Superfluid density and critical velocity near the fermionic Berezinskii-Kosterlitz-Thouless transition

TL;DR

This paper provides a theoretical analysis of superfluid density and critical velocity near the Berezinskii-Kosterlitz-Thouless transition in a 2D strongly interacting Fermi gas, proposing experimental detection methods.

Contribution

It introduces a Gaussian pair fluctuation approach to calculate superfluid properties and suggests a practical experimental setup to observe the transition.

Findings

Calculated superfluid density and critical temperature at BKT transition

Predicted a measurable jump in Landau critical velocity during experiments

Proposed stirring method to detect superfluid-normal interface

Abstract

We theoretically investigate superfluidity in a strongly interacting Fermi gas confined to two dimensions at finite temperature. Using a Gaussian pair fluctuation theory in the superfluid phase, we calculate the superfluid density and determine the critical temperature and chemical potential at the Berezinskii-Kosterlitz-Thouless transition. We propose that the transition can be unambiguously demonstrated in cold-atom experiments by stirring the superfluid Fermi gas using a red detuned laser beam, to identify the characteristic jump in the local Landau critical velocity at the superfluid-normal interface, as the laser beam moves across the cloud.

Equations

$$\mathcal{H}=\sum _{\sigma }{\bar{\psi }}_{\sigma }(\mathbf{r}){\mathcal{H}}_0{\psi }_{\sigma }(\mathbf{r})-U{\bar{\psi }}_{\uparrow }(\mathbf{r}){\bar{\psi }}_{\downarrow }(\mathbf{r}){\psi }_{\downarrow }(\mathbf{r}){\psi }_{\uparrow }(\mathbf{r}),$$

$${\Omega }_{\text{MF}}=\frac{{\Delta }^2}{U}+\sum _{\mathbf{k}}\left[{\tilde{\xi }}_{\mathbf{k}}-E_{\mathbf{k}}-\frac{2}{\beta }\ln \left(1+e^{\beta E_{\mathbf{k}}^{+}}\right)\right],$$

$${\Omega }_{\text{GF}}$$

$$\mathcal{S}\left(\mathcal{Q}\right)$$

$$\frac{1}{\beta }\sum _{|l|>l_0}{\mathcal{S}}_{\eta }\left(\mathbf{q},i{\nu }_l\right)=-\frac{1}{\pi }{\int }_{-\infty }^{+\infty }d\omega \frac{\text{Im}{\mathcal{S}}_{\eta }\left(\mathbf{q},\omega +i\gamma \right)}{e^{\beta \omega }+1},$$

$$n_s=\frac{1}{M}{\left[\frac{{\partial }^2\Omega \left({\mathbf{v}}_s\right)}{\partial v_s^2}\right]}_{v_s=0}=\frac{4M}{{\hslash }^2}{\left[\frac{{\partial }^2\Omega \left(\mathbf{Q}\right)}{\partial Q^2}\right]}_{Q=0}.$$

$$k_B{T}_c=\frac{\pi }{2}\frac{{\hslash }^2}{4M}{n}_s\left(T_c\right).$$

$${\Omega }_{\text{MF}}\left(\mathcal{Q}\right)=\frac{{\Delta }^2}{U}+\sum _{\mathbf{k}}\left[{\tilde{\xi }}_{\mathbf{k}}-E_{\mathbf{k}}-\frac{2}{\beta }\ln \left(1+e^{\beta E_{\mathbf{k}}^{+}}\right)\right],$$

$$\sum _{\mathbf{k}}\left[\frac{1-2f\left(E_{\mathbf{k}}^{+}\right)}{2E_{\mathbf{k}}}-\frac{1}{{\hslash }^2{\mathbf{k}}^2/M+{\epsilon }_B}\right]=0,$$

$${\Omega }_{\text{GF}}\left(\mathcal{Q}\right)$$

$$\mathcal{S}\left(\mathcal{Q}\right)$$

$$M_{11}$$

$$M_{12}$$

Full text

In two-dimensional (2D) many-body systems, topologically nontrivial vortex fluctuations, that are suppressed due to vortex/anti-vortex binding at low temperature, become amplified above a certain critical temperature, leading to the so-called Berezinskii-Kosterlitz-Thouless (BKT) transition Berezinskii1972 ; KT1973 ; Nelson1977 . The BKT transition has been of great importance in different branches of physics and has been observed in a range of settings Bishop1978 ; Resnick1981 ; Safonov1998 ; Hadzibabic2006 . In particular, ultracold atomic gases are an ideal candidate to understand the interaction-driven BKT physics Hadzibabic2006 , owing to the unprecedented controllability over interatomic interactions, dimensionality and species Bloch2008 . Over the past decade, the BKT transition in a 2D weakly interacting Bose gas has been extensively studied by measuring the phase coherence Hadzibabic2006 ; Clade2009 , confirming the universal equation of state Hung2011 ; Yefsah2011 , probing the superfluidity Desbuquois2012 , or observing the free vortex proliferation Hadzibabic2006 ; Schweikhard2007 ; Choi2013 .

A 2D interacting Fermi gas at the crossover from a Bose-Einstein condensate (BEC) to a Bardeen-Cooper-Schrieffer (BCS) superfluid provides a unique platform to address the universal BKT mechanism Zhang2008 ; Salasnich2013 , since the underlying character of the system changes from tightly bound composite bosons to loosely bound Cooper pairs of fermions, with decreasing attractions SchmittRink1989 . Indeed, the fermionic BKT transition is now being pursued by several cold-atom laboratories Martiyanov2010 ; Feld2011 ; Frohlich2011 ; Dyke2011 ; Orel2011 ; Koschorreck2012 ; Sommer2012 ; Zhang2012 ; Makhalov2014 ; Ong2015 ; Ries2015 ; Murthy2015 ; Dyke2016 ; Martiyanov2016 ; Fenech2016 ; Boettcher2016 ; Cheng2016 ; Turlapov2017 , and there are indications of the transition from the measurements of pair condensation and correlation function, where: (i) the center-of-mass momentum distribution of Cooper pairs, $n_{\mathbf{Q}}$, exhibits anomalous enhancement near $\mathbf{Q}=0$ below a certain temperature Ries2015 , and (ii) the first-order correlation function $g_{1}(r)$ in real space decays algebraically Murthy2015 . However, confirmation of the transition is still to be demonstrated, as these two features may be explained using a strong-coupling theory in the normal phase Matsumoto2016 . This situation marks the importance of having accurate theoretical predictions for the fermionic BKT transition.

The purpose of this Letter is to apply a strong-coupling theory, beyond mean-field, to a 2D interacting Fermi gas in the superfluid phase and present semi-quantitative predictions for the BKT critical chemical potential, critical temperature and the critical velocity at the whole BEC-BCS crossover. Through a fully microscopic calculation of both superfluid density and critical velocity, beyond the phenomenological Landau quasi-particle picture, we predict the occurrence of a significant discontinuity in the critical velocity across the transition as a result of the universal jump in superfluid density KT1973 , which would provide an unambiguous proof of the fermionic BKT transition.

The theoretical description of pairing in a 2D interacting Fermi gas at finite temperature is a long-standing challenge due to strongly enhanced quantum and thermal fluctuations. There have been intense theoretical efforts over the last thirty years, to understand the corresponding mechanism in 2D layered high-temperature superconductors SchmittRink1989 ; Loktev2001 ; Lee2006 . To a large extent, current knowledge of the fermionic BKT transition builds on mean-field approach Zhang2008 ; Salasnich2013 , which breaks down when interactions become stronger. There are a number of studies that take into account strong pair fluctuations based on the many-body $T$-matrix scheme Klimin2012 ; Watanabe2013 ; Bauer2014 ; Marsiglio2015 ; He2015 ; Mulkerin2015 ; Bighin2016 , however, these calculations typically focus on the normal state due to technical difficulties. The ab-initio quantum Monte Carlo (QMC) simulations at finite temperature encounter similar issues Anderson2015 . In this Letter, we consider a Gaussian pair fluctuation (GPF) theory Hu2006 ; Hu2007 ; Diener2008 , which is known to provide a reliable 2D equation of state at zero temperature He2015 . We generalize the GPF theory for finite temperatures below the superfluid transition, solving a crucial technical problem of removing divergences in numerics. This enables us to calculate the superfluid density, the key quantity in characterizing the BKT transition, beyond the mean-field and taking into account quantum fluctuations. Our main results, as shown in Fig. 3 and Fig. 4(b), are of significant importance for further BKT experiments with cold fermions.

The GPF theory at finite T. — A 2D interacting Fermi gas is well-described by the Hamiltonian SchmittRink1989 ,

[TABLE]

where $\psi_{\sigma}(\mathbf{r})$ is the annihilation operator for the spin state $\sigma=\uparrow,\downarrow$, $\mathcal{H}_{0}=-\hbar^{2}\nabla^{2}/(2M)-\mu$ the kinetic Hamiltonian with atomic mass $M$, $\mu$ the chemical potential, and $U$ denotes the bare interaction strength of a contact interaction between unlike fermions and is related to the binding energy $\varepsilon_{B}$ via, $1/U=\sum_{\mathbf{k}}(\hbar^{2}\mathbf{k}^{2}/M+\varepsilon_{B})^{-1}$.

Technical details of the GPF theory have been extensively discussed elsewhere He2015 ; Hu2006 ; Diener2008 , here, we only present a brief overview of the key equations and refer the readers to Supplementary Material for further details supp . Within the GPF framework, we account for strong pair fluctuations at the Gaussian level, beyond the standard mean-field treatment, and consider separately their contributions to the thermodynamic potential, $\Omega=\Omega_{\textrm{MF}}+\Omega_{\textrm{GF}}$. These two parts can be represented by the BCS Green’s function $\mathscr{\mathcal{G}}_{0}(\mathbf{k},i\omega_{m})$ and the vertex function \Gamma(\mathbf{q},$$i\nu_{l}) (i.e., the Green’s function of Cooper pairs): $\Omega_{\textrm{MF}}=-k_{B}T\sum_{\mathbf{k},i\omega_{m}}\ln[-\mathcal{G}_{0}^{-1}]$, and $\Omega_{\textrm{GF}}=(k_{B}T/2)\sum_{\mathbf{q},i\nu_{l}}\ln[-\Gamma^{-1}]$. That is, the expressions of the thermodynamic potentials for ideal fermions and bosons, where $\omega_{m}=(2m+1)\pi k_{B}T$ and $\nu_{l}=2\pi lk_{B}T$ are the fermionic and bosonic Matsubara frequencies with integers $m$ and $l$, respectively. In other words, the system may be viewed as a non-interacting mixture of fermions and pairs. Though the picture is simple, it captures the essential physics for weak and strong interactions. Indeed, at zero temperature, the GPF theory provides a quantitative description of the BEC-BCS crossover in both 3D Hu2006 ; Hu2007 ; Diener2008 and 2D He2015 ; Shi2015 . This can be extended straight forwardly to the general situation where the condensed pairs flow with a wavevector $\mathbf{Q}$, as represented by a pairing gap $\Delta e^{i\mathbf{Q}\cdot\mathbf{r}}$ Taylor2006 ; supp . In this case,

[TABLE]

where $\smash{\tilde{\xi}_{\mathbf{k}}\equiv\hbar^{2}\mathbf{k}^{2}/(2M)-[\mu-\hbar^{2}\mathbf{Q}^{2}/(8M)]}$, $E_{\mathbf{k}}\equiv\sqrt{\tilde{\xi}_{\mathbf{k}}^{2}+\Delta^{2}}$, $\beta=1/(k_{B}T)$ and $E_{\mathbf{k}}^{\pm}\equiv E_{\mathbf{k}}\pm\hbar^{2}\mathbf{k}\cdot\mathbf{Q}/(2M)$. The expression for the thermodynamic potential of pair fluctuations is more subtle Hu2006 ; Diener2008 ,

[TABLE]

and the matrix elements $M_{11}\left(\mathcal{Q}\right)$ and $M_{12}\left(\mathcal{Q}\right)$ are given in supp . The density $n$ of the system can be calculated using $n=-\partial(\Omega_{\textrm{MF}}+\Omega_{\textrm{GF}})/\partial\mu$, which determines the Fermi wavevector $k_{F}=(2\pi n)^{1/2}$, energy $\varepsilon_{F}=\pi n\hbar^{2}/M$ and temperature $T_{F}=\varepsilon_{F}/k_{B}$.

Despite the simplicity and elegance of the GPF theory, it is not easy to solve numerically in general. The technical difficulty comes from the sum over the bosonic Matsubara frequency $i\nu_{l}$ in Eq. (3), which is divergent. For an interacting 2D Fermi gas at zero temperature the problem may be solved by utilizing an additional function which has no singularities or zeros in the left hand-plane He2015 ; Diener2008 . At finite temperature, however, the GPF has only been approximately treated by taking into account the effects of low-energy phonon modes Klimin2012 ; Bighin2016 . Here, we overcome the divergence by writing Tempere2008 ,

[TABLE]

where $\mathcal{S}_{\eta}(\mathbf{q},i\nu_{l})\equiv\mathcal{S}(\mathbf{q},i\nu_{l})e^{i\nu_{l}\eta}$ and $\gamma=(2l_{0}+1)\pi/\beta$ for arbitrary positive integer $l_{0}$. Thus, the contribution to $\Omega_{\textrm{GF}}$ at a given $\mathbf{q}$ can be calculated by using Eq. (4) and taking the remaining discrete sum with $\left|l\right|<l_{0}$, in the limit of $\eta\rightarrow 0^{+}$. We have confirmed that this numerical procedure is robust and independent of the choice of $l_{0}$.

To illustrate the importance of our full treatment of the GPF, we show in Fig. 1 the results for the pressure and density equations of state at interaction strength $\smash{\beta\varepsilon_{B}=0.5}$ with $\smash{\mathbf{Q}=0}$, compared with the predictions from the mean-field theory, above $T_{c}$ calculations with the self-consistent Luttinger-Ward theory Bauer2014 ; Mulkerin2015 and lattice QMC simulation Anderson2015 , and with recent experimental measurements Fenech2016 . It is reasonable from the comparison of results in Fig. 1 that the GPF theory is semi-quantitatively reliable over the whole temperature regime. For a superfluid 2D Fermi gas, the GPF theory provides the best description to date, as current mean-field theories strongly under-estimate the interaction effects Salasnich2013 and there are no superfluid QMC calculations at finite temperature. Alternative $T$-matrix theories have so far focused on the normal state only and predicted a 2D superfluid transition at zero temperature Marsiglio2015 .

Superfluid density and phase diagrams. — We now consider the case that the condensed pairs flow with superfluid velocity $\mathbf{v}_{s}=\hbar\mathbf{Q}/(2M)$. Treating $\mathbf{v}_{s}$ as small, the superfluid density $n_{s}$ of the system can be calculated from the lowest-order change in the thermodynamic potential, i.e., $\Delta\Omega=\Omega(\mathbf{v}_{s})-\Omega(0)\simeq Mn_{s}\mathbf{v}_{s}^{2}$/2, due to the added kinetic energy of the superfluid flow Taylor2006 , thus, we obtain,

[TABLE]

The BKT critical temperature $T_{c}$ can then be estimated by self-consistently solving the KT criterion Nelson1977 ; Salasnich2013 ,

[TABLE]

Figure 2 reports the superfluid density $n_{s}$ at the interaction strength $\beta\varepsilon_{B}=0.5$, as a function the dimensionless chemical potential $\beta\mu$. The main figure shows $n_{s}$ in units of the density of an ideal Fermi gas $n_{0}=2\lambda_{T}^{-2}\ln(1+e^{\beta\mu})$, where $\lambda_{T}\equiv\sqrt{2\pi\hbar^{2}/(Mk_{B}T)}$ is the thermal wavelength, while the inset shows the superfluid fraction $n_{s}/n$. For comparison, we also plot the mean-field results (dot-dashed). By dividing both sides of the KT criterion, Eq. (6), by $n_{0}$, we find that the dimensionless critical chemical potential, $(\beta\mu)_{c}$, may be obtained by plotting $n_{s}/n_{0}$ and looking for the intercept with $8/\ln(1+e^{\beta\mu})$. Towards the low-temperature regime, $\beta\mu\rightarrow\infty$, the superfluid density calculated using the mean-field theory is typically under-estimated, although the superfluid fractions from both mean-field and GPF theories saturate to unity. Consequently, the mean-field theory predicts a larger critical chemical potential.

By repeating the calculations at different interaction strengths we obtain a phase diagram for the critical chemical potential, as shown in Fig. 3. This phase diagram is particularly useful for current cold-atom experiments, where the Fermi gas is confined in a harmonic trapping potential, $V(\mathbf{r})$, and is inhomogeneous. A section of the cloud is locally superfluid if its local chemical potential $\mu_{\textrm{loc}}=\mu-V(\mathbf{r})$ is larger than $\mu_{c}$. Therefore, experimentally, once the chemical potential at the trap center, $\mu$, and the temperature, $T$, are measured by fitting the density equation of state at the edge of the cloud with the known virial expansion Fenech2016 , one can then determine the superfluid radius of the Fermi cloud from our phase diagram, Fig. 3. To make a close connection with experiments, in the figure we show the largest chemical potential achieved in recent equation of state measurements Fenech2016 ; Boettcher2016 . It is encouraging to see that the experiment was approaching the BKT transition.

On the theoretical side, it is of interest to determine the phase diagram for the parameter space of $T_{c}/T_{F}$ and $\varepsilon_{B}/\varepsilon_{F}$, where, we calculate the superfluid fraction as a function of $T/T_{F}$. A typical prediction at $\varepsilon_{B}/\varepsilon_{F}=0.1$ is illustrated in Fig. 4(a) by solid circles, contrasted with the mean-field result (dot-dashed line). The superfluid density of a 2D interacting Fermi gas has been recently calculated by Bighin and Salasnich Bighin2016 using Landau’s phenomenological formulation for the normal density and the quasiparticle spectrum based on the zero-temperature GPF equation of state Baym2013 . Their result is plotted in Fig. 4(a) for comparison. We find that the prediction of Landau’s formulation agrees well with our full GPF calculation at low temperatures, where $n_{s}/n\sim 1$, but significantly over-estimates the superfluid fraction when the temperature becomes larger. According to the KT criterion, the critical temperature $T_{c}/T_{F}$ can be extracted by locating the intercept point between the curves $n_{s}/n$ and $8T/T_{F}$. The resulting phase diagram is reported in Fig. 4(b). Our result shows a significant improvement on the BCS side over the previous theoretical predictions Salasnich2013 ; Bighin2016 . While on the BEC side (i.e., $\varepsilon_{B}>0.5\varepsilon_{F}$), our result follows closely to the approximate prediction from Landau’s formula, since in the latter, the superfluid fraction at low temperatures $T\sim 0.1T_{F}$ is reasonably approximated. In the deep BEC regime our GPF result approaches the anticipated BKT critical temperature of a weakly interacting Bose gas Bighin2016 ; Petrov2003 , since, the molecular scattering length is correctly reproduced in the GPF theory He2015 ; Salasnich2015 . In this respect, the phase diagram Fig. 4(b) gives a coherent picture across the whole BEC-BCS crossover.

Probing the fermionic BKT transition. — We now consider way to unambiguously identify the fermionic BKT transition. Due to strong interactions, measurements of both phase coherence and free vortex proliferation, which are efficient for a weakly interacting 2D Bose gas, do not work well. Instead, we follow the idea of the recent superfluidity measurement Desbuquois2012 and propose to observe the superfluid behavior of an interacting 2D Fermi gas by stirring the cloud with a red detuned laser beam. When the Fermi cloud is in the superfluid state, we anticipate that the measured critical velocity will have a sudden jump as the position of the stirred beam moves across a critical radius $r_{c}$, which corresponds to the critical chemical potential $\mu_{c}=\mu-V(r_{c})$. This sudden increase is caused by the universal jump in the superfluid density, since just below (above) the BKT critical temperature (chemical potential), the finite superfluid density is able to support nonzero superfluid flow footnote .

Theoretically, we calculate the critical velocity from the velocity dependence of the thermodynamic potential $\Omega(\mathbf{v})$ at a given temperature, $T$. With increasing superfluid flow, the loss of stability of the system is indicated by the appearance of a local maximum in the thermodynamic potential, as illustrated in the inset of Fig. 5. The determined critical velocity at the interaction strength $\beta\varepsilon_{B}=0.5$ is presented in the main figure. The apparent discontinuity at $(\beta\mu)_{c}\sim 8$ serves as a smoking-gun signature for the BKT transition. To give some realistic numbers, consider a single 2D cloud of $N=40,000$ neutral 6Li atoms in a hybrid optical/magnetic trap with frequency $\omega_{x}\simeq\omega_{y}\sim 2\pi\times 25$ Hz at temperature $T\sim 20$ nK and at binding energy $\varepsilon_{B}=10$ nK (satisfying $\beta\varepsilon_{B}\sim 0.5$), which is within the regime attainable at Swinburne Fenech2016 . The chemical potential at the trap center is estimated to be $\mu\sim 240$ nK. Thus, the superfluid radius is about $r_{c}\sim 100$ µm, and from Fig. 5, the anticipated jump in the critical velocity would be about $\Delta v_{c}\simeq 0.6v_{T}\sim 4.5$ mm/s, which is readily detectable Weimer2015 .

Conclusions. — We have established reliable estimates for the superfluid transition temperature of a strongly interacting 2D Fermi gas. This is done by developing a Gaussian pair fluctuation theory that provides semi-quantitatively accurate predictions on the superfluid density at any interaction strength and temperature. Our results support on-going cold-atom experiments to unambiguously observe the fermionic Berezinskii-Kosterlitz-Thouless transition. Our approach may also be useful for understanding the superfluid phases of the 2D Hubbard model Paiva2004 .

Supplemental Material: Details of the thermodynamic potential

In greater detail we can derive the matrix elements of the pair fluctuation thermodynamic potential, and for completeness we also show the mean-field thermodynamic potential. Let us consider the general situation where the condensed pairs flow with a wavevector $\mathbf{Q}$, or superfluid velocity $\mathbf{v}_{s}=\hbar\mathbf{Q}/(2M)$, as represented by a pairing gap $\Delta e^{i\mathbf{Q}\cdot\mathbf{r}}$ Taylor2006 . In this case, the mean-field thermodynamic potential is given by Taylor2006 ,

[TABLE]

where $\tilde{\xi}_{\mathbf{k}}\equiv\hbar^{2}\mathbf{k}^{2}/(2M)-[\mu-\hbar^{2}\mathbf{Q}^{2}/(8M)]$, $E_{\mathbf{k}}\equiv\sqrt{\tilde{\xi}_{\mathbf{k}}^{2}+\Delta^{2}}$, $\beta=1/(k_{B}T)$ and $E_{\mathbf{k}}^{\pm}\equiv E_{\mathbf{k}}\pm\hbar^{2}\mathbf{k}\cdot\mathbf{Q}/(2M)$, and to ensure the gapless Goldstone mode, the pairing gap $\Delta$ should be calculated using the mean-field gap equation,

[TABLE]

with the Fermi distribution function $f(x)\equiv 1/(e^{\beta x}+1)$. The expression for the thermodynamic potential of pair fluctuations is more subtle Hu2006 ; Diener2008 :

[TABLE]

where the matrix elements of $-\Gamma^{-1}(\mathcal{Q})$ are given by Taylor2006 ,

[TABLE]

Here, we use the short-hand notations $i\tilde{\nu_{l}}\equiv i\nu_{l}-\hbar^{2}\mathbf{q}\cdot\mathbf{Q}/(2M)$, $E_{\pm}\equiv E_{\mathbf{k}\pm\mathbf{q}/2}$, $f_{\pm}^{\left(\pm\right)}\equiv f(E_{\mathbf{k}\pm\mathbf{q}/2}^{\pm})$, $u_{\pm}^{2}=(1+\tilde{\xi}_{\mathbf{k}\pm\mathbf{q}/2}/E_{\mathbf{k}\pm\mathbf{q}/2})/2$ and $v_{\pm}^{2}=1-u_{\pm}^{2}$ .