Collective mode contributions to the Meissner effect: Fulde-Ferrell and pair-density wave superfluids

TL;DR

This paper emphasizes the importance of including collective mode contributions, especially the amplitude (Higgs) mode, in calculating the Meissner effect for non-uniform superconductors like Fulde-Ferrell and pair-density wave states, revealing their significant impact on superfluid density.

Contribution

It demonstrates that collective mode contributions, particularly the amplitude mode, are crucial for accurately describing the Meissner effect in non-uniform superconductors, a factor often neglected in prior work.

Findings

Collective modes significantly affect the superfluid density in non-uniform superconductors.

The amplitude (Higgs) mode plays a key role in pair-density wave superconductors.

Collective mode effects can destroy superfluidity before the order parameter vanishes.

Abstract

In this paper we demonstrate the necessity of including the generally omitted collective mode contributions in calculations of the Meissner effect for non-uniform superconductors. We consider superconducting pairing with non-zero center of mass momentum, as is relevant to high transition temperature cuprates, cold atoms, and quantum chromodynamic superconductors. For the concrete example of the Fulde-Ferrell phase we present a quantitative calculation of the superfluid density, showing the collective mode contributions are not only appreciable but that they derive from the amplitude mode of the order parameter. This latter mode (related to the Higgs mode in a charged system) is generally viewed as being invisible in conventional superconductors. However, our analysis shows that it is extremely important in pair-density wave type superconductors, where it destroys superfluidity well before the mean-field order parameter vanishes.

Equations

$$\Delta [A]={\Delta }^{(0)}[A=0]+{\Delta }^{(1)}[A]+\mathcal{O}(A^2).$$

$${\frac{\partial j^z}{\partial Q}|}_{\mu ,h}=\frac{1}{2}\left(\frac{n_s^{zz}}{m}\right),$$

$${\frac{\partial j^z}{\partial Q}|}_{\mu ,h}=2\left[{\frac{{\partial }^2\Omega }{\partial Q^2}|}_{\mu ,h,\Delta }-{\left(\frac{{\partial }^2\Omega }{\partial \Delta \partial Q}\right)}^2/{\frac{{\partial }^2\Omega }{\partial {\Delta }^2}|}_{\mu ,h,Q}\right],$$

$$\mathcal{H}_{\mathrm{FF}}=\left(\begin{array}[]{cc}\xi_{\mathbf{k},\uparrow}&-\Delta\\ -\Delta^{*}&-\xi_{\mathbf{k-Q},\downarrow}.\end{array}\right).$$

$$P^{\mu \nu }(q)=\sum _{\sigma }\sum _k{G}_{\sigma }(k_{+}){\Gamma }_{\sigma }^{\mu }(k_{+},k_{-})G_{\sigma }(k_{-}){\gamma }_{\sigma }^{\nu }(k_{-},k_{+}).$$

$${\Delta }^{\ast }{\Pi }^z(0)+\Delta {\bar{\Pi }}^z(0)=P_0^z/M_0=2{(\partial |\Delta {|}^2/\partial Q)|}_{\mu ,h}.$$

$$(n_s^{ij}/m)=(n/m){\delta }^{ij}+P^{ij}(\omega =0,\mathbf{q}\to 0).$$

$$\left(\frac{n_s^{ij}}{m}\right)$$

$$-{\delta }^{iz}{\delta }^{jz}(P_0^z{)}^2/M_0.$$

Full text

There is a strong current interest in collective modes in superconductors in large part stimulated by the excitement surrounding the discovery of the Higgs boson PhysicsToday . Nevertheless, there is a widespread belief that observing these modes, directly or indirectly, is particularly challenging Parks1969 ; Varma_review_2015 . As a result they only infrequently appear in condensed matter physics Littlewood&Varma_1981 ; Browne&Levin_1983 ; Endres_2012 ; Pollet_2012 ; Auerbach_2014 ; Brandon_pathintegral_2016 ; Matsunaga_2013 . In this paper we show that in a class of very topical superconductors, collective mode effects associated with the amplitude of the order parameter play an essential role in the most fundamental quantity, the superfluid density tensor $n_{s}^{ij}$. The superconductors in question are those which have a “pair-density wave” order parameter. These are a large class of superconductors exhibiting pairing of electrons at non-zero center of mass momentum ${\bf{Q}}$. Much attention has focused on these systems from the perspective of high temperature superconductivity (in condensed matter physics Fradkin_2015 ; SeamusDavis_2016 ) and quantum chromodynamics (in particle physics Casalbuoni_2004 ).

For this class of superfluids, the collective mode contribution to the superfluid density has been largely ignored in previous literature Torma_FF_2014 ; Samokhin_FFLO_2010 , with the exception of the original calculation of the electromagnetic current by Larkin and Ovchinnikov Larkin&Ovchinnikov_FFLO_1965 . Discussion of this effect can also be found in Ref. Millis_1987 for a different situation involving non $s$-wave superconductors. In both cases the size and specific nature of the collective mode contributions was not accessible.

We provide two different, but related, derivations of the superfluid density for the tractable case of the Fulde-Ferrell (FF) superfluid Fulde&Ferrell_1964 . Importantly, this enables us to compute numerical values for the sizeable collective mode effects in $n_{s}^{ij}$. The first method is based on using the Ward-Takahashi identity in the Kubo formalism, while the second method is based on studying the equilibrium current. In both approaches particle number is manifestly conserved and gauge invariance is maintained. Through the first approach we find that amplitude collective modes drive the superfluid density (along the direction parallel to $\mathbf{Q}$) to zero at temperatures lower than those associated with the vanishing of the mean-field order parameter.

Before giving these more complete calculations, here we provide a general argument for the necessity of including collective mode effects in non-uniform superconductors. The origin of collective mode contributions to the Meissner effect Larkin&Ovchinnikov_FFLO_1965 ; Millis_1987 lies in the fact AGD_1975 that, in the presence of a vector potential $A^{\mu}$, the order parameter $\Delta$ will depend on $A^{\mu}$ through the gap (saddle-point) equation Brandon_pathintegral_2016 . A series expansion of $\Delta[A]$, in powers of $A^{\mu}$, is thus

[TABLE]

Here $\Delta^{(0)}$ is the order parameter in the absence of $A^{\mu}$ and $\Delta^{(1)}$ is a correction linear in $A^{\mu}$. It is this term which gives rise to the rarely discussed collective mode contributions to the superfluid density. Since $\Delta^{(1)}$ is a scalar quantity, it can depend on only scalar, linear functions of $A^{\mu}$. Therefore, in a uniform superfluid $\Delta^{(1)}$ is a function of only $\nabla\cdot{\bf A}$ 111In general, for a uniform superfluid $\Delta^{(1)}$ will be a function of $\partial_{\mu}A^{\mu}$. For the superfluid density, however, only the vector component of $A^{\mu}$ is of interest.. Thus, if one chooses the (“transverse”) gauge such that $\nabla\cdot{\bf A}=0$, the collective mode contribution $\Delta^{(1)}$ vanishes identically AGD_1975 .

However, for a non-uniform system there are other scalar, linear functions of $A^{\mu}$. In particular, for a pair-density wave superfluid with pairing vector ${\bf Q}$, $\Delta^{(1)}$ can depend on other scalar, linear quantities such as $\mathbf{A}\cdot\mathbf{Q}$. Hence, for this non-uniform superfluid, even in the gauge where $\nabla\cdot{\bf A}=0$, $\Delta^{(1)}$ may still be non-zero. In principle, this allows for a collective mode contribution to the superfluid density. [For future use in the discussion below, we define $\Delta^{(1)}=\left.(d\Delta[A]/dA_{\mu})\right|_{A=0}A_{\mu}=\int dq\ \Pi^{\mu}(q)A_{\mu}(q)$.]

To illustrate this effect, we implement this argument for the specific case of the Fulde-Ferrell Fulde&Ferrell_1964 superfluid. For simplicity the FF pairing vector is assumed to be $\mathbf{Q}=Q\hat{z}$. In the FF phase specifically, both a continuous rotational and global gauge symmetry are spontaneously broken. Similarly discrete time-reversal symmetry is also spontaneously broken. However, gauge invariant observables are translationally invariant Radzihovsky_FFLO_2011 . Due to the underlying rotational symmetry of the FF state, the superfluid density vanishes along the directions transverse to $\mathbf{Q}$. Hence $n_{s}^{xx}=n_{s}^{yy}=0$, and thus only $n_{s}^{zz}$ needs to be considered Radzihovsky_FFLO_2011 . As has been posited Fulde&Ferrell_1964 , and will be shown in more detail below, the superfluid density can be written as

[TABLE]

where $j^{z}(Q)$ is the equilibrium current.

It is useful to express Eq. (2) in terms of the mean-field thermodynamic potential $\Omega$, where $j^{z}(Q)=2\left.(\partial\Omega/\partial Q)\right|_{\mu,h,\Delta}$. The mean-field values of the chemical potential, gap, magnetic field, and pairing vector are denoted by $\mu_{0},\Delta_{0},h_{0},$ and $Q_{0}$ respectively. The saddle-point condition which determines $Q_{0}$ is then $j^{z}(Q_{0})=0$. Similarly, the saddle-point condition which determines $\Delta_{0}$ is $\left.(\partial\Omega/\partial\Delta)\right|_{\mu,h,Q}=0$. In terms of $\Omega$, Eq. (2) becomes

[TABLE]

where both saddle-point equations, and the symmetry of mixed partial derivatives has been used.

Equation (3) indicates that there are two contributions to the superfluid density. The first is the conventional “bubble” term (which is usually assumed to be sufficient) and the second represents the collective mode contribution required for gauge invariance. Importantly, a stability inequality for the FF superfluid based on the thermodynamic potential curvature ChihChun_FFLO_2006 ; Yan_FFLO_2007 ; Qijin_FFLO_2017 is equivalent to requiring that both $n^{zz}_{s}$, as derived above, and $\left.\left(\partial^{2}\Omega/\partial\Delta^{2}\right)\right|_{\mu,h,Q}$ are positive. From this, it follows that for a stable FF superfluid the collective mode contribution always acts to reduce the overall size of the superfluid density. The above arguments, however, still do not indicate how large the magnitude of this effect is.

In this paper the collective-mode contribution will be shown to be appreciable; this underlines the inadequacy of including only the so-called bubble term Torma_FF_2014 ; Samokhin_FFLO_2010 . Equally important is the nature of these collective mode corrections. For the FF superfluid we will show that they derive from the amplitude mode of the order parameter. This mode is thought to be rather invisible in conventional superconductors Parks1969 . Nevertheless we demonstrate how it arises to ensure the electromagnetic (EM) response is manifestly gauge invariant. In this way, it importantly affects the superfluid density in pair-density wave superconductors. Readers uninterested in the technical details can skip to the numerical results for a simple understanding of our main results.

Mean-field formalism.– The FF mean-field Hamiltonian, in the $\psi^{\mathsf{T}}_{\mathbf{k}}~{}=(c_{\mathbf{k},\uparrow},c_{\mathbf{-k+Q},\downarrow}^{\dagger})$ basis, is $H_{\mathrm{FF}}~{}=\sum_{\mathbf{k}}\psi_{\mathbf{k}}^{\dagger}\mathcal{H}_{\mathrm{FF}}\psi_{\mathbf{k}}$, where Qijin_FFLO_2007

[TABLE]

Here an irrelevant constant $-\sum_{\mathbf{k}}\xi_{\mathbf{k-Q},\downarrow}$ has been ignored. The notation is as follows: the dispersion relation is defined by $\xi_{\mathbf{k},\sigma}=\mathbf{k}^{2}/2m-\mu_{\sigma}$, where $\mu_{\sigma}$ is the fermionic chemical potential for a species with spin $\sigma=\uparrow,\downarrow$, $m$ is the fermion mass, and $\Delta$ denotes an $s$-wave pairing gap. It is useful to define $\mu=\tfrac{1}{2}\left(\mu_{\uparrow}+\mu_{\downarrow}\right)$ and $h=\tfrac{1}{2}\left(\mu_{\uparrow}-\mu_{\downarrow}\right)$. The dispersion relations are then written compactly as $\xi_{\mathbf{kQ}}=(1/2m)[\mathbf{k}^{2}+(\mathbf{Q}/2)^{2}]-\mu$, $E_{\mathbf{kQ}}^{2}=\xi_{\mathbf{kQ}}^{2}+\Delta^{2}$, $h_{\mathbf{kQ}}=h-\mathbf{k}\cdot{\mathbf{Q}}/2m$. Throughout the paper $\hbar=k_{B}=c=1$.

The inverse Nambu Green’s function is then $\mathcal{G}^{-1}=i\omega_{n}-\mathcal{H}_{\mathrm{FF}},$ where $i\omega_{n}$ is a fermionic Matsubara frequency. The inverse bare Green’s function is defined by $G^{-1}_{0,\sigma}(k)=i\omega_{n}-\xi_{\mathbf{k}}$. Thus, the off diagonal Gorkov function is $\mathcal{G}_{12}(k)=\Delta G_{0,\downarrow}(-k+Q)G_{\uparrow}(k)$, where the (spin-up) Green’s function is $G_{\uparrow}(k)=\mathcal{G}_{11}(k)$. Note that Greek indices denote spacetime coordinates: $\mu=(t,x,y,z)$; whereas Roman indices denote spatial coordinates: $i=(x,y,z)$. Here $Q^{\mu}=(0,\mathbf{Q})$. Explicit calculation then gives the full Green’s function which has appeared in the literature Qijin_FFLO_2007 ; Torma_FF_2014 . Using this, the particle number is $n=\sum_{\sigma}\sum_{k}G_{\sigma}(k)$, where $\sum_{k}\equiv\beta^{-1}\sum_{i\omega_{n}}\sum_{\mathbf{k}}$ with $\beta$ being inverse temperature.

From Dyson’s equation, $G^{-1}_{\sigma}(k)=G^{-1}_{0,\sigma}(k)-\Sigma_{\sigma}(k),$ the self energy is $\Sigma_{\sigma}(k)=-|\Delta|^{2}G_{0,\bar{\sigma}}(-k+Q)$. An important identity is then $\mathcal{G}_{12}(\widetilde{k})=\Delta G_{0,\downarrow}(-\widetilde{k}_{-})G_{\uparrow}(\widetilde{k}_{+})=\Delta G_{0,\uparrow}(\widetilde{k}_{+})G_{\downarrow}(-\widetilde{k}_{-})$. [For convenience, we have defined $\widetilde{k}^{\mu}_{\pm}\equiv k^{\mu}\pm Q^{\mu}/2$.]

We now study the EM response of this superfluid. For the issue of primary concern in this paper (the superfluid density) the distinction between neutral and charged superfluids is irrelevant. For the purposes of simplicity, our general equations are for neutral superfluids. We apply linear response theory, where, a fictitious vector potential $A^{\mu}$ is applied, and at the end of the calculation $A^{\mu}\rightarrow 0$. The EM current is $j^{\mu}(q)=K^{\mu\nu}(q)A_{\nu}(q)$, where $K^{\mu\nu}(q)$ is the EM response kernel. The response kernel can also be expressed as $K^{\mu\nu}(q)=P^{\mu\nu}(q)+(n/m)\delta^{\mu\nu}(1-\delta_{\mu,0})$ (with $\mu$ and $\nu$ not summed over) where the EM response functions are denoted by $P^{\mu\nu}(q)$. In the Kubo formalism the EM response functions for a superfluid are

[TABLE]

Here $q^{\mu}=(i\Omega_{m},\mathbf{q})$, with $i\Omega_{m}$ a bosonic Matsubara frequency. The quantity $\Gamma^{\mu}(k_{+},k_{-})$ denotes the full EM vertex, where the incoming (outgoing) momentum is $k_{+}$ $(k_{-}$), with $k^{\mu}_{\pm}\equiv k^{\mu}\pm q^{\mu}/2$ . To determine the full vertex $\Gamma^{\mu}(k_{+},k_{-})$, we apply the Ward-Takahashi identity (WTI) Ryder : $q_{\mu}\Gamma^{\mu}_{\sigma}(k_{+},k_{-})=G^{-1}_{\sigma}(k_{+})-G^{-1}_{\sigma}(k_{-})=q_{\mu}\gamma^{\mu}_{\sigma}(k_{+},k_{-})+\Sigma_{\sigma}(k_{-})-\Sigma_{\sigma}(k_{+}).$ This is an exact relation in quantum field theory which relates the single particle Green’s function to the full vertex. It is a gauge invariant statement and here it reflects the underlying global $U(1)$ gauge symmetry. Furthermore, satisfying the WTI ensures conservation of particle number (or charge in the charged superfluid case). The bare WTI, $q_{\mu}\gamma^{\mu}_{\sigma}(k_{+},k_{-})=G^{-1}_{0,\sigma}(k_{+})-G^{-1}_{0,\sigma}(k_{-})$, is satisfied by the bare vertex $\gamma^{\mu}_{\sigma}(k_{+},k_{-})=(1,\textbf{k}/m)$.

In the limit $q^{\mu}\rightarrow 0$, the WTI reduces to the Ward identity: $\Gamma^{\mu}_{\sigma}(k,k)=\gamma^{\mu}_{\sigma}(k,k)-(\partial\Sigma_{\sigma}(k)/\partial k_{\mu})$. The second term diagrammatically represents a vertex insertion in the self energy. This relation then importantly shows that the full vertex can be obtained by performing all possible vertex insertions in the full Green’s function Ryder ; Nambu . Due to the spontaneous symmetry breaking of the global $U(1)$ symmetry, a collective mode vertex $\Pi^{\mu}(q)$ $(\bar{\Pi}^{\mu}(q))$ can be inserted into the order parameter $\Delta$ $(\Delta^{*})$. After these vertex insertions, the full vertex can be determined exactly. Importantly, the collective mode effects enter in the superfluid density via the combination $\left(\Delta^{*}\Pi^{\mu}(0)+\Delta\bar{\Pi}^{\mu}(0)\right)$.

Now we show that it is the amplitude rather than phase mode which is important to the calculation of the FF superfluid density. The collective mode vertices are self-consistently determined from the gap equation and, for $q^{\mu}\neq 0$, satisfy $q_{\mu}\Pi^{\mu}(q)=2\Delta,q_{\mu}\bar{\Pi}^{\mu}(q)=-2\Delta^{*}$. Note that, as $q^{\mu}\rightarrow 0$, the right hand side of these expressions remains finite, and so must the left hand side. It follows that $\Pi^{\mu}(q)$ and $\bar{\Pi}^{\mu}(q)$ are singular in the $q^{\mu}\rightarrow 0$ limit. This pole corresponds to the phase mode of the gap; equivalently it reflects the Nambu-Goldstone boson associated with the spontaneous symmetry breaking of global $U(1)$. On the other hand, note that $q_{\mu}\left(\Delta^{*}\Pi^{\mu}(q)+\Delta\bar{\Pi}^{\mu}(q)\right)=0$. This identity is non-singular in the $q^{\mu}\rightarrow 0$ limit, and reflects the fact that $\Delta^{*}\Pi^{\mu}(q)+\Delta\bar{\Pi}^{\mu}(q)$ does not have a zero momentum pole. Thus, this quantity corresponds to the amplitude mode of the gap.

The gap equation for an FF superfluid is $\Delta/g=\Delta\sum_{\sigma}\sum_{k}G_{0,\downarrow}(-k+Q)G_{\uparrow}(k)=\sum_{\sigma}\sum_{k}\mathcal{G}_{12}(\widetilde{k})$ Qijin_FFLO_2007 . Expressing this equation diagrammatically allows one to perform all vertex insertions on both the gaps ($\Delta$, $\Delta^{*}$) and on the bare and full Green’s functions. This procedure then leads to the explicit form of the collective mode vertices $\Pi^{\mu},\bar{\Pi}^{\mu}$, as shown explicitly in the Supplemental Material. From this analysis one can then obtain the zero momentum limit of the quantity $\Delta^{*}\Pi^{z}+\Delta\bar{\Pi}^{z}$ of importance here.

Note that $\Delta$ is a function of the FF pairing vector $Q$, and by differentiating the gap equation with respect to $Q$ (at fixed $\mu$ and $h$) one can obtain $\left.(\partial|\Delta|^{2}/\partial Q)\right|_{\mu,h}$. An explicit calculation then gives the following important identity (for $\Delta\neq 0$):

[TABLE]

The order of limits in which frequency and momentum are taken to zero is important; frequency $i\Omega_{m}$ and $q^{z}$ are set to zero, and then $q^{x},q^{y}\rightarrow 0$. In the following section this will be clarified. The quantities $P^{z}_{0}$ and $M_{0}$ are generalized three-particle and four-particle Green’s functions, respectively, which are defined in the next section. The generalized Green’s functions in Eq. (6) also appear in a similar form in the work of Larkin and Ovchinnikov Larkin&Ovchinnikov_FFLO_1965 and Millis Millis_1987 . Finally, note that when $Q=0$, $P^{z}_{0}=0$, and thus this collective mode term does not contribute for a homogeneous superfluid.

Superfluid density derivation via Kubo formula.– In this section we use the Kubo formula and Eq. (5) to derive the superfluid density tensor:

[TABLE]

Note that, the order of limits in the above expression is crucial. To compute $n^{ij}_{s}$, first set $\omega=q^{i}=q^{j}=0$, then take $q^{k}\rightarrow 0$, where $k\neq i,j$. The collective modes are contained within the second term.

Evaluating this expression we find

[TABLE]

where we define $X_{\mathbf{k}}\equiv D^{-1}\sinh(\beta E_{\mathbf{kQ}})$, and $Y_{\mathbf{k}}\equiv D^{-2}(1+\cosh(\beta E_{\mathbf{kQ}})\cosh(\beta h_{\mathbf{kQ}}))$ with $D\equiv\cosh(\beta E_{\mathbf{kQ}})+\cosh(\beta h_{\mathbf{kQ}})$.

The first term in Eq. (Collective mode contributions to the Meissner effect: Fulde-Ferrell and pair-density wave superfluids) represents the usual Torma_FF_2014 ; Samokhin_FFLO_2010 “bubble” contribution, due to bubble terms in both $(n/m)\delta^{ij}$ and $P^{ij}(0)$. The second term represents the collective mode contribution arising solely from $P^{ij}(0)$. As an important check we note that Eq. (Collective mode contributions to the Meissner effect: Fulde-Ferrell and pair-density wave superfluids) is identical to Eq. (2) and Eq. (3), where explicit calculation shows that the bubble term is $4\left.(\partial^{2}\Omega/\partial Q^{2})\right|_{\mu,h,\Delta}$ and $(\partial^{2}\Omega/\partial\Delta\partial Q)=-\Delta P^{z}_{0},\left.(\partial^{2}\Omega/\partial\Delta^{2})\right|_{\mu,h,Q}=4\Delta^{2}M_{0}$. Here $\Omega=\Delta^{2}/g-\beta^{-1}\sum_{\mathbf{k}}\left\{\mathrm{log}[2\cosh(\beta E_{\mathbf{kQ}})+2\cosh(\beta h_{\mathbf{kQ}})]-\beta\xi_{\mathbf{kQ}}\right\}$ is the mean-field thermodynamic potential Torma_FF_2014 ; Qijin_FFLO_2017 .

Note that the collective mode contribution is only along the direction of the FF pairing vector, in agreement with the general arguments presented earlier. Direct calculation shows that $n^{ij}_{s}$ is diagonal, with $n^{xx}_{s}=n^{yy}_{s}=0$, as required by symmetry.

Superfluid density derivation via equilibrium current.– A verification of this Kubo analysis and the collective mode contributions can be made in a slightly simpler fashion. Here we derive the superfluid density in the direction along the FF pairing vector using only the equilibrium current and its partial derivative with respect to $Q$. The equilibrium current in the $z$-direction is $j^{z}(Q)=\sum_{\sigma}\sum_{k}(\widetilde{k}^{z}_{+}/m)G_{\sigma}(\widetilde{k})$. This expression follows from $j^{z}=2\left.(\partial\Omega/\partial Q)\right|_{\mu,h,\Delta}$. By symmetry the mean-field currents in the other directions vanish: $j^{x}=j^{y}=0$.

In what follows it will be important to fix $\mu$ and $h$, and to consider the $Q$-dependence of only the gap: $\Delta(Q)$. The following lemma, whose proof is given in the Supplemental Material, will also be required: $\left.(\partial G^{-1}_{\sigma}(\widetilde{k}_{+})/\partial Q)\right|_{\mu,h}=-(1/2)\Gamma^{z}_{\sigma}(\widetilde{k}_{+},\widetilde{k}_{+}).$ The partial derivative of $j^{z}$ can now be computed. Using the number equation $n=\sum_{\sigma}\sum_{k}G_{\sigma}(k)$, along with the aforementioned Lemma, the partial derivative of $j^{z}$ is then $\left.(\partial j^{z}/\partial Q)\right|_{\mu,h}=(n/2m)-\sum_{\sigma}\sum_{k}(\widetilde{k}^{z}_{+}/m)G^{2}_{\sigma}(\widetilde{k})\left.(\partial G^{-1}_{\sigma}(\widetilde{k}_{+})/\partial Q)\right|_{\mu,h}=(n^{zz}_{s}/2m)$. Note that the above expression, which reproduces Eq. (2) and Eq. (Collective mode contributions to the Meissner effect: Fulde-Ferrell and pair-density wave superfluids), includes collective mode contributions arising through $\Gamma^{z}(\widetilde{k}_{+},\widetilde{k}_{+})$.

Numerical Results.– In Fig. (1) the superfluid density with collective mode effects (blue curve) is plotted as a function of temperature for the case of a polarized superfluid with $p=0.75$ and interaction strength (via the scattering amplitude) $1/k_{F}a=0$. These parameters were chosen because there are independent calculations in the literature Qijin_FFLO_2017 claiming to establish a regime of stability for this FF phase. There it was argued that the thermodynamic potential must satisfy two conditions: $\left.(\partial^{2}\Omega/\partial\Delta^{2})\right|_{\mu,h,Q}>0$ as well as the condition requiring that the right hand side of Eq. (3) is positive. Importantly, this latter stability criterion is precisely equivalent to $n^{zz}_{s}$ being positive. Our independent calculations yield a largest temperature for which the superfluid is stable to be $T_{c}/T_{F}\sim 0.6-0.65$, in rough agreement with Ref. Qijin_FFLO_2017 for the same input parameters. It should be noted that, although we are in the strong interaction regime, for quantitative purposes strict mean-field parameters are used in these plots. For numerical checks we have verified that our mean-field solutions are global minima of the thermodynamic potential Radzihovsky&Sheehy_2010 and that the blue curve computed via Eq. (Collective mode contributions to the Meissner effect: Fulde-Ferrell and pair-density wave superfluids) is numerically equivalent to that computed via the equilibrium current using Eq. (2).

The red curve in Fig. (1) is the bubble contribution which is usually Torma_FF_2014 ; Samokhin_FFLO_2010 all that is considered. The green curve plots the transverse superfluid density. As required by symmetry, $n^{xx}_{s}=0$ for all $T<T_{Q}$ for which the FF pairing vector $Q$ persists. When $T\geq T_{Q}\sim 0.2T_{F}$, the FF phase ceases to exist and instead a Sarma superfluid (with $Q\equiv 0$) exists. In this regime $n^{zz}_{s}$ has no collective mode contribution so that $n^{zz}_{s}=n^{xx}_{s}=n^{yy}_{s}$. This explains the small regime above $T_{Q}$ where $n^{xx}_{s}>0$, which persists until $\Delta=0$ at $T^{*}$. It can be noted that the effects of the collective modes are quite appreciable in this plot. This follows because the bubble term is proportional to $(\Delta/E_{F})^{2}$, whereas the collective mode term is proportional to $(Q/k_{F})^{2}$. (Note though the integrands in both expressions are somewhat different.) Near zero temperature, with $p=0.75$ and $1/k_{F}a=0$, $\Delta/E_{F}\sim 0.16$ whereas $Q/k_{F}\sim 0.71$. Thus qualitatively the collective mode contribution is expected to be an important contribution.

This figure encapsulates the important point that collective modes of the order parameter can substantially reduce the transition temperature from its mean-field value. In this way a pairing gap persists above the temperature at which the superfluid density disappears. This is a variant on a particular pseudogap scenario in the high $T_{c}$ cuprates Emery_1995 which invokes phase rather than amplitude collective mode contributions to suppress $T_{c}$ relative to its mean field value.

Conclusions.– In this paper we have computed the superfluid density tensor $n^{ij}_{s}$ for the FF superfluid phase. Importantly, we have shown (using multiple, distinct theoretical frameworks) that widely neglected collective (amplitude) mode contributions cannot be ignored. In general they will affect $n^{ij}_{s}$ for the broad class of $\mathbf{Q}\neq 0$ pair-density wave superconductors. Indeed, while Fig. (1) was obtained using the specific microscopic approach of Fulde and Ferrell, we believe its qualitative features (except for the behavior of the transverse superfluid density) are more generic. This figure suggests that a conventional Landau Ginsburg expansion Larkin&Ovchinnikov_FFLO_1965 may be problematic. In a temperature range near but slightly above $T_{c}$, the mean-field gap persists so that $\Delta$ may not be sufficiently small. Similarly, at temperatures near $T^{*}$ where this parameter is appropriately small, the stable phase has vanishing $\mathbf{Q}$. This poses a challenge for future work.

Given the intense interest in condensed matter observations of a Higgs mode, one can inquire as to what is the relation between the amplitude mode evident in pair-density wave superconductors and the Higgs mode in condensed matter Littlewood&Varma_1981 ; Browne&Levin_1983 ; Auerbach_2014 ; Brandon_pathintegral_2016 ; Endres_2012 ; Matsunaga_2013 . The Higgs mechanism is associated with a charged system and the present theory of the superfluid density, which depends on the amplitude mode, is applicable to both charged and neutral superfluids. However, because we are considering the case of zero frequency and zero wave number the Higgs mode is not observed as a collective resonance. Nevertheless, we have ascertained in this paper that its very existence has important consequences for readily accessible physical quantities in pair-density wave superconductors.

Acknowledgements.– We thank Qijin Chen and Matthew Roberts for helpful conversations. This work was supported by NSF-DMR-MRSEC 1420709.