Inhomogeneity simplified
Abstract
We study models of translational symmetry breaking in which inhomogeneous matter field profiles can be engineered in such a way that blackbrane metrics remain isotropic and homogeneous. We explore novel Lagrangians involving square root terms and show how these are related to massive gravity models and to tensionless limits of branes. Analytic expressions for the DC conductivity and for the low frequency scaling of the optical conductivity are derived in phenomenological models, and the optical conductivity is studied in detail numerically. The square root Lagrangians are associated with linear growth in the DC resistivity with temperature and also lead to minima in the optical conductivity at finite frequency, suggesting that our models may capture many features of heavy fermion systems.
1 Introduction
Holographic modelling of strongly coupled condensed matter systems has generated a great deal of interest over recent years; for reviews see [1, 2]. It is remarkable that many features of strongly coupled matter can be captured by static, isotropic solutions of Einstein–Maxwelldilaton models. Nonetheless as one tries to develop more realistic models it is clear that such holographic geometries cannot adequately capture many important features of strongly interacting systems.
The focus of this paper will be on modelling systems with broken spatial translational symmetry. Realistic condensed matter systems never have perfect translational symmetry: the symmetry is explicitly broken both by lattice effects and by the presence of inhomogeneities. This breaking of translational invariance is necessary for particles to dissipate momentum, without which there would be a delta function in the conductivity at zero frequency.
From this Ward identity it is evident that one can generically violate momentum conservation, while preserving energy density conservation, by introducing background sources in the field theory which depend on the spatial coordinates. (Note that spontaneous breaking of the translational symmetry on its own is not enough to dissipate momentum.) The introduction of such sources is rather natural: a source for \(A_i\) with periodicity in the spatial directions represents an ionic lattice while other lattice effects can be captured by a periodic scalar field.
Holographically, spatially dependent sources for the conserved current can be modelled by a dual gauge field which is spatially modulated. The backreaction of this field onto the metric and other fields gives rise to fields which are stationary but inhomogeneous. In \((d+1)\) bulk dimensions one therefore has to solve partial differential equations in the radial coordinate and the spatial coordinates which are only tractable numerically. Numerical analysis has shown that such explicit breaking of translational invariance indeed removes the delta function in the conductivity at zero frequency [3, 4].
Clearly it would be interesting to understand the origin of this scaling behaviour better but the scaling emerges from the numerical analysis and does not make evident which ingredients are crucial to obtain a scaling regime. For example, it is known that one can obtain scaling behaviour for the AC conductivity without explicitly breaking translational invariance; scaling with the correct exponent arises in Einstein–Maxwelldilaton models, although solutions with the required value of \(\gamma \) appear to be thermodynamically unstable [7]. While one expects that the scaling is associated with an underlying quantum critical state, the scaling itself emerges at finite temperature and, from the holographic viewpoint, is therefore not associated not only with the spacetime region immediately adjacent to the horizon but also with regions further from the horizon. From this perspective it is not obvious to what extent the scaling should be sensitive to the details of the far IR or the mechanism of translational symmetry breaking.
As explored in [8, 9, 10], simplified models of translational symmetry breaking can be obtained by imposing symmetries on the bulk solutions: one can tune matter field profiles such that the metrics for the equilibrium configurations are homogeneous but anisotropic. The resulting equations of motion therefore simplify, reducing to ordinary differential equations in the radial coordinates, although these equations nonetheless still need to be solved numerically. In such models one does not find scaling behaviour of the AC conductivity, which indicates that this behaviour is nongeneric. An interesting feature of these models is that one finds transitions between metallic and insulator behaviour as parameters are adjusted; see also [11, 12, 13] for related discussions on metal–insulator transitions.
In this paper we will explore the simplest possible models of translational symmetry breaking, namely those for which the inhomogeneous matter field profiles are chosen such that the metrics for the equilibrium configurations remain both homogeneous and isotropic. The equations of motion for the equilibrium blackbrane solutions can therefore be solved explicitly analytically. The presence of inhomogeneous matter field profiles nonetheless guarantees that momentum can be dissipated by fluctuations propagating around these equilibrium solutions, and therefore one obtains finite DC conductivities.
Massive gravity models [14, 15, 16, 17] have been proposed as translational symmetry breaking models of this type. However, massive gravity is a bottom up phenomenological theory and it is not clear that it is well defined at the quantum level. The holographic dictionary between the background metric used in massive gravity and the dual field theory is obscure. It is therefore preferable to work with models whose topdown origin can be made more manifest.
As discussed above, switching on any operator source with spatial dependence triggers momentum dissipation. Moreover, any scalar field action with shift symmetry admits solutions for which the scalar field is linear in the spatial coordinates and thus the scalar contributions to the stress energy tensor are homogeneous. As shown in [18], by choosing an action with a number of massless scalar fields equal to the number of spatial directions one can engineer scalar field profiles such that the bulk stress energy tensor and hence the resulting blackbrane geometry are both homogeneous and isotropic. See also the earlier work in [19] in which homogeneous and isotropic black branes supported by fluxes were classified; it would be interesting to find AdS/CFT applications for these solutions.
Square root actions are unconventional but have arisen in several related contexts. For example, time dependent profiles of scalar fields associated with the cuscuton square root action have been proposed in the context of dark energy [20, 21]. The same action arose in the context of holography for Ricci flat backgrounds: the holographic fluid on a timelike hypersurface outside a Rindler horizon has properties consistent with a hydrodynamic expansion around a \(\phi = t\) background solution of the cuscuton model [22, 23].
It was shown in [18] that the \(\alpha _2\) term of massive gravity is related to massless scalar fields: the background brane solutions are completely equivalent and certain transport properties (shear modes) agree. Note that not all transport properties agree, since the linearised equations are only equivalent for a subset of fluctuations, those with constrained momenta in the spatial directions. In Sect. 2 we will show that the \(\alpha _1\) term of massive gravity is related to the square root terms (1.4). Again, the background brane solutions are completely equivalent and DC conductivities also agree but as in [18] the models are not completely equivalent; even at the linearised level the equations of motion for fluctuations with generic spatial momenta do not agree. The inequivalence between the models is made manifest when one uses a Stückelburg formalism for massive gravity.
There has been considerable debate about stability and ghosts in massive gravity, as well as the scale at which nonlinear effects occur and effective field theory breaks down; see for example [24, 25, 26, 27, 28, 29, 30, 31]. Clearly all such issues are absent in models based on massless scalar fields but related issues occur in the square root models (1.4): perturbation theory around the trivial background \(\phi _I = 0\) is illdefined. From the holographic perspective, it is not a priori obvious that the bulk fields \(\phi _I\) are dual to local operators in the conformal field theory whose dimensions are real and above the unitary bound and whose norms are positive.
In Sect. 3 we show that the fields \(\phi _I\) are dual to marginal operators in the conformal field theory. The bulk field equations admit a systematic asymptotic expansion near the conformal boundary for any choice of nonnormalisable and normalisable modes of these scalar fields, in which all terms in the asymptotic expansion are determined in terms of this data. The bulk action can be holographically renormalised in the standard way. This analysis provides evidence that the action (1.4) is physically reasonable.
We also show in Sect. 3 that correlation functions of the operators dual to the square root scalar fields \(\phi _I\) of (1.4) can be computed in any holographic background in which there are nonvanishing profiles for these fields. These operators indeed behave as marginal operators and the norms of their two point functions are positive for \(a_{1/2} > 0\). However, the expressions we obtain for the two point functions are not analytic as the background profiles for the scalar fields are switched off.
The action (1.4) is reminiscent of the volume term in a brane action. In Sect. 3 we show that such actions can indeed arise as tensionless limits of brane actions: the fields \(\phi _I\) then correspond to transverse positions of branes.
We show that such models have a finite DC conductivity, as expected, and analyse the temperature dependence of the DC conductivity. The parameter \(\tilde{\alpha }\), which is nonzero whenever there are background profiles for the square root fields, leads to a linear increase in the resistivity with temperature at low temperature in a field theory in three spacetime dimensions. In dimensions greater than three the DC conductivity increases with temperature for all values of the parameters \((\tilde{\alpha }, \tilde{\beta })\).
We explore the low frequency behaviour of the optical conductivity at low temperature, finding that for all values of our parameters there is a peak at zero frequency, indicating metallic behaviour. However, we show that our models do not fit Drude behaviour even at very low temperature: the effective relaxation constant is complex, indicating that momentum not only dissipates but oscillates.
Perhaps unsurprisingly, we see no signs of scaling behaviour of the optical conductivity at intermediate frequencies but our numerical analysis indicates minima can arise in the conductivity at intermediate frequencies and low temperatures (in three spacetime dimensions). The behaviour of the optical conductivity in our models is similar to that of heavy fermion compounds: these also have a DC conductivity which increases linearly with temperature at low temperature and they exhibit a transition to a decoherent phase at low temperature in which the conductivity has a minimum at finite frequency. In heavy fermions the origin of this minimum is a hybridisation gap, caused by felectrons hybridising with conduction electrons, while the dip in the conductivity in our model is a strongly coupling phenomenon, associated with the mixing between scalar and gauge field perturbations.
The plan of this paper is as follows. In Sect. 2 we explore models for translational symmetry breaking based on shift invariant scalar field actions and we show how such models are related to massive gravity and to scaling limits of branes. In Sect. 3 we analyse square root models, demonstrating that a welldefined holographic dictionary can be constructed. In Sect. 4 we build phenomenological models and compute DC and AC conductivity in these models, showing that features reminiscent of heavy fermions are obtained. In Sect. 5 we analyse generalisations of our models. We conclude in Sect. 6.
2 The simplest models of explicit translational symmetry breaking
In summary, given any Lagrangian functional built out of \((d1)\) scalar fields with shift symmetry, one can construct solutions for which the stress energy tensor preserves spatial isotropy and homogeneity. The backreaction on the blackbrane metric therefore preserves the usual blackbrane form for the metric, with a different blackening factor. The breaking of translational invariance by the scalar fields ensures that the momenta of fluctuations can be dissipated. In the remainder of this section we will consider the physical interpretations of various types of functionals.
2.1 Polynomial Lagrangians
2.2 Relation to massive gravity
Another conceptual difference between our model and massive gravity is the following. In our models the scalar fields \(\phi _{I}\) and \(\chi _{I}\) are treated as independent fields but in massive gravity they are identified as the same field. As we discuss in Sect. 5, it is, however, straightforward to restrict to the case in which these fields are identified.
2.3 Relation to branes
3 Square root models
While one can obtain solutions for any polynomial functional, one would usually restrict to the case of \(m=1\), i.e. massless scalar fields. In AdS/CFT the operators dual to these scalar fields are marginal scalar operators and the bulk scalar profiles are therefore immediately interpretable in the dual theory as linear profiles for the associated couplings.
For integer \(m > 1\) the action is higher derivative and for noninteger \(m\) the action would be considered nonlocal. In this section we will argue that both cases may in some limits nonetheless be relevant in bottom up models.
3.1 Holographic renormalisation for square root models
3.2 Thermodynamics of brane solutions
3.3 Two point functions
Let us now consider perturbations about a background solution with nonvanishing scalar fields. The linearised problem is perfectly well defined when the square root terms are expanded around any nonvanishing background. However, the metric and scalar field fluctuations are coupled and the equations of motion need to be diagonalised. If the scalar field fluctuations were decoupled from those of the metric, \(a_{1/2}\) would need to be positive for the fluctuations to have the correct sign kinetic term (and hence, correspondingly, positive norm correlation functions in the holographically dual theory). Since the metric and scale fluctuations are coupled one cannot immediately conclude that the sign of the coefficient \(a_{1/2}\) in (3.13) must be positive.
4 Phenomenological models
Systematic holographic renormalisation would be required to determine the terms in ellipses in the (4.11) and the free energy. The scalar field profiles (4.7) are nonnormalisable modes, associated with deformations of the dual field theory, and therefore the thermodynamically preferred state is that with lowest free energy at fixed \((c_{1/2},c_1)\). It is possible that the homogeneous black branes (4.4) are not the thermodynamically preferred state, particularly at low temperatures, but we will not investigate phase transitions here. Note that the nearhorizon geometry remains \(AdS_2 \times R^{d1}\), as in Reissner–Nordström, and the entropy does not vanish at zero temperature.
4.1 Linearised perturbations
To progress further we need to work out the asymptotic expansions near the conformal boundary for the various fluctuations fields under consideration. The three sets of fields \(a_I, \zeta _I, \xi _I\) have both homogeneous and inhomogeneous contributions. Since the field equations (4.19)–(4.21) are second order linear ODEs we expect each field to have two homogeneous contributions: one corresponding to a normalisable mode, and one nonnormalisable. Since we are primarily interested in computing the conductivity we will turn off the nonnormalisable modes for the scalar fields, which correspond to perturbing the sources for the dual operators in the field theory. (Note that the background solution still has sources for these operators.)
It is clear from equation (4.39) that \(\dot{\bar{\Pi }}_{1I} / \bar{\lambda }_{I} \sim O(\bar{\omega }^2)\) and \(\dot{\bar{\Pi }}_{1/2 I}/\bar{\lambda }_I \sim O(\bar{\omega }^2)\). Similarly we know that \(\bar{\Pi }_{1I}/\bar{\lambda }_I \sim O(\bar{\omega })\) and \(\bar{\Pi }_{1/2 I}/\bar{\lambda }_I \sim O(\bar{\omega })\): these conditions are satisfied at the horizon due to ingoing boundary conditions and are conserved throughout the bulk by the field equations. We will use these properties in deriving the DC conductivity below.
4.2 DC conductivity
The background brane solutions coincide between our model and massive gravity. The DC conductivities agree since the fluctuation equations also coincide for homogeneous fluctuations carrying no spatial momenta. We show in Appendix B that the fluctuation equations in our model and in massive gravity are completely equivalent at zero frequency.
4.3 Parameter space restrictions
Positivity of the norms of the two point functions of the scalar operator dual to the massless scalar field (or, equivalently, absence of ghosts) requires that \(a_1 \ge 0\). Since \(c_1\) and \(\mu \) are real, \(\tilde{\beta } \ge 0\) and hence \(P^2 \ge 0\). The sign of \(\tilde{\alpha }\) is more subtle, as it depends on \(a_{1/2}\), \(c_{1/2}\) and \(\mu \). The nonlinearity of the square root terms however prevents us from placing restrictions on the sign of \(a_{1/2}\). Previously we showed that \(a_{1/2} c_{1/2}\) should be positive when \(\mu = 0 = a_1\). This suggests \(\tilde{\alpha }\) should be positive, for positive \(\mu \), and negative for negative \(\mu \).

\(T \ge 0\): The system has a nonnegative temperature.

\(z_0 > 0\): The blackbrane horizon location is at a real and positive position in the holographic bulk direction.

\(\sigma _\mathrm{DC} \ge 0\): The system has a nonnegative conductivity.

\(f(z) > 0\) for \(z \in (0,z_0)\): The point \(z=z_0\) is indeed the true horizon location, no other horizons exist between this and the boundary.
\(\mu > 0\)  \( \mu < 0\) 

\(a_1 > 0\)  \(a_1 > 0\) 
\(a_{1/2} c_{1/2} > 0\)  \(a_{1/2} c_{1/2} > 0\) 
\(\tilde{\beta } > 0\)  \(\tilde{\beta } > 0\) 
\(\tilde{\alpha } \ge 0\)  \(\tilde{\alpha } \le 0\) 
\(\mu z_0 = \mu z_0^{} > 0\)  \(\mu z_0 = \mu z_0^{+} < 0\) 
Note that the restrictions discussed in this section do not ensure complete thermodynamic stability as other possible phases have not been investigated here.
4.4 DC conductivity temperature dependence
Recall that for the \(\mu < 0\) plots decreasing \(\tau \) corresponds to increasing \(T\). The symmetry between the \(\mu > 0\) and the \(\mu < 0\) branches is easily understood because \(\tilde{\alpha }(\mu z_0)^{1}\) is invariant under \(\mu \rightarrow  \mu \), \(\tilde{\alpha } \rightarrow  \tilde{\alpha }\), \(\tau \rightarrow  \tau \).
4.5 Finite frequency behaviour at low temperature
The low frequency behaviour of the AC conductivity at low temperature can be obtained by rewriting the fluctuation equations as Schrödinger equations and matching asymptotics between the IR and UV regions. This technique has been applied to a number of AdS/CMT models; see for example [7, 12, 40, 41, 42, 43].
4.6 Relation to Drude behaviour
4.7 AC conductivity numerics
In this section we explore the behaviour of the AC conductivity by numerically solving the linearised perturbations equations. To find the values of \(\sigma (\omega )/\mu ^{d3}\) numerically we use a Mathematica code to solve the shooting problem of solving these ODEs with the desired nearboundary asymptotics and ingoing boundary conditions at the horizon. The code calculates the \(r\) series expansions of the dimensionless perturbations near the horizon and the boundary with some randomly chosen initial data. This initial data is then used in Mathematica’s NDSolve function to integrate the ODEs to some predetermined point in the bulk. At that point the difference between the perturbations and their first derivatives coming from the two ends is computed. The process is then repeated for some initial data that is close to the randomly chosen data to construct an approximation to the Jacobian. We then proceed via the multivariate secant method of root finding to find initial data that is a better approximation to the true data that causes the difference function to vanish. We analysed the case of \(d=3\) but a qualitatively similar behaviour is likely to occur in other dimensions.
Using the numerical results one can also investigate the fit to a Drude peak at low frequency. The numerics show that one can only fit to a Drude formula using a relaxation time \(\tau _r\) which is complex; therefore our system does not behave as a Drude metal even at very low temperature.
Unlike [3, 4], we see no clear signs of scaling behaviour of the optical conductivity at intermediate frequencies, \(T < \omega < \mu \). The AC conductivity displays several features similar to that of heavy fermion compounds. Heavy fermion materials also have a DC resistivity which increases with temperature, with a transition from normal metal behaviour to hybridised behaviour occurring below the decoherence temperature. In the hybridised phase felectrons hybridise with conduction electrons, leading to an enhanced effective mass and a hybridisation gap. Figure 5 shows that the peak in the conductivity sharpens at low temperatures, and a minimum in the conductivity develops for \(\tau \lesssim 0.2\) at intermediate frequencies \(\omega /\mu \sim 0.5\). The minimum is enhanced by increasing \(\tilde{\alpha }\) and decreasing \(\tilde{\beta }\) (i.e. increasing the amplitudes of the square root scalar fields and decreasing the amplitudes of the massless scalar fields). In our models the minima in the conductivity are strong coupling phenomena, with the reduced conductivity being associated with increased amplitudes of the scalar field fluctuations at these frequencies.
5 Generalised phenomenological models
In this section we consider other phenomenological models based on actions with massless scalar fields and square root terms.
5.1 Scalar fields identified
5.2 Other square root models
6 Conclusions
In this paper we have focussed on simple models of explicit translational symmetry breaking. The main advantage of these models is that the brane backgrounds are isotropic and homogeneous and can therefore be constructed analytically. The holographic duals to the bulk symmetry breaking can also be explicitly identified, unlike in massive gravity models, and correspond to switching on spatial profiles for marginal couplings in the field theory.
Couplings growing linearly with spatial directions represent a qualitatively different mechanism for momentum dissipation than lattice and phonon effects in an ordinary metal. It is therefore perhaps unsurprising that our models do not exhibit ordinary metal behaviour. Nonetheless these models do show a peak in the optical conductivity at zero frequency; the DC resistivity increases linearly in temperature at low temperature in three boundary dimensions and by tuning the parameters one obtain minima in the optical conductivity at finite frequency. These features are reminiscent of strange metals and heavy fermion systems and suggest that it may be interesting to explore such models further.
The novel phenomenology is associated with the square root actions (1.4): when this term is switched off one does not find linear growth of the DC resistivity with temperature, for example. Despite the apparent nonlocality of this action, we showed in Sect. 3 that the holographic dictionary is well defined and one can work perturbatively about any background solution for this action. Moreover, we can view (1.4) as a scaling limit of a brane action (2.61). Brane actions exhibit no nonanalytic behaviours when the background field profiles vanish and should give a qualitatively similar phenomenological behaviour to (1.4). It would therefore be interesting to develop topdown phenomenological models based on branes, which capture the desirable features of (1.4).
One issue with our blackbrane backgrounds is that they have finite entropy at zero temperature, indicating that they may not be the preferred phase at very low temperatures. Generic Einstein–Maxwelldilaton models admit Lifshitz and hyperscaling violating solutions whose entropy scales to zero at zero temperature; see [8, 41, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53], and it would be straightforward to extend our discussion of translational symmetry breaking using massless and square root scalar fields to such models.
Footnotes
Notes
Acknowledgments
Both M.T. and W.W. acknowledge support from STFC. M.T. acknowledges support from a grant of the John Templeton Foundation. The opinions in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.
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