Normal charge densities in quantum critical superfluids

We show that, in contrast to previously appeared arguments, the zero temperature limit of translation invariant superfluid hydrodynamics does not imply a vanishing normal charge density. This conclusion follows from consistently coupling the hydrodynamic theory to external sources. In addition, we investigate the normal density in holographic models of quantum critical superfluids. Here, models with an emergent IR Lorentz symmetry lead to a vanishing normal density with the velocity of second sound obeying Landau's conjecture in terms of the emergent infrared lightcone velocity, $c_2^2\simeq c_{IR}^2/d$. On the other hand, models with an emergent dynamical Lifshitz exponent $z>1$ may have a non-vanishing normal density, depending on the spectrum of irrelevant deformations around the underlying quantum critical groundstate. In this case, the velocity of fourth sound $c_4^2 \simeq \rho_s/d\rho$ offers a crisp diagnostic of a non-vanishing normal density. We comment on the relevance of our results to recent measurements of the superfluid density and low energy spectral weight in overdoped cuprates.


INTRODUCTION
Much of traditional superfluid and BEC superconductor phenomenology can be explained by Landau and Tisza's simple two-fluid hydrodynamical model [1,2] and its relativistic generalizations [3][4][5][6][7][8]. These relativistic invariant equations of superfluidity were shown to hold in holographic models where the transport coefficients can be derived from the gravitational dual to the boundary fluid [9][10][11][12][13]. Though the early holographic models focused on the original holographic superfluid [14][15][16], the bulk action can be generalized to include running couplings and bounded scalar potentials [17][18][19][20] which give rise to emergent Lorentz symmetry in the IR. We find that the two-fluid hydrodynamic model still works well in describing these models.
The Landau-Tisza model describes the superfluid as a mixture of two fluid components, the normal state with charge density ρ n and velocity u µ and the superfluid with charge density ρ s and flow velocity v µ . The total charge density is the sum of both components, ρ = ρ n +ρ s . From this simple set-up, even at the non-dissipative level, translation invariance and equilibrium thermodynamics were argued to give the remarkable prediction [21,22], This prediction aligned nicely with experimental realizations of Bose-Einstein condensation in 3 He, 4 He, cold atom experiments, and traditional BEC superconductors.
In this Letter, we show, in contrast to the argument presented in [21,22], that consistently coupling external sources to translation invariant superfluid hydrodynamics relaxes the constraint (1) and therefore that determining ρ (0) n requires knowledge of the IR equation of state. Along these lines, Ref. [22] also showed that a Galilean, time-reversal invariant, single species superfluid must obey (1), irrespective of superfluid hydrodynamics. Here, using holographic models with quantum critical dynamics in the infrared as examples, we recover a similar result for phases with an emergent Lorentz symmetry, also consistent with the superfluid effective field theory, [23,24]. On the other hand, we find that non-relativistic quantum critical systems with dynamical critical exponent z > 1 can have ρ (0) n = 0. Hence, we conclude that a non-vanishing ρ (0) n is not a result of the breakdown of the two-fluid model but rather a result of the quantum critical nature of the IR of these superfluids. Even after explicitly breaking translations, we show this conclusion does not change.
Our results may help shed light on recent experimental reports of anomalously low superfluid densities in overdoped high-T c superconductors [25] (see [26,27] for commentary). Subsequent spectroscopic studies [28] revealed very little loss of low energy spectral weight at low temperatures in the superconducting phase, suggesting a nonvanishing ρ (0) n . While the authors of [29][30][31] attributed this to disorder effects that can be captured in the so-called 'dirty BCS' theory, fitting the experimental data relies on an ad hoc renormalization of the Drude weight [30]. Thus, no theoretical consensus has been reached on the experimental findings of [25,28], see also [32,33]. The findings we report in this Letter suggest this might rather be a consequence of the quantum critical properties of the superconducting phase. In this section, we rederive the thermodynamic argument in the case of relativistic superfluids. Our results apply to any theory with translation invariance, including Galilean invariant theories. Relativistic symmetry leads to simpler notation and aligns nicely with our holographic example. A more thorough derivation can be found in Appendix A.
We start with the equations of relativistic charged superfluid hydrodynamics, following [9]. It is sufficient to work at the non-dissipative level. The system is described by the following equations (setting the speed of light c = 1) The first line expresses the local conservation laws: the conservation of the fluid stress tensor and the conservation of the U (1) symmetry current, respectively. The last line states the constraints from gauge invariance and thermodynamics; respectively, a "Josephson equation" which relates the time component of the background gauge field to the phase of the superfluid, ϕ, and the statement that in equilibrium, the entropy density is conserved. Here, Hydrodynamics states that these equations can be solved in terms of a derivative expansion of local thermodynamic variables and the fluid velocity. At non-dissipative order, we may write There are no contributions from F µν since these are first order in derivatives. This form of the current and stress tensor is frame invariant. The total charge density is the sum of the normal, ρ n , and superfluid, ρ s , densities. The normal energy density, n , and pressure, P , satisfy the Smarr and Gibbs relations, We perturb about equilibrium, writing The fluctuation equations can be massaged into the form If s → 0 as T → 0, consistency of this equation requires If δu i and δϕ were allowed to fluctuate independently, we would conclude ρ (0) n = 0, as in [21,22].
However, introducing an external source for ϕ through δH = d 2 xs ϕ · ∂ϕ leads to s i ϕ = ρ s (∂ i ϕ − µu i ), [34] (see also Appendix A). Setting the external source to zero, the superfluid velocity v i ≡ ∂ i ϕ/µ is aligned with the fluid velocity u i and equation (6) is automatically satisfied. Therefore, consistent coupling of the hydrodynamics to external sources evades the conclusion that ρ (0) n = 0. The fluctuation equations lead to an electrical conductivity at non-dissipative order [35], Importantly, lim T →0 ωIm[σ] = ρ/µ, irrespective of whether ρ (0) n = 0 or not. The Kramers-Kronig relations require that Re[σ] also has a delta function as ω → 0 with the same weight. Eq. (7) applies equally well to superconductors with a dynamical gauge field, as the conductivity is measured with respect to the total electric field, which relates it to the unscreened retarded Green's function.
If we explicitly break translations weakly, the momentum relaxes slowly with an inverse lifetime Γ and the conductivity becomes The imaginary pole is now proportional only to the superfluid density, though this says nothing about ρ

HOLOGRAPHIC QUANTUM CRITICAL SUPERFLUIDS
Holography relates the low energy dynamics of a finite temperature strongly interacting gauge theory with a large number of colors in d + 1 spacetime dimensions to the dynamics of a classical gravitational system in d + 2 dimensions with a black hole [36,37]. While explicit examples are known from string theory which fix the action of the gravitational theory, applied holography posits that a consistent set of a small number of fields, such as scalars and U (1) gauge fields, coupled to gravity in (d + 2) anti-de Sitter spacetime is able to capture the universal low energy dynamics of a large number of strongly interacting quantum systems near a quantum critical point or phase [38].
In particular, these quantum critical theories should be characterized by the dependence of correlation functions on certain universal exponents, for instance the dynamical critical exponent, z, the hyperscaling violation parameter θ, and the spatial dimension d. Holographically, these exponents are captured by an extremal (zero temperature) horizon of the form [39,40] where the horizon is at r → ∞ when z ≥ 1. The radial coordinate r functions as a renormalization scale so that under scale transformations, A very general gravitational model which can lead to these extremal solutions is the following, [19], When |η| = 0, the U (1) symmetry is broken and the dual theory can be thought of as a superfluid [15,16]. The field φ is a neutral scalar called the dilaton which has a source on the boundary φ s . The fields ψ I are chosen to have linear dependence on the spatial dimensions, ψ i = mx j δ ij so that when Y = 0, they explicitly break translation but not rotation invariance [41]. The gauge field is chosen only to have a background time component whose value at the boundary of AdS sets the chemical potential, µ, which sources a charge density, ρ. We have set 16πG = 1. See Appendix B for further details.
The solutions (9) are found for potentials which behave in the IR as The gauge field and translation breaking scalars can be engineered to be marginal or irrelevant deformations of the IR critical phase, [40,42]. We will be concerned with phases where the charged scalar is irrelevant in the IR, taking the asymptotic value η 0 [19]. This implies that the scaling exponents are the same in the superfluid as in the normal phase, so that many of the scaling properties at low temperature are inherited from the normal phase.

NORMAL DENSITIES IN HOLOGRAPHIC SUPERFLUIDS
To find the normal density, we perturb our system by turning on a spatially homogeneous infinitesimal external electric field in the x-direction, E x e −iωt , sourcing both an electric and a momentum current (see Appendix C). As ω → 0, the equation for the momentum current where δα x is the gauge-invariant electric field, δα x = E x /(iω). This response requires that µδu x = δα x as we argued earlier.
In the companion paper [43], we explore transport in the superfluid phases of the holographic model (11) for general potentials in greater detail. Here, for illustrative purposes, we present an explicit example in d = 2 that leads to ρ (0) n = 0 and one that leads to ρ (0) n = 0, including when translations are broken. Specifically, we use the model of [19] with Upon varying the dilaton source, this model has two IR phases characterized by critical exponents, In the first case, we first need to redefine r → r 1/z before sending z → +∞ in (9). The IR behavior of φ in the two phases is φ = ± √ 3 ln (r). In the first case, Z(φ) diverges and leads to a finite electric flux, ρ in , from the extremal horizon, suggesting a "fractionalization" of charged degrees of freedom into a subset confined in the condensate and subset deconfined in the thermal bath hidden by the horizon [44,45]. In the second case, Z(φ) → 0 causing the flux to vanish in the IR and all charged degrees of freedom are confined into the charged condensate in a "cohesive" phase.
These two cases can also be distinguished by the vanishing of ρ in are not immediately related. The first is a quantity defined in the two-fluid hydrodynamic model while the second is a microscopic measurement of the uncondensed degrees of freedom. This is analogous to BEC superconductivity where not all electrons condense into Cooper pairs, yet ρ n → 0 [46]. In fact, in [43], we discuss pure Lifshitz superfluid solutions [17,18] in which ρ After solving for the bulk α x (r), we combine (7) with the knowledge of the total background charge density ρ = ρ s + ρ n to extract both ρ n and ρ s . Our numerical results are shown in Fig.1. In [43], we show analytically that where ρ (0) n depends on UV parameters, for instance, the source, φ s , and c IR ≡ L t /L x r 1−z h ∼ T 1−1/z is the lightcone velocity in the IR. The ... indicate terms from more irrelevant deformations of the IR geometry. Interestingly, the leading order temperature dependencies of the normal density behave as power-laws with exponents determined by the underlying IR phase, characteristic of quantum critical systems. This is in contrast to BCS superconductivity, in which it is found that ρ n is exponentially suppressed [21]. On the other hand, in 4 He, the normal (mass) density is controlled by phonons so that ρ n = (sT ) where the coefficient is the phonon speed of sound, c p [47]. This is identical to (16), trading c IR → c p and taking the limit c p 1.
In [43], we find that fractionalized phases with ρ n depends on the competition between two terms proportional to sT and c 2 IR , respectively, [43]. If c 2 IR dominates at low T , then ρ (0) n = 0. Otherwise, ρ (0) n = 0 and to leading order ρ n is given by (16). This result is consistent with the relativistic superfluid effective field theory [24], but is also true for z = 1. Thus, for the quantum critical superfluids considered here, cohesive: Generically, many irrelevant deformations of the criticial IR geometry compete to drive the system toward the UV. In particular, while a universal deformation proportional to sT always exists, dangerously irrelevant operators may control the temperature dependence of thermodynamic or transport observables [40,48,49]. It is then remarkable that the criteria in (17) leads to the universal temperature dependence (16) for cohesive phases.
As a final illustration that ρ (0) n = 0 is a signature of criticality rather than, for instance,  The curves are the real part of (8).
disorder, we explicitly break translations in (11), with Y (φ) = exp ∓φ/ √ 3 , where the minus sign is for fractionalized phases and the plus for cohesive phases. This choice ensures that translation breaking is sufficiently irrelevant to not destabilize the IR geometry. We omit the detailed accounting of gauge invariant fluctuations which can be found, for instance, in [50]. Due to the introduction of broken translations, we confirm lim ω→0 ωIm[σ] = ρ s /µ (see also [51][52][53]) as in (8)

LOW TEMPERATURE BEHAVIOR OF HYDRODYNAMIC MODES
Eq. (16) has interesting consequences on the spectrum of hydrodynamic modes at low temperatures. The superfluid second sound velocity is given by [10,12] Using (16) and s ∼ T (d−θ)/z , we find c 2 2 = zc 2 IR /(d−θ). This is the generalization of Landau's conjecture [54] to critical IR geometries. For fractionalized phases, on the other hand, we find c 2 2 ∼ sT , which decays parametrically faster with temperature than c IR when (17) holds. In both cases, the superfluid sound velocity vanishes at T = 0. This is in marked contrast to the relativistic case z = 1 and the superfluid effective field theory [24], which lead to a non-vanishing T = 0 superfluid velocity. We expect the Goldstone mode should interpolate to a dispersion relation ω ∼ k z in the limit T k. It would be interesting to work this out in our model.
Fourth sound is defined as the sound mode which propagates when the normal velocity vanishes [54], given by In the second equality, we have used the low temperature behavior, ρ ∼ µ d . Thus, fourth sound provides a direct measure of whether ρ (0) n = 0, since then c 2 4 = 1/d. This result explains some observations reported in previous literature, [10,55]. In dirty superfluids with broken translations, only fourth sound survives. In particular, (19) matches the expressions in [56].

DISCUSSION
In this Letter, we have shown that a non-vanishing ρ  n , perhaps in cold atom experiments, which we expect would be a generic feature of quantum criticality. n does and does not vanish. Together, these observations give further evidence that a transition between two types of quantum critical phases may explain the phenomenology in the overdoped cuprates, see also [60][61][62].
As a final remark, we observe that in a Lifshitz quantum critical fractionalized phase, our result (15) implies ρ s ρ (0) s + #T 1−θ/z . Setting θ = 0, the superfluid density displays a universal T -linear scaling for all z > 1. A similar observation was reported by recent experiments in overdoped La 2−x Sr x CuO 4 [25]. For z = 2, the heat capacity will also receive a T -linear contribution. The value z = 2 has appeared previously in theoretical models of high T c superconductors, see e.g. [63][64][65].
Acknowledgments. We would like to thank Nigel Hussey, Catherine Pepin, and Steve Kivelson for useful discussions. In addition, we would like to thank Tomas Andrade and Richard Davison for initial collaboration at an early stage of this project. We would also like to thank Luca Delacrétaz for many discussions on superfluid hydrodynamics. We are grateful to Richard Davison, Sean Hartnoll, Chris Herzog and Jan Zaanen for helpful comments on a previous version of this manuscript. We would especially like to thank Jan Zaanen who first brought the results of [25] to our attention at the Aspen Center for Physics, where this work was initiated and which is supported by National Science Foundation grant PHY-1607611.

This work was supported by the European Research Council (ERC) under the European
Union's Horizon 2020 research and innovation programme (grant agreement No.758759).

Appendix A: Details of the hydrodynamics
Here, we go into more detail about the linearized hydrodynamic fluctuations used to derive (6), keeping the discussion self-contained.
As pointed out in the main text, the system is described by the following equations The first two equations express the local conservation laws: the first is the conservation of the fluid stress tensor and the second is the conservation of the U (1) symmetry current. The last two equations are required by gauge invariance and thermodynamics: the third equation Hydrodynamics states that these equations can be solved in terms of a derivative expansion of local thermodynamic variables and the fluid velocity. At non-dissipative order, we may write There are no contributions from F µν since these are first order in derivatives. Here we have defined the superfluid velocity v µ = 1 µ ∂ µ ϕ and the Josephson equation becomes u µ v µ = −1. This form of the current and stress tensor is frame invariant. The total charge density is ρ = ρ n + ρ s and the normal energy density, n , and pressure, P , satisfy the Smarr relation and Gibbs relations, Note that the true energy density, ≡ T 00 = n + µρ s .
We now look at fluctuations about equilibrium. Here it is useful to choose a frame, for instance one in which the normal and superfluid components are at rest. We emphasize that this choice does not affect the argument. We write T = T 0 + δT , µ = µ 0 + δµ, u µ = (1, δu i ) µ , The fluctuations in the fluid stress tensor and current become.
Hence, the set of equations (20) now become, Using the Gibbs relation, the second equation can be rewritten Finally, if s → 0 as T → 0, consistency of these equations requires In [21], supposing independence of δu i and δϕ requires ρ n = 0. As we show in the main text, ∂ t (µδu i − ∂ i δϕ) generically vanishes. This follows from consistently considering sources for the thermodynamic variables in the Hamiltonian [34].
An external source deforms the Hamiltonian in linear response as A small constant source applied in the infinite past and suddenly switched off at t = 0, leads to a response in A given by the matrix of static susceptibilities (Fourier transformed), Here ε → 0 + leads to nice analyticity properties. Notably, if the source is a hydrodynamic variable, then χ AB = χ BA . Now, the first law tells us that the Hamiltonian must contain a term Fluctuations in v i due to a source s v,i are obtained via the Hamiltonian deformation, which, at linear order, implies from (23), On the other hand, we must have so in fact In the absence of external sources, s i v = 0 ⇒ v i = u i so that the superfluid and normal velocities are aligned and we have in linear response. We further note that within the context of linear response, the equations (23) also imply the electric conductivity (at non-dissipative order) Since ρ = ρ n + ρ s , the pole in the imaginary frequency can be used to directly find ρ n Next, consider fluctuations with a spatially varying source with momentum k µ = (0,k i ) µ .
Fluctuations in the superfluid velocity are parallel tok. Hence, if we look at transverse fluctuations,∂δϕ does not contribute. In particular, the transverse momentum fluctuations obey δT 0⊥ = (µρ n + sT )δu ⊥ ⇒ χ P ⊥ P ⊥ = µρ n + sT (38) where χ P ⊥ P ⊥ follows from a discussion similar to χvP . So far, we have omitted dissipative terms in our hydrodynamic discussion. They can be found, for instance, in [12,34]. A general result from their inclusion is the Einstein relation which follows from general properties of retarded Green functions for conserved quantities, see for instance the discussion below Eq. (4.21) of [34]. Here, η is the shear viscosity which relaxes gradients in the fluid velocity and D ⊥ is the momentum diffusion constant which controls the rate of relaxation of the conserved transverse momentum. For cohesive holographic phases, as we will discuss below, η = s/4π and D ⊥ c 2 IR /(4πT ) giving the relation we found in the main text.
The holographic action in (11) gives the following field equations where G µν = R µν − 1 2 Rg µν is the Einstein tensor. We use the following ansatz for the metric and matter fields consistent with the staticity and rotational symmetry, ds 2 = −D(r)dt 2 + B(r)dr 2 + C(r)(dx 2 + dy 2 ), which gives the equations of motion The scalar ψ I equations are trivially satisfied by our ansatz.
From these equations, we defined the flux from the black hole horizon as where r = r h is the radial location of the black hole horizon. When lim T →0 ρ in ≡ ρ (0) in = 0, we are in a cohesive phase and when ρ (0) in = 0, we are in a fractionalized phase. The equations can be combined to the simple equation which is seen to be conserved for m = 0. The term inside the square brackets gives (−sT ) when evaluated on the horizon.
In the UV (r → 0), we require that the metric functions and matter fields have an expansion, Here, η (1) is a source for the complex scalar. When η (1) = 0 and O 2 = 0, the U (1) symmetry is spontaneously broken. The factor of √ 2 is a normalization convention [15]. Next , φ s is a source term for the dilaton which we can use to vary the condensation temperature T c and drive a phase transition between fractionalized and cohesive phases. Notably, sourcing φ breaks conformal invariance as reflected in the trace of the stress tensor. When the asymptotics are inserted into the conservation equation (44), we derive the Smarr relation, Inserting this expansion into the equations of motion gives Variation of this pressure shows that η (1) = 0 means the superfluid velocity is not sourced.
The that appears here is the true energy density, rather than the normal energy density n . The two are related by = n + µρ s .
Appendix C: Holographic computation of the conductivity and normal/superfluid densities The conductivity is obtained by sourcing a fluctuating spatial component of the gauge field, δa x = a x (r)e −iωt . For m = 0, this requires sourcing δg tx = g tx (r)e −iωt and for m = 0 we must also source δg xr = g xr (r)e −iωt and a fluctuation in ψ x = ψ x + δψ x e −iωt , see for instance [41]. For ease of reading, we will only write the translation invariant equations.
The applied electric field is F xt = iωa (0) x and if we do not apply a temperature gradient g (0) tx = 0. The frequency-dependent conductivity is, following the holographic renormalization procedure [15,66], We are interested in extracting the normal and superfluid charge densities, which can be read off from the low frequency behavior of the ac conductivity through (36) and (37). As we now explain, they can be computed more simply by solving the ω = 0 limit of (48).
This equation has two independent solutions, one regular at the horizon and another which is singular there, given by the Wronskian. At low frequencies, a matching argument shows that the singular solution does not contribute to the imaginary part of the conductivity [67]: a x (r) = a (reg) x (r) + iωZ(φ(r h )) a (reg) Thus, it is enough to compute a (reg) x to read off the weight of the imaginary pole of the conductivity, which together with the relation ρ = ρ n + ρ s , gives access to both the normal and superfluid densities.
At zero temperature, the fluctuation equations reduce to imply that a (reg) x = a (0) x µ A t , where A t is the background solution for the gauge field. Plugging into (52), this gives lim ω→0 Im[σ(ω)] = ρ µ (53) as expected.
Furthermore, the constraint equation (49) reads Gauge invariance allows us to trade a (0) x ↔ ∂ x ϕ and inspecting (23), we find that this equation requires δu x = ∂ x δϕ/µ at T = 0 which is consistent with our discussion of the static susceptibility matrix. * blaise.gouteraux@polytechnique.edu † eric.mefford@polytechnique.edu