6

Symmetry broken phases

A mass divergence in a cuprate material (measured from quantum oscillations) occurs at the same dopings where superconductivity is most robust against a large magnetic field. The interplay of quantum criticality and superconductivity will be discussed in this chapter. Image taken from [648] with permission.

6.1 Condensed matter systems

The metallic Fermi liquid states with quasiparticles, reviewed in §4.2, are well-known to be unstable to BCS superconductivity in the presence of an arbitrarily weak attractive interaction. This is a consequence of the finite density of quasi-particle states at the Fermi surface, and the consequent logarithmic divergence of the Cooper pair propagator. Moreover, this instability is present even in systems with a bare repulsive interaction: it was argued [491] that the renormalized interaction eventually becomes negative in a high angular momentum channel, leading to superconductivity by the condensation of Cooper pairs with non-zero internal angular momentum, albeit at an exponentially small temperature.

Turning to non-Fermi liquid metallic states, discussed in §4.2.1–4.2.3, the instability to superconductivity is usually present, and often at a reasonably high temperature. All of these theories have critical bosonic excitations which are responsible for the disappearance of quasiparticle excitations at the Fermi surface. The same bosons can also induce a strong attractive interaction between the Fermi surface excitations, leading to Cooper pair formation and the appearance of superconductivity. However, the absence of quasiparticles also implies that the logarithmic divergence of the Cooper pair propagator is not present. Therefore, there is a subtle interplay between the strong critical attractive interaction, which promotes superconductivity, and the absence of quasiparticles, which is detrimental to superconductivity: the reader is referred to the literature [2; 690; 579; 512; 647; 748] for studies examining this interplay in a renormalization group framework. One of these works [647] has studied a T = 0 BKT transition between a superconductor and a critical metal, similar to that found in holography in §6.6.

The case for high temperature superconductivity in a non-quasiparticle metal is best established for the case of the spin density wave critical point examined in §4.2.2. Here the attractive interaction induced by the boson fluctuations leads to d-wave superconductivity for Fermi surfaces with the topology of those found in the cuprates. Sign-problem-free quantum Monte Carlo simulations find d-wave superconductivity [85; 680; 523; 747] at temperatures in reasonable agreement with those predicted by the quantum critical theory [3; 748].

The quantum critical metal with nematic fluctuations, studied in §4.2.1, is unstable to pairing in all spin-singlet even parity channels [579; 512; 647]. This insensitivity to angular momenta arises from the fact that the nematic boson is near zero lattice momentum, and so only couples electrons independently on small antipodal patches of the Fermi surface. In practice, other UV features of the model will select a particular angular momentum as dominant. We have already mentioned above the strong superconducting instability seen in Monte Carlo studies of a closely related quantum critical point [513].

Finally, the critical metals with emergent gauge fields in §4.2.3 are also usually unstable to superconductivity via a route similar to that for the nematic fluctuations. With non-Abelian gauge fields, there is always a channel with attractive interactions, and this has been discussed in the context of color superconductivity in the quark-gluon plasma [697]. With a U(1) gauge field, superconductivity appears if there are charged excitations with opposite gauge charges (as was the case with the model of Figure 4.3). The single exception is the case with a U(1) gauge field in which all the fermions carry the same gauge charge: now the singular interaction between antipodal fermions is repulsive, and a non-superconducting critical state can be stable [579].

6.2 The Breitenlohner-Freedman bound and IR instabilities

The holographic IR scaling geometries that we have discussed have a built-in mechanism for instability towards ordered phases. In the best studied case of AdS spacetimes, this is called the Breitenlohner-Freedman bound [112; 580; 487]. A field becomes unstable in the IR scaling geometry if its scaling dimension ΔIR becomes complex. For minimally coupled scalar fields with dimension given by (2.26), this occurs if1

We had noted in §1.6.3 above that a negative mass squared does not in itself lead to an instability of these geometries. That is because the negative curvature of the background effectively acts like a box that cuts off long wavelength instabilities. However, the criterion (6.1) shows that the mass squared cannot be too negative. Note that the Breitenlohner-Freedman bound is a bound on the bulk mass squared, and is different from the unitarity bound on the scaling dimension (also discussed in §1.6.3, the unitarity bound is the lower limit of ΔIR for which alternate quantization is allowed). In cases with hyperscaling violation, or with a logarithmically running scalar field that violates true scale invariance, the criterion for instability can be more complicated. That is because the correlation functions of fields need not have a simple scaling form in the IR regime. Some issues arising in these cases are discussed in [154]. In general, the most robust indication of an instability will be from the behavior of correlations functions in the leading far IR limit ω → 0. A complex scaling exponent in that limit indicates an instability.

If the instability (6.1) were to occur in the near-boundary region of the spacetime, it would indicate a sickness of the underlying theory. However, if the instability occurs in the interior of the spacetime, it can be resolved by condensation of the unstable mode. The backreaction of the condensate then alters the interior geometry, self-consistently removing the instability. We shall discuss some examples shortly. There are various holographic mechanisms that can lead to an IR mass that is unstable due to (6.1) while the UV mass is stable. Indeed, understanding the different such mechanisms amounts to understanding the different types of ordering instability that can arise in holography. The canonical mechanism is that of a charged scalar field. The minimal Lagrangian for such a field is

Here q is the charge of the field. The effective mass squared of the field gets a negative contribution from the coupling to a background Maxwell scalar potential At, so that [320]

6.1), verified explicitly in [356; 325; 192].

The physics at work behind the imaginary scaling dimension for a charged scalar field is the same as that discussed for charged fermions in §4.6.3 above. That is to say, the competition between m and q in (6.3) is between electromagnetic screening and gravitational anti-screening. If the electromagnetic term wins out, then pair production in the near-horizon geometry will discharge the event horizon. See Figure 4.7 above. The endpoint of the instability will be a charged condensate in the near horizon geometry, illustrated in Figure 6.1.

Figure 6.1
The zero temperature endpoint of the instability: a charged bosonic condensate in the spacetime. Figure taken with permission from [351].

The charged condensate ‘hair’ on the black hole is held in by the gravitational potential of the asymptotically anti-de Sitter spacetime and is pushed away from the black hole by any remaining charge behind the horizon (or, strictly at T = 0, by the spatial geometry collapsing to zero size in the interior). This is the manner in which such backgrounds evade the ‘no hair’ theorems of conventional general relativity. Usually, one expects any matter to either fall into the black hole or radiate out to infinity, but both avenues of escape are closed off here. The resulting solutions – black holes with a hovering charged condensate outside the horizon – are of course closely analogous to the electron stars depicted in Figure 4.8 above. There is no need to take a WKB limit with bosons in order to obtain classical soltuions.

There are two important differences between bosonic and fermionic instabilities. These are both due to the fact that bosons can macroscopically occupy quantum states. Firstly, the bosonic pair production instability can be seen at the level of solutions to the classical equations of motion following from (6.2). See the discussion below and in [265]. Secondly, the condensate will have a definite phase and hence spontaneously break the electromagnetic symmetry [320; 355; 356]. Such spontaneous symmetry breaking is the main topic of this section.

The discussion above pertains to T = 0. That is where there is a true IR fixed point that can be discussed without reference to the UV completion of the geometry. As temperature is turned on, the horizon appears as an IR cutoff on the spacetime. As temperature is increased, more and more of the interior scaling regime is swallowed up by the black hole horizon, and eventually most of the background geometry is given by the asymptotic regime where the negative Maxwell contribution to (6.3) is small. The scalar field will therefore be stable at such high temperatures. It follows that whenever (6.1) is satisfied, so that the T = 0 interior scaling geometry is unstable, there exists a critical temperature Tc above which symmetry is restored. Increasing the temperature effectively makes the box in which the scalar propagates narrower in the radial direction, ultimately cutting off the long wavelength instability entirely. This restoration of symmetry with temperature is of course what one would expect to find on general grounds.

Symmetry can also be restored remaining at T = 0 by tuning sources in the boundary field theory, such as magnetic fields. This leads to a very interesting class of infinite order quantum phase transitions that will be discussed in §6.6 below.

Our focus in the following will be on instabilities of low energy compressible phases, of the sort outlined above. A few qualifications are in order. Firstly, there can also be instabilities for which the unstable mode is not localized in the near horizon region [260]. These are bosonic analogues of Fermi surfaces located away from the IR geometry, discussed in §4.5.3. While such instabilities certainly break the symmetry, they are not generated by the universal low energy dynamics and are probably best understood semi-holographically, as in §4.5.3. Secondly, continuous phase transitions triggered by the zero mode instability can sometimes be pre-empted by first order transitions to the symmetry broken phase [278]. We will discuss the computation of the relevant free energy below, but will mostly focus on the case of continuous phase transitions. Thirdly, neutral fields can also condense by violating the IR Breitenlohner-Freedman bound but not the UV bound [356]. This leads to a phase transition that is not necessarily an ordering transition. A proper physical understanding of these transitions has not been achieved at the time of writing. It is possible that they are simply large N ‘artifacts’, in which a thermal vacuum expectation value changes in a non-analytic way at some temperature Tc. If decorated with additional indices or with a 2 symmetry, such neutral fields can act as proxies for magnetization transitions [436; 120].

6.3 Holographic superconductivity

6.3.1 The phase transition

The first step is to diagnose the instability of the normal, symmetric phase. Following the above discussion we start with a zero temperature perspective and explain why a complex imaginary scaling dimension Δ in the IR scaling geometry leads to an instability. An instability is found by showing that there is a pole in the upper half complex frequency plane of the retarded Green’s function of the charged scalar operator 𝒪 in the boundary field theory, dual to the charged field ϕ with action (6.2) in the bulk. Such poles are disallowed by causality and lead to exponentially growing perturbations upon taking the Fourier transform. The low frequency, zero temperature retarded Green’s function of 𝒪 can be computed using the same matching procedure described in §4.4.1 and §4.5.2. Thus, as in (4.75) and (4.108), we have

Recall again that Deff. = 1 for the case of AdS2 × d. Here – unlike in the earlier sections – we are interested in zero momentum, k = 0 modes, and hence the above expression holds also for IR scaling geometries with z < ∞. With a complex dimension ΔIR, the exponent ν is pure imaginary. For scalar fields, it can be shown that the Green’s function (4.75) then has an infinite number of poles in the upper half complex frequency plane, accumulating at ω = 0 [265; 433]. This is the superconducting instability.

For example, for the minimal charged scalar field (6.2) in the extremal AdS2 × d background (4.37), one has

4.70) above. It is analogous to the exponent (4.109) found earlier for charged fermions. From (6.5) we see that the IR geometry becomes unstable whenever the charge of the boson is sufficiently large compared to the mass

4.133) for fermionic pair production in the IR spacetime. Similar inequalities relating the charge and mass of the field will exist for more general IR scaling geometries. The key point is to know the effective mass (6.3) of the charged field in the IR, as well the condition (6.1) for instability.

Zero temperature is typically deep in the regime where the normal phase is unstable. We have discussed in §6.2 how increasing the temperature works to stabilize the solution. At the critical temperature, the unstable mode becomes a zero mode. The critical temperature Tc itself is therefore found by looking for a normalizable solution to the scalar equation of motion with ω = k = 0. In a general background of the form (4.44), the linearized zero frequency and momentum equation for the scalar is, from the action (6.2),

A normalizable mode implies that the field is not sourced at the asymptotic boundary, i.e. ϕ(0) = 0 in (1.28), and is regular at the event horizon, so that ϕ → const. on the horizon as in §4.4.2 above. The equation is to be solved through the entire spacetime, not just in the IR region. Equation (6.7) has the form of a Schrödinger equation. The zero mode occurs when there is a zero energy bound state of the Schrödinger equation. The critical temperature Tc is the highest temperature at which such a bound state appears. To find the bound state of the Schrödinger equation, one must typically use numerics. Shooting or spectral methods have both been widely used.

Because the zero temperature instability is localized in the scaling regime, one expects that the normal phase will become stable once the nonzero temperature horizon has ‘eaten up’ the scaling regime. In the simplest holographic phases, obtained by doping a strong interacting CFT as described in Chapter 4 above, this will imply that Tc ∼ μ is set by the chemical potential. For the example of a charged scalar field (6.2) in the Reissner-Nordström-AdS background of Einstein-Maxwell theory, discussed in §4.3.1 above, Tc is a function of the UV scaling dimension Δ and charge γq of the scalar field [192]. The dependence is shown in Figure 6.2.

Figure 6.2
Critical temperature as a function of UV scaling dimension Δ and charge γq of the operator that condenses. Contours show values of γTc. Figure adapted with permission from [192].

The boundary of the unstable region in this plot is precisely described by the curve (6.6), as we should expect. A second feature of the plot is that as the UV unitarity bound is approached, the critical temperature diverges. See [192] for a possible interpretation of this fact.

When the critical temperature is low, Tc μ, then the dynamics of the instability is fully captured by the emergent IR scaling theory. In particular, this means that the low energy (ω μ) spectral weight of the order parameter at temperatures Tc < T μ is sensitive the IR scaling dynamics and can be described using the matching procedures of §4.4 so that

4.122) above.

As TTc from above it can be shown that a quasinormal mode of the system moves upwards in the complex frequency plane, towards ω = 0 [30; 630]. Close to the critical temperature this leads to a pole in the retarded Green’s function

where a is complex, such that the pole is in the lower half complex frequency plane for T > Tc, while b and c are real. The divergence as TTc is the generic behavior expected from time-dependent Landau-Ginzburg theory, with dissipation of the order parameter.

6.3.2 The condensed phase

So far we have characterized the onset of the instability from the point of view of the normal phase. We can now turn to the ordered dynamics at temperatures T < Tc.

Once the scalar field has condensed, a new background geometry must be found in which ϕ is nonzero. This requires solving the coupled equations of motion following from adding the charged scalar (6.2) to the Einstein-Maxwell-dilaton (or more general) theory (4.43) that described the normal state. This task is not substantially different to the construction of the Einstein-Maxwell dilaton backgrounds of §4.3.2 or the electron star of §4.6.3. A characteristic of the spontaneous symmetry breaking case is that a solution with the appropriate boundary conditions for the scalar field (an expectation value but no source) will only exist below the critical temperature.

As always, the universal low energy and low temperature dissipative dynamics of the symmetry broken phase is captured by the zero temperature IR geometry. The bulk physics is very similar to that of the electron star discussed in §4.6.3. The only real difference is that the charge-carrying field in the bulk is a boson rather than a fluid of fermions. In particular, as with the electron star in (4.137), a common scenario is that at zero temperature all of the charge is outside of the horizon, enabling an emergent Lifshitz scaling

The simplest instance of this behavior may be Einstein-Maxwell-charged scalar theory, where the scalar has m2 > 0 and no potential [407; 324], but it has also been found more widely in e.g. [351; 369; 308]. The value of z depends on the details of the theory. An interesting phenomenon that appears to be fairly common is that the charge density operator can become irrelevant in the new IR scaling theory. Irrelevance of the charge density operator leads to an emergent IR Lorentz invariance with z = 1 [329; 288; 327; 306]. This means that the bulk scalar potential vanishes with a faster power of r towards the interior than shown in (6.10), i.e.

This behavior is less exotic than the anomalous dimension of a conserved current discussed in §4.3.4, because now the U(1) symmetry is spontaneously broken at the IR fixed point. The Maxwell field is Higgsed in the near horizon geometry by the charged scalar condensate. Equation (6.11) is just the behavior of a massive vector field in anti-de Sitter spacetime.

More generally if the charged scalar runs in the IR – playing a role similar to the logarithmically running dilaton in (4.50) above – rather than tending to a constant as in (6.10), hyperscaling violation is also possible [308; 306]. This more general class of behaviors allows for zero temperature transitions between solutions will all, some or none of the charge in the scalar field condensate outside the black hole [7]. Such transitions are analogous to the fermionic fractionalization transitions discussed in §4.6.3. A particular feature of bosonic fractionalization transitions in large N holographic models is that the fractionalized phase cannot break the symmetry spontaneously. That is, if all the charge is behind the horizon, the symmetry is unbroken.

Emergent scaling geometries such as (6.10) mean, as we have explained in §2.3.1, that in general the specific heat at low temperatures in the symmetry broken phase goes like

We have added the subscript S to distinguish the superconducting values of the exponents θS and zS from their values in the unstable normal state. We will refer to those as θN and zN in the next few sentences. The power law behavior (6.12) shows that, contrary to a conventional s-wave superconductor, the ordered state is not gapped. The persistence of a neutral T = 0 ‘horizon’ in the IR shows that while the charge has become cohesive, neutral degrees of freedom remain deconfined. This may connect with some of the topologically ordered phases discussed in §6.1. The neutral degrees of freedom can be gapped by triggering a confinement transition, as in §2.4. However, gapping the deconfined critical excitations in the normal state will also remove the IR superconducting instability. The interplay of holographic superconductivity and confinement is studied in [605; 415].

While general results do not exist at the time of writing, in various examples with Lifshitz scaling in the normal and superconducting states (i.e. θS = θN = 0), it was found that zS < zN [369]. This implies that while a gap has not formed, the ordered state does have fewer low energy degrees of freedom than the unstable normal state. This is broadly to be expected from the perspective of ‘entropy balance’ as follows. At the critical temperature Tc of a second order phase transition the free energy and entropy of the two phases are equal fS(Tc) = fN(Tc), sS(Tc) = sN(Tc), but the specific heat jumps so that cN(Tc) < cS(Tc). The sign of the jump is fixed by the fact that the free energy of the superconducting state must be higher just below Tc. Using these facts and integrating up the relation c = T∂s/∂T in both phases

We have allowed for a zero temperature entropy density in the normal phase. This is the entropy balance equation. The fact that cS > cN close to the upper limit of these integrals, i.e. the ordered state has more degrees of freedom at energies just below the transition temperature, must then be balanced by the normal state having more low energy degrees of freedom. This balance was explored in various holographic models in [369], which also discusses experimental realizations of the balance in unconventional superconductors.

So far we have discussed the universal low temperature dynamics of the ordered phase. A further universal regime emerges at temperatures just below Tc. Here, on general grounds, the system should be described by Landau-Ginzburg theory. Indeed this is the case, and the only input of the full bulk solution is to fix the various phenomenological parameters in the Landau-Ginzburg action [552; 389; 630]. These coefficients then determine various properties of the state, such as the response to an external magnetic field. At leading order in large N there are no quantum corrections to these mean-field results for the phase transition, although unconventional exponents are possible [278; 279], even in low space dimension. In §6.4.3 below we show how bulk quantum fluctuations, suppressed by large N, lead to algebraic long range order in two boundary space dimensions.

There is a large literature characterizing the non-universal regimes at intermediate temperatures in the ordered phase.2 By numerically constructing the backgrounds one can obtain thermodynamic quantities and the expectation value of the charged operator as a function of temperature. One can study the Meissner effect and the destruction of superconductivity by a magnetic field. The main conclusion of these works is that, away from the gapless low temperature limit (6.12) and away from the unconventional nature of the superconducting instability as revealed in e.g. (6.8), the basic phenomenology at intermediate temperatures is just that of a conventional s-wave superconductor [733]. See e.g. [356; 402; 593]. By constructing backgrounds with inhomogeneous sources, which requires solving PDEs, one can furthermore recover the conventional physics of Josephson junctions [412] and vortices [19; 583; 214; 471; 432; 209]. Unconventional and universal low energy physics does emerge in the zero temperature limit of holographic vortices. The vortices interact with the emergent gapless degrees of freedom described by the interior scaling geometry (6.10), appearing as a defect in the low energy scale invariant theory. This interaction leads to a ‘rigorous’ computation of drag forces on a vortex [209]: a feat which may prove useful in our discussion of superfluid turbulence in § 7.4.

6.4 Response functions in the ordered phase

6.4.1 Conductivity

The most characteristic response of a superconductor is, of course, the infinite dc conductivity. Formally speaking, this arises in the same way as the infinite conductivities due to translational invariance that we encountered in formulae such as (5.43). As explained in §5.4.4, whenever an exactly conserved operator overlaps with the total electrical current operator 5.56)

. This operator can be called the total supercurrent or total superfluid velocity. 3

As with the weight of the translational invariance delta function in (5.43), the strength of a superconducting delta function is determined by thermodynamic susceptibilities that appear in hydrodynamics. In the case in which momentum is not exactly conserved, so that the supercurrent is the only infinitely long-lived operator, then from §5.4.4

Here we defined the superfluid density ρs as well as the mass scale m. See e.g. [174] for more details (and a slightly different normalization of ρs). In the case of superconductivity emerging from a compressible phase obtained by doping a CFT, as in Chapter 4 above, the translational and superconducting delta functions combine to give [172]

Here ρn and ρs are the normal and superfluid components of the charge density. These can be defined from the overlap of momentum and supercurrent operators with the total current operator, respectively. The total charge density ρ = ρn + ρs. In the zero temperature limit, therefore, (6.16) simplifies to 𝒟 = ρ/μ. In general, the denominators in (6.16) follow from relativistic superfluid hydrodynamics [392; 90; 172], in an analogous way to how the denominators in (5.43) followed from ordinary relativistic hydrodynamics. In particular, the relation m = μ going from (6.15) to (6.16) is fixed by Lorentz invariance.

To obtain the weight of the delta function in a holographic superconductor it is therefore sufficient to compute the thermodynamic susceptibilities above. These are purely static quantities, and so they can be obtained without needing to solve any nonzero frequency equations. Furthermore, despite controlling the response at zero frequency, the susceptibilities are not IR dissipative variables but will instead depend on the spectrum of the theory at all energies. Thus the zero frequency computation to be done will depend on the entire bulk geometry, not just the near horizon region. To obtain 𝒟 using the definition (5.57) above one must solve the perturbation equation for the vector potential at ω = k = 0, that is δAx = ax(r). This perturbation will mix with metric perturbations δgtx. We saw this, for instance, in (5.145) above. Consider the case of the Einstein-Maxwell-charged scalar model. After solving for δgtx(r),4 and eliminating it from the equations, the resulting equation for ax(r) is [356]

4.44), and the charged background scalar field profile is given by ϕ(r). Note there are two contributions to the final ‘mass’ term for ax. One comes from the background charge density – proportional to p2 – and the other comes from the background condensate – proportional to ϕ2. Roughly speaking, these lead to the two contributions in the weight of the delta function in (6.16). To calculate the weight 𝒟 we most solve (6.17), imposing regularity on the horizon. In practice, ax will have two behaviors towards the horizon as rr+ and we must keep the less singular one. With the solution ax(r) at hand, the weight of the delta function is then given by

For the above Einstein-Maxwell model, z = 1. Here we have used the holographic dictionary (3.48b) and (3.49b) and also the formula (5.57) for the ‘Drude’ weight with χJxJx = 0.

The weight of the delta function has been computed numerically in many models of holographic superconductivity. Of particular interest are cases in which translational symmetry has been broken. In these cases the delta function is purely due to superconductivity and the expression (6.15) holds. Early papers that computed the superfluid density in non-translationally invariant setups are [409; 774; 528]. While one expects that the superfluid density is associated to the symmetry-breaking charge that is condensed outside of the event horizon, a quantitive relation between 𝒟 and the total charge outside the horizon has not been established at the time of writing. With weak momentum relaxation, the superconducting delta function sits on top of a Drude peak of the type described in Chapter 5. More generally, with weak or strong momentum relaxation, one expects that in the low frequency limit the divergent zero frequency conductivity is accompanied by a constant ‘incoherent’ conductivity, analogous to σQ that appears in (5.43) and (5.54). The temperature dependence of this ‘normal’ component will then depend on the properties of the near horizon scaling geometry as well as the relevance or irrelevance of translational symmetry breaking.

A second feature of conventional BCS superconductors is a gap in the optical conductivity σ(ω). This gap is the energy required to excite a conducting particle-hole pair from the Cooper pair condensate. Thus, while a conventional superconductor has infinite conductivity at ω = 0, at zero temperature it has vanishing conductivity for small but nonzero ω. At nonzero temperature, σreg e−2Δ/T, where Δ is the energy gap for a single particle excitation. The low frequency behavior of the optical conductivity is dissipative and determined by low energy degrees of freedom. Therefore σreg(ω) should depend only on the near horizon scaling geometry (6.10). The absence of a gap suggests that this contribution will be power law in temperature or frequency rather than exponentially suppressed. Indeed the matching arguments of Chapter 4, here applied in the simpler case of k = 0, imply that in general at zero temperature [329; 407; 306]

Recall that the exponent Φ was discussed in §4.3.4. One similarly expects the temperature dependence σreg ∼ T(d−2−θ+2Φ)/z as in (4.66). The appendix of [178] is useful for relating various different scaling exponents that have been used to describe the conductivity in scaling geometries.

Equation (6.19) shows that generally there is a soft gap in the optical conductivity. On general grounds one expects a reduction of low energy spectral weight at low and zero temperature in the superconducting phase, relative to the normal phase. This is because of the Ferrel-Glover-Tinkham sum rule:

Here the normal and superconducting optical conductivities can be at any temperature (in the normal and superconducting phases, respectively). The sum rule can be explicitly verified in particular calculations, for a general holographic discussion see [332]. It is clear in (6.20) that the presence of the superfluid density in 𝒟 means the nonzero frequency spectral weight is depleted in the superconducting state. Of course, at low but nonzero temperatures, in addition to the behavior (6.19) there may also be a Drude peak at low frequencies if translations are weakly broken. These behaviors are illustrated in Figure 6.3 for a holographic model of superconductivity with broken translation invariance.

Figure 6.3
Optical conductivity in a holographic superconductor with broken translation invariance. Dashed line is above Tc, topmost solid line is at Tc, subsequent lines are at progressively lower temperatures. In the left plot, translations are weakly broken, and the sharp Drude peak in the normal state persists into the superconducting state. In the right plot, translations are strongly broken. The weak Drude peak in the normal state eventually disappears at low temperatures in the superconducting state. All plots exhibit the loss of spectral weight into the superconducting delta function as temperature is lowered. Figures taken with permission from [474].

6.4.2 Superfluid hydrodynamics

In the ordered phase, the new dynamical ingredient is the superfluid velocity or supercurrent, which is the gradient of the phase of the Goldstone boson ξμ = μϕ. A homogeneous supercurrent flows without decay and is responsible for the divergent conductivity discussed in the previous subsection. The conserved supercurrent is a new thermodynamic variable and can be taken to be large, leading to new thermodynamic states. In a Lorentz invariant theory the first law becomes, see e.g. [385],

Here ρs is the superfluid charge density, that appeared in the previous §6.4.1.

Bulk gravitational solutions with a nonlinear supercurrent have been constructed in the probe limit in [81; 385; 50; 49] and with full backreaction on the metric in [706]. This last paper derives the equations of non-dissipative superfluid hydrodynamics from gravity. Because the solution involves a flow, it is stationary but not static. This requires a more general form of the background fields than we have considered so far

Here ua is a timelike boost velocity and na is spacelike vector. A gauge choice has been made to put the supercurrent into the Maxwell field A rather than as an x-dependent condensate ϕ. The various radial functions we have introduced above are not all independent. See [706] for further details.

The papers in the previous paragraph all show that if the supercurrent becomes large, then eventually the charged condensate is driven to zero and the normal state is recovered. Energetic computations in those papers furthermore suggest that sometimes the continuous transition is pre-empted by a first order transition to the normal state. However it is subtle to compare the free energies of the normal state and the superfluid state with a supercurrent because the supercurrent is a thermodynamic variable that does not exist in the normal state. It has been noted in [29] that before the condensate is driven to zero by the supercurrent, the Goldstone boson mode develops an instability at a nonzero wavevector k > 0. The endpoint of this instability is not known at the time of writing. It may restore the normal state or it may lead to a spatially modulated superfluid state. The second option would add a further example to spatially modulated phases discussed in §6.5.2 below.

A finite non-dissipative supercurrent can furthermore flow at zero temperature. In this case, the destruction of superfluidity caused by increasing the supercurrent causes a quantum phase transition [51].

In a strongly coupled, non-quasiparticle, superfluid state, the appropriate framework to understand transport is superfluid hydrodynamics. This amounts to adding an additional hydrodynamic field, the superfluid velocity ∂ϕ, to the various hydrodynamic theories of Chapter 5. The most important consequence of the new hydrodynamic variable is the appearance of a new dispersive mode, second sound. All of the modes can be found and characterized from perturbations of the bulk Einstein equations about the ordered state, much as we have done previously for some of the normal state hydrodynamic modes. In particular, properties such as the second sound speed and parameters characterizing dissipative effects in inhomogeneous superfluid flow can be computed from the bulk. Important papers in this endeavor include [30; 393; 396; 392; 90; 91; 89]. As with other hydrodynamic modes, these excitations appear as quasinormal modes with frequencies that become small at low momenta. Certain non-hydrodynamic quasinormal modes close to the real frequency axis have also been calculated and shown to influence the late time relaxation to equilibrium in a holographic superfluid state [89]: see §7.3.

6.4.3 Destruction of long range order in low dimension

In 1+1 dimensions at zero temperature, or in 2+1 dimensions at nonzero temperature, global continuous symmetries cannot be spontaneously broken. This is known as the Coleman-Mermin-Wagner-Hohenberg theorem. Quantum or thermal fluctuations of the phase of the order parameter lead to infrared divergences that wash out the classical expectation value of the order parameter. There is, however, a remnant of the ordering in the form of ‘algebraic long-range order’. This means that the correlation function of the order parameter has the large distance behavior

In a truly long range ordered state, this correlator would tend to a constant, while in a conventional disordered state it would decay exponentially rather than as a power law.

The ordered holographic phases we are discussing do not exhibit this expected low dimensional behavior at leading order in large N. This is because there are order N2 (say) degrees of freedom, but only a single Goldstone boson. Fluctuation effects of the Goldstone mode will then only show up at subleading order at large N. The upshot is that the decay power K in (6.23) goes like e.g. 1/N2 (cf. [762]). Thus the large N and long distance limits do not commute. This is part of our comment in §1.6.4 that such Goldstone modes are ‘non-geometrized’ low energy degrees of freedom in holographic models. To see the destruction of long range order, one-loop effects need to be considered in the bulk. This can be done using the methods we have already discussed in §3.5.2 and §4.6.2. We now outline the computation, following [46].

The first step is to identify a quasinormal mode in the bulk that corresponds to the Goldstone or ‘second sound’ mode in the superfluid phase. This mode will then give the dominant low energy contribution to the Green’s function for the scalar field, using the formula (3.95). It will be this Green’s function running in a loop that causes the quantum disordering of the phase. The mode is found starting from a rotation of the profile ϕ(r) of the background scalar field

𝜖 is a small number. This just a gauge transformation of the background: ϕ → eiλϕ, with λ constant. However, it is a ‘large’ gauge transformation that doesn’t vanish on the boundary (because λ is constant in r). This means that the mode generated in (6.24) is a normalizable and physical ω = k = 0 excitation. Starting from this solution, one can solve the bulk equations of motion perturbatively in small ω and k. Doing this turns out to be essentially independent of the form of the background solution. One finds that a normalizable solution continues to exist so long as [436; 46]

Here 6.21) above. This final result can be derived directly from superfluid hydrodynamics [385].

The above paragraph establishes the second sound quasinormal mode (actually a normal mode to the order we have worked, dissipation will appear at higher orders in the dispersion relation (6.25)). The algebraic decay (6.23) is now obtained as follows. In the bulk we can calculate

The last equality is a standard result following from Wick contracting. Here ϕ(r) is the classical background profile and θ is the phase. The phase is related to the perturbation (6.24) as δϕ = (r)θ. Using (3.95), then, the singular part of the phase two point function is

The numerical prefactor Θ is obtained in [46]. The prefactor scales like an inverse power of large N. This singular contribution does not depend on r or r′. The one loop correlation function in the exponent of (6.27) is given, using standard methods to turn the sum over Matsubara frequencies into an integral along the real frequency axis, by

and hence, taking (6.27) to the boundary r, r′ → 0, we obtain (6.23) with

6.4.4 Fermions

The spectral functions of fermionic operators computed in §4.5 can be revisited in the ordered background. It was quickly recognized that there are two important new ingredients in the superconducting phase [259; 331]. Firstly, the zero temperature emergent IR scaling geometry (6.10) in the superconducting phase typically has z < ∞. As explained in §4.5.3, low energy fermionic excitations near some nonzero kkF are kinematically unable to decay into such a quantum critical bath and hence will have exponentially long lifetimes. Secondly, there are couplings that can appear generically in the bulk action between the charged fermion and charged scalar field. In the presence of a scalar condensate, certain of these couplings have the effect of gapping out low energy fermionic excitations.

The combination of the two effects outlined in the previous paragraph can then lead to well defined (long lived) fermionic excitations with a small gap in the superconducting phase, emerging from a normal z = ∞ phase in which there were gapless but very short lived fermions, as described in §4.5.2 above. We will now explain this scenario in more detail. The scenario itself is semi-holographic in the sense of §4.5.3. It can be described in terms of conventional field theoretic fermions coupled to a large N quantum critical bath, with the nature of the bath changing between the normal (z = ∞) and superconducting (z < ∞) states.

If the charge of the condensed scalar is twice that of the fermion operator, then the following ‘Majorana’ coupling [259] can be added to the fermion action (4.102)

In fact this coupling only works for even d; for odd d two different bulk Dirac fields are needed. This coupling is the relativistic version of the conventional coupling between Bogoliubov quasiparticles in the superconducting state with the condensate. In fact, we have encountered this coupling already in (4.125), where we defined Γ5 and the charge conjugate fermion Ψc. The difference is that whereas Δ previously was a condensate of a bulk fermion bilinear, ϕ is dual to a single-trace scalar operator in the dual field theory.

A nonzero η5 coupling gaps any low energy fermionic excitations in the superconducting state [259]. This is illustrated in Figure 6.4, which shows the zero temperature spectral density of the fermionic operators as a function of ω and k at different values of η5. This figure also shows the effects of the emergent z = 1 scaling, as described in the figure caption. ‘Effective lightcones’ similar to those seen in Figure 6.4 – i.e. within which fermions are inefficiently scattered by the large N critical bath – also arise for 1 < z < ∞ scaling geometries [360].

Figure 6.4
Zero temperature fermionic spectral densities as a function of ω and k. Top left: Normal state extremal RN-AdS background, exhibiting gapless but strongly scattered fermions. Top right: Zero temperature superconducting state (with z = 1) with η5 = 0. Gapless fermions still exist but are now very weakly scattered within the z = 1 lightcone. Bottom left: Small nonzero η5 in the superconducting state. The fermions are now gapped, but still long lived within the lightcone. Bottom right: Larger nonzero η5. The fermions are both gapped and short lived. Figures taken with permission from [259].

Fermionic spectral functions have been studied in geometries with a charged scalar background that originate from a microscopic string theoretic construction [196; 197]. The effective bulk theories that arise in this case are more complicated, involving several coupled fermionic fields. With the full microscopically determined interactions, all fermions are found to be gapped, some by couplings analogous to (6.32).

Fermion spectral functions can also be computed in backgrounds with a p-wave or d-wave superconducting condensate, to be discussed in §6.5.1 below. In these cases couplings analogous to (6.32) lead to anisotropic gapping of the Fermi surface, due to the anisotropy of the underlying condensate. This leads to versions of ‘Fermi arcs’ [84; 741].

6.5 Beyond charged scalars

The minimally coupled charged scalar field (6.2) is perhaps the simplest model that can lead to a sufficiently negative effective mass squared (6.3) in the interior scaling geometry such that an instability is triggered according to (6.1). In this case, the global U(1) symmetry of the boundary theory is spontaneously broken by condensation of the charged operator dual to the bulk field. This general mechanism can be greatly generalized to allow for other patterns of symmetry breaking. We will first discuss more general instabilities that preserve spatial homogeneity in §6.5.1, while §6.5.2 will discuss the spontaneous breaking of spatial homogeneity. With many possible types of instability, it is natural to expect competition between the different orders to lead to rich phase diagrams. We will not discuss these phase diagrams here but refer the reader to works such as [79; 233; 232; 118; 481]. One also encounters ‘intertwined’ orders such as pair density waves, in which spatial and internal symmetries are broken simultaneously [218; 221; 155; 156; 119].

6.5.1 Homogeneous phases

6.5.1.1 p-wave superconductors from Yang-Mills theory

A rather constrained model is given by gravity coupled to a non-Abelian gauge field in the bulk [321; 326; 658; 80; 33]:

is the field strength of an SU(2) gauge field. That is

𝜖ABC. Take the U(1) subgroup of SU(2) generated by τ3 to be the electromagnetic U(1). The action (6.33) together with the expression for the field strength (6.34) implies that the A1 and A2 components of the gauge potential, the ‘W-bosons’, are charged under this U(1). Therefore, this Einstein-Yang-Mills theory has a similar to structure to the Einstein-Maxwell-charged scalar theory discussed in the previous few sections. In particular, in a background carrying U(1) charge, the W-bosons will acquire a negative mass squared analogous to (6.3). They can therefore be expected to condense below some Tc if the gauge coupling e is large compared to the gravitational coupling κ.

There are several new ingredients with a Yang-Mills condensate relative to the charged scalar case. Firstly, the condensate is a spatial vector, and hence should be called a p-wave (rather than s-wave) superconductor [326]. Secondly, the condensate is the spatial component of gauge potential, which is dual to a conserved current in the boundary field theory, which therefore breaks time reversal invariance. Thirdly, the zero temperature limit of the backgrounds turns out to have qualitative differences with the scalar cases discussed above [80; 78], realizing a hard rather than soft gap towards charged excitations. Let us discuss these in turn.

Generically, the vector condensate means that the condensed phase will be anisotropic. The gauge potential [321; 326] and metric [80; 33] take a form such as

is analogous to the charged scalar. The anisotropy arises because, in this case, the x direction is singled out by the condensed vector. In fact, in d = 2 space dimensions it is also possible to achieve isotropic condensates. This occurs if the U(1) gauge and U(1) spatial rotational symmetries are locked, so that

6.35). The form of the condensate is determined by finding the configuration that minimizes the free energy. In the Einstein-Yang-Mills theory, the anisoptropic phase (6.35) is dominant [326].

The breaking of time reversal is most interesting in the isotropic p + ip case. We explained in §4.7.1 above how in an isotropic system, breaking time reversal allowed a nonzero Hall conductivity σxy. Indeed, such a Hall conductivity arises in symmetry broken holographic p + ip states [658]. No external magnetic field is needed. The Hall conductivity remains finite in the ω = 0 limit.

In the zero temperature limit, the holographic p-wave superconductor exhibits an emergent z = 1 scaling symmetry in the far interior, similarly to the discussion between equations (6.10) and (6.11) above. That is, the charge density operator is irrelevant in the emergent IR scaling theory. However, in contrast to equation (6.11) for the s-wave case, at least for sufficiently large gauge field coupling e, the electrostatic potential now vanishes exponentially fast towards the IR [80; 78]

The technical reason for this different behavior is that in the emergent IR Anti-de Sitter spacetime, the Higgsing of the electric field by a charged vector is stronger than the Higgsing by a charged scalar. This is because the effective mass for the photon involves the contraction 6.37) will lead to an exponentially small amount of electric flux through the horizon at low temperatures. Relatedly, while the emergent z = 1 scaling geometry means that there are gapless neutral degrees of freedom in the system, the regular contribution to the conductivity is exponentially small at low frequencies and temperatures σreg e−Δ/T. That is, the charged sector is gapped. Indeed at zero temperature the regular part of the conductivity satisfies

.

6.5.1.2 Challenges for d-wave superconductors

A d-wave condensate is described by a charged, spin two field in the bulk [134]. While a charged spin one field admits a simple description via Yang-Mills theory, the same is not true for a charged spin two field. The dynamics of spin two fields is highly constrained by the need to reduce the degrees of freedom of a complex, symmetric tensor field φab, which has (d + 2)(d + 3)/2 components, down to the d(d + 3)/2 dimensions of the irreducible representation of the relevant little group, SO(d− 1). If the extra modes are not eliminated by suitable constraints, they tend to lead to pathological dynamics. In the context of applied holography, these issues are discussed in [83; 477]. Those papers include references to the earlier gravitational physics literature.

With a dynamical complex symmetric tensor φab at hand, a d-wave condensate is described by a nonvanishing profile for φxx(r) = −φyy(r) or φxy(r) = φyx(r). Unlike a p-wave condensate, a single component d-wave condensate does not break time reversal invariance. However, time reversal is broken in the presence of a complex superposition of the two profiles above [135].

A theory of a charged spin two field that is consistent on a general background is not known. The current state of the art is the following quadratic action for φab that can be added to Einstein-Maxwell theory [83]

Here φ = φaa and Da = ∇a − iAa. There is a parameter g, the gyromagnetic ratio [477]. This Lagrangian is ghost free and has the correct number of degrees of freedom, but only with a fixed background Einstein geometry, satisfying Rab = −(d + 1)/L2 gab and with g = 1/2. Even in such cases, for large enough values of the electromagnetic field strength, or gradients thereof, the equations of motion are either non-hyperbolic or lead to acausal propagation. Therefore, while the model can capture some of the expected dynamics of d-wave superconductivity close to the transition temperature, it will not be able to access the universal low temperature regimes where backreaction on the metric and often large Maxwell fluxes are important.

It has been emphasized in [477] that a natural way to obtain fully consistent dynamics for a massive, charged spin two degree of freedom is to consider Kaluza-Klein modes in the bulk, discussed in §1.8. In a Kaluza-Klein construction there is an internal manifold that has a U(1) symmetry. A perturbation of the bulk metric with ‘legs’ in the non-compact holographic dimensions (the ‘AdS’ directions) but that depends on the coordinates of the internal manifold will be charged under this U(1) symmetry. It will be a massive, charged spin two field. In the language of §1.8, the difficulty of constructing a theory for just this single excitation is the difficulty of finding a consistent truncation that includes this mode. This single mode instead couples to the infinite tower of Kaluza-Klein modes with increasing mass and charge. That is to say, one must solve the full problem of the inhomogeneities in the internal dimension, rather than restricting to a single Fourier component. While technically challenging, this is possible in principle and will lead to a completely well-behaved holographic background with d-wave condensate(s).

At the time of writing it is not known whether the additional bulk structure that seems to be required for a d-wave condensate is a reflection of interesting field theoretic aspects of such symmetry breaking. The presence of internal manifolds in the bulk is something of an undesirable feature of holographic models, as it implies the emergence of an additional locality in the dual field theory that is presumably an artifact of large N. The extra locality is hidden from sight in consistent truncations. On the other hand, the ability at large N to focus on a single operator 𝒪 without having to worry about the multitrace operators 𝒪2 etc, is also an artifact of large N. So, the need to consider many operators (the Kaluza-Klein modes) in the discussion of d-wave condensates is also, in some sense, more realistic.

Finally, we can note that while several experimentally observed unconventional superconductors are d-wave, the underlying effective low energy dynamics is often s-wave. Specifically, for superconductivity emerging from a spin density wave critical point, mentioned in §6.1, the d-wave condensate arises from pairing between fermions in distinct hot spots. In the effective theory, the different hot spots appear as a ‘flavor’ label for the fermions, as in §4.2.2. Thus, the microscopic orientation dependence of the pairing is simply described by the flavor index structure of an s-wave interaction.

6.5.2 Spontaneous breaking of translation symmetry

Spontaneous breaking of translation symmetry is common in condensed matter physics. This subsection will review holographic models where translation symmetry is spontaneously broken. The underlying mechanism is the same as that described in §6.2 and the dynamics of the transition are again those described in §6.3.1. The important difference with cases we have considered so far is that the modes that satisfy the instability criterion (6.1) have a nonzero spatial momentum, k ≠ 0.

Recall that the instability criterion (6.1) corresponds to an operator with a complex exponent in the IR scaling geometry. In §4.4.1 and elsewhere above we have emphasized a special feature of z = ∞ scaling: the resulting semi-local criticality means that operators typically have k-dependent scaling exponents. This allows, in principle, operators with a range of momenta with k ≠ 0 to acquire complex exponents while the homogeneous k = 0 operators remain stable. This will be the origin of all the inhomogeneous instabilities discussed below. In §4.4.1 we also emphasized that z = ∞ scaling shared an important property of Fermi surface physics: the presence of low energy spectral weight at nonzero momenta. The instabilities we are about to discuss depend on the presence of this spectral weight in z = ∞ IR scaling geometries. In this loose sense they can be considered strongly coupled analogues of the nonzero wavevector instabilities of Fermi surfaces.

6.5.2.1 Helical instabilities

Helical order breaks translation symmetry in a homogeneous way, as we have noted in our discussion of transport in §5.9. A simple holographic model leading to helical order is Einstein-Maxwell theory in d = 3 with a Chern-Simons term of coupling constant α [596]:

We have encountered this theory previously in §4.7.2. We will now see that the symmetric background of the theory has an instability. As we explained in §4.7.2, the Maxwell field itself becomes charged due to the nonlinear Chern-Simons term. Heuristically, this is why the theory can have instabilities of the ‘pair production’ kind that we have discussed above.

To investigate possible instabilities of the Chern-Simons theory, it is instructive to start with five dimensional Maxwell-Chern-Simons theory in flat space. Since the Chern-Simons theory is non-linear, we will need to turn on a background electromagnetic field for the term to have an effect on linearized perturbations. Denote the 5 dimensions with (t, r, x, y, z). Here r will become the bulk radial dimension shortly. Turning on an electric field Frt = E, the linearized equations of motion for the transverse modes

with ifi = 0. Looking for modes with only ω and kx non-zero, we find a pair of circularly polarized modes fy ±ifz with dispersion relation

Clearly, for 0 < ±kx < 4|αE| (with appropriate choice of sign), there is an instability. In particular, the most unstable mode is at |kx| = 2|αE|.

The story is similar in a holographic context, though a little more complicated. The background is the AdS5-RN black hole, which has a bulk electric field with r-dependence, as in (4.30). At low enough temperatures, and for large enough values of |α| > αc there is a continuous phase transition to a phase with spontaneous spatial modulation, analogous to that described in the previous paragraph [596].5 As with superfluid instabilities, these low temperature instabilities correspond to an “effective mass” of the perturbations that is below the bound (6.1) for the near-horizon, zero temperature AdS2 × 3 spacetime. This analysis gives the critical Chern-Simons coupling to be αc = (2κ2/e2)3/2 × 0.2896…. The unstable modes have the same form as those found in the previous paragraph – the gauge field will break translation symmetry in a boundary direction (x, without loss of generality), and the spatial components of the gauge field will be of the form

In the boundary, there will be a similar helical pattern of electric current. The condensates Jy and Jz appear at T < Tc, similarly to in the holographic superconductors above. The critical temperature associated with such instabilities appears to be reduced by external magnetic fields [34]. The condensed phase was constructed numerically in a probe limit in [608]; the backreaction of gravity was accounted for in [220]. The backreaction is tractable without solving PDEs because the helical condensate (6.43) leads to a homogeneous geometry with Bianchi VII symmetry [425]. The zero temperature limit of the fully backreacted solution exhibits an emergent anisotropic IR scaling symmetry with entropy density vanishing like s ∼ T2/3 [236].

Supergravity fixes the value of [333]. As was noted in [596], this happens to be just below the critical value for this spatially modulated instability.

6.5.2.2 Striped order

This section describes the development of “striped” order which breaks translation symmetry in a single spatial direction, in an inhomogeneous fashion. We focus on the endpoint of an instability identified in [219]. This task requires solving coupled nonlinear PDEs and has only been possible numerically. The first such models were of the form [660; 661; 216; 760]

⋯. Like (6.40), this action breaks parity and time-reversal. The important point about this action is that, in a background electric field, the final term gives a derivative coupling between the pseudoscalar ϕ and magnetic fluctuations of the Maxwell potential. This term has a similar structure to the Chern-Simons coupling in (6.40).

The symmetric nonzero density background is the AdS4-RN charged black hole. Below a critical temperature Tc, there is a continuous phase transition to a striped phase. As in the previous subsection, the existence of the instability – supported at nonzero wavevector – can be seen by analyzing the effective mass squared of fluctuations about the zero temperature AdS2 × 2 near horizon geometry. In numerics, one looks for striped black holes with a fixed wavevector k; in practice, the black hole which forms will have whatever k leads to the lowest free energy. Numerical results confirm that the striped phase is thermodynamically preferred below the critical temperature.

Figure 6.5 shows surface plots of the curvature of the resulting striped geometries. Here x denotes the boundary direction along which stripes form and r is the holographic direction. As T/Tc becomes small, the curvature modulations become more pronounced. In the boundary theory, in addition to the operator dual to ϕ picking up a spatially modulated expectation value, one finds an electric current Jy which flows normal to the modulation direction. The resulting striped phase appears to have vanishing zero-temperature entropy. The identification of possible inhomogeneous zero temperature, near horizon geometries that would control the low energy and low temperature physics, potentially exhibiting emergent scaling of some kind, is a challenging open problem. It was emphasized in [374] that z = ∞ scaling is in principle compatible with strong spatial inhomogeneities.

Figure 6.5
The profile of the Ricci scalar (normalized to the AdS value) in the striped phase for T = 0.87Tc (left) and T = 0.04Tc (right). Figure taken from [760] with permission.

In all the examples mentioned above, the bulk models which spontaneously break translation symmetry have broken either parity or time-reversal symmetry. [222] presents a linear stability analysis of two systems preserving these symmetries, in which spontaneous translation symmetry breaking is possible. More complicated models than (6.44) likely exhibit first order transitions to striped phases [760]. String theoretic constructions of holographic models with closely related spatially modulated instabilities were studied in [231; 219].

Stripe instabilities can also arise in holographic CFTs at zero charge density but in background magnetic fields [231; 230; 157]. These instabilities do not require the final interaction in (6.44) but rather occur already in the EMD type models (4.43) that we have studied in detail. The only requirement is that Z′(0) ≠ 0 or V′(0) ≠ 0 in (4.43) so that the dilaton is forced to be nonzero. In four dimensions, fluctuations about a magnetic background with a Z(Φ)FabFab interaction are the same as fluctuations about an electric background with the Y(ϕ)𝜖abcdFabFcd interaction of (6.44). Hence the magnetic instabilities are closely related to the instabilities above. Backreacted solutions of the corresponding ordered phase showing magnetization density waves were constructed in [237].

6.5.2.3 Crystalline order

In addition to striped phases, one can also look for “crystalline” phases in holography, where translation symmetry is broken in all spatial directions. As a first step, very close to the critical temperature, we can imagine superimposing stripes of specific wavevectors to form a Bravais lattice. Of course, we wish to determine whether or not the true endpoint of these instabilities corresponds to stripes or lattices (and if so, what kind of lattice). A first step towards such a calculation occurred in the probe limit of a bulk SU(2) Yang-Mills theory with a spatially modulated instability. This model suggested that the triangular lattice was indeed the endpoint of this instability [115]. More recently, the gravitational backreaction has also been taken into account using models similar to (6.44). As in the discussion on stripes, we focus on boundary theories in two spatial dimensions. In [761], black holes with a rectangular “lattice” of stripes were constructed; later in [228], it was shown that triangular lattices are often preferred in a background magnetic field. Figure 6.6 shows the formation of ‘magnetization’ currents in the boundary theory in the triangular lattice phase.

Figure 6.6
Triangular lattice. The expectation value of the current in the boundary theory, in a black hole with a triangular lattice ‘ground state’. The largest currents flow in circular patterns, and the smallest current is in the interstitial regions between them. Figure taken from [228] with permission.

Perhaps surprisingly, it seems to be the case that often the crystalline phases are thermodynamically disfavored compared to the simple striped phase [761]. It is possible to choose boundary conditions which make the rectangular lattice preferred [761], but we do not have a clear understanding of when or why different symmetry breaking patterns are preferred. As with the inhomogeneous striped geometries discussed in the previous section, the construction of inhomogeneous zero temperature IR scaling geometries – if such geometries exist – remains a challenging open problem.

In doped CFT3s, the spontaneous formation of a lattice admits an S-dual description. We discussed this S-duality (or, particle-vortex duality) of CFT3s in §3.4.6 and §4.7. The charge density dualizes to a background magnetic field and hence the lattice can be dually understood as a generalized Abrikosov flux lattice. In confining phases obtained by condensing vortices, these lattices obey a commensurability relation on the magnetic flux (and hence charge density in the original desciption) per unit cell of the lattice [677]. Such commensuration is not seen in the holographic models discussed above. This pictured motivated the study of an Abrikosov lattice in [74], following a perturbative technique used for vortex lattices in holographic superconductors [553]. It was argued in [553] that this lattice will be triangular, based on Ginzburg-Landau arguments, but as noted in [71], these arguments do not correctly predict the order of the holographic phase transition, so it is unclear whether such arguments can reliably predict the final shape of the lattice.

6.5.2.4 Conductivity

In the presence of spontaneous symmetry breaking the dc electrical conductivity remains infinite, even along the direction of symmetry breaking. This is easily understood from a field theoretic point of view. Suppose that translation symmetry is spontaneously broken in such a way that the local density is ρ = ρ0 + ρ1 cos(k(xx0)). This is a valid solution for any value of x0. So we expect phonon or ‘phason’ modes – analogous to the Goldstone bosons of a superfluid – which globally translate x0. If we apply an electric field to this system, we will excite such zero wave number phonons, and the pattern of symmetry breaking will globally translate without any momentum relaxation, leading to an infinite conductivity.6

There is a quick way to see how an infinite conductivity arises in holography. Let us return to the derivation of §5.6.5. We introduced disorder perturbatively via a sourced bulk scalar field φ0(k, r) ∼ rd+1−Δ near the boundary. For simplicity, take Δ > (d + 1)/2. Since the source is fixed, the finite momentum perturbations in the bulk δφ(k, r) ∼ rΔ near the boundary. With these scalings one can check that δ𝒫x ∼ r0 near the boundary, which is consistent with our claim that we may impose the boundary condition δ𝒫x = ρ/Ld at r = 0. However, suppose that φ0 ∼ rΔ as well near the boundary. Then we find that δφ/φ0 = a0r0 + acrc + near the boundary, with c > 0 describing the next order correction, and δ𝒫x ∼ r2Δ−d−1+c, implying that δ𝒫x(r = 0) = 0. Hence, it is impossible to excite the bulk mode (5.148). Following the discussion in §5.6.5, the only bulk mode we can excite is the boost mode. The perturbation of the gauge field in the bulk is hence

to leading order in ω, away from the horizon. But this is sufficient to compute the conductivity perturbatively, and we find

5.43).

6.6 Zero temperature BKT transitions

We saw in §6.3.1 above that nonzero temperature holographic phase transitions were described by conventional Landau-Ginzburg physics. Fluctuations effects at the transition are suppressed at large N, because they involve fluctuations of only a single mode, and hence the transitions are well captured by mean-field theory. In contrast, when holographic compressible phases are driven into ordered phases as a function of some source at zero temperature, even the large N physics is strongly non-mean-field. This subsection will show how the emergent IR scaling regimes and the fact that the instability occurs when an operator is driven to have complex scaling exponents lead to certain ‘quantum BKT’ infinite order quantum phase transitions.

What happens if, by varying some auxiliary sources in the theory, we continuously push the effective IR mass squared of an operator through the instability bound (6.1)? It was explained in [453] that this should be understood as the annihilation of two RG flow fixed points. In §1.6.3 we noted that for the range of masses just above the instability (6.1) – here we generalize the discussion therein to general z, see [41; 468] –

two different normalizable boundary conditions were possible at the asymptotic boundary. These correspond to choosing the operator 𝒪 dual to the bulk scalar field to have the different dimensions

As noted in §1.6.3, the different boundary conditions then correspond to different scaling theories that are related to each other by RG flow triggered by a double trace deformation ∫dd+1x 𝒪2. This deformation is irrelevant about the ‘standard’ quantization with dimension Δ+ but relevant about the ‘alternate’ quantization with dimension Δ. These two fixed points merge as 𝒪2 coupling is marginal.

The merger of fixed points is described by the beta function [453]

𝒪2 interaction and α is the parameter that tunes the theory across the transition. For α > α the β = 0 fixed points are at Figure 6.7. The lack of a fixed point indicates an IR instability. Such unstable flows typically result in the condensation of an operator.

Figure 6.7
Annihilation of two RG fixed points. RG flows from (6.49) with (from top to bottom) α > α, α = α and α < α.

The renormalization group flow equation (6.49) is familiar from the classical BKT transition. In particular, this means that as the transition point is approached, a low energy scale is generated that is non-analytic in α. For instance, just below the critical α, integrating the flow equation gives

⋆. This scale can be seen directly from the scaling geometry as follows [453]: solve the wave equation for a field with , and cut off the geometry in the UV and IR by imposing Dirichlet boundary conditions at r = rUV and r = rIR, respectively. Then an unstable, growing in time solution to the wave equation appears when

This shows that the scale rIR is where the geometry will need to be modified by a condensate in order to remove the instability.

Explicit holographic models realizing these quantum BKT phase transitions were first studied in [443; 436]. The tuning parameter in both cases is a magnetic field. In [436] the zero temperature destruction of holographic superconductivity by a magnetic field was shown to be in this class.

The quantum BKT transitions can be understood semi-holographically [439; 435]. In fact we have already used this perspective in the formula (6.4) above for the Green’s function of the scalar field, obtained by matching with the IR geometry. The dynamics of quantum BKT transitions are characterized by coupling the gapless Goldstone mode (6.24), that is supported at intermediate radial scales in the spacetime and hence is not geometrized, as we noted in §6.4.3 above, to the large N quantum critical bath that undergoes a merger with another critical point as we have just described above.

We noted above that the operator 𝒪2 that drives the RG flow between the two fixed points is necessarily marginal at the quantum critical coupling where the two fixed points merge. This has some interesting phenomenological consequences, especially at large N where the single trace operator 𝒪 then has dimension Δ = Deff./2 at the critical point (more generally operator dimensions do not add). Such an operator can be coupled to fermions to obtain marginal Fermi liquid fermionic Green’s functions [435; 742] or can be coupled to currents to directly obtain T-linear resistivity [235].

Exercises

6.1.  Superconductivity of the Gubser-Rocha black hole. Consider the EMD action (4.43), in 4 bulk spacetime dimensions, with

For simplicity set L = e = 2κ2 = 1.

a) Show that the following is an exact asymptotically AdS solution of the EMD equations:

with all non-listed components of gab and Aa vanishing. We recommend using RGTC or a similar computer program to evaluate the EMD equations in this coordinate system. Note that in these coordinates, r → ∞ is the AdS boundary; we generally do not use this convention, but this geometry is simpler in such coordinates. (6.53) is called the Gubser-Rocha black hole [330], and it is an elegant example of a top-down solution with hyperscaling violation.

b) Show that in the IR, there is an emergent hyperscaling geometry as T → 0. What are the values of z and θ? Hint: First find the relationship between r+ and 𝒬 and the temperature T and charge density ρ. It may help to convert the IR geometry to a more familiar coordinate system.

c) Now, let us add a charged scalar

Although φ = 0 on the background, there may be an instability to superconductivity. Follow the logic of §6.3.1 to show that we may consider real-valued fluctuations of φ, and that a superconducting instability will arise whenever there is a regular, normalizable solution to

d) Numerically find Tc(Q) both for M2 = −2 and M2 = −5/4. You should find that, in the latter case, there is no superconductor at all below a critical Qc. [154]

e) To understand the phenomenon of a critical Qc, compute meff given by (6.3). Show that for small Q, meff always remains below the BF bound. Show that this effect is rather general [543], by considering the IR limit of meff for the scalar fields of §4.3.2, with B(Φ) given in (4.63).

6.2.  Instability of a neutral scalar. The discussion in §6.2 focused on bulk instabilities that can be interpreted as pair production. In this exercise you will instead investigate the instability of a neutral scalar field. You will also look at the role of the full geometry in determining the critical temperature. This exercise follows [356]. Re-inserting the frequency dependence into (6.7) and setting the charge q = 0, the equation for neutral scalar fluctuations about the charged black hole background is, c.f. (4.80),

for some potential V which you should determine.

b) In (6.58), ω2 plays the role of the energy. Convince yourself that the black hole background will therefore be unstable to condensation of ϕ if the Schrödinger equation admits a negative energy bound state.

c) Restrict to the Reissner-Nordström-AdS background of Einstein-Maxwell theory. Then a = 1 and b = f, with f given by (4.28). Restrict also to d = 2 and (mL)2 = −2. Show that the potential V is positive everywhere if . What does this imply for any bound state energy? What can you conclude about any potential critical superconducting temperature Tc?

d) A lower bound on Tc can be obtained from the variational principle. According to the variational principle, the ground state energy is upper bounded by

where ψ should be normalized and satisfy an appropriate boundary condition. You can impose ψ ∼ r0 as r → 0, corresponding to the physical falloff (1.28) of normalizable modes with (mL)2 = −2 and Δ = 1 (recall ϕ = ). Alternate quantization is being used here (see §1.6.3), as this makes the instability easier to find. There should technically be a boundary term in (6.59) in this case, but that will vanish so long as we restrict to modes with ψ(0) = 0. In order to establish the presence of a negative energy bound state at some temperature, it is sufficient to find a variational mode ψ such that S[ψ] < 0 at that temperature. Normalization is unimportant as you only care about the sign of S. Using the class of variational wavefunctions

where α is a parameter to optimize over, show that a superconducting instability exists over a range of temperatures. In this way obtain a lower bound on Tc. How does it compare with the actual value quoted for Δ = 1 and q = 0 in Figure 6.2 in the text?

6.3.  Striped instability. In this exercise we will consider in more detail the onset of an instability to a striped phase [219]. We consider the bulk action (6.44) with

with m, a and b constants.

a) This bulk action has an exact AdS2 × 2 solution, as described in §4.3.1, describing a T = 0 locally critical theory. Confirm that (4.37) with L2 = 1 is an exact solution of these equations of motion with for the remainder of the problem.

b) Consider a perturbation

Define ψxy = iωhxy. Show that the resulting equations of motion can be written in terms of three modes: v = (ψxy, a, w), obeying a set of coupled scalar wave equations in AdS2:

c) Plot the smallest eigenvalue of the effective 3 × 3 mass matrix above as a function of m2a, b, and of k. Where are there instabilities when k = 0? Show that there are further instabilities when k ≠ 0, and that for large m2a, these are in fact the only instabilities.

Notes

1 As noted below (4.70), in z = ∞ geometries such as AdS2 × d one must take Deff. = 1, counting the number of scaling boundary dimensions.

2 In approaching the literature, one should be aware that many computations are done in a ‘probe limit’ in which the backreaction of the Maxwell field and charged scalar on the metric can be neglected [355; 321]. While this limit has the advantage that some interesting behavior can be exhibited with minimal computational effort, and was historically important, it is unable to capture the universal low temperature dynamics in which the metric plays a key role, and also cannot describe the interplay of superconductivity with momentum conservation.

3 Superconductors and superfluids differ by the presence of a dynamical gauge field in superconductors. In this respect, what we are describing is a superfluid, as there is no dynamical Maxwell field in the boundary QFT. However, for many purposes including computing σ, we can pass back and forth between the language of superfluidity and superconductivity without problem. This is because in any charged system coupled to a dynamical Maxwell field, when we write Ohm’s law j = σE, E is the total electric field, the external field plus that generated by polarization of the medium (screening), and hence σ is given by the current-current Green’s functions of the ungauged theory, without a dynamical Maxwell field.

4 In order to obtain the first order constraint equation for δgtx(r) it is necessary to allow the fields to be time dependent and then set the time dependence to zero after integrating the constraint equation in time. This is important because we will ultimately want to calculate 5.57), while the susceptibility χJxJx vanishes. To obtain the equation in the text it is also necessary to use the equations of motion for the background.

5 It is possible that this phase transition is actually first order once quantum effects in the bulk are accounted for – see the arguments in [608].

6 In actual condensed matter systems, these phason modes are pinned by disorder so that, for example, Wigner crystals are insulators [189].