Thermodynamic contacts and breathing mode physics of 1D p-wave Fermi gases in the high temperature limit

One fundamental type of dynamics in a many-body system is the collective mode. Most often collective modes are described using linearized hydrodynamic equations, while their finite lifetime arises from higher-order effects [1, 2]. Although they are not exact excitations as they possess a finite lifetime, collective modes provide insights on how the dynamics of the system depend on the interactions. An excellent platform for studying this physics is an atomic gas system. The experimental control over atomic gases [3] allows one to study thoroughly the collective mode physics for different particle statistics, dimensions, interactions, and even in different phases [4, 5, 6, 7, 8, 9, 11, 12, 10, 13, 14].

A particular atomic gas system of recent interest is a spin-polarized Fermi gas confined to one spatial dimension (1D) [15, 16, 19, 20, 21, 23, 24, 22, 17, 18]. Since the gas is spin-polarized, the s- or even-wave interactions are suppressed, and the leading interactions are p- or odd-wave in character. From the effective-range expansion, one expects that this gas is universal and can be described by a zero-range theory [15]. In other words, the physics can be described by a single low-energy scattering volume, $\ell$, while the contribution to the energetics and dynamics from the effective range, $r$, should be small and perturbative. This is in contradistinction to a 3D p-wave Fermi gas where the effective range although small, is pivotal for understanding both the energetics and dynamics [27, 28, 25, 26, 29].

For a harmonically trapped 1D spin-polarized p-wave Fermi gas, the most basic collective mode is the breathing mode. The breathing mode is unique in that it can be shown to occur at exactly twice the harmonic trap frequency and to be undamped when the interactions are scale-invariant [30, 31]. For a zero-range 1D spin-polarized p-wave Fermi gas, scale invariance arises when the gas is at resonance, that is when the 1D scattering volume is divergent, or when the gas is non-interacting, $\ell=0$.

When one explicitly breaks the scale invariance either by a finite $\ell$ or by the presence of an effective range, $r$, the frequency of the breathing mode will deviate from its scale invariant value. As discussed for s-wave interactions [32, 8, 14] and for p-wave interactions in 2D [11, 33], the shift in the breathing mode frequency in fact depends on the thermodynamic contacts, which characterize the change in the energy of the system with respect to the low-energy scattering parameters, $\ell$ and $r$.

In this article I evaluate both the thermodynamic contacts, as well as the shift in the breathing mode frequency for a p-wave Fermi gas confined in 1D. In the high-temperature limit, one can obtain analytical results to leading order in the virial expansion, which is to leading order in $n\lambda_{th}$, where $n$ is the density and $\lambda_{th}=\sqrt{2\pi\hbar^{2}/(mk_{B}T)}$ is the thermal de Broglie wavelength for a gas of atoms with mass $m$ and temperature $T$. The shift in the breathing mode frequency depends on both the scattering length and effective range contacts, $C_{\ell}$ and $C_{r}$, respectively. I then apply these results to experimentally applicable situations, in the hopes of future experimental studies.

The remainder of this article is organized as follows. Sec. II provides the basics of the 1D p-wave Fermi gas using a two-channel model. Sec. III evaluates the thermodynamic contacts using the virial expansion. Sec. IV then proceeds with a calculation of the frequency shift of the breathing mode within the virial expansion. The experimental applicability of these results is then discussed in Sec. V. Finally, I conclude in Sec. VI.

In this work I use a two-channel model for the 1D p-wave Fermi gas. The Hamiltonian is given by:

Technically this theory has a ultraviolet (UV) divergence. As is custom, one can remove this divergence by examining the two-body scattering properties in the vacuum. The two-body scattering can be calculated according to Fig. (1). Consider two spin polarized fermions with momenta, $Q/2\pm k$, and total energy $Q_{0}=Q^{2}/4+k^{2}$. As shown in Appendix A, the $T$-matrix depends on the dressed molecular propagator, $D(Q,Q_{0})$, via:

From Eq. (2) one can then identify the scattering volume, $\ell$, and the effective range, $r$:

As discussed previously [32, 8, 14, 11, 33], the shift in the breathing mode frequency depends on the thermodynamic contacts. The thermodynamic contacts are related to the change in the energy with respect to the low-energy scattering parameters, $\ell$ and $r$:

Eq. (6) can be evaluated using finite temperature field theory [34]. Since the contact operators are bilinear in the molecular degrees of freedom, they will be functions of the dressed molecular propagator, calculated in the presence of the many-body background:

In the high temperature limit the chemical potential is large and negative, and thus the fugacity, $e^{\beta\mu}$, is very small: $e^{\beta\mu}\ll 1$. The expansion of Eqs. (7-8) in terms of the fugacity is known as the virial expansion. As the contacts are related to the change in energy with respect to the scattering parameters, the contacts are proportional to $e^{2\beta\mu}$ at leading order in the virial expansion. Performing the virial expansion to this order, one obtains the following expressions for the contacts:

Eq. (9) only accounts for the scattering states contribution to the contacts and ignores the bound state contribution which only exists for $\ell>0$. As discussed in Appendix C, the bound-state contribution is only relevant near resonance, where the bound state energy is small but finite. Near resonance, the wavefunctions of the closed channel molecules have substantial overlap with that of the scattering states. Therefore one expects a finite number of closed channel molecules, and a finite contribution to the contacts from the bound-state. In the opposite limit of weak interactions, the two-body bound state is quite deep. If one is concerned with the upper-branch physics, the relaxation rate for pairs of individual atoms to form closed-channel molecules is quite small [3]. The number of closed-channel molecules is then small and the bound-state contribution to the contacts is negligible. For the simplicity of the presentation, I have relegated the molecular contribution to the contacts to Appendix C.

Eq. (9) also suggests a relationship between the two contacts:

For arbitrary $\ell$ and $r$, Eq. (9) does not have a closed solution, and has to be determined numerically. However, there are a number of limits which allow for analytical results. First consider the gas near resonance, and with an effective range that is small compared to the thermal de Broglie wavelength: $r/\lambda_{th}\ll 1$. After restoring $\hbar$ and the atomic mass $m$, one obtains:

Similarly in the weakly interacting limit the contacts are given by:

A final important limit is to consider exactly at resonance, when $\ell^{-1}=0$. In this case the scale invariance is solely broken by the effective range. In this limit the contacts are given by:

Eqs. (11-13) constitute one of the main results of this paper, and characterize the dependence of the thermodynamic contacts on the scattering parameters, $r$ and $\ell$.

With the contacts in hand, it is now possible to examine the shift in the breathing mode frequency. Therefore, consider the spin-polarized 1D Fermi gas in the presence of a harmonic potential with frequency $\omega$. For this discussion I integrate out the molecular degrees of freedom, and focus only on the fermions. The total Hamiltonian is then equivalent to a one-channel model of interacting spin-polarized fermions inside a harmonic trap:

The breathing mode for this system is an oscillatory motion associated with the moment of inertia:

From Eq. (16) and the Heisenberg equation of motion, one can obtain the following dynamic equation for the moment of inertia:

If one neglects the contacts, Eq. (18) has the analytical solution:

The difficulty in evaluating Eq. (18) in general is in determining the time-dependence of the thermodynamic contacts. To this end it is important to note that the breathing mode motion is dilatory in nature and can be described with a time-dependent density of the form:

From Eq. (20), the dynamics of the moment of inertia are also equivalent to a time-dependent rescaling:

This conclusion ought to be contrasted with the arguments presented in Ref. [37]. In Ref. [37], the dilatory motion of the contacts, Eq. (22), follows from an expansion of the contacts around a scale invariant point, and is only valid to leading order in the breaking of scale invariance. In this work the dilatory motion of the contacts follows from the trivial dynamics of the density, which is valid to leading order in the virial expansion. Thus, in this work the scaling ansatz is valid for arbitary interaction strengths provided $n\lambda_{th}\ll 1$.

Substituting Eqs. (21) and (22) into Eq. (18), one obtains a second order differential equation for $\lambda(t)$:

Eq. (26) is the second main result of this work. It relates the shift of the breathing mode frequency to the thermodynamic contacts. It is straightforward to see that in the case of resonance and zero effective range, i.e. when $\ell^{-1}=0$ and $r=0$, and in the non-interacting limit, i.e. when $\ell=0$, $\omega_{B}=2\omega$, which is consistent with non-relativistic conformal symmetry.

In order to evaluate Eq. (26), I employ a local density approximation (LDA) in the high-temperature limit. The details of the LDA are shown in Appendix D. Combining the LDA with the high-temperature expansion of the contacts, Eq. (9), one obtains the following formula for the shift of the breathing mode frequency:

Eq. (27), has to be evaluated numerically, but one can obtain analytical expressions for the shift of the breathing mode frequency near the resonantly interacting limit:

For a more complete picture, consider Fig. (2). In this figure the shift in the breathing mode frequency is presented as a function of $\tilde{\ell}^{-1}$ for various values of the effective range, $\tilde{r}$. One prominent feature of Fig. (2) is that the shift in the breathing mode becomes more negative as the effective range becomes larger. Conversely, the contribution due to the scattering volume, $\tilde{\ell}$, depends on the sign of $\tilde{\ell}$. This can be qualitatively understood by examining Eq. (18) and the pressure, $P$. As shown in Appendix B, the pressure is given by:

One can also see that the shift in the breathing mode frequency is finite at resonance due to the presence of the effective range. Away from resonance, the the shift in the breathing mode frequency is more substantial for $\tilde{\ell}<0$, as some cancellation occurs for $\tilde{\ell}>0$. In the weakly interacting limit, the contribution from the effective range becomes minuscule in comparison to the leading contribution governed by $\tilde{\ell}$, which reinforces the idea that the effective range merely gives a perturbative correction.

The results obtained thus far are for strictly one-dimensional systems. Realistically, however, the one-dimensional physics described above is a low-energy approximation for a three-dimensional spin-polarized Fermi gas placed in a cylindrical trapping potential with tight radial confinement. The low-energy scattering volume, $\ell$, and the effective range, $r$, in the ground state of the radial confinement are related to their 3D counterparts, $v$ and $R$ respectively, via [16, 39, 24]:

Since the 3D scattering volume can be tuned via a p-wave Feshbach resonance [28], the confinement strength can tune both $\ell$ and $r$; a phenomenon called the confinement induced resonance [40, 41]. Across the confinement induced resonance $\ell$ can be tuned continuously from positive to negative infinity via the confinement induced resonance. The 1D effective range is more or less constant, but its value can be enhanced when $a_{\perp}$ is larger than the 3D effective range, $R$. Hence the energy dependence of the 1D scattering can be enhanced by the confinement. For example the effective range for spin-polarized ${}^{40}K$ is [28]: $R\approx 3.8$ nm. For a radial confinement of $\omega_{\perp}\approx 2\pi*400$ kHz, or equivalently $a_{\perp}\approx 25$ nm, the 1D effective range is $r\approx 18$ nm, which is larger than the 3D effective range by almost a factor of $5$. In fact this effect becomes stronger for weaker radial confinement. Although the 1D effective range is an irrelevant quantity to the energetics and dynamics in the renormalization group sense [38], the effective range may be important to the dynamics due to its enhancement from the radial confinement. The importance of the effective range then depends not only on the typical energy scale in the problem, according to the effective range expansion Eq. (4), but also this enhancement from the radial confinement.

To further examine the experimental applicability of these results, the shift in the breathing mode frequency for a 1D spin polarized Fermi gas confined in an axial harmonic potential with frequency $\omega\approx 2\pi*200$ Hz is presented in Fig. (3). For a radial confinement with frequency $\omega_{\perp}\approx 2\pi*400$ kHz, the quasi 1D regime is defined via: $\hbar\omega_{\perp}\gg k_{B}T$, while the high temperature limit is given by: $T\gg N\hbar\omega$, see Appendix D. For the parameters under consideration this leads to $40\ \mu K>T>50\ nK$, where I have used $N\approx 10$.

The results in Fig. (3) are for temperatures $5\ \mu K>T>0.4\ \mu K$, which sits within the quasi 1D-regime and high-temperature limits. At resonance, the shift in the contact is small and only a few percent. However slightly away from resonance and for $\ell<0$, the shift can be about $3-5\%$ depending on the temperature. The asymmetry in Fig. (3) also implies that the effective range is important in describing the shift in the breathing mode frequency.

The main results of this work are the evaluation of the thermodynamic contacts and the shift of the breathing mode frequencies in the high temperature limit, see Eqs. (11-13, 27-29). These results are valid to lowest non-trivial order in the virial expansion, i.e. to second order in $\left(n\lambda_{th}\right)^{2}$.

Although the above results extend to arbitrary interactions strengths, it is important to note that this collective mode does possess a finite lifetime when the scattering volume is finite. The finite lifetime of the breathing mode can be shown to be related to the finite bulk viscosity, $\zeta$, and scales as $\zeta^{-1}$ [37]. In the high-temperature limit, the bulk viscosity will also occur at order $O\left(n\lambda_{th}\right)^{2}$ [29, 43, 44, 45]. Moreover, since the bulk viscosity must be positive, as it is related to entropy production, the bulk viscosity ought to be proportional to $\ell^{2}$ and $r^{2}$ for a weakly interacting and resonantly interacting gas, respectively. Practically one expects the breathing mode picture breaks down around $\tilde{\ell}\approx 1$. A more in-depth study of the bulk viscosity is beyond the scope of this work and will be presented in the future.

One interesting part of this study is that the effective range for a 1D spin-polarized Fermi gas can be enhanced by the presence of radial confinement, see Eq. (31). This is due to the competition between the validity of the low-energy three-dimensional scattering, set by $R$, and the validity of the quasi-1D regime, set by $a_{\perp}$. This increase of the effective range can lead to deviations from the zero-range limit at smaller energy scales. This is evident in Fig. (3). There the contribution to the shift in the breathing mode frequency due to the effective range is important for $\ell<0$. A similar phenomenon has also been emphasized for the case of quasi two-dimensional Fermi superfluids in Ref. [46].

This situation ought to be contrasted to the spin 1/2 s-wave Fermi gas in 3D. This system has the same renormalization group structure as the spin-polarized 1D Fermi gas with p-wave interactions [38]. In both cases, the effective range is an irrelevant quantity, and leads to perturbative effects; a conclusion consistent with effective field theory [25, 26]. As the effective range is irrelevant, the 3D s-wave Fermi gas has a well defined zero-range limit, where the physics are universally described by the s-wave scattering length. In this case, the effective range can often be ignored as it is generally quite small. For the spin-polarized 1D Fermi gas, although a well-defined zero-range limit exists, the effective range can be important due to its enhancement from the transverse confinement, as evidenced by Figs. (2-3).

I would like to thank Shizhong Zhang and Joseph H. Thywissen for useful discussions. This work was supported through the Research Grants Council of the Hong Kong Special Administrative Region, China (General research fund, HKU 17304719, 17304820 and collaborative research fund C6026-16W and C6005-17G), and the Croucher Foundation under the Croucher Innovation Award.