Magnetic properties of a Fermi gas in a noncommutative phase space

Abstract

Motivated by the strong constraints to noncommutative quantum theories coming from Gamma Ray Bursts and the precision attained by SQUID devices in measuring magnetic fields, we study in this article the thermodynamic behaviour of a Fermi gas subject to the action of a background magnetic field, in two and three dimensional space with noncommutative coordinates and momenta. An explicit expression is obtained for the magnetization and magnetic susceptibility of the gas, both for Landau’s diamagnetism and Pauli’s paramagnetism. We are lead in this way to an upper bound $\theta \lesssim (10 \,\text {Gev})^{-2}$ for the noncommutative parameter.

Appendices

The fugacity in the $\beta \sim 0$ regime

In the high temperature regime –i.e. in the inverse temperature $\beta \sim 0$ regime– we may obtain the asymptotic expansion of the fugacity z in terms of $\beta$ and the density of particles ${\bar{n}}$. As a first step, we compute ${\bar{n}}$ as the derivative of the partition function with respect to $\mu _c$ as $\beta$ remains constant:

$$\begin{aligned} {\bar{n}}&= \rho \,z\sum _{n=0}^{\infty } \frac{e^{-\beta \omega _c\left( n+\frac{1}{2}\right) }}{1+z\,e^{-\beta \omega _c\left( n+\frac{1}{2}\right) }}. \end{aligned}$$

(A.1)

From this expression it can be seen that as the temperature becomes larger z must tend to zero in order for ${\bar{n}}$ to remain finite. Using this fact, one may expand the denominator⁹ in (A.1) and then recast the density of particles by performing a resummation:

$$\begin{aligned} {\bar{n}}&=- \rho \,\sum _{r=1}^{\infty }\ \frac{\left( -ze^{-\beta \omega _c/2}\right) ^{r}}{1-{e^{-\beta \omega _c r}}}. \end{aligned}$$

(A.2)

Now, it is known that every series like (A.2) may be inverted by using the Lagrange inversion theorem. The final result is

$$\begin{aligned} z=-e^{\beta \omega _c/2}\sum _{n=1}^{\infty } \frac{1}{n!} \left( \sum _{k=1}^{n-1}\frac{(-1)^{k} (n+k-1)!}{(n-1)!} B_{n-1,k}({\hat{f}}_1,\cdots ,{\hat{f}}_{n-k}) \right) \left( -\frac{f\,{\bar{n}}}{\rho } \right) ^n, \end{aligned}$$

(A.3)

where we have made use of the Bell polynomials $B_{n,k}$, and defined the following functions of $\beta$:

$$\begin{aligned} \begin{aligned}{\hat{f}}_k:&=\frac{1-e^{-\beta \omega _c}}{(k+1)(1-e^{-\beta \omega _c(k+1)})},\\ f:&= 1-e^{-\beta \omega _c}. \end{aligned} \end{aligned}$$

(A.4)

It should be noticed that expression (A.3) is well-behaved in the limit $\beta \rightarrow 0$ and provides us with the following expansion for the fugacity z:

$$\begin{aligned} z \sim \frac{{\bar{n}}\beta \omega _c}{\rho } +O(\beta ^2). \end{aligned}$$

(A.5)

On using Euler-Maclaurin formula to compute Pauli’s paramagnetism in the low temperature regime

In this App. we will show how to obtain an analytical result for the partition function considered in Sect. 4 for the Pauli paramagnetism’ problem,

$$\begin{aligned} -\beta \phi =\rho \sum _{s=\pm 1}\sum _{n=0}^{\infty } \log (1+e^{-\beta \omega _c n+\alpha _s}), \end{aligned}$$

(B.1)

where the parameter $\alpha _s$ has been defined in Eq. (4.2).

For a while we will forget about the index s which labels the spin up and down components—if we just focus on the sum over the n index we can write according to Euler-Maclaurin Formula

$$\begin{aligned} -\frac{\beta \phi }{\rho }&= \int _0^{\infty }dx \log (1+e^{-\beta \omega _c x+\alpha _s}) + \frac{1}{2} \log (1+e^ {\alpha _s}) \nonumber \\&\quad +\frac{B_2}{2} \frac{\beta \omega _c}{1+e^{-\alpha _s}}-\frac{B_4}{4!}\frac{(\beta \omega _c)^3}{4}\sinh \left( \frac{\alpha _s}{2}\right) \text {sech}^3\left( \frac{\alpha _s}{2}\right) \nonumber \\&\quad +\frac{1}{2}\sum _{k=1}^{\infty }\int _0^{\infty } dt\frac{\cos (2\pi k\,t)}{(2\pi k)^4} \frac{(\beta \omega _c)^4}{8}\frac{\sinh (\beta \omega _c t-\alpha _s)-2}{\cosh ^4\left( \frac{\alpha _s-\beta \omega _c t}{2}\right) }, \end{aligned}$$

(B.2)

where $B_n$ is the n-th Bernoulli number.

The first integral on the RHS of (B.2) may be splitted in the following two, which may be easily computed in the low temperature regime, viz. when $\beta \epsilon _F\gg 1$ and $\epsilon _F \gg \omega _c$ (or equivalently $\alpha _s\gg 1$):

$$\begin{aligned} \begin{aligned}\int _0^1 dx \frac{\alpha }{\beta \omega _c} \log (1+e^{(1-x)\alpha })&=\frac{6\alpha ^2+\pi ^2+12{\text {Li}}_2(-e^{-\alpha })}{12\beta \omega _c},\\ \int _0^{\infty }dx \log (1+e^{-\beta \omega _c x})&=\frac{\pi ^2}{12\beta \omega _c}. \end{aligned} \end{aligned}$$

(B.3)

In this expression we have made use of ${\text {Li}}_2(\cdot )$, the polylogarithm function of order 2.

On the other hand, the last term in the RHS of (B.2) can be neglected in our regime since it is $O(e^{-\alpha _s})$. In effect, the cosine is bounded above by one so the entire term is bounded above by

$$\begin{aligned} \frac{(\beta \omega _c)^3}{4}\sinh \left( \frac{\alpha _s}{2}\right) \text {sch}^3\left( \frac{\alpha _s}{2}\right) , \end{aligned}$$

(B.4)

proving thus our assertion.

Adding expressions (B.2) and (B.3), and reintroducing the spin’ index s, we finally obtain for the partition function in the low temperature regime

$$\begin{aligned} -\beta \phi = \frac{\rho }{2}\sum _{s=\pm } \left( \frac{\alpha _s^2}{\beta \omega _c}+\alpha _s+\frac{\pi ^2}{3\beta \omega _c}+B_2\beta \omega _c+O(e^{-\alpha _s})\right) . \end{aligned}$$

(B.5)