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  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  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  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 .
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 . 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 .
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 . 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 
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 × ℝd. 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.
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 . 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 . 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 , 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 . 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  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 : a feat which may prove useful in our discussion of superfluid turbulence in § 7.4.
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)
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. ,
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 . 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  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  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 .
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 : 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. ). 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 .
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
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
220.127.116.11 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]:
18.104.22.168 Challenges for d-wave superconductors
A d-wave condensate is described by a charged, spin two field in the bulk . 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 .
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 
Here φ = φaa and Da = ∇a − iAa. There is a parameter g, the gyromagnetic ratio . 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  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.
22.214.171.124 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 :
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 .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 × ℝ3 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 . The condensed phase was constructed numerically in a probe limit in ; the backreaction of gravity was accounted for in . The backreaction is tractable without solving PDEs because the helical condensate (6.43) leads to a homogeneous geometry with Bianchi VII symmetry . The zero temperature limit of the fully backreacted solution exhibits an emergent anisotropic IR scaling symmetry with entropy density vanishing like s ∼ T2/3 .
Supergravity fixes the value of . As was noted in , this happens to be just below the critical value for this spatially modulated instability.
126.96.36.199 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 . 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]
188.8.131.52 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 . 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 , black holes with a rectangular “lattice” of stripes were constructed; later in , 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.
Perhaps surprisingly, it seems to be the case that often the crystalline phases are thermodynamically disfavored compared to the simple striped phase . It is possible to choose boundary conditions which make the rectangular lattice preferred , 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 . Such commensuration is not seen in the holographic models discussed above. This pictured motivated the study of an Abrikosov lattice in , following a perturbative technique used for vortex lattices in holographic superconductors . It was argued in  that this lattice will be triangular, based on Ginzburg-Landau arguments, but as noted in , 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.
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(x−x0)). 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
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  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 
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 .
6 In actual condensed matter systems, these phason modes are pinned by disorder so that, for example, Wigner crystals are insulators .