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Demystifying the Holographic Mystique

D. V. KhveshchenkoDepartment of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599

Thus far, in spite of many interesting developments, the overall progress towards a systematic studyand classification of various ’strange’ metallic states of matter has been rather limited. To that end,it was argued that a recent proliferation of the ideas of holographic correspondence originating fromstring theory might offer a possible way out of the stalemate. However, after almost a decade ofintensive studies into the proposed extensions of the holographic conjecture to a variety of condensedmatter problems, the validity of this intriguing approach remains largely unknown. This discussionaims at ascertaining its true status and elucidating the conditions under which some of its predictionsmay indeed be right (albeit, possibly, for a wrong reason).

Condensed Matter Holography: The Promise

Among the outstanding grand problems in condensedmatter physics is that of a deeper understanding and clas-sification of the so-called ’strange metals’ or compressiblenon-Fermi liquid (NFL) states of the strongly interactingsystems. However, despite all the effort and a plethoraof the important and non-trivial results obtained withthe use of the traditional techniques, this program stillremains far from completion.

As an alternate approach, over the past decade therehave been numerous attempts inspired by the hypo-thetical idea of holographic correspondence which origi-nated from string/gravity/high energy theory (where it isknown under the acronym AdS/CFT ) to adapt its mainconcepts to various condensed matter (or, even more gen-erally, quantum many-body) systems at finite densitiesand temperatures1,2.

In its original context, the ’bona fide’ holographic prin-ciple postulates that certain d + 1-dimensional (’bound-ary’) quantum field theories (e.g., the maximally super-symmetric SU(N) gauge theory) may allow for a dual de-scription in terms of a string theory which, upon a propercompactification, amounts to a certain d+2-dimensional(’bulk’) supergravity. Moreover, in the strong couplinglimit (characterized in terms of the t’Hooft coupling con-stant λ = g2N ≫ 1) and for a large rank N ≫ 1 of thegauge symmetry group, the bulk description can be fur-ther reduced down to a weakly fluctuating gravity modelwhich can even be treated semiclassically at the lowest(0th) order of the underlying 1/N -expansion.

In the practical applications of the holographic con-jecture, the partition function of a strongly interactingboundary theory with the Lagrangian L(φa) would thenbe approximated by a saddle-point (classical) value ofthe bulk action described by the Lagrangian L(gµν , . . .)which includes gravity and other fields dual to theirboundary counterparts1,2

Z[J ] =

∫ N∏

a=1

Dφa exp(−∫

dtdd~xL(φa)) ≈

exp(−∫

drdtdd~xL(gµν , . . .)) (1)

evaluated with the use of a fixed background metric

ds2 = gttdt2 + grrdr

2 +∑

ij

gijdxidxj (2)

while any quantum corrections would usually be ne-glected by invoking the small parameter 1/N .Thus, considering that the task of solving a system of

coupled Einstein-type differential equations can be fairlystraightforward conceptually (albeit not necessarily tech-nically), the holographic approach could indeed becomea novel powerful tool for studying the strongly corre-lated systems and a viable alternative to the practicallyimpossible problem of summing the entire perturbationseries. Specifically, if proved valid, some of the broad’bottom-up’ generalizations of the original holographicconjecture known as ’AdS/CMT’ (which, in many in-stances, should have been more appropriately called ’non-AdS/non-CFT’) could indeed provide an advanced phe-nomenological framework for discovering new and classi-fying the already known types of the NFL behavior.Thus far, however, a flurry of the traditionally de-

tailed (hence, rarely concise) publications on the topichave generated not only a good deal of enthusiasm butsome reservations as well. Indeed, the proposed ’ad hoc’generalizations of the original string-theoretical construc-tion involve some of its most radical alterations, wherebymost of its stringent constraints would have been aban-doned in the hope of still capturing some key aspects ofthe underlying correspondence. This is because the tar-get (condensed matter) systems generically tend to beneither conformally, nor Lorentz (or even translationallyand/or rotationally) invariant and lack any supersym-metric (or even an ordinary) gauge symmetry with some(let alone, large) rank-N non-abelian group.Moreover, while sporting a truly impressive level

of technical profess, the exploratory ’bottom-up’ holo-graphic studies have not yet helped to resolve such cru-cially important issues as:• Are the conditions of a large N , (super)gauge sym-metry, Lorentz/translational/rotational invariance of theboundary (quantum) theory indeed necessary for estab-lishing a holographic correspondence with some weaklycoupled (classical) gravity in the bulk?• Are all the strongly correlated systems (or only a pre-

2

cious few) supposed to have gravity duals?• What are the gravity duals of the already documentedNFLs?• Given all the differences between the typical condensedmatter and string theory problems, what (other than thelack of a better alternative) justifies the adaptation ’adverbatim’ of the original (string-theoretical) holographic’dictionary’? and, most importantly:• If the broadly defined holographic conjecture is indeedvalid, then why is it so?

Considering that by now the field of CMT hologra-phy has grown almost a decade old, it would seem thatanswering such outstanding questions should have beenconsidered more important than continuing to apply theformal holographic recipes to an ever increasing numberof model geometries and then seeking some resemblanceto the real world systems without a good understandingas to why it would have to be there in the first place. Incontrast, the overly pragmatic ’shut up and calculate’ ap-proach prioritizes computational tractability over phys-ical relevance, thus making it more about the method(which readily provides a plethora of answers but maystruggle to specify the pertinent questions) itself, ratherthan the underlying physics.

On the other hand, there exist of course importantongoing efforts towards, both, constructing various ’top-down’ holographic models3 as well as trying to deriveholography from the already known concepts such as arenormalization procedure on the information-relatedtensor networks4. However, the former approach formu-lated in terms of such objects as D-branes remains tobe rather exotic and somewhat hard to connect to fromthe CMT perspective, whereas the latter one (which, inpractice, amounts to a massive use of the Stratonovichand Trotter transformations combined with numericalsolutions of the resulting flow equations of the functionalRG-type) has yet to deliver a well-defined bulk geometry,other than the basic AdS with the dynamical z = 1(or its Lifshitz modification with z = 2), that wouldbe reminiscent of those metrics which are extensivelyutilized in the ’bottom-up’ studies (see below).

Condensed Matter Holography: The Evidence

The circumstantial evidence that would be typicallyinvoked in support of the general idea of holography in-cludes such diverse topics as thermodynamics of blackholes, hydrodynamics of quark-gluon plasma and unitaryultracold Fermi-gases, geometrization of quantum entan-glement entropy, etc. However, while possibly attestingto the validity of some aspects of the holographic conceptin general, those arguments may not be immediately per-tinent to the specific condensed matter systems.

Therefore, for the broadly generalized holographic con-jecture to prove relevant and gain a predictive power inthe latter context its predictions would have to be sys-tematically contrasted against experimental data as wellas the results of other, more conventional, techniques pro-

viding a preliminary insight. Also, the holographic cal-culations would have to be carried out for (and allow fora cross-check between) a whole range of the thermody-namic and transport quantities, including specific heat,DOS, charge and spin susceptibilities, electrical, thermal,and spin conductivities, electron spectral function, etc.By some (admittedly, risky) analogy with any evidence

claimed to support, e.g., the ’science’ of UFO and otherparanormal phenomena - most (but not all) of which canbe readily dismissed - in order to ascertain the true sta-tus of the CMT holography one would need to focus on(and identify the physical origin(s) of) those cases of fac-tual agreement that can be deemed reliable and repro-ducible. It especially concerns those instances where theholographic predictions were reported to agree quantita-tively with the results of some exact analytic5 or almostexact (e.g., Monte Carlo)6 calculations (see below).At the ’ad hoc’ level the holographic approach has

already been opportunistically applied to a great va-riety of condensed matter systems which includes the’strange’ Fermi and Bose metals describing quantum-critical spin liquids, supersolids, quantum smectics andnematics, Mott insulators, (in)coherent conductors, itin-erant (anti)ferromagnets, Quantum Hall effect, grapheneand other Dirac/Weyl metals, multi-channel Kondo andother quantum impurity models, etc.Routinely, the holographic calculations would be per-

formed in the ’mean-field’ approximation (i.e., at the 0th

order of the would-be 1/N -expansion) and then com-pared to some selected sets of experimental data on thesystems that tend to lack any supersymmetry, are char-acterized by the number of species N ∼ 1 (such as spin,orbital, and/or valley components) and have only moder-ate (as opposed to very strong) interactions. The abovecaveats notwithstanding, however, such studies would of-ten seek nothing short of (and occasionally claim to havefound) a quantitative agreement with the data.Also, many of the early CMT holographic calculations

were carried out by ’seeking where the lights are’ and uti-lizing just a handful of the historic black-hole solutions,the central among which is the Reissner-Nordstrom (RN)one that asymptotically approaches the AdSd+2 (anti-de-Sitter) andAdS2×Rd geometries in the UV and IR limits,respectively1,2

gtt = −f(r)/r2, grr = 1/r2f(r), gij(r) = δij/r2 (3)

and is accompanied by the scalar potential A0 = µ(1 −r/rh). The emblackening factor f(r) = 1 − (1 +µ2)(r/rh)

d+1 + µ2(r/rh)2d incorporates the chemical po-

tential µ and vanishes at the horizon of radius 1/rh pro-portional to the Hawking temperature T which is sharedby the bulk and boundary degrees of freedom. Notably,this radius remains finite even when the temperatureT = (d + 1 − (d − 1)µ2)/4πrh vanishes, thereby givingrise to the non-vanishing entropy S(T → 0) 6= 0 and sug-gesting that the corresponding boundary theory couldprovide a description of some isolated ’quantum impu-rity’, rather than a correlated many-body state with a

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non-trivial spatial dispersion.Accordingly, the boundary fermion propagatorG(ω, k)

demonstrates the behavior dubbed as ’semi-locallycritical’7 which is characterized by a non-trivial fre-quency, yet a rather mundane momentum, dependence

G(ω,k) = 1/(ak + bkω2νk) (4)

where ak, bk, and νk are smooth functions of the mo-mentum k. In the space-time domain, the correspondingbehavior

G(τ, x) ∼ exp(−S(τ, x)) (5)

is governed by the (semi)classical action S(τ, x) =√

x2 + (1− 2νkF)2 ln2 τ and is consistent with that of

the spatially (almost) uncorrelated ’impurities’, each ofwhich exhibits a characteristic d = 0 quantum-criticalscaling. Such NFL behavior indeed bears some superficialresemblance to that found in a certain class of the heavy-fermion compounds8 and DMFT calculations9. However,it is also plagued with such spurious features as poten-tially multiple Fermi surfaces, dispersionless peaks, andlog-oscillating ω-dependence7.Given the multitude of the experimentally discovered

NFLs, the ’locally-critical’ scenario would seem to bemuch too limited to encompass more general types ofthe real-life NFL scenaria where, both, the x- and τ - (or,correspondingly, ω- and k-) dependencies of the propa-gator would be distinctly non-trivial. In light of thatrealization, the focus of the early holographic studies hasgradually shifted towards a broader class of geometries,including such intrinsically non-Lorentz-invariantmetricsas the Lifshitz, Shroedinger, helical Bianchi, etc. A par-ticular attention has been paid to the static, diagonal,and isotropic ’hyperscaling-violating’ (HV) metrics withthe radial dependence of the form (up to a conformalequivalence),

gtt ∼ −r2θ/d−2z, grr = gii ∼ r2θ/d−2 (6)

its finite-T version featuring the additional factor f =1− (r/rh)

d+z−θ. The dynamical exponent z controls theboundary excitation spectrum ω ∝ qz, while θ quantifiesa non-trivial scaling of the interval ds → λθ/dds, thescaling-(albeit not Lorentz-) invariant case of the Lifshitzmetric corresponding to θ = 0. The physically sensiblevalues of z and θ must satisfy the ’null energy conditions’

(d− θ)(d(z − 1)− θ) ≥ 0, (z − 1)(d+ z − θ) ≥ 0 (7)

signifying a thermodynamic stability of the correspond-ing geometry.A proper choice of θ determining the ’effective di-

mension’ deff = d − θ was discussed in the context offermionic entanglement entropy which singles out thevalue θ = d− 1, consistent with the notion of the Fermisurface as a d−1-dimensional membrane in the reciprocal(momentum) space10.

The HV metrics arise among the solutions of variousgeneralized gravity theories, including those with mas-sive vector (Proca) fields, Horava gravity, as well as theEinstein-Maxwell-dilaton (EMD) theory which includesan additional neutral scalar (’dilaton’) field11,

LEMD =1

2(R+

d(d+ 1)

L2)− (∂φ)2

2−U(φ)−V (φ)

2F 2µν (8)

, where the dilaton potential U(φ) and the effective gaugecoupling V (φ) are given by some (typically, exponential)functions of φ. This model is believed to be dual toa strongly interacting boundary theory at finite densityand temperature which is deformed away from the hyper-scaling limit by a relevant neutral scalar operator dual tothe dilaton. The HV solutions of the coupled Einstein-Maxwell equations have also been obtained by taking intoaccount a back-reaction of the fermionic matter fields onthe background geometry12. It should be noted, though,that apart from a few exceptions13 such analyses werelimited to the hydrodynamic (Thomas-Fermi) descrip-tion of the fermions, while leaving out their more subtle(exchange and correlation) effects.One purported success of the theory (8) (with U(φ) ∼

φ2 and V (φ) = const) was a numerical fit to the ex-perimentally observed power-law behavior of the mid-infrared optical conductivity14

σ(ω) ∼ ω−2/3 (9)

in the normal state of the superconducting cuprates suchas BSCY CO15. Notably, though, such an agreement wasfound over less than half of a decade (2 < ωτ < 8) whilethe later studies did not confirm it16. Besides, it was alsoargued to be intermittent with the alternate ’universal’,∼ ω−1, behavior.Another notable example is the holographic

calculation17 of the numerical prefactor C in theempirical Homes relation between the superfluid density,critical temperature, and normal state conductivity18

ρs = CTcσ(T0+c ) (10)

This relation is obeyed by a large variety of supercon-ductors, including the weakly coupled (yet, sufficientlydisordered) conventional ones where a justification forthe use of holography would be rather hard to come byand where the Homes’ relation can be shown to holdwithin the scope of the traditional theory19. The holo-graphic theory of Ref.17 claimed its success with findingthe prefactor C ≈ 6.2 to be close to the experimental val-ues in the cuprates which happen to be C ≈ 8.1 and 4.4for the ab- and c-axes transport, respectively. Somewhatironically, though, this would seem like a case where, inthe absence of a compelling underlying reason, a perfectquantitative agreement could do more harm than goodto the cause.Arguably, a stronger case could have been made if the

range of the possible values of C were unusually nar-row, thus potentially hinting at some universality of the

4

result. This, however, does not seem to be the case.Moreover, the holographic calculation of Ref.17 does notreadily reproduce any other empirical relations such asthe Schneider’s one20

TcT ∗c

= 2

√

ρsρ∗s

(1 − 1

2

√

ρsρ∗s

), (11)

thereby suggesting that the reported agreement with ex-periment may have been largely fortuitous.Also, to further strengthen the case for holography,

some works tend to use increasingly more and more com-plicated models with a larger number of fields while op-erating under the assumption that the IR physics shouldbe fairly universal and insensitive to such details. Theresults, on the contrary, indicate that there is little uni-versality, as by varying the contents of the bulk theoryone can dramatically alter the boundary behavior - e.g.,reproducing on demand the entire phase diagram of thecuprates with all the four main phases as well as thedomains of their coexistence in the extended EM modelwith the additional vector and scalar fields which repre-sent the competing orders21.There has also been some effort on the experimental

side, such as the report of measuring the shear viscosity-to-entropy density ratio in the cuprates22 which wasfound to be close to its celebrated holographic lowerbound value23

η/S = 1/4π (12)

It should be noted, though, that instead of such genuinetwo-particle characteristic as the correlation function ofthe stress tensor’s component Txy in Ref.22 the viscos-ity η was deduced from the single-particle electron spec-tral function. Furthermore, even the bound (12) itselfis known not to be totally universal, its value becom-ing lower, e.g., in the absence of the spatial rotationalinvariance24

ηxzxz/S = (1/4π)(gxx/gzz) (13)

Yet another example of a successful application ofthe holographic approach was presented in Ref.6

where an impressive agreement was found between theholographically computed frequency-dependent (finite-temperature) conductivity σ(ω) and the Monte Carlo re-sults for the Bose-Hubbard model whose critical behaviorbelongs to the universality class of the O(2)-symmetricalWilson-Fisher critical point. Upon analytically continu-ing to the real frequencies this agreement gets progres-sively worse for ω < 2πT , though25. Most intriguingly,the original work of Ref.6 utilized the EM holographicmodel with a rather special higher-derivative term

∆L = CabcdFabF cd (14)

where Cabcd is the Weyl tensor, thus making one won-der as to the reasons behind a seemingly unique role of

this model as the potential Bose-Hubbard’s dual. How-ever, in the later Ref.25 nearly identical results were ob-tained with the use of a much less exotic EMD modelwith U(φ) = φ2 and V (φ) = 1 + αφ. Thus, the previ-ously reported agreement with the MC results (limitedto ω > 2πT ) appears to be rather common, which ob-servation takes away much of the intrigue surroundingthe holographic model equipped with the term (14) andmakes less pressing the need for understanding its other-wise inexplicably serendipitous success.

In light of the above, one would be led to the con-clusion that at its current stage the CMT holographystill lacks a true ’smoking gun’ that could lend a firmsupport for this approach (albeit, possibly, in somereduced, rather than its most broad, overreaching,form). Therefore, it is perhaps not too surprisingthat there is still no consensus, neither on the exactimplications of the reports of some apparent agreementbetween the holographic predictions and the results ofother approaches and/or experimental data, nor thegeneral applicability and the principal limitations of theholographic approach itself.

Emergent Geometry and ’Holography Light’

Regardless of those practices in the field of CMT holog-raphy that make it prone to criticism, its general idea isundeniably appealing. Indeed, it falls very much in linewith some of the most profound paradigms in quantummany-body physics, including those of the bulk-edge cor-respondence and the notion of running couplings in theprocess of renormalization alongside the changing scale ofenergy/length/information. The former has recently re-ceived a lot of attention with the advent of topological in-sulators/superconductors and Dirac/Weyl (semi)metals,while the latter provides a standard framework for relat-ing the bare (UV) to the effective (IR) physics.

Arguably, of all the holographic studies the most im-portant is the quest into its possible underlying physicalcause(s). This concerns, first and foremost, those situa-tions where some precise - analytic or numerical - agree-ment has been found between the results of the holo-graphic and some conventional approaches. One suchrecent example is provided by the random infinite-rangefermion-hopping models that were studied in Ref.5, draw-ing from the previous analyses of the random infinite-range spin-coupling systems26.

These models also share a number of common featureswith the Kondo lattice27 and matrix28 ones, their uni-fying physical theme being that of a single ’quantumimpurity’ interacting self-consistently with a local bath.This behavior makes them potentially amenable to thedescription based on the holographic ’semi-locally criti-cal’ scenario of Refs.7.

Indeed, in Refs.5 an emergent invariance under thegroup of reparametrizations was discovered and its rela-tion to the ’semi-local’ AdS2 ×Rd geometry was demon-strated by establishing an exact agreement between the

5

holographically computed single-particle propagators inthis (non-fluctuating) geometry7 and the two- and four-point correlation functions of certain exactly solvablerandom fermion-hopping models.Notably, such a perfect agreement was achieved with-

out invoking a small 1/N parameter, thereby suggestingthat the 1/N -corrections (that would have been presentin the original string-theoretical AdS/CFT correspon-dence) may, in fact, be absent altogether. More precisely,although the large-N limit indeed had to be taken in therandom spin-coupling model of Ref.26 (after taking thelimit of a large number N of the lattice sites ), the ’semi-local’ behavior of its random fermion-hopping counter-parts sets in already at N = 1 (but still for N >>∞)5.Generalizing the results of Refs.5, one can construct

a whole family of ’semi-local’ NFL regimes described bythe propagator G(τ) which obeys the integral equations(n = 2 in Refs.5)

∫

G(τ1, τ)Σ(τ, τ2)dτ = δ(τ1 − τ2),

Σ(τ1, τ2) = λGn(τ1, τ2)Gn−1(τ2, τ1) (15)

that are manifestly invariant under an arbitrary changeof variables τ = f(σ)

G(τ1, τ2) = [f ′(σ1)f′(σ2)]

−1/2nG(σ1, σ2) (16)

To derive these equations from the (disorder-averaged)Hamiltonian of a random infinite-range fermion-hoppingmodel

H =

n∑

k=1

ck∑

ai,bi

ψ†a1 . . . ψ

†akψb1 . . . ψbk (17)

where the parameters ck need to be fine-tuned29, whilefor the their arbitrary values the system conforms to thegeneric behavior described by the n = 2 case. Thisis somewhat reminiscent of the situation with the ex-actly soluble 1d spin-S chain model whose behavior ismarkedly non-generic for all S > 1/230.Specifically, the solution of Egs.(15) demonstrates an

expressly NFL behavior

G(τ) ∼ τ−1/n (18)

or, equivalently, Σ(ω) ∼ ω1−1/n which can also be re-produced holographically with the use of a massive bulkfermion subject to the AdS2×Rd geometry and describedby the action Lf = iψγµ∂µψ −mψψ provided that thefermion mass m is chosen as follows

∆ψ =1

2− (m2R2 − q2E2)1/2 = 1/2n (19)

where R is the AdS radius, q is the fermion charge, and Emeasures the particle-hole asymmetry5. Thus, regardlessof whether or not the ’semi-locally critical’ scenario ofRef.7 is immediately applicable to any real-life materials,its predictions might still prove quite useful, as far as the

properties of some properly crafted random spin-couplingand fermion-hopping models are concerned.

Notably, though, the dimension (19) arises in the so-called ’unstable’ boundary CFT which is dual to AdS2

7,while in the ’stable’ one (to which the former is supposedto flow under a double-trace deformation) the operatordimension would be ∆+ = 1

2 + (m2R2 − q2E2)1/2 > 1/2,thus making the value (19) unattainable for any integern > 1.

It is also worth noting that the behavior similar toEq.(18) can be envisioned in the context of, e.g., sub-Ohmic spin-bath models which, incidentally, can be for-mulated in terms of the localized Majorana fermions.Among other things, it gives rise to the ubiquitousLorentzian (’Drude-like’) frequency dependencies of thevarious observables, their width being given by the in-elastic phase relaxation rate31. It might be interesting toexplore such a possible connection further29, as well asto reproduce Eq.(18) in the holographic models aspiringto describe the Kondo physics32.

Also, while being characteristic of the purely classi-cal (non-fluctuating) asymptotic near-horizon geometry7

the emergent reparametrization symmetry (15) does notrise to the same level as that in the elaborate constructsof the original (string-theoretical) holographic correspon-dence where the bulk supports a full-fledged quantum(super)gravity theory that only becomes classical in thelarge-N limit1. Nevertheless, it is quite remarkable thatthe quantum systems in question allow for some of theirproperties to be expressed in purely geometrical terms.In that regard, emergent classical metrics and con-

comitant effective gravity-like descriptions are not thatuncommon, the most remarkable example being pro-vided by the intriguing relationship between quantumphysics and classical geometry in the form of the fa-mous extremal area law of entanglement entropy33. Toa somewhat lesser extent, same can be said about suchconnections found in thermodynamics of phase transi-tions (’Fisher-Ruppeiner metric’)34, quantum informa-tion and tensor network states35, QHE hydrodynamics(’Fubini-Study metric’)36, Chern classes of the Blocheigenstates of momentum37, Berry phase of the adiabatictime evolution38, etc.Importantly, the emergent geometric structures such

as effective metrics, curvatures, spin connections, etc.can indeed create the appearance of a limited form ofsome bulk-boundary correspondence (’holography light’)that can be inadvertently mistaken for manifestations ofthe hypothetical full-fledged CMT holography.

Holographic Phenomenology: The Cuprates

Obtaining verifiable holographic predictions can beparticularly instructive in those situations where exper-imental data exhibit robust scaling dependencies, thushinting at a possible (near-)critical regime that mightbe amenable to a holographic scaling analysis39. Onesuch example is provided by the holography-inspired phe-

6

nomenologies of the cuprates which focused on the robustpower-law behaviors of the longitudinal electrical conduc-tivity, Hall angle, and magnetoresistivity that violatesthe conventional Kohler’s law40

σ ∼ T−1

tan θH ∼ T−2,

∆ρ/ρ ∼ ρ2 (20)

In the previously proposed scenaria, Eqs.(20) were ar-gued to indicate a possible existence of two distinctly dif-ferent scattering times: τtr ∼ 1/T and τH ∼ 1/T 2 whichcharacterize the longitudinal vs transverse41 or charge-symmetric vs anti-symmetric42 currents. Yet another in-sightful proposal was put forward in the framework ofthe marginal Fermi liquid phenomenology43.In contrast to the anomalous transport properties of

the cuprates, their thermodynamic ones are more stan-dard, including the Fermi-liquid-like specific heat C ∼ T(except for a possible logarithmic enhancement44). A re-cent attempt to rationalize such experimental findingswas made in Refs.45,46. Under the assumption of anunderlying one-parameter scaling the dimensions of theobservables were expressed in terms of the minimal setwhich includes the dynamical critical exponent z along-side the dimensions of the (mass) density ∆n and effectivecharge ∆e, the two accounting for, roughly speaking, thewave function renormalization and vertex corrections, re-spectively.The linear thermoelectric response is then described in

terms of a trio of the fundamental kinetic coefficients

J = σE− α∇TQ = T αE− κ∇T (21)

where the Onsager’s symmetry is taken into account andthe off-diagonal entries in the 2× 2 matrices σ, α, κ rep-resent the Hall components of the corresponding conduc-tivities.To set up the scaling relations that reproduce Eqs.(20)

and other experimentally observed algebraic dependen-cies one has to properly account for the time reversal andparticle-hole symmetries. Also, one should be alerted tothe fact that the exponents governing the temperaturedependencies in the kinetic coefficients would be the sameas those appearing in their frequency-dependent opticalcounterparts39. However, the previous analyses based onthe semiclassical kinetic equation show that such leading(minimal) powers of T may or may not actually survive,depending on whether or not the quasiparticle disper-sion and Fermi surface topology conspire to yield com-parable rates of the normal and umklapp inelastic scat-tering processes47. It would appear, though, that in thecuprates, both, the multi-pocketed (in the under- andoptimally-doped cases) as well as the extended concave(in the over-doped case) hole Fermi surfaces comply withsuch necessary conditions.As an additional consistency check, the kinetic coeffi-

cients are expected to be consistent with the Fermi liq-uid relations S ∼ (T/eσ)dσ/dµ, νN ∼ (T/eB)dθH/dµ

where the r.h.s. are proportional to the Fermi surfacecurvature but do not contain such single-particle charac-teristics as scattering time or effective mass and, there-fore, might be applicable beyond the coherent quasipar-ticle regime.It was found in46 that, somewhat surprisingly, most of

the experimental data favor the rather mundane solution

z = 1, ∆e = 0, ∆n = 1 (22)

which imposes the following relation on the Seebeck (S),Hall Lorentz (LH), and Nernst (νN ) coefficients (barringany Sondheimer-type cancellations)

[S] + [νN ]− [LH ] = 0 (23)

In the optimally doped cuprates, the experimentallymeasured thermopower, apart from a finite offset term,demonstrates a (negative) linear T -dependence48. Asregards the Hall Lorentz and Nernst coefficients, thedata on the untwinned samples of the optimally dopedY BaCuO were fitted into a linear dependence, LH ∼ T ,whereas νN was generally found to decrease with in-creasing T , thus suggesting [νN ] < 049. Moreover, theNernst signal increases dramatically with decreasing tem-perature, which effect has been attributed to the super-conducting fluctuations and/or fluctuating vortex pairswhose (positive) contribution dominates over the quasi-particle one (the latter can be of either sign, depend-ing on the dominant type of carriers) upon approachingTc. Besides, νN turns out to be strongly affected by aproximity to the pseudogap regime and can even becomeanisotropic.The ’Fermi-surfaced’ solution (22) should be compared

with the significantly more exotic one obtained in Ref.45

z = 4/3, ∆e = −2/3, ∆n = 2 (24)

where the thermoelectric coefficients S ∼ T 1/2, LH ∼T, νN ∼ T−3/2 do not obey Eq.(23). Also, in this casean agreement with the linear behavior of LH reported inRef.49 is guaranteed by imposing it as one of the con-stitutive relations. In spite of this predestined success,though, this scheme fails to reproduce the linear (up toa constant) thermopower, although it claimed the alter-nate S = a−bT 1/2 dependence to provide an even betterfit to the data48.Also, the solution (24) predicts C ∼ T 3/2 and, there-

fore, appears to be at obvious odds with the observedthermodynamic properties as well44. Moreover, it pre-dicts the linear in T (longitudinal) Lorentz ratio L =κ/Tσ similar to its Hall counterpart, contrary to a con-stant L, as suggested by the scenario (22).Although an additional experimental effort is clearly

called for in order to discriminate more definitively be-tween the above predictions, it should be mentioned thatmore recently a slower-than-linear temperature behaviorof the (electronic) Lorentz ratio has been reported50.Lastly, the solution (24) features θ = 0 and strong (in-

frared) charge renormalization (∆e < 0). Since in the

7

presence of a well-defined d-dimensional Fermi surfacethe ’hyperscaling-violation’ parameter is expected to co-incide with its co-dimension, hence θ = d−110, the valueθ = 0 hints at a point-like (’Dirac’) Fermi surface (ifany). In contrast, Eq.(22) suggests a rather simple phys-ical picture where neither the Fermi liquid-like disper-sion, nor the effective charge demonstrate any significantrenormalization. Thus, to construct a viable phenomeno-logical description of the optimally doped cuprates onemight be able to do away without introducing the ad-ditional charge exponent ∆e, contrary to the assertionsmade in Refs.45,51.It might also be worth mentioning that another long-

time baffling, the underdoped, phase of the cuprates hasrecently been downgraded to a potentially simpler orga-nized state of matter with some of its properties beingFermi-like52.Also, the recent attempt to match the exponents (24)

with the predictions of the semi-phenomenological ’un-particles’ model showed that it can only be achieved atthe expense of a further, multi-flavored, extension of thatscenario53. However, the sheer complexity of the pro-posed construction seems to indicate rather clearly thatthe latter can hardly provide a natural (let alone, mini-mal) theory of the anomalous transport in the cuprates.Moreover, in the last of Refs.53 the method of fractional

derivatives was employed to generalize the Maxwell’s andcontinuity equations,

∂αµFµν = Jν

∂αi Ji + ∂0ρ = 0 (25)

However, it can be readily seen that Eqs.(25) are onlyconsistent when the dimension of the vector potential[Ai] = [∂αi ] = α equals z (which was chosen as z = 1 inRef.53 ), thus prohibiting it from taking any anomalousvalue.In fact, a gradual realization of how strained can be

the attempts to squeeze the holographic phenomenol-ogy of the cuprates into the ’Procrustes bed’ of thetwo-parametric HV geometries may have already startedmaking its way into the holographic community54.Another viable candidate for applications of the

holographically-inspired scaling theory is provided bythe self-consistent strong-coupling solutions aimed at de-scribing the properties of certain 2d and 3d antiferro-magnetic metals. Analyzed under the assumption of amomentum-independent NFL self-energy, Σ(ω) ∼ ω1−α,it yields the critical exponents as sole functions of thespatial dimension55

α = 1/2− 1/zb, zb = 4d/3, zf = 1/(1− α),

ν = 1/(2 + zbα) (26)

Notably, the hyperscaling relations are still obeyed, re-sulting in the anomalous dependencies

C ∼ T 1−α, σ ∼ ωα−1, χs ∼ Tα−1 (27)

In the presence of disorder these results are likely to bemodified, though56.Another recent work57 utilized the functional

RG technique, thus obtaining a different solutionwhich might be pertinent to the AFM metals:zf = zb = 3/2, θ = 0, ν = γ = 1. Reproducingthese solutions and searching for the new ones29 posesan interesting challenge and presents a important testfor the holographically inspired scaling theory of thesematerials.

From AdS/CMT to Holographic Transport

A great many of the early works on CMT holographyutilized the standard holographic recipe for computingelectrical conductivity and other kinetic coefficients asboundary limits of the ratios between the convergent (as-sociated with a response) and divergent (associated witha source) terms in the solutions of the radial classicalequations1. This prescription would result in, e.g., theexpression for the electrical conductivity

σ(ω) =1

iω

B

A=

1

iωr2−d

d lnFridr

|r→0 (28)

where the component of the field tensor Fri = A +Brd−1/(d − 1) is a bulk dual of the boundary electriccurrent.Moreover, in the absence of any dilaton accounting for

the scale dependent couplings and bringing about a de-pendence on the radial holographic coordinate, the physi-cal observables become anomaly-free and scale-invariant,hence unaffected by renormalization. In that case theycan be cast in a purely algebraic form in terms of thehorizon metric, as per the ’membrane paradigm’58. Forinstance, the electrical conductivity and charge suscepti-bility read

χ = (

∫ rh

0

drgrrgtt(−g)1/2 )

−1

σ = (−ggrrgtt

)1/2gii|rh , (29)

where g is the determinant of the metric. Per the Ein-stein’s relation he two are related via the diffusion coef-ficient, σ = χcD.However, while it might be possible to predict such

general features as the exponents in the power-law fre-quency dependences of the optical electrical and thermalconductivities39 by using the formal prescriptions similarto Eq.(28,29), the latter do not properly account for theactual physical contents of the boundary theory and apotentially intricate interplay between its different scat-tering mechanisms.A list of the proposed sources of current and/or mo-

mentum relaxation (which is often confusingly referredto as ’dissipation’) employed in holography includes ran-dom boundary and horizon potentials, helical (BianchiV II0-type) and spatially periodic (’Q-lattice’) geome-tries, massive gravity, axion fields, etc.59.

8

Common to all the different approaches, though, is theunifying fact that the conductivity and other kinetic co-efficients would be typically given by a sum of two terms

σ(ω) = σ0 +e2n2

χPP (γ − iω)(30)

although in the ’top-down’ constructions using the DBIaction3 Eq.(30) would be replaced by σ =

√

σ20 + σ2

1 .In Eq.(30) the momentum susceptibility χPP = E+P

is equal to the enthalpy density. and the first term cor-responds to the (potentially, universal) diffusion-limitedcontribution to the conductivity that survives in theparticle-hole symmetric limit of zero charge density (n→0). It is often associated with the processes of ’paircreation’ by which a neutral system develops a non-vanishing (yet, finite) conductivity in the absence of mo-mentum relaxation (γ → 0), while the DC (ω → 0) con-ductivity of a finite density system is governed by thesecond term in (30), becoming infinite in the limit γ → 0.Moreover, in the case of Q-lattices and 1d periodic po-

tentials (but not those that break translational invariancein all the spatial dimensions) the conductivity can stillbe expressed in terms of the horizon data, thus general-izing the case of zero charge density and no momentumrelaxation. Interestingly, though, even for generic (’non-Q’) lattices which break translational symmetry in allthe spatial dimensions the conductivity can be obtainedby solving some linearized, time-independent, and forcedNavier-Stokes-type equations of an effective (charged andincompressible) fluid living on the horizon61, thus ex-tending the ’membrane paradigm’ along the lines of thegeneral concept of a fluid-gravity correspondence62. Thelatter allows for a dual description of the bulk gravity the-ory in terms of the hydrodynamics of a certain boundaryfluid whose stress-energy tensor acts as a source for theboundary metric. However, while unveiling yet anotherintrinsically geometric aspect of an underlying quantumdynamics, the fluid-gravity correspondence is clearly notidentical to (albeit, possibly, far more general than) theoriginal string-theoretical holography.In certain axion models63, the elastic rate can be

readily computed although the resulting behavior γ ∼max[T 2,m2]/T can hardly remain physical in the T → 0

limit. Remarkably, though, at m =√2rh =

√8πT the

linearized (classical gravitational) equations of motionappear to be exactly solvable, yielding the frequency-and momentum-dependent response functions G(ω, k) ina closed form and also signaling an emergent SL(2, R)×SL(2, R) symmetry63.

In the more general situation, momentum conservationwould be broken by some operator O with the dimension∆O and the corresponding elastic scattering rate γ canbe expressed in terms of the spectral density D(ω, k) =Im < O(ω, k)O(−ω,−k) >59

γ =1

dχPP

∫

dkk2ImD(ω, k)

ω|ω→0 (31)

In the EMD context, the following non-universal behav-ior was obtained by treating the pertinent ’random Isingmagnetic field’ disorder perturbatively64,

σ(T ) ∼ T 2(z−1−∆O)/z (32)

Also, the generalized Harris criterion for the relevance ofdisorder

∆ < d+ z − θ/2 ∆ < (d− θ)/2 + z (33)

gets saturated for ∆O = (d − θ)/2 + z, resulting in theuniversal, yet θ-dependent, behavior64

σ ∼ T (θ−2−d)/z ∼∼ 1/(Sm2) (34)

where the concomitant scaling of the entropy density,S ∼ T (d−θ)/z, has been taken into account. In Eq.(34)the second factor can be interpreted as the ’thermalgraviton mass’, m ∼ T 1/z. Thus, it is only in the ex-treme ’locally-critical’ limit z → ∞ where the famousprediction σ ∼ 1/S of Ref.65 could be confirmed withoutany sleight of hands.Moreover, should the relevant bosonic modes happen

to be transverse and, therefore, protected from develop-ing a mass on the grounds of unbroken gauge invariance(as it would be the case in any SU(2)- or U(1)-symmetricspin liquid state), the problem of computing the conduc-tivity would require a full non-perturbative solution.Notably, even in the HV model with z = 3/2 and θ = 1

(which values satisfy the relation z = 1 + θ/d saturatingthe second of Eqs.(7)) proposed as a viable candidate forthe bulk dual of the theory of 2d fermionic matter cou-pled to an overdampted bosonic (e.g., gauge) field10,66

the conductivity exponent in Eq.(34) ((θ−2−d)/z = −2)would be different from its target value (−1). This obser-vation, too, might be pointing at some deeper problemwith reproducing the linear normal state resistivity of thecuprates in the fremework of the HV holographic models.More recently, the main focus of the CMT holographic

phenomenology began to shift towards developing ageneral ’holography-augmented’ transport theory63,67,68.This commendable effort strives to establish general re-lations between the transport coefficients and find theirbounds (if any) that could remain valid regardless ofthe (in)applicability of the generalized holographic con-jecture itself. Such relations are expected to hold inthe hydrodynamic regime governed by strong inelasticinteractions where the rates of momentum relaxationdue to elastic disorder, lattice-assisted inelastic (Umk-lapp) electron-electron and electron-phonon (outside ofthe phonon drag regime) scattering are all much smallerthan the universal inelastic rate Γ ∼ T of the normalcollisions controlling the formation of a thermalized hy-drodynamic state itself23,65.Notably, the onset of hydrodynamics is a distinct prop-

erty of the strongly correlated systems, as it is unattain-able in the standard FL regime where all the local (in thek-space) quasiparticle densities nk are nearly conserved.

9

The hydrodynamic regime does not set in for d = 1 ei-ther due to the peculiar 1d kinematics which facilitatesthe emergence of infinitely many (almost) conserved den-sities.In the hydrodynamic regime, the holographic results

have also been systematically compared to those of thehydrodynamic67 and memory matrix68 formalisms whichdo not rely on the existence of well-defined quasiparti-cles. The kinetic coefficients obtained by means of thesealternate techniques appear to be similar to the holo-graphic expressions (30), identifying the density n withthe current-momentum susceptibility χJP =< J |P >= nwhich quantifies the contribution to the current from’momentum drag’. Such formalism provides an elegantand physically transparent language for discussing, e.g.,a decoupling of the electric current onto the coherentcomponent parallel (in the functional sense) to the mo-mentum P and the orthogonal, incoherent, one whichhas no overlap with P and is responsible for the univer-sal conductivity of the neutral (particle-hole symmetric)system

J =χJPχPP

P + Jincoh (35)

Although under a closer inspection the hydrodynamic re-sults were found to be subtly different (beyond the lead-ing order) from the earlier holographic ones, such differ-ences have been reconciled by including the correctionsto σo and the residue of the pole in Eq.(30) at ω = −iγ(’Drude weight’)63,67. This agreement between the holo-graphic and hydrodynamic/memory matrix analyses wasargued to provide an additional evidence supporting (atleast, the transport-related aspects of) the former.It should be noted, though, that the AC kinetic coeffi-

cients akin to Eq.(30) appear to have a strictly monotonicfrequency dependence which is governed by the coherentzero-frequency Drude peak whose presence reflects theexistence of a (nearly) conserved momentum which over-laps with the electrical current for n 6= 0.On the other hand, a non-monotonic - or, else, strictly

universal (frequency-independent) - behavior would benecessary in order for the conductivity to comply withthe holographic sum rules proposed in Refs.6:

∫ ∞

0

(σ(ω)− σ(∞))dω = 0

∫ ∞

0

(1

σ(ω)− 1

σ(∞))dω = 0 (36)

These sum rules account for a transfer of the spectralweight from the coherent Drude peak to the incoherenthigh-frequency tail. Such a non-monotonic behavior de-velops at ω ∼ Γ and may or may not be detectable bythe hydrodynamic/memory matrix analyses, though.One of the most striking predictions of the memory

matrix calculations is that of the reported absence of(many-body) localization and diffusion-dominated metal-lic transport in strongly interacting ’strange metals’69. It

would seem rather surprising, though, that while impos-ing robust conductivity bounds in d = 1 and 2 (wherein the latter case the lower bound even seems compara-ble to the upper one), this approach fails to find suchbounds in d > 2 where, according to the general wisdom,localization would be even less likely to set in.At a deeper level, a fully quantum description of lo-

calization (or a lack thereof) still seems to remain out ofreach in all the existing holographic treatments of disor-der. In fact, any ’mean-field’ approximation is likely to bein principle incapable of taking into account not only thelocalization-related phenomena (multiple acts of elasticscattering) but also the interference ones (multiple actsof consecutive elastic and inelastic scattering).For one thing, the former type of the conductivity cor-

rections would depend not only on the elastic scatter-ing rate but also on some rather special inelastic ones(Cooperon/Diffuson phase-breaking), whereas the latterwould typically acquire its temperature dependence viathe fermion occupation factors. Neither source of the T -dependence can be readily envisioned in the variationalapproach of Ref.69, though. While suitable for analyzingclassical percolation, this approach operates in terms ofthe Kirchoff’s law for a random network of classical re-sistors and employs the quadratic Thomson’s variationalaction akin to that for maximum entropy production. Inlight of such potential caveats it does not seem surpris-ing that the results of Ref.69 suggest the diffusive metallicbehavior even at weak repulsive interactions strengths in2d - which would then be in conflict even with the per-turbative Altshuler-Aronov theory.Leaving out the remaining possibility of finding a bet-

ter variational ansatz that could yield a lower conductiv-ity than the simplest one of a constant current/densityemployed in Ref.69 and resulting in the above lowerbound, it is worth noting that the scenario of Ref.69 mightalso be limited to the rather special class of models whereno coupling between the translation symmetry-breaking(Stueckelberg) degrees of freedom and the Maxwell fieldis allowed.To that end, a further investigation into the issue70

showed that, in contrast, no finite (let alone, universal)lower bound seems to exist in the more general mod-els which include generic couplings between the axionswith the expectation values Φi ∼ xµδ

µi and vector and/or

scalar fields

∆L =∑

n=1

(gµν∂µΦi∂νΦi)n[anF

2µν + bnφ

2] (37)

Thus, conceivably, the scenario of ’many-body delocal-ization’ proposed in Ref.69 is neither generic, nor gen-eralizable beyond the scopes of the variational analysis.However, the results of Refs.70, too, should be taken cau-tiously, as, e.g., the second reference finds the conductiv-ity to be potentially negative, depending on the strengthof the couplings an, bn.In that regard, although these and other predictions

would certainly benefit from clear disclaimers about the

10

limited range of the holographic parameters for whichphysically sensible predictions (e.g., positive conduc-tivity) can be made, conspicuously, such limits do notreadily follow from the phenomenological ’bottom-up’approach itself. In particular, it would be premature toapply any of the aforementioned holographic scenaria tothe analysis of, e.g., the 2DEG systems demonstratingthe apparent metal-insulator transition, complete withits peculiar scaling properties71.

Holographic Transport: Dirac/Weyl materials

An important example of the systems whose transportproperties can be studied systematically and then com-pared to the holographic predictions is provided by theinteracting Dirac/Weyl (semi)metals. In the past, thetransport properties of (pseudo)relativistic systems havealready been addressed in the context of, both, interact-ing 2d bosons and fermions72. Such studies utilized thestandard method of quantum kinetic equation which of-fers a viable insight that can be compared to the specificpredictions of the general theory of holographic trans-port. The fermionic variant of this problem pertinentto the case of Coulomb-interacting graphene was stud-ied in the so-called two-mode approximation where theemphasis was made on the conservation of charge andenergy densities, as well as momentum density which isequivalent to the energy current up to the higher-orderdissipative corrections (which feature is specific to the(pseudo)relativistic dispersion)73.

In the case of a weak Coulomb coupling α = e2/hv <<1 the equilibration rate of all other, non-conserved, modeswould be of order Γnon−cons ∼ α2| lnα|T which ishigher than that of the conserved ones, Γcons ∼ α2T ,thanks to the 2d kinematic (logarithmic) divergence ofthe Coulomb collision integral. Thus, the conservedmodes can be singled out, while neglecting all those thatundergo a faster relaxation73.

It turns out, however, that the system possesses an-other (nearly)conserved density which is the ’imbalance’mode corresponding to the total number of electronsand holes (as opposed to their difference related to thecharge density)74. Away from the neutral (particle-holesymmetric) regime its relaxation takes place through theAuger-type processes with the still slower rate of orderΓimb ∼ T 4/µ3. Close to the neutrality point the results ofthe two- and three-mode analyses appear to differ some-what, thus showing that the imbalance mode can indeedimpact the low-energy hydrodynamic behavior74.

It would, therefore, be desirable to carry out the three-mode calculations for the SU(N)-symmetric fermionswith N ≫ 1, focusing on the regime 1/N ≪ α2 ≪ 1where a comparison with the holographic predictions canbe made29.

Notably, the kinetic equation tends to yield the con-ductivity which develops the Drude peak followed by adip at ω ∼ Γ, while no other peaks (nor zeros) emerge, incontrast to the holographic/hydrodynamic/memory ma-

trix result (30). Besides, it remains to be seen as to howthe explicitly computed corrections to the optical con-ductivity of graphene75

σ(ω)

σ(∞)= 1 +

C

| lnω| (38)

match with the general prediction25

σ(ω)

σ(∞)= 1 +

∑

n

Cn(T

ω)∆n (39)

where the sum is taken over the operator expansion ofthe product of two current operators.Another quantity of special interest would be the

Lorentz ratio (ordinary, as well as the Hall one) as afunction of frequency. In the DC limit and near the neu-trality point this ratio is dramatically enhanced becausethe heat and charge currents are controlled by two differ-ent mechanisms: momentum relaxation vs electron-holepair creation. This prediction was found to be in a goodagreement with the recent data on graphene76 showing a20-fold enhancement (although the ratio η/s ∼ 10 es-timated in Ref.76 seems to indicate that even a free-standing graphene may not be that strongly coupled,after all).It would also be interesting to generalize the analysis

to d = 2+ ǫ dimensions where the effects of the Coulombinteractions are expected to be less dramatic and the sep-aration between the slowly vs rapidly decaying modesbecomes less pronounced29.Besides, one would need to introduce yet another

nearly conserved density describing an imbalance be-tween the numbers of quasiparticles in the vicinity of thetwo different Weyl points. This ’chiral’ density incorpo-rating (for ǫ = 1) the effects of the 3d chiral anomaly givesrise to such concomitant transport phenomena as a nega-tive magnetoresistance77. Some of its underlying geomet-ric aspects can be observed even at the semiclassical levelwhere the phase-space dynamics is naturally describedin terms of the Bloch curvature37 Ωk = ǫij <

∂Ψ∂ki

| ∂Ψ∂kj >which affects the semiclassical equations of motion

dr

dt=∂ǫ

∂k−Ωk × dk

dt,

dk

dt= − ∂ǫ

∂r+ΩB × dr

dt(40)

The current relaxation is then described by the equations

∂νTiν = F iνJν − γT it,

∂νJν = αǫµνλρF

µνFλρ (41)

Reproducing these effects in their entirety would provideanother check point for (and further advance) the holo-graphic transport theory78.In that regard, one should also mention a series of

works aimed at connecting holography to the more con-ventional field theoretical analysis79. It seems, though,that the actual graphene and 3dWeyl/Dirac metals alike

11

can hardly provide a viable playground for the CMTholography, as the effects of interactions in these ma-terials are either mild or even outright weak. It would,therefore, be quite interesting to find some realization oftheir counterparts ’on steroids’ in the condensed matterrealm.

The extended multi-mode kinetic analysis of graphenemay also allow for a deeper understanding of the role ofviscosity in electron transport. The viscous terms be-come relevant for η > (enw)2/σ where w is the width ofthe graphene sample and such analysis can even be per-formed in the framework of the Navier-Stokes equation,this time in the real space80. Incidentally, this regimehas recently been studied in the ultra-pure 2d metalPdCoO2

81, although the estimated ratio η/S ∼ 106 canhardly allow one to view it as ideal fluid.

Also, the problem of a hot spot relaxation ingraphene74 presents a specific example of the general’quench’-type phenomena which, too, have been exten-sively addressed by the CMT holography82. The 1dvariant of this problem involves two reservoirs of differ-ent temperatures TL and TR which are brought into athermal contact, its solution featuring a non-equilibriumstationary state characterized by a definite temperature(TLTR)

1/2 and constant energy flux (this solution doesnot seem to readily extend to the limit of either van-ishing temperature, though). Moreover, this stationarystate was shown to spread outward in the form of shockwaves and was related to the Lorentz-boosted black-holemetric in the bulk, thus suggesting yet another explicitholographic connection.

To that end, it was conjectured in Refs.82 that asimilar behavior would occur in higher dimensions aswell. However, the recent analyses of the hot spotrelaxation in 2d graphene74 suggest that a more involvedscenario where, both, the resistive and viscous effectsinterfere with one another might be realized. Notably,the recent holographic work of Refs.83, too, shows amore involved dynamics of the expanding stationarystate.

Analogue Holography Demonstrators

The above examples of ’emergent geometry’ prompt asystematic quest into the apparent holography-like rela-tionships that would be governed by the already known,rather than some hypothetical, physics. To that end, thegeneral holographic concept can benefit from the possi-bility of being simulated in various controlled ’analogue’environments.

One prospective design of a ’holography simulator’ wasproposed for implementation in flexible graphene andother semi-metallic monolayers (silicene, germanine, sta-nine, etc.)84. Such stress-engineered desktop realizationsof the system of 2d Dirac fermions in a curved geometry85

can also be grown on commensurate substrates (e.g.,h − BN) in order to endow the bulk fermions with afinite mass via hybridization.

In Ref.84 a number of situations were discussed wherea physical edge of the curved graphene flake which sup-ports (almost) non-interacting massive Dirac fermionspropagating in a curved 2d space exhibits a behavior thatwould be typically attributed to the effects of some phan-tom 1d interactions. In contrast to the aforementioned’semi-local’ scenario of Refs.7, though, it is the momen-tum dependence of the 1d propagator G(ω, k) that tendsto become non-trivial.Specifically, in graphene the artificial gauge and metric

fields represent the elastic (phonon) degrees of freedom,their effective vector and scalar potentials

A0 ∼ uxx + uyy, Ax ∼ uxx − uyy, Ay ∼ uxy (42)

being composed of the components of the strain tensor86,uij =

12 (∂iuj + ∂jui + ∂ih∂jh).

Here ui(x) and h(x) are the local in- and out-of-planedisplacements of the monolayer, respectively, while thevalley-specific fields Aµ have opposite signs at the twodifferent Dirac points. Computing the massive free Diracfermion propagator in a curved space and taking itsboundary limit one obtains the large-scale asymptoticbehavior (4) governed by the geodesic action Shol(τ, x)for the background metric. For instance, in the case of agraphene flake shaped as a surface of rotation with theline element (here r and φ are the normal polar coordi-nates)

dl2 = dr2[1 + (∂h(r)

∂r)2] + r2dφ2 (43)

with h(r) ∼ (R/r)η for r ≥ a the ’warp factor’ g(r) ∼1/r2η+2 diverges at small r. One then obtains thegeodesic action

S(τ, x) = m√

τ2 + (Rxη)2/(η+1) (44)

which reveals an unconventional behavior of the bound-ary propagator as a function of the distance along theedge, thus suggesting the ’holographic’ dynamical criti-cal exponent z = η/(η + 1).It is instructive to compare the asymptotic (4) with the

action given by (44) with the propagator of 1d fermionsinteracting via a pairwise potential U(x) ∼ 1/xζ whereζ < 1. Using the standard 1d bozonization techniqueand matching the large-x asymptotics, one finds that (44)can mimic the spatial decay of the 1d propagator in thepresence of such interactions, provided that η = (1 −ζ)/(1 + ζ).Another instructive example is provided by the line

element

dl2log = dr2 +R2 exp(−2(r/R)λ)dφ2 (45)

in which case the geodesic action reads

S(τ, x) = m√

τ2 +R2(ln x/a)2/λ (46)

12

For λ = 1 and at large x the propagator decays alge-braically, G(0, x) ∼ 1/xmR which is reminiscent of thebehavior found in the 1d Luttinger liquids, while forλ 6= 1 Eq.(46) yields a variety of stretched/compressedexponential asymptotics which decay faster (for λ < 1)or slower (for λ > 1) than any power-law. For instance,by choosing λ = 2/3 one can simulate a faster-than-

algebraic spatial decay, G(0, x) ∼ exp(−const ln3/2 x),in the 1d Coulomb gas (σ = 1) which is indicative of theincipient 1d charge density wave state.For other values of λ Eq.(46) reproduces the behav-

ior in the boundary theory governed by the interactionU(x) ∼ (lnx)(2/λ)−3/x. Although the physical origin ofsuch a bare potential would not be immediately clear,multiplicative logarithmic factors do routinely emerge inthose effective 1d interactions (’double trace operators’)that are associated with various marginally (ir)relevanttwo-point operators. On the experimental side, theboundary correlations can be probed with such estab-lished techniques as time-of-flight, edge tunneling, andlocal capacitance measurements, thus potentially helpingone to hone the proper analogue-holographic ’dictionary’.This holography-like correspondence once again sug-

gests that some limited form of a bulk-boundary relation-ship might, in fact, be quite robust and hold regardless ofwhether or not the systems in question possess any par-ticular symmetries, unlike in the original AdS/CFT con-struction. Naively, this form of correspondence can evenbe related to the Einstein’s equivalence principle (i.e.,’curvature equals force’), according to which free motionin a curved space should be indistinguishable from theeffect of a physical interaction (only, this time around, inthe tangential direction).As an alternate platform for doing analogue hologra-

phy, optical metamaterials have long been considered ascandidates for simulating such effects of general relativ-ity as event horizons, redshift, black, white, and wormholes, inflation, Hawking radiation, dark energy, multi-verse, Big Bang and Rip, metric signatures transitions,’end-of-time’, and other cosmological scenaria.However, the earlier proposals focused on the effective

3 + 1-dimensional metrics which requires some intricateengineering of the locked permittivity (ǫ) and permeabil-ity (µ) tensors87

ǫij = µij = gij√−g/|g00| (47)

More recently, it was proposed to use extraordinary (TM-polarized) monochromatic photons with the dispersionω2 = k2z/ǫxy + k2xy/ǫzz for simulating the effective 2 + 1-dimensional metrics

gττ = −ǫxy, grr = gφφ/r2 = −ǫzz (48)

in the media with µ = 188.

In the hyperbolic regime, ǫxy > 0 and ǫzz < 0, themomentum component kz can then be thought of as aneffective frequency, whereas ω plays the role of mass.In Ref.89 it was shown that the experimentally attain-able two-component metamaterial configuration consist-ing of alternating metallic and insulating (flat or cylin-drical) layers can simulate some of the HV geometries.The boundary propagator describing static spatial corre-lations of the optical field on the interface between themetamaterial and vacuum was found to behave as

< Eω(x)E−ω(0) >∼ exp[−|ωx|θ/z] (49)

which should be contrasted to its counterpart in anisotropic medium with a (negative) dielectric constant,< Eω(x)E−ω(0) >∼ exp(−ω|x|). Experimentally, suchcorrelations can be studied by analyzing the statistics ofa non-local optical field distribution with the use of holo-graphic and speckle interferometry. Albeit still requir-ing a careful engineering of the dielectric media, somehallmark features of the analogue holography could, inprinciple, be detected and then compared with, e.g., cor-relators of the vertex operators in the theory of a stronglyself-interacting 2d bosonic field, akin to that describingthe thermodynamics of a fluctuating elastic membrane.Lastly, one could further elaborate on yet another (his-

torically, far more extensively studied) potential play-ground for analogue holography which is provided by theacoustic realizations of ’emergent gravity’90. By imple-menting such proposals one might also hope to estab-lish a putative bulk-boundary correspondence (alongsideits own ’dictionary’) for a broader variety of the com-bined (pseudo)gravitational backgrounds, thereby gain-ing a better insight into the physical origin(s) of the ap-parent holography-like properties of various CMT andAMO systems.Together with the systematic comparison between the

predictions of the CMT holography and other, more tra-ditional, approaches and/or experimental data it wouldbe a necessary step towards vetting the intriguing,yet speculative, holographic ideas before the latter canbe rightfully called a novel technique for tackling the’strange metals’ and other strongly correlated materials.In any event, though, it would seem rather unlikely that ahands-on expertise in string theory will become a manda-tory prerequisite for understanding the cuprates.The author gratefully acknowledges the support from

and/or hospitality at the Aspen Center for Physics(funded by the NSF under Grant 1066293), the GalileoGalilei Institute (Florence), the Kavli Institute for thePhysics and Mathematics of the Universe (Kashiwa), theKavli Institute for Theoretical Physics (Beijing), the In-ternational Institute of Physics (Natal), and NORDITA(Stockholm) during the various stages of compiling thesenotes.

13

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