Geometric contribution to the Goldstone mode in spin-orbit coupled Fermi superfluids

TL;DR

This paper investigates how the quantum metric tensor influences the Goldstone mode in spin-orbit coupled Fermi superfluids, revealing that geometric effects cause nonmonotonic behavior of sound velocity across the BCS-BEC crossover.

Contribution

It demonstrates that the quantum metric tensor's geometric contribution significantly affects collective excitations, specifically the Goldstone mode, in spin-orbit coupled Fermi superfluids.

Findings

Quantum metric influences superfluid properties related to effective mass.

Geometric effects cause nonmonotonic sound velocity in BCS-BEC crossover.

Numerical analysis for Weyl and Rashba couplings supports the geometric impact.

Abstract

The so-called quantum metric tensor is a band-structure invariant whose measure corresponds to the quantum distance between nearby states in the Hilbert space, characterizing the geometry of the underlying quantum states. In the context of spin-orbit coupled Fermi gases, we recently proposed that the quantum metric has a partial control over all those superfluid properties that depend explicitly on the mass of the superfluid carriers, i.e., the effective-mass tensor of the corresponding (two- or many-body) bound state. Here we scrutinize this finding by analyzing the collective phase and amplitude excitations at zero temperature. In particular to the Goldstone mode, we present extensive numerical calculations for the Weyl and Rashba spin-orbit couplings, revealing that, despite being small, the geometric contribution is solely responsible for the nonmonotonic evolution of the sound velocity in the BCS-BEC crossover.

Equations

$$H$$

$$-U\sum _{\mathbf{k}{\mathbf{k}}^{\prime }\mathbf{q}}{a}_{\uparrow ,\mathbf{k}+\mathbf{q}/2}^{†}{a}_{\downarrow ,-\mathbf{k}+\mathbf{q}/2}^{†}{a}_{\downarrow ,-{\mathbf{k}}^{\prime }+\mathbf{q}/2}{a}_{\uparrow ,{\mathbf{k}}^{\prime }+\mathbf{q}/2},$$

$$S_0=\frac{{\Delta }_0^2}{TU}+\sum _{s\mathbf{k}}\left\{\frac{{\xi }_{s\mathbf{k}}-E_{s\mathbf{k}}}{2T}+\ln [f(-E_{s\mathbf{k}})]\right\},$$

$$S_2^{\theta }$$

$$S_2^{\lambda }$$

$$M_{q,E}^{11}$$

$$M_{q,O}^{11}$$

$$M_q^{12}$$

$$Q_{ij}^{\mathrm{intra}}$$

$$C_{ij}^{\mathrm{intra}}$$

$$Q_{ij}^{\mathrm{inter}}$$

$$C_{ij}^{\mathrm{inter}}$$

$$\displaystyle\mathbf{G}_{k}^{-1}=\left[\begin{array}[]{cc}(\mathrm{i}\omega_{\ell}-\xi_{\mathbf{k}})\sigma_{0}-\mathbf{d}_{\mathbf{k}}\cdot\boldsymbol{\sigma}&\mathrm{i}\Delta_{0}\sigma_{y}\\ -\mathrm{i}\Delta_{0}\sigma_{y}&(\mathrm{i}\omega_{\ell}+\xi_{\mathbf{k}})\sigma_{0}-\mathbf{d}_{\mathbf{k}}\cdot\boldsymbol{\sigma}^{*}\\ \end{array}\right],$$

$$G_k^{11}$$

$$G_k^{22}$$

$$G_k^{12}$$

$$G_k^{21}$$

$$M_q^{11}$$

$$\displaystyle\times\bigg{[}(1-f-f^{\prime})\left(\frac{u^{2}u^{\prime 2}}{\mathrm{i}\nu_{n}-E-E^{\prime}}-\frac{v^{2}v^{\prime 2}}{\mathrm{i}\nu_{n}+E+E^{\prime}}\right)$$

$$+$$

$$M_q^{12}$$

$$\displaystyle\times\bigg{[}(1-f-f^{\prime})\left(\frac{uvu^{\prime}v^{\prime}}{\mathrm{i}\nu_{n}+E+E^{\prime}}-\frac{uvu^{\prime}v^{\prime}}{\mathrm{i}\nu_{n}-E-E^{\prime}}\right)$$

$$+$$

$$\displaystyle S_{2}=\frac{1}{2T}\sum_{q}\mathrm{\Theta}_{q}^{\dagger}\left(\begin{array}[]{cc}M_{q,E}^{11}+M_{q}^{12}&\mathrm{i}M_{q,O}^{11}\\ -\mathrm{i}M_{q,O}^{11}&M_{q,E}^{11}-M_{q}^{12}\end{array}\right)\mathrm{\Theta}_{q},$$

Full text

I Introduction

The quantum geometric tensor provost80 ; berry89 ; thouless98 lies at the heart of modern solid-state and condensed-matter physics: while it describes the local geometry of the underlying Bloch states, it can be related to the global properties of some systems in most peculiar ways. For instance, its imaginary part corresponds to the Berry curvature measuring the emergent gauge field in momentum space, and it can be associated with the topological invariants that are used to classify quantum Hall states or topological insulators xiao10 ; qi11 . In contrast, its real part corresponds to the quantum metric tensor measuring the quantum distance/fidelity between nearby Bloch states, and it can be associated with, e.g., the density and the effective-mass tensor of the superfluid (SF) carriers in spin-orbit coupled Fermi gases iskin18a ; iskin18b , as well as with many other systems resta11 ; tan19 ; yu18 .

In the context of multi-band superconductivity, the new surge of theoretical interest in quantum-metric effects dates back only to 2015, soon after its direct association with the geometric origin of superfluidity in topologically nontrivial flat bands peotta15 . It was found that the SF weight (also known as the SF stiffness tensor) has two distinct contributions based on the physical mechanisms involved. In addition to the conventional contribution that is controlled by the electronic spectra of the Bloch bands, there is also the so-called geometric contribution coming from the interband processes peotta15 ; liang17 . It later turns out that the quantum geometry has a partial control not only over the SF weight but also over all those SF properties that depend explicitly on the effective-mass tensor of the corresponding (two- or many-body) bound state iskin18b . There have been many studies on the subject exploring a variety of multi-band Hamiltonians, including most recently that of the twisted bilayer graphene hu19 ; julku19 ; xie19 .

Motivated by the ongoing experimental progress wang12 ; cheuk12 ; williams13 ; huang16 ; meng16 and given our recent findings iskin18a ; iskin18b ; earlierworks , here we consider a two-component Fermi gas with an arbitrary spin-orbit coupling (SOC), and analyze its collective excitations in the BCS-BEC crossover. We show that the velocity of the Goldstone mode has two contributions as well: in addition to the conventional intraband one that is controlled by the helicity spectrum, the geometric interband contribution is controlled by the quantum metric. We present extensive numerical calculations for the Weyl and Rashba SOCs revealing that, despite being much smaller than the conventional one, the geometric contribution is solely responsible for the nonmonotonic evolution of the sound velocity in the BCS-BEC crossover.

The rest of this paper is organized as follows. In Sec. II, after introducing the many-body Hamiltonian, we first describe the effective-Gaussian action for the phase- and amplitude-only degrees of freedoms, and then derive analytical expressions for the collective modes with a special emphasis on the distinction between the conventional and geometric contributions. Much of the theoretical details on the effective-action approach are deferred to the Appendix A. The resultant Goldstone (phase) mode is analyzed Sec. III, where we present numerical results for a 3D Fermi gas with a Weyl or a Rashba SOC, and for a 2D Fermi gas with a Rashba SOC for completeness. We end the paper with a brief summary of our conclusions in Sec. IV.

II Collective Excitations

Having a two-component ($\uparrow$ and $\downarrow$) Fermi gas with an arbitrary SOC that is driven across an $s$-wave Feshbach resonance in mind, we consider the Hamiltonian

[TABLE]

where $a_{\sigma\mathbf{k}}^{\dagger}$ ($a_{\sigma\mathbf{k}}$) creates (annihilates) a pseudospin-$\sigma$ fermion with momentum $\mathbf{k}$, i.e., in units of $\hbar\to 1$ the Planck constant. In the first line, $\xi_{\mathbf{k}}=\epsilon_{\mathbf{k}}-\mu$ is the usual free-particle dispersion $\epsilon_{\mathbf{k}}=k^{2}/(2m)$ shifted by the chemical potential $\mu$, $\mathbf{d}_{\mathbf{k}}=\sum_{i}d_{\mathbf{k}}^{i}\boldsymbol{\widehat{i}}$ is the SOC field with $\boldsymbol{\widehat{i}}$ denoting the unit vector along the $i=(x,y,z)$ direction, $\sigma_{0}$ is a $2\times 2$ identity matrix, and $\boldsymbol{\sigma}=\sum_{i}\sigma_{i}\boldsymbol{\widehat{i}}$ is a vector of Pauli spin matrices. Our notation is such that $d_{\mathbf{k}}^{i}=\alpha_{i}k_{i}$ corresponds to the Weyl SOC when $\alpha_{i}=\alpha$ for all $i$, and to the Rashba SOC when $\alpha_{z}=0$. We choose $\alpha\geq 0$ without the loss of generality. Thus, the one-body problem is described by the Hamiltonian density $h_{\mathbf{k}}^{0}=\epsilon_{\mathbf{k}}\sigma_{0}+\mathbf{d}_{\mathbf{k}}\cdot\boldsymbol{\sigma},$ determining the helicity spectrum $\epsilon_{s\mathbf{k}}=\epsilon_{\mathbf{k}}+sd_{\mathbf{k}}$ and the helicity eigenstates $|s\mathbf{k}\rangle$ via the wave equation $h_{\mathbf{k}}^{0}|s\mathbf{k}\rangle=\epsilon_{s\mathbf{k}}|s\mathbf{k}\rangle.$ Here, $s=\pm$ labels the helicity bands, $d_{\mathbf{k}}=|\mathbf{d}_{\mathbf{k}}|$, and the unit vector $\widehat{\mathbf{d}}_{\mathbf{k}}=\mathbf{d}_{\mathbf{k}}/d_{\mathbf{k}}$ plays an important role throughout the paper.

In the second line of Eq. (4), $U\geq 0$ corresponds to the strength of the zero-ranged density-density interactions between $\uparrow$ and $\downarrow$ particles, and we tackle this term by making use of the imaginary-time functional path-integral formalism for the paired fermions. This formalism offers a systematic approach to go beyond the mean-field theory as the fluctuation effects can be incorporated into the effective action order by order as a power-series expansion around the stationary saddle-point carlos97 ; iskin05 ; he12 ; shenoy12 . As shown in the Appendix, the zeroth-order (i.e., the saddle-point) action coincides with the mean-field one, and can be written as

[TABLE]

where $\Delta_{0}=U\langle a_{\uparrow\mathbf{k}}a_{\downarrow,-\mathbf{k}}\rangle$ is the saddle-point order parameter with $\langle\dots\rangle$ denoting a thermal average, $T$ is the temperature in units of $k_{B}\to 1$ the Boltzmann constant, $E_{s\mathbf{k}}=\sqrt{\xi_{s\mathbf{k}}^{2}+\Delta_{0}^{2}}$ is the quasiparticle energy spectrum associated with $\xi_{s\mathbf{k}}=\epsilon_{s\mathbf{k}}-\mu,$ and $f(x)=1/(e^{x/T}+1)$ is the Fermi-Dirac distribution. Here, we take $\Delta_{0}$ to be a real parameter without the loss of generality, and determine its value by imposing the saddle-point condition $\partial S_{0}/\partial\Delta_{0}=0$ on the order parameter, leading to $1=U\sum_{s\mathbf{k}}[1-2f(E_{s\mathbf{k}})]/(4E_{s\mathbf{k}}).$ Furthermore, $\mu$ is determined by the thermodynamic relation $N_{0}=-T\partial S_{0}/\partial\mu$ for the number of particles, leading to $N_{0}=\sum_{s\mathbf{k}}\{1-(\xi_{s\mathbf{k}}/E_{s\mathbf{k}})[1-2f(E_{s\mathbf{k}})]\}/2.$ These saddle-point equations constitute the self-consistent BCS-BEC crossover theory for pairing, and they coincide with those of the mean-field approximation. The coupled solutions for $\Delta_{0}$ and $\mu$ have served as a reliable guide for the cold-atom community, providing a qualitative description of the SF state at sufficiently low temperatures across an $s$-wave Feshbach resonance carlos97 ; iskin05 ; he12 ; shenoy12 .

Going beyond $S_{0}$ in the Appendix, we first introduce the bosonic field $\Delta_{q}=\Delta_{0}+\Lambda_{q},$ where $\Lambda_{q}$ corresponds to the fluctuations of the order parameter around its stationary value $\Delta_{0}$ with the collective index $q=(\mathbf{q},\mathrm{i}\nu_{n})$ denoting both the pair momentum $\mathbf{q}$ and the bosonic Matsubara frequency $\nu_{n}=2\pi nT$, and then derive the effective-Gaussian action $S_{\mathrm{Gauss}}=S_{0}+S_{2}$. Note that the first order term $S_{1}$ vanishes thanks to the saddle-point condition on the order parameter. Furthermore, it is physically insightful to separate the phase and amplitude degrees of freedoms through the unitary transformation $\Lambda_{q}=(\lambda_{q}+\mathrm{i}\theta_{q})/\sqrt{2},$ leading to a phase-only action $S_{2}^{\theta}$ and to a amplitude-only one $S_{2}^{\lambda}$ that are of the form

[TABLE]

Here, since $M_{q,O}^{11}$ couples the phase and amplitude modes, the phase-only action is derived by integrating out the amplitude fields, and vice versa iskin05 . While the thermal expressions for the matrix elements $M_{q}^{ij}$ of the inverse-fluctuation propagator $\mathbf{M}_{q}$ are given in the Appendix, here we restrict ourselves to the collective excitations on top of the thermal ground state. Thus, setting $T=0$, and denoting $\xi_{s,\mathbf{k}+\mathbf{q}/2}$ by $\xi$; $\xi_{s^{\prime},-\mathbf{k}+\mathbf{q}/2}$ by $\xi^{\prime}$; $E_{s,\mathbf{k}+\mathbf{q}/2}$ by $E$; $E_{s^{\prime},-\mathbf{k}+\mathbf{q}/2}$ by $E^{\prime}$; and $\widehat{\mathbf{d}}_{\pm\mathbf{k}+\mathbf{q}/2}$ by $\widehat{\mathbf{d}}_{\pm}$, a compact way to write the matrix elements is iskin05

[TABLE]

Here, $M_{q,E}^{11}$ and $M_{q}^{12}$ are even functions of $\mathrm{i}\nu_{n}$, but $M_{q,O}^{11}$ is an odd one.

The dispersions $\omega_{\mathbf{q}}$ for the collective modes are determined by the poles of the propagator matrix $\mathbf{M}_{q}^{-1}$ for the pair fluctuation fields, i.e., $\omega_{\mathbf{q}}^{\theta}$ for the phase mode and $\omega_{\mathbf{q}}^{\lambda}$ for the amplitude one are determined, respectively, by setting $[\dots]=0$ in Eqs. (6) and (7) after an analytic continuation $\mathrm{i}\nu_{n}\to\omega+i0^{+}$ to the real axis carlos97 ; iskin05 ; he12 ; shenoy12 . For this purpose, it is sufficient to retain terms that are up to quadratic orders in their small $\mathbf{q}$ and $\omega$ expansions collnote , i.e., $M_{q,E}^{11}+M_{q}^{12}=A+\sum_{ij}C_{ij}q_{i}q_{j}-D\omega^{2}+\cdots;$ $M_{q,E}^{11}-M_{q}^{12}=\sum_{ij}Q_{ij}q_{i}q_{j}-R\omega^{2}+\cdots;$ and $M_{q,O}^{11}=-B\omega+\cdots.$ This calculation leads to a gapless phase (Goldstone) mode $(\omega_{\mathbf{q}}^{\theta})^{2}=\sum_{ij}x_{ij}q_{i}q_{j}$ with $x_{ij}=Q_{ij}/(R+B^{2}/A)$ for the phase-only action, and to a gapped amplitude (Higgs) mode $(\omega_{\mathbf{q}}^{\lambda})^{2}=\omega_{0}^{2}+\sum_{ij}y_{ij}q_{i}q_{j}$ with $\omega_{0}=\sqrt{(A+B^{2}/R)/D}$ and $y_{ij}=C_{ij}/D+Q_{ij}/(R+R^{2}A/B^{2})$ for the amplitude-only action. Notice that the velocities of the modes have a tensor structure in general, and can be anisotropic depending on the $\mathbf{k}$-space structure of the SOC. We are interested only in the low-energy Goldstone mode at $T=0$ since this mode is not damped for all $U\neq 0$ provided that the quasiparticle excitations $E_{s\mathbf{k}}$ are gapped for all $\Delta_{0}\neq 0$.

It turns out that the general expressions for the nonkinetic-expansion coefficients can simply be written as a sum over their well-known counterparts for the usual one-band problem, i.e., $A=\sum_{s\mathbf{k}}\Delta_{0}^{2}/(4E_{s\mathbf{k}}^{3}),$ $B=\sum_{s\mathbf{k}}\xi_{s\mathbf{k}}/(8E_{s\mathbf{k}}^{3}),$ $D=\sum_{s\mathbf{k}}\xi_{s\mathbf{k}}^{2}/(16E_{s\mathbf{k}}^{5}),$ and $R=\sum_{s\mathbf{k}}1/(16E_{s\mathbf{k}}^{3})$ are due entirely to intraband mechanisms carlos97 ; iskin05 ; he12 ; shenoy12 . Motivated by the recent findings iskin18a ; iskin18b ; earlierworks , and to gain more physical insight, we divide the kinetic-expansion coefficients into two distinct contributions $Q_{ij}=Q_{ij}^{\mathrm{intra}}+Q_{ij}^{\mathrm{inter}}$ and $C_{ij}=C_{ij}^{\mathrm{intra}}+C_{ij}^{\mathrm{inter}},$ based on their physical origins, i.e., depending on whether intraband or interband processes are involved. For instance, a compact way to write their intraband (i.e., conventional) contributions is

[TABLE]

We note that these terms can be transformed into their more familiar but more involved forms carlos97 ; sdMforms through first writing $\partial[(\partial\xi_{s\mathbf{k}}/\partial k_{j})/E_{s\mathbf{k}}^{3}]/\partial k_{i}=[\partial^{2}\xi_{s\mathbf{k}}/(\partial k_{i}\partial k_{j})]/E_{s\mathbf{k}}^{3}-3(\partial\xi_{s\mathbf{k}}/\partial k_{i})(\partial\xi_{s\mathbf{k}}/\partial k_{j})\xi_{s\mathbf{k}}/E_{s\mathbf{k}}^{5},$ and then identifying $\sum_{\mathbf{k}}\xi_{s\mathbf{k}}\partial[(\partial\xi_{s\mathbf{k}}/\partial k_{j})/E_{s\mathbf{k}}^{3}]/\partial k_{i}\equiv-\sum_{\mathbf{k}}(\partial\xi_{s\mathbf{k}}/\partial k_{i})(\partial\xi_{s\mathbf{k}}/\partial k_{j})/E_{s\mathbf{k}}^{3}$ after an integration by parts, and hence the name conventional. On the other hand, a compact way to write their interband (i.e., geometric) contributions is

[TABLE]

where $g_{\mathbf{k}}^{ij}=(1/2)\lim_{\mathbf{q}\to\mathbf{0}}\partial^{2}(\widehat{\mathbf{d}}_{+}\cdot\widehat{\mathbf{d}}_{-})/(\partial q_{i}\partial q_{j})$ or equivalently $g_{\mathbf{k}}^{ij}=(\partial\widehat{\mathbf{d}}_{\mathbf{k}}/\partial k_{i})\cdot(\partial\widehat{\mathbf{d}}_{\mathbf{k}}/\partial k_{j})/2$ is the total quantum metric tensor of the helicity bands iskin18a ; iskin18b . Alternatively, the quantum metric can be written as $g_{\mathbf{k}}^{ij}=[\sum_{\ell}(\partial d_{\mathbf{k}}^{\ell}/\partial k_{i})(\partial d_{\mathbf{k}}^{\ell}/\partial k_{j})-(\partial d_{\mathbf{k}}/\partial k_{i})(\partial d_{\mathbf{k}}/\partial k_{j})]/(2d_{\mathbf{k}}^{2}),$ reducing to $g_{\mathbf{k}}^{ij}=\alpha_{i}\alpha_{j}(d_{\mathbf{k}}^{2}\delta_{ij}-d_{\mathbf{k}}^{i}d_{\mathbf{k}}^{j})/(2d_{\mathbf{k}}^{4})$ for the specific case when $d_{\mathbf{k}}^{i}=\alpha_{i}k_{i}$. Since the interband contributions are directly controlled by the geometry of the underlying quantum states in the parameter ($\mathbf{k}$) space, we refer to them as the geometric ones.

All of the coefficients $A$, $B$, $C_{ij}$, etc., are derived at $T=0$ for arbitrary $U$ and $\mathbf{d}_{\mathbf{k}}$, and can be used both in two and three dimensions. Next we present their specific application to three systems that are of immediate experimental and/or theoretical interest: (i) a 3D Fermi gas with a Weyl SOC, (ii) a 3D Fermi gas with a Rashba SOC, and (iii) a 2D Fermi gas with a Rashba SOC.

III Numerical Results

In this paper, our primary interest is in the cooperation and/or competition between the conventional intraband contribution $x_{ij}^{\mathrm{intra}}=Q_{ij}^{\mathrm{intra}}/(R+B^{2}/A)$ and the geometric interband one $x_{ij}^{\mathrm{inter}}=Q_{ij}^{\mathrm{inter}}/(R+B^{2}/A)$ on the velocity of the Goldstone mode. For this purpose, one first needs to obtain a self-consistent solution for $\Delta_{0}$ and $\mu$ by coupling the saddle-point condition and the saddle-point number equation, and then plug these solutions into the coefficients.

For instance, in Fig. 1, we show $x_{ij}=x_{0}\delta_{ij}$ for a 3D Fermi gas with a Weyl SOC, where $\mathbf{d}_{\mathbf{k}}=\alpha\mathbf{k},$ and therefore, $x_{0}=x_{0}^{\mathrm{intra}}+x_{0}^{\mathrm{inter}}$ is isotropic in the entire $\mathbf{q}$ space with $\delta_{ij}$ a Kronecker-delta. In accordance with the cold-atom literature, the two-body $s$-wave scattering length $a_{s}$ in vacuum is substituted in this figure for $U$ via the usual relation $V/U=-mV/(4\pi a_{s})+\sum_{\mathbf{k}}1/(2\epsilon_{\mathbf{k}}),$ where $V$ is the volume of the system. Furthermore, we choose $N=k_{F}^{3}V/(3\pi^{2})$ for the number of particles, and use the Fermi momentum $k_{F}$ and the Fermi velocity $v_{F}=k_{F}/m$ as the relevant length and velocity scales in our numerical calculations. This is such that increasing $U$ from [math] changes the dimensionless parameter $1/(k_{F}a_{s})$ continuously from $-\infty$ to [math] to $+\infty$ in the BCS-BEC crossover.

In the weak-SOC limit when $\alpha/v_{F}\ll 1$, we recover the usual problem where $x_{0}/v_{F}^{2}$ is known to be a monotonically decreasing function of $1/(k_{F}a_{s})$ with the well-known limits, i.e., $1/3$ in the BCS regime and $k_{F}a_{s}/(3\pi)$ in the BEC regime carlos97 ; iskin05 . Note that since the geometric contribution $x_{0}^{\mathrm{inter}}$ is originating from the interband processes, it plays a negligible role when $\alpha/v_{F}\to 0$. On the other hand, for intermediate $\alpha/v_{F}$ values, we find that $x_{0}/v_{F}^{2}$ is a nonmotonic function of $1/(k_{F}a_{s})$, exhibiting a noticeable peak in the crossover region around the unitarity he12 ; shenoy12 . This result is quite a surprise given that all of the ground-state properties are found to be monotonic functions of $1/(k_{F}a_{s})$ for the usual BCS-BEC crossover problem with $s$-wave interactions carlos97 . In comparison to the dominant contribution from $x_{0}^{\mathrm{intra}}$, our study reveals that, despite being small, the contribution from $x_{0}^{\mathrm{inter}}$ is sizeable enough to cause such a significant difference in $x_{0}$. Thus, even though a direct observation of $x_{0}^{\mathrm{inter}}$ may not be possible, realization of a nonmonotonic $x_{0}$ may be considered as the ultimate evidence for its subleading role.

Similarly, in Fig. 2, we show $x_{ij}=x_{ii}\delta_{ij}$ for a 3D Fermi gas with a Rashba SOC, where $\mathbf{d}_{\mathbf{k}}=\alpha\mathbf{k}_{\perp}$ with $\mathbf{k}_{\perp}=k_{x}\boldsymbol{\widehat{x}}+k_{y}\boldsymbol{\widehat{y}},$ and therefore, $x_{\perp}=x_{\perp}^{\mathrm{intra}}+x_{\perp}^{\mathrm{inter}}$ is still isotropic in $q_{x}q_{y}$ plane but it differs from that $x_{zz}=x_{zz}^{\mathrm{intra}}$ of the $q_{z}$ direction with $x_{zz}^{\mathrm{inter}}=0$. Note that since $g_{\mathbf{k}}^{iz}=0$ for a Rashba SOC, all of the matrix elements $x_{iz}^{\mathrm{inter}}$ trivially vanish, leading to the aforementioned anisotropy. For intermediate $\alpha/v_{F}$ values, we again find that $x_{\perp}/v_{F}^{2}$ is a nonmonotonic function of $1/(k_{F}a_{s})$, exhibiting a noticeable peak in the crossover region around the unitarity. However, $x_{zz}/v_{F}^{2}$ turns out to be a monotonic one no matter what $\alpha/v_{F}$ is. Thus, the dramatic difference between $x_{\perp}$ and $x_{zz}$ may be used as an ultimate evidence for the presence of an interband contribution to the former, reflecting the geometry of the underlying quantum states.

For completeness, in Fig. 3, we show $x_{ij}=x_{0}\delta_{ij}$ for a 2D Fermi gas with a Rashba SOC, where $\mathbf{d}_{\mathbf{k}}=\alpha\mathbf{k},$ and therefore, $x_{0}=x_{0}^{\mathrm{intra}}+x_{0}^{\mathrm{inter}}$ is isotropic in the entire $\mathbf{q}$ space. In accordance with the cold-atom literature, the two-body binding energy $\varepsilon_{b}\geq 0$ in vacuum is substituted in this figure for $U$ via the usual relation $A/U=\sum_{\mathbf{k}}1/(2\epsilon_{\mathbf{k}}+\varepsilon_{b}),$ where $A$ is the area of the system. Furthermore, we choose $N=k_{F}^{2}A/(2\pi),$ and use the Fermi energy $\epsilon_{F}=k_{F}^{2}/(2m)$ as the relevant energy scale in our numerical calculations. This is such that increasing $U$ from [math] increases $\varepsilon_{b}/\epsilon_{F}$ continuously from [math] to $\infty$ in the BCS-BEC crossover. In the weak-SOC limit when $\alpha/v_{F}\ll 1$, we again recover the usual problem where $x_{0}/v_{F}^{2}$ is known to be $1/2$ for all $\varepsilon_{b}/\epsilon_{F}$, i.e., it remains a constant in the entire BCS-BEC crossover iskin05 . In contrast to the 3D results shown in Figs. 1 and 2 where the geometric contribution $x_{0}^{\mathrm{inter}}$ plays important roles only in the crossover region with a rapid decay in the BCS and BEC regimes, here $x_{0}^{\mathrm{inter}}$ retains its strength for most $\varepsilon_{b}/\epsilon_{F}$ values except for the extreme BCS limit when $\varepsilon_{b}/\epsilon_{F}\ll 1$. Even then we find that the fraction $x_{0}^{\mathrm{inter}}/x_{0}$ does not grow much, and saturates around a quarter.

IV Conclusions

In summary, here we considered a two-component Fermi gas with an arbitrary SOC, and analyzed its collective excitations in the BCS-BEC crossover. Motivated by the recent findings iskin18a ; iskin18b , we divided the velocity of the Goldstone mode into two distinct contributions based on the physical mechanisms involved. While the conventional (intraband) contribution is controlled by the $\mathbf{k}$-space derivatives of the helicity (one-body) spectrum, the geometric (interband) one is controlled by the $\mathbf{k}$-space derivatives of the helicity eigenstates or the so-called quantum metric tensor characterizing the geometry of the underlying quantum states. We presented extensive numerical calculations for the Weyl and Rashba SOCs revealing that, despite being much smaller than the conventional one, the geometric contribution is solely responsible for the nonmonotonic evolution of the mode velocity in the BCS-BEC crossover.

These findings for the Goldstone mode in spin-orbit coupled Fermi gases are in complete agreement with our recent works that the quantum metric tensor has a subleading control over all those SF properties that depend explicitly on the effective-mass tensor of the corresponding (two- or many-body) bound state iskin18a ; iskin18b . We note that whether the quantum-metric contribution can be isolated from the total is far from being clear with the current experimental techniques involving spin-orbit coupling. However, in the same spirit, given that the starting Hamiltonian Eq. (4) is quite generic, our analytical expressions for the Goldstone mode may also find direct applications in other contexts that exhibit a two-band band structure. In particular, it is desirable to study the Goldstone mode in a narrow- or a flat-band system iskin19a ; iskin19b , for which the geometric contribution is expected to take the lead. As a final remark, our work suggests that an analogous contribution to the collective excitations must be present in many other multi-band systems including the twisted bilayer graphene.

Acknowledgements.

This work is supported by the funding from TÜBİTAK Grant No. 1001-118F359.

Appendix A Effective-Action Approach

The imaginary-time functional path-integral formalism allows a systematic derivation of the effective action in the form of a power-series expansion of the fluctuations of the order parameter around its saddle-point value carlos97 ; iskin05 ; he12 ; shenoy12 . In this approach, to decouple the interaction term that is quartic in the fermionic degrees of freedom, one first uses a Hubbard-Stratanovich transformation at the expense of introducing a bosonic field $\Delta_{q}$, and then integrates out the remaining terms that are quadratic in the fermionic degrees of freedom. Finally, by decomposing $\Delta_{q}=\Delta_{0}+\Lambda_{q}$ in terms of a $q$-independent stationary field $\Delta_{0}$ and $q$-dependent fluctuations around it, one may in principle obtain an effective action at the desired order in $\Lambda_{q}$. Here we include the first nontrivial term and obtain the effective-Gaussian action $S_{\mathrm{Gauss}}=S_{0}+S_{2}$, as the first-order term $S_{1}$ trivially vanishes thanks to the saddle-point condition given in the main text.

For our interests, the zeroth-order action in $\Lambda_{q}$ can be written as $S_{0}=\Delta_{0}^{2}/(TU)+\sum_{s\mathbf{k}}\xi_{s\mathbf{k}}/(2T)-(1/2)\sum_{k}\ln[\det(\mathbf{G}_{k}^{-1}/T)],$ where the collective index $k=(\mathbf{k},\mathrm{i}\omega_{\ell})$ denotes both the particle momentum $\mathbf{k}$ and the fermionic Matsubara frequency $\omega_{\ell}=(2\ell+1)\pi T$ carlos97 ; iskin05 ; he12 ; shenoy12 . Here, $\mathbf{G}_{k}^{-1}=\mathrm{i}\omega_{\ell}\mathbf{1}-H_{\mathbf{k}}^{0}$ is the inverse Green’s function for the saddle-point Hamiltonian $H_{\mathbf{k}}^{0}$, i.e.,

[TABLE]

determining the quasiparticle and quasihole energies through $\det\mathbf{G}_{k}^{-1}=[(\mathrm{i}\omega_{\ell})^{2}-E_{+,\mathbf{k}}^{2}][(\mathrm{i}\omega_{\ell})^{2}-E_{-,\mathbf{k}}^{2}]=0,$ leading to $E_{s\mathbf{k}}=\sqrt{\xi_{s\mathbf{k}}^{2}+\Delta_{0}^{2}}$ for the former. Performing the summation over the fermionic Matsubara frequency leads to the saddle-point action given in Eq. (5) of the main text.

The second-order term in $\Lambda_{q}$ can be written as $S_{2}=\sum_{q}|\Lambda_{q}|^{2}/(TU)+(1/4)\mathrm{Tr}\sum_{kq}\mathbf{G}_{k}\mathbf{\Sigma}_{q}\mathbf{G}_{k+q}\mathbf{\Sigma}_{-q},$ where $\mathrm{Tr}$ denotes a trace over the particle/hole and spin sectors, and the collective index $q=(\mathbf{q},\mathrm{i}\nu_{n})$ denotes both the pair momentum $\mathbf{q}$ and the bosonic Matsubara frequency $\nu_{n}=2\pi nT$ iskin05 . Here, the matrix elements of the Green’s function $\mathbf{G}_{k}$ can be written as

[TABLE]

and those of the fluctuations $\mathbf{\Sigma}_{q}$ are $\Sigma_{q}^{11}=\Sigma_{q}^{22}=0,$ $\Sigma_{q}^{12}=\mathrm{i}\Lambda_{q}\sigma_{y}$ and $\Sigma_{q}^{21}=-\mathrm{i}\Lambda_{-q}^{*}\sigma_{y}$  edelstein95 ; gorkov01 . After a lengthy but a straightforward summation over the fermionic Matsubara frequency, we find $S_{2}=[1/(2T)]\sum_{q}\mathbf{\bar{\Lambda}}_{q}^{\dagger}\mathbf{M}_{q}\mathbf{\bar{\Lambda}}_{q},$ where $\mathbf{\bar{\Lambda}}_{q}^{\dagger}=(\Lambda_{q}^{*}\,\Lambda_{-q})$ and $\mathbf{M}_{q}$ is the inverse fluctuation propagator carlos97 ; iskin05 ; he12 ; shenoy12 . Simplifying the notation by denoting $\xi_{s,\mathbf{k}+\mathbf{q}/2}$ by $\xi$; $\xi_{s^{\prime},-\mathbf{k}+\mathbf{q}/2}$ by $\xi^{\prime}$; $E_{s,\mathbf{k}+\mathbf{q}/2}$ by $E$; $E_{s^{\prime},-\mathbf{k}+\mathbf{q}/2}$ by $E^{\prime}$; and $\widehat{\mathbf{d}}_{\pm\mathbf{k}+\mathbf{q}/2}$ by $\widehat{\mathbf{d}}_{\pm}$, and defining the functions $u^{2}=(1+\xi/E)/2$; $u^{\prime 2}=(1+\xi^{\prime}/E^{\prime})/2$; $v^{2}=(1-\xi/E)/2$; $v^{\prime 2}=(1-\xi^{\prime}/E^{\prime})/2$; $f=1/(e^{E/T}+1)$; and $f^{\prime}=1/(e^{E^{\prime}/T}+1)$, a compact way to write the matrix elements of $\mathbf{M}_{q}$ is carlos97 ; iskin05

[TABLE]

The remaining elements are related via $M_{q}^{22}=M_{-q}^{11}$ and $M_{q}^{21}=M_{q}^{12}$. We note that while $M_{q}^{12}$ is even both under $\mathbf{q}\to-\mathbf{q}$ and $\mathrm{i}\nu_{n}\to-\mathrm{i}\nu_{n}$, $M_{q}^{11}$ is even only under $\mathbf{q}\to-\mathbf{q}$.

While calculating the collective excitations, it is useful to represent $\Lambda_{q}=\alpha_{q}e^{\mathrm{i}\gamma_{q}}$ in terms of real functions $\alpha_{q}$ and $\gamma_{q}$, and associate $\lambda_{q}=\sqrt{2}\alpha_{q}\cos(\gamma_{q})$ with the amplitude degrees of freedom and $\theta_{q}=\sqrt{2}\alpha_{q}\sin(\gamma_{q})$ with the phase ones in the small $\gamma_{q}$ limit. This procedure corresponds to a unitary transformation $\Lambda_{q}=(\lambda_{q}+\mathrm{i}\theta_{q})/\sqrt{2},$ where $\lambda_{q}$ and $\theta_{q}$ are real functions. Furthermore, assuming $\lambda_{-q}=\lambda_{q}^{*}$ and $\theta_{-q}=\theta_{q}^{*}$, we find

[TABLE]

where we denote $\mathrm{\Theta}_{q}^{\dagger}=(\lambda_{q}^{*}\,\theta_{q}^{*})$, and separate $M_{q}^{11}=M_{q,E}^{11}+M_{q,O}^{11}$ in terms of an even function $M_{q,E}^{11}=(M_{q}^{11}+M_{q}^{22})/2$ in $\mathrm{i}\nu_{n}$ and an odd one $M_{q,O}^{11}=(M_{q}^{11}-M_{q}^{22})/2,$ and use $M_{q}^{12}=M_{q}^{21}$  carlos97 ; iskin05 ; he12 ; shenoy12 . The phase-only action given in Eq. (6) is derived by integrating out the amplitude fields, and vice versa for the Eq. (7). We note that since the quasiparticle-quasihole terms with the prefactor $(f-f^{\prime})$ have the usual Landau singularity for $q\to(\mathbf{0},0)$ causing the collective modes to decay into the two-quasiparticle continuum, a small $q$ expansion is possible only in two cases: (i) just below the critical SF transition temperature provided $\Delta_{0}\to 0\ll\mathrm{i}\nu_{n}\to\omega$ iskin18b , and (ii) at $T=0$. In this work, we are interested in the latter case earlierworks .