CCompressible Matter at a Holographic Interface
Moshe Rozali
Department of Physics and Astronomy,University of British Columbia,Vancouver, BC V6T 1Z1, Canada
We study the interface between a fractional topological insulator and an ordinary insulator, bothdescribed using holography. By turning on a chemical potential we induce a finite density of matterlocalized at the interface. These are gapless surface excitations which are expected to have afermionic character. We study the thermodynamics of the system, finding a symmetry preservingcompressible state at low temperatures, whose excitations exhibit hyperscaling violation. Theseresults are consistent with the expectation of gapless fermionic excitations forming a Fermi surfaceat finite density.
I. INTRODUCTION
Holography is a fairly recent approach to the studyof strongly interacting systems which have gravitationalduals. As such, it is ideal to model and gain qualitativeunderstanding of otherwise ill-understood phenomena incondensed matter systems. The approach has been ap-plied successfully to the study of homogeneous phases ofholographic matter, and has provided insight into diversephenomena such as superconductivity, quantum phasetransitions, and non-Fermi liquid behavior.Many interesting phenomena in modern condensedmatter physics involve boundary excitations at the in-terface of two different bulk materials. Under suitablecircumstances, those boundary excitations are gaplessby virtue of topological considerations. Those consid-erations protect the excitations from developing a gapin the presence of noise and other small perturbations.These localized excitations are particularly interestingwhen they are predicted to be purely fermionic, as theyare a natural candidate for an holographic non-Fermi liq-uid [1].In this note we initiate the study of gapless surfaceexcitations living on interfaces of different holographicmaterials. To that end, we consider the holographicmodel constructed in [2] (for related development see [3–8]) which describes the interface of a fractional topologi-cal insulator and an ordinary (topologically trivial) insu-lator. Briefly, the model describes a fermion coupled tomassless gauge fields, with spatially varying mass profile m ( x ); index theorems then guarantee the existence of afermionic bound state on the surface . Since the fermionsare expected to be strongly correlated (by virtue of theircoupling to deconfined gauge fields), it is interesting tostudy the physics of these boundary excitations usingvarious probes. The model also has charged bosonic excitations, but they are notexpected to form bound states at the interface. We find belowthat there is no symmetry breaking in the model, i.e. the bosonsare not condensed.
Here we start such an investigation by studying thethermodynamics of the system. We find that the matterlocalized at the interface is compressible – the density d varies smoothly with the chemical potential µ , with d (cid:48) ( µ ) (cid:54) = 0 for all values of the chemical potential (and T (cid:28) µ ). Furthermore, the excitations around this stateseem to exhibit a scaling symmetry with hyperscalingviolation, within the precision of our numerics. Theseresults suggest the existence of a Fermi surface for theboundary excitations.The plan of this note is as follows: in section II weintroduce the holographic setup and describe the inho-mogeneous backgrounds we construct, where the surfaceexcitations are put at finite density. In section III wediscuss the thermodynamics of the system. We end bydiscussing directions for further exploration of the natureof these and other boundary excitations. II. THE HOLOGRAPHIC SETUP
We are interested in describing the surface excitationsat non-zero temperature T and finite chemical potential µ . The starting point is then a black hole in AdS × S .We use the conventions of [9–11], work in the Poincarecoordinates, and set the radius of curvature of AdS tounity: ds = 12 π T (cid:18) − f ( ρ ) h ( ρ ) dt + h ( ρ ) d (cid:126)x (cid:19) + dρ ρ + d Ω d Ω = dθ + sin θ d Ω + cos θ dφ f ( ρ ) = 1 − ρ h ( ρ ) = 1 + 1 ρ We are interested in 7-branes embedded in
AdS andwrapping the 3-sphere inside S , localized at an angle θ on the five-sphere. The 7-brane embedding is character-ized by the variation of θ as function of a radial coor-dinate ρ in AdS, and in our case one of the boundarycoordinates x . The embedding is a minimal surface sub-ject to asymptotic boundary conditions which encode theparameters of the problem: temperature, chemical poten-tial and the mass parameter m . The precise definition of a r X i v : . [ h e p - t h ] N ov those asymptotic quantities appears below.The embedding can end smoothly when the 3-sphereshrinks to zero size [12], ρ sinθ = 0, which can happen inone of two ways: • If ρ (cid:54) = 0 and sin θ = 0, the 7-brane ends ata finite radial coordinate in AdS, away from thehorizon, whose value depends (monotonically) onthe mass parameter m . Such embeddings aredubbed ”Minkowski embeddings”; they correspondto states with no density . For zero temperatureand µ < | m | these are the only possible embeddingsand therefore in this parameter range the system isnecessarily at zero density. • On the other hand, if ρ = 0, then − < sin θ < µ > | m | .In order to describe an interface we study the systemwith a mass profile m ( x ) depending on a single boundarydirection x , interpolating between m ( x ) = − M , a casewhich by convention we call the topological insulator, and m ( x ) = M which corresponds in our sign conventions toa regular insulator. The system is insulating in the bulk,but supports gapless excitations on the interface (where m ( x ) = 0). We are interested in probing the physics ofthose excitations.To that effect we turn on a constant chemical po-tential µ . The effect of the chemical potential is easiestto understand at zero temperature. In that case, if wechoose to satisfy µ < M , the chemical potential would in-duce no finite density in either one of the bulk insulators,but it would induce finite density at the interface, in theregion where m ( x ) < µ . The embedding then interpo-lates between the Minkowski embedding (ending at somefinite radial coordinate), and the black hole embedding(ending at the black hole horizon).In practice, in order to avoid numerical instabilitiesassociated with the abrupt transition between the twotypes of embeddings , we choose the mass profile suchthat a small density is induced at the edges of the spa-tial interval. While the embedding is then of the ”blackhole” type throughout the spatial interval, the quantitiescalculated are largely independent of the mass profile, aslong as the density induced is at least moderately peakedat the interface. Minkowski embeddings with strings attached are possible in prin-ciple, but they give subleading contributions to the thermody-namics for any finite density. For a system in diffusive thermal equilibrium the chemical po-tential is constant even for inhomogeneous configurations. Though it is not necessary for our purposes, it is interesting toconstruct solutions which correspond to phase boundaries in theholographic context, along the lines of [13].
Ansatz and Equations of Motion
To describe the embedding we choose to parametrizethe worldvolume in terms of the radial coordinate ρ andthe spatial coordinate x . In our parametrization, the em-bedding is characterized by a single function χ ( ρ, x ) =cos θ ( ρ, x ). The asymptotic behavior of the coordinate χ at large ρ , χ ( ρ, x ) → m ( x ) ρ , determines the mass pro-file m ( x ), we choose to approximate a step function as m ( x ) = M e − ax − M for a (cid:29)
1. As long as the mass pro-file is sufficiently narrow and steep, the parameters a, M do not significantly influence the physical quantities wecalculate .Having chosen a parametrization, we add a worldvol-ume gauge field A (choosing 2 πα (cid:48) = 1). It is straight-forward to find the following DBI action: S DBI = N T ρ (1 − χ ) f h (cid:112) I χ + I A + I int I χ = 1 − χ + ( ∂ ρ χ ) + ( ∂ x χ ) ρ hI A = (1 − χ ) (cid:18) − h ( ∂ ρ A ) f − ∂ x A ) ρ f (cid:19) I int = − ρ ( ∂ ρ χ∂ x A − ∂ ρ A ∂ x χ ) where the form of I χ and I A reveals the elliptic nature ofthe equations. The field A and the coordinate x are cho-sen to be dimensionless, like all other quantities appear-ing in the action. With this choice we measure dimen-sionful quantities in units of the temperature T . There-fore, the asymptotic value of A at large ρ determinesthe chemical potential in units of T , A → ˜ µ = µT .The regularity of the solution near the boundary of theembedding is discussed in [14]. For a black hole embed-ding we need to satisfy ∂ ρ χ = 0 and A = 0 at ρ = 0.Asymptotically in the ρ direction, we set a UV cutoff ρ = Λ and choose A ( ρ = Λ , x ) = µ to set the chem-ical potential, and χ ( ρ = Λ , x ) = m ( x ) ρ to specify themass profile. Finally, we choose homogeneous Neumannboundary conditions for both functions at the boundariesof the spatial interval. The Solutions
We solve the equations of motion numerically usingpseudospectral methods implemented in Matlab, utiliz-ing a Chebyshev grid to discretize the equations, andsolving the resulting set of algebraic equations usinga Newton-Raphson-Kantorovich iteration. The conver-gence of the solution with the linear size of the grid isexponential. The results shown are for a grid of 900 We checked that changing those parameters by two orders ofmagnitude changes the calculated quantities by less than onepart in 10 . Radial DirectionSpatial Direction E m bedd i ng F un c t i on FIG. 1: The embedding function χ near the horizon.The physical parameter is µ/T = 2.points (giving accuracy of about one part in 10 for thevalues of the fields).Before extracting physical properties, we briefly dis-cuss the features of the solutions. In figures (1-3) wedisplay a few characteristic solutions, at various valuesof the parameters.In figure 1 we display the embedding function χ . Thespatial variation of χ increases towards the horizon,where its profile is correlated with the boundary den-sity: the density is maximal for χ = 0 and decreases tozero as χ → ±
1. The value χ = ± S pa t i a l D i r e c t i on FIG. 2: Contour plot of the gauge potential A thatshows the spatial inhomogeneity. The horizontaldirection is a section of the the radial coordinate. Thephysical parameter is µ/T = 3 .
4. sity.
III. THERMODYNAMICS
The thermodynamics of the system is completely spec-ified when expressing the number density n in terms ofthe chemical potential µ and the temperature T . Equiv-alently one can specify the free energy F ( µ, T ), then n = ∂F∂µ . We choose to calculate the density n , whichis read off the subleading fall-off of the gauge field, A → ˜ µ − T ρ n (˜ µ ) as ρ → ∞ , where ˜ µ = µT .The subleading term in the expansion of χ , related tothe chiral condensate in e.g. [10], is found to vanish at theinterface for all our solutions. This is a result of a paritysymmetry preserved by our mass profile, and indicatesthe absence of symmetry breaking and the existence ofnon-trivial fluid at the interface. The physics of that fluiddepends only on mu and T , leading to the dimensionalanalysis below.Since our background is inhomogeneous, there is sep-arate information in densities per unit volume (at fixedspatial position x ), and the corresponding densities perunit area (when integrating over the spatial direction x ).Here we mainly discuss the free energy and number den-sities per unit volume , at the center of the interface x = 0,ending with comments on the integrated quantities.Based on dimensional analysis in 3+1 dimensions, F = T F (˜ µ ), since T, µ are the only scales in the problem [15].For relativistic systems which form a Fermi surface at lowtemperatures, for example for free massless fermions , F (˜ µ ) ∼ ˜ µ + ... . The leading term in this expansioncorresponds to n ∼ µ , the contribution to the densityfrom a putative Fermi surface.The subsequent terms in the small ˜ µ expansion giveinformation on low energy excitations of the system. Inparticular, the second term in the expansion gives theleading temperature dependence of the free energy at lowtemperatures. Thus, this term is responsible for the tem-perature scaling of the entropy S in that regime. Theentropy scales differently for systems with or without aFermi surface. For degenerate relativistic fermions, onehas S f ∼ T at low temperatures, whereas for relativisticbosons S b ∼ T (see for example [16]). The larger en-tropy at low temperatures reflect the fact that systemswith a Fermi surface have more low energy excitations inthe degenerate limit .The difference between bosons and fermions can alsobe parametrized in terms of the hyperscaling violation The normalization of this expression depends on the normaliza-tion of the fields in the microscopic theory, for example for usthis would involve the normalization of the fermion bound statewavefunction. Note that the scaling of the entropy depends only the densityof low energy states and does not assume those excitations areweakly interacting.
Log(chemical potential)
Log ( den s i t y ) FIG. 3: Interface density (normalized to unity at ˜ µ = 1)as function of the chemical potential, on a log-log scale.The power law behavior of n (˜ µ ) at large ˜ µ is apparent,as are deviations from this behavior at lower values of ˜ µ .exponent θ , such that S ∼ T − θ with θ = 0 for bosonsand θ = 2 for fermions. The origin of the differencein scaling is the nature of the excitations of the Fermisurface. The low energy excitations are constrained tobe close to the Fermi surface, thus reducing the effectivedimensionality of the system.After parametrizing the possible small temperature be-havior, we can present the results of our model. Figure(4) displays a log-log plot of the density as function of thechemical potential (both measured in units of tempera-ture). The small temperature expansion discussed in thissection corresponds to the large ˜ µ limit. The functionaldependence encoded in figure (4) reveals a few interestingfeatures of the system studied.The most robust feature is the leading behavior at large˜ µ , n (˜ µ ) ∼ ˜ µ α , with α (cid:39) n (˜ µ ), α is fairlyrobust, changing between 2.95 and 3.1 (for the displayedgraph α = 2 . d with respect to the chemical potential is non-zero atlow temperatures, i.e. the state at the interface is com-pressible . A Low temperature compressible state, in theabsence of symmetry breaking, can be taken as indicativeof a Fermi surface (see e.g. the discussion in [17]).Reading off the subleading terms in the ˜ µ expansion isless robust. Nevertheless, one can develop an asymptoticexpansion in µ , a low temperature expansion. To probenature of the low energy excitations, parametrized by thevalue of θ , we fit the asymptotic data to the form n (˜ µ ) ∼ ˜ µ + c ˜ µ . We find that the value for c is non-zero with great A quadratic term would correspond to θ = 3, violtaing the bound θ ≤ d −
1, based on the asymptotic scaling of the entanglemententropy expected to hold in all local quantum field theories [18]. confidence , as c = 0 (the value required for hyperscalingto hold) is about 3 sigma away from the central valuefor the fit for c . This fits the expectation of hyperscalingviolation exponent θ = 2, the value corresponding tosmall excitations of a Fermi surface.In summary, we have examined the density per unitvolume induced on the interface, revealing structure con-sistent with the fermionic nature of the boundary excita-tions. Unfortunately, the densities per unit area cannotbe reliably computed using our solutions. Tracking thechange in the boundary layer width as function of the pa-rameter ˜ µ is necessary for that purpose. However, in ourappraoch the width of the domanin wall is determined by a rather than by dynamical properties of the interface.It would be interesting to improve on our discussion andcalculate reliably the densities per unit area; we leavethat for future work. IV. CONCLUSIONS AND OUTLOOK
In this note we constructed inhomogeneous solutionsdescribing the holographic dual of an interface between afractional topological insulator and an ordinary insulator.We have used these solutions to start the investigation ofthe gapless strongly correlated fermions expected to liveon that interface. Indeed, by studying the low tempera-ture expansion of the thermodynamics of the system wefound results consistent with these expectations.A few directions for further study are immediately ap-parent. It is interesting to study more general modelswith fermions bound to an interface, for example mod-els with a non-trivial dynamical exponent z , or modelswith fermionic excitations only – though bosons are notexpected to form gapless bound states at the interface,they are likely to limit the precision of the analysis. It isalso desirable to study such systems at zero temperatureand finite density. In that limit the effects of a degen-erate Fermi gas are likely to be more pronounced andcan be more cleanly studied. As mentioned in the text,moving away from the ”thick” domain wall regime is aninteresting, albeit challenging, problem.Most interestingly, since the boundary fermions are ex-pected to be strongly correlated (i.e. coupled to masslessgauge fields), these types of systems are natural candi-dates for an holographic dual for a non-Fermi liquid. It istherefore particularly interesting to search for further sig-natures of the fermionic nature of the interface, for exam-ple oscillations in response functions, or non-analyticityin the fermionic spectral densities of the system. We ex-pect that general enough models will exhibit characteris-tics of strange metallic behavior, and we hope to return Study of subsequent terms in the ˜ µ expansion should reveal con-tributions from bosonic degrees of freedom. However, the sce-nario of having only bosonic contributions is excluded. to some of these questions in the near future.Finally, beyond that particular example of an holo-graphic interface, there is a whole range of fascinatinginterface physics to be explored holographically. We ex-pect that the methods used here will be useful in explor-ing this research direction. Acknowledgments
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