right) -\mu \right] \right\} +\chi } \nonumber \\&+\frac{gV}{\left( 2\pi \right) ^{3}}\int d^{3}\varvec{p}\left\{ -\left[ \frac{1}{\exp \left\{ \beta \left[ \epsilon _{r}\left( \varvec{p}\right) +\delta _{r}\left( \varvec{p}\right) +u^{\prime }\left( \bar{n}\right) -\mu \right] \right\} +\chi }\right] ^{2}\right\} \nonumber \\&\times \exp \left\{ \beta \left[ \epsilon _{r}\left( \varvec{p}\right) +\delta _{r}\left( \varvec{p}\right) +u^{\prime }\left( \bar{n}\right) -\mu \right] \right\} \beta \left. \frac{\partial u^{\prime }\left( \bar{n} \right) }{\partial V}\right| _{\mu ,T}. \end{aligned}$$

(A3)

In this sense, we identify the first term as $\bar{N}/V$; the second term we rewrite using

$$\begin{aligned} \left. \frac{\partial u^{\prime }\left( \bar{n}\right) }{\partial V} \right| _{\mu ,T}=u^{\prime \prime }\left( \bar{n}\right) \left. \frac{ \partial \bar{n}}{\partial V}\right| _{\mu ,T}=\frac{u^{\prime \prime }\left( \bar{n}\right) }{V}\left( \left. \frac{\partial \bar{N}}{\partial V} \right| _{\mu ,T}-\frac{\bar{N}}{V}\right) , \end{aligned}$$

(A4)

as

$$\begin{aligned} -\sum _{r}\bar{n}_{r}\frac{\beta }{V}u^{\prime \prime }\left( \bar{n}\right) \left( \left. \frac{\partial \bar{N}}{\partial V}\right| _{\mu ,T}-\frac{ \bar{N}}{V}\right) \exp \left\{ \beta \left[ \epsilon _{r}\left( \varvec{ p}\right) +\delta _{r}\left( \varvec{p}\right) +u^{\prime }\left( \bar{n} \right) -\mu \right] \right\} , \end{aligned}$$

and to finish we obtain

$$\begin{aligned} \left. \frac{\partial \bar{n}}{\partial V}\right| _{\mu ,T}= & {} \frac{1}{V }\left( \left. \frac{\partial \bar{N}}{\partial V}\right| _{\mu ,T}- \frac{\bar{N}}{V}\right) \nonumber \\= & {} \frac{1}{V}\left( -\sum _{r}\bar{n}_{r}\frac{\beta }{V}u^{\prime \prime }\left( \bar{n}\right) \exp \left\{ \beta \left[ \epsilon _{r}\left( \varvec{p}\right) +\delta _{r}\left( \varvec{p}\right) +u^{\prime }\left( \bar{n}\right) -\mu \right] \right\} \right) \nonumber \\&\times \left( \left. \frac{\partial \bar{N}}{\partial V}\right| _{\mu ,T}-\frac{\bar{N}}{V}\right) . \end{aligned}$$

(A5)

Additionally, this relation is verified if

$$\begin{aligned} 1=-\sum _{r}\bar{n}_{r}\frac{\beta }{V}\exp \left\{ \beta \left[ \epsilon _{r}\left( \varvec{p}\right) +\delta _{r}\left( \varvec{p}\right) +u^{\prime }\left( \bar{n}\right) -\mu \right] \right\} u^{\prime \prime }\left( \bar{n}\right) . \end{aligned}$$

(A6)

Nevertheless, in a general case, this is not true for the reason that $u\left( n\right)$ is an absolutely arbitrary interaction potential density. As a matter of fact, it must hold that

$$\begin{aligned} \left. \frac{\partial \bar{N}}{\partial V}\right| _{\mu ,T}=\frac{\bar{N} }{V}, \end{aligned}$$

(A7)

i.e., $\left. \frac{\partial \bar{n}}{\partial V}\right| _{\mu ,T}$ must vanish. Finally, we should notice that, since $\delta _{r}\left( \varvec{p} \right)$ is a function only of $\varvec{p}$, it does not mess up the extensive property. Therefore, for such thermal properties, even in the presence of Lorentz violation, the extensive characteristic of the system is maintained as well.

Appendix B: Numerical analyses

See Tables 3, 4, 5, 6, 7 and 8.

Table 3 The scalar operator concerning the fermion sector

Table 4 The vector operator concerning the fermion sector

Table 5 The pseudovector operator for fermions

Table 6 The tensor operator for fermions

Table 7 The vector operator in the boson sector

Table 8 The tensor operator for the boson sector

Here, we provide such Appendix to exhibit a concise explanation for the numerical calculations encountered throughout this manuscript. We show the thermal quantities, namely the energy, the mean particle number and the entropy per volume, for different values of $\beta$. Besides, $\mathcal {E}$, $\mathfrak {N}$ and $\mathfrak {S}$ are quantities representing the energy, the mean particle number as well as the entropy per volume, respectively. The outputs for fermions and bosons modes are displayed as follows: