Effective theory for ultracold strongly interacting fermionic atoms in two dimensions

TL;DR

This paper introduces a minimal theoretical model for 2D strongly interacting ultracold Fermi gases, accurately predicting their equation of state and breathing mode frequency, and resolving experimental puzzles related to quantum anomalies.

Contribution

The authors develop a minimal model requiring two interaction parameters to explain experimental observations in 2D Fermi gases, advancing understanding of quantum anomalies and phase transitions.

Findings

Accurate predictions for equation of state and breathing mode frequency.

Resolution of experimental puzzles regarding quantum anomaly.

Identification of conditions to observe quantum anomaly conclusively.

Abstract

We propose a minimal theoretical model for the description of a two-dimensional (2D) strongly interacting Fermi gas confined transversely in a tight harmonic potential, and present accurate predictions for its equation of state and breathing mode frequency. We show that the minimal model Hamiltonian needs at least two independent interaction parameters, the 2D scattering length and effective range of interactions, in order to quantitatively explain recent experimental measurements at nonzero filling factor $N/N_{2D}$, where $N$ is the total number of atoms and $N_{2D}$ is the threshold number to reach the 2D limit. We therefore resolve in a satisfactory way the puzzling experimental observations of reduced equations of state and reduced quantum anomaly in breathing mode frequency, due to small yet non-negligible $N/N_{2D}$. We argue that a conclusive demonstration of the much-anticipated quantum anomaly is possible at a filling factor of a few percent. Our establishment of the minimal model for 2D ultracold atoms could be crucial to understanding the fermionic Berezinskii-Kosterlitz-Thouless transition in the strongly correlated regime.

Equations

$$f_{Q2D}(k;a_{3D},a_z)=\frac{4\pi }{\sqrt{2\pi }{a}_z/a_{3D}+\varpi \left(k^2{a}_z^2/2\right)},$$

$$f_{2D}\left(k;a_s,R_s\right)=\frac{4\pi }{-2\ln \left(k{a}_s\right)-R_s{k}^2+i\pi },$$

$$\mathcal{H}$$

$$E\left[\ln \left(k_F{a}_s\right),k_F^2{R}_s\right]=E_{\text{MF}}+\Delta E_{\text{GPF}}+\Delta E_{\text{c}},$$

$${\hslash }^2{\omega }_B^2=-2⟨r^2⟩{\left[\frac{d⟨r^2⟩}{d\left({\omega }_{\perp }^2\right)}\right]}^{-1},$$

Full text

Two-dimensional (2D) quantum many-body systems are of great interest, due to the interplay of reduced dimensionality and strong correlation, which leads to enhanced quantum and thermal fluctuations Randeria1989 and a number of ensuing quantum phenomena such as Berezinskii–Kosterlitz–Thouless (BKT) physics Berezinskii1972 ; Kosterlitz1973 . In this respect, the recently realized 2D Fermi gas of ultracold 6Li and 40K atoms under a tight axial confinement provides a unique platform Levinsen2015 ; Turlapov2017 , with unprecedented controllability particularly on interatomic interactions. To date, many interesting properties of ultracold 2D Fermi gases have been thoroughly experimentally explored Turlapov2017 , including the equation of state (EoS) at both zero temperature Makhalov2014 ; Martiyanov2016 and finite temperature Fenech2016 ; Boettcher2016 , radio-frequency spectroscopy Frohlich2011 ; Sommer2012 ; Zhang2012 , pair momentum distribution Ries2015 , first-order correlation function and BKT transition Murthy2015 , and quantum anomaly in breathing mode frequency Vogt2012 ; Holten2018 ; Peppler2018 . These results may shed light on understanding other important strongly correlated 2D systems, such as high-$T_{c}$ layered cuprate materials Loktev2001 , 3He submonolayers Ruggeri2013 , exciton-polariton condensates Deng2010 and neutron stars Pons2013 .

The present theoretical model of ultracold 2D Fermi gases is simple Levinsen2015 ; Turlapov2017 . Under a tight harmonic confinement with trapping frequency $\omega_{z}$ along the axial $z$-axis and a weak confinement $\omega_{\perp}$ in the transverse direction, the kinematic 2D regime is reached when the number of atoms $N$ is smaller than a threshold $N_{2D}\simeq(\omega_{z}/\omega_{\perp})^{2}$, so all the atoms are forced into the ground state of the motion along $z$ Turlapov2017 . The interatomic interactions are then described by a single $s$-wave scattering length $a_{2D}$ Makhalov2014 , which is related to a 3D scattering length $a_{3D}$ via the quasi-2D scattering amplitude Petrov2001 . Various experimental data have been compared and benchmarked with different theoretical predictions of the simple 2D model Bertaina2012 ; Orel2011 ; Hofmann2012 ; Taylor2012 ; Bauer2014 ; Barth2014 ; He2015 ; Shi2015 ; Mulkerin2015 ; Anderson2015 . For EoS, i.e., the chemical potential and pressure at essentially zero temperature, good agreements were found Makhalov2014 ; Boettcher2016 . But, at the quantitative level the experimental data somehow lie systemically below the accurate predictions from auxiliary-field quantum Monte Carlo (AFQMC) simulations Makhalov2014 ; Boettcher2016 . The discrepancy is not so serious and might be viewed as an indicator of small deviation from the 2D kinematics Turlapov2017 , in spite of the fact that the 2D condition $N\ll N_{2D}$ is well satisfied. However, a serious problem does arise when two experimental groups measured the breathing mode frequency in the deep 2D regime most recently Holten2018 ; Peppler2018 . The observed frequency turned out to be much smaller than the well-established theoretical prediction in the strongly interacting regime Hofmann2012 ; Taylor2012 . This discrepancy is at the qualitative level, suggesting that the simple 2D model with a single parameter $a_{2D}$ may not be sufficient for the description of ultracold 2D Fermi gases Hu2019 .

The purpose of this Letter is to provide a minimal theory of ultracold 2D Fermi gases, with the inclusion of a properly defined effective range of interactions (see Fig. 1). The significant role played by effective range was realized in our previous work Hu2019 . However, the determination of the effective range there turns out be problematic. We solve the proposed model Hamiltonian at zero temperature by taking into account strong pair fluctuations at Gaussian level and beyond (Fig. 2), with the help of a correlation energy from AFQMC in the zero-range limit Shi2015 . This enables us to predict accurate EoS (Fig. 3 and Fig. 4), as well as reliable breathing mode frequency (Fig. 5). The puzzling quantitative and qualitative discrepancies, observed in the previous comparisons between experiment and theory Turlapov2017 ; Makhalov2014 ; Boettcher2016 ; Holten2018 ; Peppler2018 , are therefore naturally resolved in a satisfactory way.

Effective range of interactions. We start by considering the collision of two fermions with mass $M$ and unlike spin in a highly anisotropic harmonic trapping potential, described by a quasi-2D scattering amplitude Petrov2001 ,

[TABLE]

where $a_{z}\equiv\sqrt{\hbar/(M\omega_{z})}$ is the harmonic oscillator length along the $z$-axis and the function $\varpi(x)$ has the expansion $\varpi(x\rightarrow 0)\simeq-\ln(2\pi x/\mathcal{B})+(2\ln 2)x+i\pi$ with $\mathcal{B}\simeq 0.905$ Petrov2001 . In the simplest treatment, one may parameterize the quasi-2D collision using a 2D scattering length $a_{2D}$ Turlapov2017 ; Makhalov2014 , by setting the 2D scattering amplitude $f_{2D}(k;a_{2D})=-2\pi/\ln[ka_{2D}(k)/i]=f_{Q2D}(k;a_{3D},a_{z})$. In general, one thus obtains a momentum-dependent $a_{2D}(k)$, which in the zero-energy limit takes the form $a_{2D}(k\rightarrow 0)=a_{s}\equiv a_{z}\sqrt{\pi/\mathscr{\mathcal{B}}}\exp(-\sqrt{\pi/2}a_{z}/a_{3D})$ Petrov2001 . The advantage of this simple treatment is that the description universally depends on a single parameter $a_{2D}$, to be evaluated at a characteristic collision momentum $k_{0}$, i.e., $k_{0}=\sqrt{2M\tilde{\mu}}/\hbar$, where $\tilde{\mu}$ is the chemical potential that does not include the two-body binding energy Turlapov2017 ; Makhalov2014 .

A more adequate parametrization of the 2D collision is to include an effective range of interactions $R_{s}$ in the 2D scattering amplitude Adhikari1986 ,

[TABLE]

whose pole gives a two-body bound state with binding energy $\varepsilon_{B}=\hbar^{2}\kappa^{2}/M$, where the wavevector $\kappa$ satisfies $R_{s}=2\ln(\kappa a_{s})/\kappa^{2}$. The same two-body bound state should be supported by the pole of the quasi-2D scattering amplitude in Eq. (1) as well. By setting $k\rightarrow i\kappa$ there, we find $\sqrt{2\pi}a_{z}/a_{3D}+\varpi[-\varepsilon_{B}/(2\hbar\omega_{z})]=0$ NoteEB . Therefore, we can directly calculate the effective range $R_{s}$, once $\varepsilon_{B}$ or $\kappa$ is solved at a given $a_{z}/a_{3D}$.

The effective range obtained in this way is reported in Fig. 1. It decreases monotonically from $R_{s}^{(0)}\equiv(-\ln 2)a_{z}^{2}$ with increasing $a_{z}/a_{3D}$ (main figure) or binding energy $\varepsilon_{B}$ (inset). We note that $R_{s}^{(0)}$ can be easily derived from the second expansion term in $\varpi(x\rightarrow 0)$ and its magnitude, i.e., $R_{s}^{(0)}\sim a_{z}^{2}$, is a clear indication of the quasi-2D nature of atom collisions Turlapov2017 ; Petrov2001 . As the wavefunction of two colliding atoms at distance within $a_{z}$ is set by the full 3D contact interaction potential, these collisions can never be purely 2D. They can only be approximately treated as 2D, out of the range $\sim a_{z}$.

Many-body theory. To account for the effective range $R_{s}$, it is useful to adopt a two-channel model Hu2019 ; Liu2005 ; Schonenberg2017 :

[TABLE]

where $\xi_{\mathbf{p}}\equiv\hbar^{2}\mathbf{p}^{2}/(2M)-\mu$, and $c_{\mathbf{k}\sigma}$ and $b_{\mathbf{q}}$ are the annihilation operators of atoms and molecules in the open- and closed-channel, respectively. The channel coupling $g$ is related to $R_{s}$, via $R_{s}=-4\pi^{2}\hbar^{4}/(M^{2}g^{2})$, the detuning $\nu$ of molecules is tuned to reproduce the binding energy $\varepsilon_{B}$, i.e., $\nu=-\varepsilon_{B}+(g^{2}/\mathcal{S})\sum_{\mathbf{k}}[\hbar^{2}\mathbf{k}^{2}/M+\varepsilon_{B}]^{-1}$ Hu2019 ; Schonenberg2017 , and $\mathcal{S}$ is the area.

We solve the model Hamiltonian at different orders of approximation at zero temperature. Formally, the ground-state energy $E$ may be decoupled as,

[TABLE]

where $k_{F}=(2\pi n)^{1/2}$ is Fermi wavevector and $\varepsilon_{F}=\hbar^{2}k_{F}^{2}/(2M)$ is Fermi energy for a system with number density $n$. The mean-field (MF) theory provides the leading term $E_{\textrm{MF}}$, while the major correction arising from strong pair fluctuations at Gaussian level can be obtained by using the Gaussian pair fluctuation (GPF) theory He2015 ; Hu2019 ; Hu2006 ; Hu2007 ; Diener2008 , i.e., $\Delta E_{\textrm{GPF}}=E_{\textrm{GPF}}-E_{\textrm{MF}}$. The effect of pair fluctuations beyond Gaussian level may be characterized by a correlation energy $\Delta E_{\textrm{c}}$, which is anticipated to be much smaller than $\Delta E_{\textrm{GPF}}$. To see this, in Fig. 2(a) we plot the ground-state energy in the zero-range limit ($R_{s}=0$), predicted by mean-field theory, GPF theory He2015 and AFQMC simulation Shi2015 . Indeed, the correlation energy given by the difference between the GPF and AFQMC energies is notably smaller than $\Delta E_{\textrm{GPF}}$. In particular, $\Delta E_{\textrm{c}}$ becomes vanishingly small in the tight-binding limit of $\ln(k_{F}a_{s})\rightarrow-\infty$ He2015 . It is then useful to define a beta function $\beta=\Delta E_{\textrm{c}}/\Delta E_{\textrm{GPF}}\ll 1$, which varies as functions of the two dimensionless interaction parameters $\ln(k_{F}a_{s})$ and $k_{F}^{2}R_{s}$. For small $k_{F}^{2}R_{s}$, however, it seems plausible to assume that $\beta$ relies on $\varepsilon_{B}/\varepsilon_{F}$ only, whose dependence can be readily extracted in the zero-range limit using the AFQMC data, as shown in the inset of Fig. 2(a). Other possible choice of the $\beta$-function is considered in Supplemental Material SM .

We thus establish a viable procedure to calculate the ground-state energy at nonzero effective range. For a given set ($k_{F}a_{s}$, $k_{F}^{2}R_{s}$), we first calculate the binding energy $\varepsilon_{B}/\varepsilon_{F}$ and determine the value of $\beta$. Both mean-field and GPF theories are then applied to obtain $E_{\textrm{MF}}$ and $\Delta E_{\textrm{GPF}}$, and consequently $\Delta E_{\textrm{c}}=\beta\Delta E_{\textrm{GPF}}$. In Fig. 2(b), we present $E=E_{\textrm{GPF}}+\Delta E_{\textrm{c}}$ in black line for a fixed ratio $R_{s}/a_{s}^{2}\simeq-0.2511$, at which we may benchmark our prediction against available high-precision diffusion Monte Carlo (DMC) data (i.e., the single green dot) Schonenberg2017 ; NoteDMC . We find that the correction $\Delta E_{\textrm{GPF}}$ becomes smaller at nonzero effective range. Towards the non-interacting limit ($a_{s}\rightarrow\infty$) and hence large $k_{F}^{2}R_{s}$, $\Delta E_{\textrm{GPF}}$ vanishes quickly. This is understandable, since pair fluctuations become weaker with decreasing channel coupling $g$ and even mean-field theory may provide accurate prediction at sufficiently large $k_{F}^{2}R_{s}$ Schonenberg2017 . The correlation energy also significantly reduces at finite effective range and we find $\left|\Delta E_{\textrm{c}}\right|<0.02N\varepsilon_{F}$ at all interaction strengths for $R_{s}/a_{s}^{2}\simeq-0.2511$. The agreement between our theory with DMC is excellent, with a difference less than $0.01N\varepsilon_{F}$.

Equation of state. Once the ground-state energy $E$ of a uniform 2D Fermi gas is determined, we calculate directly the chemical potential $\mu$ and pressure $P$ using standard thermodynamic relations. Experimentally, these homogeneous EoS can be extracted from a low-temperature trapped Fermi gas, by using the local density approximation Butts1997 , which assigns a local chemical potential $\mu(r)=\mu_{\textrm{peak}}-V(r)$ to each position $r$ in the potential $V(r)=M\omega_{\perp}^{2}r^{2}/2$. Both the peak chemical potential $\mu_{\textrm{peak}}$ and the* in situ* density distribution $n(r)$ can be experimentally measured Makhalov2014 ; Fenech2016 ; Boettcher2016 , from which one deduces the homogeneous density EoS $n(\mu)$. By further using the force balance condition Makhalov2014 , $\nabla P(r)=-n(r)\nabla V(r)$, the homogeneous pressure EoS $P(n)$ can also be determined.

In Fig. 3, we show the experimental data for the peak chemical potential $\mu_{\textrm{peak}}$, measured at different magnetic fields (i.e., $a_{3D}$) and hence at different $\ln(k_{F}a_{s})$ Boettcher2016 ; NoteEoS . Our predictions for the peak chemical potential, calculated under the same experimental condition, are plotted by the black solid line. We find a good agreement between theory and experiment at $\ln(k_{F}a_{s})>0$. Due to the large effective range of interactions in the experiment (i.e., $k_{F}^{2}R_{s}\lesssim-1.2$ at $N\simeq N_{2D}$ Boettcher2016 ), the zero-range predictions from AFQMC appear to strongly over-estimate the chemical potential. The use of an effective $a_{2D}$ can not fully explain the discrepancy (see the inset and also Fig. 1 in Ref. Boettcher2016 ), as we mentioned earlier. The discrepancy can also be hardly understood by possible systematic effects such as finite temperature and the failure of a 2D model due to a finite filling factor $N/N_{2D}$ SM ; He2019 .

In Fig. 4, we present the comparison between our predictions and the experimental data Makhalov2014 ; Martiyanov2016 for pressure at the trap center. In this case, we have $N\simeq 0.35N_{2D}$ and therefore the effect of the effective range may become weaker. Nevertheless, we can see clearly that in the strongly interacting regime (i.e., $0<\ln(k_{F}a_{s})<2$), the experimental data lie systematically below the zero-range results from AFQMC. The model Hamiltonian with a finite effective range should be used, in order to quantitatively understand the experimental measurement. We note that, in harmonic traps the pressure at the center is fixed by the force balance condition to $P=M\omega_{\perp}^{2}N/(2\pi$) Martiyanov2016 . Using the peak density of an ideal trapped Fermi gas $n_{F}^{\textrm{HO}}=M\omega_{\perp}\sqrt{N}/(\pi\hbar)$ Turlapov2017 , we find that the peak density $n\equiv n(r=0)$ can be written in terms of the pressure at the trap center, i.e., $n/n_{F}^{\textrm{HO}}=[P/(n\varepsilon_{F}/2)]^{-1/2}$. This provides an alternative way to illustrate the data, as shown in the inset.

In both Fig. 3 and Fig. 4, the agreement between theory and experiment becomes worse at small $k_{F}a_{s}$, suggesting the inadequacy of our theory towards the limit of a Bose-Einstein condensate (BEC). This is because, experimentally the BEC regime is reached by changing $a_{3D}$ instead of $k_{F}$. For a small positive $a_{3D}$ the system is better viewed as a quasi-2D weakly interacting BEC, with a 2D scattering length $a_{2D}^{(m)}$ determined from the 3D molecular scattering length $a_{3D}^{(m)}\simeq 0.6a_{3D}$ Petrov2004 and with an effective range $R_{s}^{(m)}\sim-a_{z}^{2}$. Our two-channel model cannot fully recover this interaction-driven BEC limit. For more details, we refer to Supplemental Material SM .

Breathing mode and quantum anomaly. We now turn to consider the breathing mode frequency, which was recently measured in two experiments at $N\simeq 0.2N_{2D}$ Holten2018 ; Peppler2018 , as shown in Fig. 5 by green circles and blue squares. Theoretically, the zero-temperature breathing mode frequency can be conveniently calculated by using the sum-rule approach Menotti2002 ; Hu2014 ,

[TABLE]

where $\left\langle r^{2}\right\rangle=N^{-1}\int d^{2}\bm{r}[r^{2}n(r)]$ is the squared radius of the Fermi cloud at a given trapping frequency $\omega_{\perp}$. In the classical treatment, a 2D Fermi gas is scale-invariant Pitaevskii1997 and acquires a polytropic density EoS, $\mu(n)\propto n{}^{2}$. As a result, the mode frequency is pinned to $2\omega_{\perp}$, regardless of temperature and interactions Pitaevskii1997 . The deviation of the breathing mode frequency away from $2\omega_{\perp}$ can be viewed a quantum anomaly Hofmann2012 ; Taylor2012 , arising from strong quantum pair fluctuations in 2D Olshanii2010 .

As readily seen from Fig. 5, the observed quantum anomaly in the two experiments is far below the prediction from AFQMC for zero-range interactions with a single 2D scattering length NoteBMAFQMC . It can only be understood when we use the proposed minimal model for 2D ultracold fermions and take into account the realistic finite effective range at $N\simeq 0.2N_{2D}$. The quantitative difference between our theory and experiment at $0<\ln(k_{F}a_{s})<1$ could be caused by the finite temperature in the two experiments SM , which is in the range $[0.10-0.22]T_{F}$.

It turns out that the breathing mode frequency or quantum anomaly depends sensitively on the effective range. The zero-range result of AFQMC can hardly be asymptotically approached, even we decrease the number of atoms down to just a few percent of $N_{2D}$ (see the red dot-dashed line at $N=0.02N_{2D}$). In this case, however, the deviation from the classical limit of $2\omega_{\perp}$ is very significant and its experimental confirmation will deepen our understanding of the long-sought 2D quantum anomaly in cold atoms Olshanii2010 , which was recently observed Murthy2019 .

Conclusions. We have established a minimal model to describe ultracold interacting fermions confined in two dimensions and have solved it accurately at zero temperature with the help of existing AFQMC results. We have shown that the confinement-induced effective range of interactions has to be included, in order to understand the recent measurements on quantum anomaly in a qualitative manner and on equation of state at quantitative level. Our results pave the way to investigate the crucial role played by effective range in other two-dimensional quantum many-body systems and provide an excellent starting point to address the fermionic Berezinskii-Kosterlitz-Thouless transition with cold-atoms Murthy2015 ; Mulkerin2017 .