Adventures in Holographic Dimer Models
SSU-ITP-10/26, SLAC-PUB-14252
Adventures in Holographic Dimer Models
Shamit Kachru
Department of Physics and SLAC,Stanford University,Stanford, CA 94305, USA [email protected]
Andreas Karch
Department of Physics,University of Washington,Seattle, WA 98195-1560, USA [email protected]
Sho Yaida
Department of Physics, Stanford University,Stanford, CA 94305, USA [email protected]
We abstract the essential features of holographic dimer models, and develop sev-eral new applications of these models. First, semi-holographically coupling free bandfermions to holographic dimers, we uncover novel phase transitions between conven-tional Fermi liquids and non-Fermi liquids, accompanied by a change in the structureof the Fermi surface. Second, we make dimer vibrations propagate through the wholecrystal by way of double trace deformations, obtaining nontrivial band structure. Ina simple toy model, the topology of the band structure experiences an interestingreorganization as we vary the strength of the double trace deformations. Finally, wedevelop tools that would allow one to build, in a bottom-up fashion, a holographicavatar of the Hubbard model. a r X i v : . [ h e p - t h ] J a n I. INTRODUCTION
Holographic models of condensed matter systems have seen a recent surge in interest.While this program can by now point to a few successes, there is one feature of realisticsolids that is commonly not shared by their holographic stand-ins: in real solids, translationinvariance is broken to a discrete subgroup by formation of a lattice. In the holographicmodels with translational symmetry unbroken, momentum is strictly conserved as thereare no Umklapp processes to dissipate it. Consequently, in the background of an electricfield, energy and (at finite density) momentum are pumped into the system at a constantrate. This leads to unrealistic transport properties such as an unsmeared delta-functionDrude peak, among other things. Also, in heavy fermion materials, strong correlationsof itinerant electrons with localized spins on a lattice are believed to trigger interestingphenomena such as quantum criticality. It is desirable to find holographic models whichexhibit lattice structure.In [6], we constructed holographic models with translational symmetry explicitly bro-ken by fermionic degrees of freedom localized on lattice sites, interacting with a continuumgauge field. These systems have many appealing features. They naturally give rise to bulkfermions living on a lattice of AdS spacetimes; such spacetimes play a crucial role in holo-graphic non-Fermi liquids [10–13]. Thus they provide a natural home to study holographicnon-Fermi liquids without worrying about the physics of the asymptotically anti-de Sitter(AdS) Reissner-Nordstr¨om black brane, especially its large ground state degeneracy andpotential instabilities. It has also been argued from the field theory point of view [14] thata lattice of localized defect fermions interacting with continuum degrees of freedom is thebest candidate for a condensed matter system exhibiting the phenomenology of holographic A second, somewhat related, feature which is lacking in most holographic studies of condensed mattersystems is disorder. For early studies of how to include disorder, see [1, 2]. One way to work around this is to make use of the large number of internal degrees of freedom, which isone of the defining features of holographic models. A plasma sea with order N excitations, where N is alarge integer, can effectively act as a heat and momentum dump for charge carriers, leading to interestingDC conductivities [3–5]. An alternative proposal in which the whole gauge theory was forced to live on a lattice was put forward in[7]. A method for incorporating lattice defect fermions through semi-holographic techniques, which worksonly when the defect fermions are neutral under the large N gauge group, appears in [8]. Other latticedefect models with related features to the models we study will appear in [9]. non-Fermi liquidsIn this paper, we abstract a few essential features of the holographic dimer models con-structed in [6], and then extend our knowledge of these models in several directions. First,we add free band fermions to the boundary theory and then weakly mix them with thelarge N sector semi-holographically, following Faulkner and Polchinski [8]. When the large N sector undergoes a dimerization transition, it induces a dramatic change in the singletfermion sector. Namely, the melting of dimers turns a normal Fermi liquid into a non-Fermiliquid, with an accompanying change in the structure of its Fermi surface. This transitionmay be of some interest in relation to toy models of heavy fermion materials [14].We then turn to a study of dimer vibrations. In the original holographic dimer models,the dimers do not effectively talk to each other in the large N limit, and each has a discretevibration spectrum. By adding double trace deformations, we let the dimers communicatewith each other, allowing their vibrations to propagate through the whole crystal in the formof Bloch waves. As an illustration, we explicitly work out the band structure in a solvabletoy model of this type. Interestingly, we observe a reorganization in the topology of theband structure as we vary the strength of the double trace deformations.Finally, we also outline how one would deform the system by an actual fermion hoppingterm so as to liberate otherwise immobile localized defect fermions. The correspondingproblem in the bulk is beyond current brane technology, as one would have to know thenon-Abelian generalization of the Dirac-Born-Infeld (DBI) action describing the fluctuationsof the brane. Only the lowest few terms in a derivative expansion of this action are currentlyunderstood. However the deformation we identify can easily be implemented in a bottom-up model, where the bulk theory is not taken to be a system realized in string theorybut instead is governed by an effective action with free parameters which can be matchedagainst “experiments.” With this toolkit it is in principle possible to construct holographicavatars of one’s favorite lattice models such as the Hubbard model. This should allow one toparametrize his/her ignorance in terms of the coefficients in the bulk effective action. Whilesuch a program has been somewhat successful for QCD [15, 16], we will refrain from anysuch attempts in this work.The organization of this paper is as follows. We first abstract out the salient aspects ofour previous construction [6] in Sec. II. In Sec. III we then describe semi-holographic phasetransitions between Fermi liquids and non-Fermi liquids, much in the spirit of [8]. Changing FIG. 1: (a)The high-temperature phase with U (1) × U (1) symmetry. (b)The low-temperaturephase with U (1) × U (1) → U (1) via brane recombination. gears, in Sec. IV, we show how one can map out the band structure of dimer vibrations inthe holographic dimer models perturbed by simple double trace deformations. For a solubletoy model, we observe reorganization in the topology of the band structure as the strength ofthe double trace deformations increases. Lastly, in Sec. V, we outline how knowledge of thenon-Abelian DBI action would enable us to study a genuine fermion hopping deformationand how one could go about building holographic bottom-up models of generic stronglycoupled lattices. Detailed calculations supporting the plots of band structure in Sec. IVhave been relegated to the appendix, “Hypergeometric-ology.” II. PHYSICS OF A HOLOGRAPHIC DIMER
We first review the essential picture of a single dimer in the holographic dimer lattice mod-els constructed in [6]. We began with a pair consisting of a D5-brane and an anti-D5-brane,with the bulk spacetime being the AdS Schwarzschild black brane [see Fig.1(a)]. Asymp-totically, they wrap copies of AdS × S in the AdS × S . In the boundary field theory, theyintroduce localized defect fermions, transforming under fundamental and antifundamentalrepresentations of the SU ( N ) gauge group of the continuum gauge field, respectively. Aswe cool down the system, the size of the black brane shrinks down, and the D5-brane andanti-D5-brane pair up [see Fig.1(b)], spontaneously breaking U (1) × U (1) down to a diago-nal U (1). In the boundary field theory, this transition is characterized by a mildly nonlocalorder parameter involving fermions from neighboring sites as well as an open Wilson line toinsure gauge invariance [17]. This is the essence of the dimerization transition worked outin detail in [6].We now look, rather abstractly, at fluctuations of these probe branes. A. Undimerized phase: gapless spectrum
In this phase, we can study each probe brane separately. For every possible fluctuationof the probe brane, there exists a corresponding gauge singlet operator localized at a point, O J , bosonic or fermionic. Here J labels whether the operator is associated with a D5-braneor anti-D5-brane, and it will be promoted to a lattice index when we discuss a lattice ofdimers.By studying fluctuations in the bulk, it is in principle possible to work out (cid:90) dte iωt (cid:104)O J ( t ) O † J (cid:48) (0) (cid:105) = iδ J,J (cid:48) G ( ω ) (1)in detail. For example, had we stayed in the undimerized phase down to zero temperature,the induced metric on the (anti-)D5-brane is exactly that of AdS and (0 + 1)-dimensionalconformality would dictate G ( ω ) = cω − (2)with c a calculable complex number and ∆ the operator dimension of O J . When we turn to semi-holographic constructions, it is crucial to keep in mind that G ( ω )in our concrete holographic lattice model behaves differently from the Green’s function forAdS × R d − at finite temperature, appearing in the construction of holographic non-Fermiliquids. This is because the metric on the embedded D5-brane, induced from the AdS Schwarzschild black brane metric, is different from the near-horizon geometry of the AdSReissner-Nordstr¨om black brane. Strictly speaking, we can only trust our analysis in the undimerized phase down to temperatures of order T a ∼ √ λ/N . At this point the backreaction of the D5-branes on the background geometry can no longerbe neglected [6]. For most of this work we can safely neglect this complication as we are shielded from theseparametrically small temperatures by the dimerization phase transition which occurs at T a ∼
1. However,if we considered a lattice made purely of D5-branes instead of alternating D5- and anti-D5-branes (as wasalso discussed in [6]), there would be no dimerization transition. In that case the physics would be wellcaptured by the AdS gravity background down to this parametrically very low temperature. Nevertheless, one generic behavior of this type of model is that, since the probe brane istouching the horizon in this phase, the spectrum is gapless. This means that lim ω → G ( ω ) =0. B. Dimerized phase: gapped spectrum
In this phase, probe branes recombine and no longer stretch down to the horizon, leadingto a gapped spectrum. Still, the induced metric near the asymptotic boundary is that ofAdS . Let us write the asymptotic AdS metric as ds = 1 z J ( − dt + dz J ) . (3)Then near the boundary point where the anti-D5-brane stretches down ( z D5 = 0), a time-dependent fluctuation with frequency ω can be expanded as e − iωt (cid:104) α D5 ( ω ) (cid:110) z − ∆D5 + ... (cid:111) + β D5 ( ω ) (cid:8) z ∆D5 + ... (cid:9)(cid:105) (4)whereas near the boundary point where the connected D5-brane stretches up ( z D5 = 0), thesame fluctuation can be expanded as e − iωt (cid:2) α D5 ( ω ) (cid:8) z − ∆D5 + ... (cid:9) + β D5 ( ω ) (cid:8) z ∆D5 + ... (cid:9)(cid:3) . (5)The coefficients α D5 ( ω ), β D5 ( ω ), α D5 ( ω ), and β D5 ( ω ) are related by a frequency-dependentmatrix as follows: α D5 ( ω ) β D5 ( ω ) = t ( ω ) t ( ω ) t ( ω ) t ( ω ) α D5 ( ω ) β D5 ( ω ) ≡ T ( ω ) α D5 ( ω ) β D5 ( ω ) . (6)For a simple phenomenological toy model, T ( ω ) is worked out in detail in Appendix A.In the absence of any deformation of the theory, there exists a steady dimer vibrationwith frequency ω n if and only if there exists a consistent nontrivial solution with α D5 ( ω n ) = α D5 ( ω n ) = 0. This can happen if and only if t ( ω n ) = 0. This gives rise to a pole in (cid:82) dte iωt (cid:104)O J ( t ) O † J (cid:48) (0) (cid:105) at ω = ω n .Typically there exists a gap to the first dimer excitation and at low frequency correlationfunctions give lim ω → (cid:90) dte iωt (cid:104)O J ( t ) O † J (cid:48) (0) (cid:105) = iA J,J (cid:48) , (7)with all the components of A J,J (cid:48) generically nonzero.In passing, we note that for the top-down D3/D5 system of [6], the worldvolume gaugefield and the slipping mode scalar mix due to the Wess-Zumino terms in the action. Themixed sector gives rise to two towers of scalar fields depending on the angular momentum l on the internal sphere. As shown in [18] they behave like fields with m l = ( l + 3)( l + 4) and m l = l ( l − l = 0 we are hence effectively describing a massless scalar,presumably dual to the defect fermion bilinear, as well as a massive scalar mode dual to adimension 4 operator.Now that we have gathered essential information regarding the dimerization transitionand spectra in both phases, let us look at several physical applications. Anticipating the hugelandscape of large N dimer models, we will keep ∆ and all the other information (computablein explicit models) as undetermined free parameters. From here on, each subsequent sectionof the paper can be read independently. III. SEMI-HOLOGRAPHIC PHASE TRANSITIONS
A rich set of non-Fermi liquid behaviors has recently been discovered by studying thephysics of probe fermions in the asymptotically AdS Reissner-Nordstr¨om background [10–13]. The near-horizon AdS × R region of the black brane plays a crucial role in organizingand explaining this physics; the physics of the emergent “locally quantum critical” theorydual to the AdS region is what gives rise to the non-Fermi liquid behavior. However, thisblack brane is in some ways nongeneric. For instance, it suffers from a superconductinginstability in the presence of generic charged scalar fields in the bulk [19], and neutralscalar fields coupled to the kinetic term of the bulk U (1) gauge field deform the near-horizon geometry [20] to be of the Lifshitz form [21]. Even the backreaction of the fermionsthemselves deforms the near-horizon geometry to Lifshitz form at subleading orders in 1 /N [22] [shifting the AdS × R geometry, which has a dynamical critical exponent z = ∞ ,to instead have z ∼ N ]. While in many cases these deformations may leave the essentialphysics of the fermion spectral function unchanged (see [8] for a nice discussion), it is alsoreasonable to find other ways that the essential insights of [10–13] can be reproduced in amore robust setting. The AdS regions spanned by the D5- and anti-D5-branes in the top-down holographic dimer model of [6] provide an alternative way to obtain the same physics.Here, we explore this in a semi-holographic setting following [8], and we abstract the mainfeatures of the top-down model to include more generic possibilities.We begin with a large N field theory, governed by some action S strong , with the followingfeatures:1. There is a lattice of defect fermions which undergoes a dimerization transition as wevary the external parameters such as temperature [see Fig.2 for the (1+1)-dimensionalcase]. We will focus on the cases for which this parameter is temperature, but one caneasily generalize.
2. There exist fermionic operators O FJ localized at the J th lattice site, whose thermalcorrelation functions in the undimerized phase are known and gapless: (cid:90) dte iωt (cid:104)O FJ ( t ) O F † J (cid:48) (0) (cid:105) = iδ J,J (cid:48) G ( ω ) , with G ( ω ) ∼ ω − for ω (cid:29) T. (8)3. In the dimerized phase, the spectrum is gapped andlim ω → (cid:90) dte iωt (cid:104)O FJ ( t ) O F † J (cid:48) (0) (cid:105) = iA J,J (cid:48) . (9)Here, A J,J (cid:48) is nonzero (generically if and) only if J = J (cid:48) or J and J (cid:48) are paired up viadimerization.For example, for the literal D5 probe theory in AdS × S , we can take O FJ = χ † J λ N =4 ( J ) χ J (10)and work out G ( ω ) and A J,J (cid:48) as a function of external parameters. Here λ N =4 ( J ) is the N = 4 gaugino evaluated at the J th lattice site, and χ J is the probe fermion associated For instance, one can consider driving such a transition by going to finite chemical potential for the large N gauge fields at T = 0, at the cost of introducing Reissner-Nordstr¨om black branes. At sufficientlylarge chemical potential, even at zero temperature, the horizon of the extremal Reissner-Nordstr¨om blackbrane grows large, and the probe branes will transition back to a configuration where they stretch to thehorizon instead of reconnecting. It would be interesting to determine the order of this phase transition atzero temperature. with the J th site. There is also an infinite tower of similar operators of higher conformaldimension. We will, however, keep our discussion abstract.Note that we are only guaranteed of the scaling form (8) governed by the (0+1)-dimensional conformal invariance when ω (cid:29) T . Therefore, looking forward for a moment(to the stage where we mix the O F s with semi-holographic fermions) this behavior of theGreen’s function will be relevant when studying excitations close to the Fermi surface, onlyif the disconnected phase persists to very low temperatures (compared to the Fermi mo-mentum k F ). This is achievable in our models, because the temperature of the dimerizationtransition is T c ∼ a defect [6], where a defect is the lattice spacing for defect fermions, and can bedialed freely; while k F ∼ a itinerant is another free parameter, where a itinerant is the lattice spac-ing for semi-holographic itinerant free fermions, which can also be adjusted independently.Thus we make a hierarchy a defect (cid:29) a itinerant .We now semi-holographically couple this large N field theory to the free band fermion inthe spirit of [8]: S = S strong + (cid:88) J,J (cid:48) (cid:90) dt (cid:104) c † J ( iδ J,J (cid:48) ∂ t + µδ J,J (cid:48) + t J,J (cid:48) ) c J (cid:48) (cid:105) + g (cid:88) J (cid:90) dt (cid:104) c † J O FJ + (Hermitian conjugate) (cid:105) . (11)Here t J,J (cid:48) characterizes the band structure of the originally free fermion c sector, which nowmixes with the large N dimer model through the coupling constant g .The key insight of [8] is that large N factorization of the field theory (which would workeven at small ’t Hooft coupling) can be used to infer the modifications to the two-pointfunctions of the conducting c fermions arising from the coupling g . The g = 0 Green’sfunction for the c fermions is G ( k , ω ) ≡ − i N l . s . (cid:88) J,J (cid:48) (cid:90) dte iωt − i k · ( x J − x J (cid:48) ) (cid:104) c J ( t ) c † J (cid:48) (0) (cid:105) g =0 ∼ ω − v | k − k F ( k ) | (12) At very low frequency G ( ω ) will still approach zero on general grounds, but it may do so with a differentscaling dimension ∆ (cid:48) or in even more complicated ways. For notational simplicity, we made the free c fermions live on the same lattice sites as the defect fermionsdo. As just mentioned, however, we should really make the c fermions live on a much finer lattice to getthe hierarchy a itinerant ∼ k F (cid:29) T c ∼ a defect . Also as in § c and O F in (11) to be matrices inspin space. k F ( k ) the point on the Fermi surface, closest to the argument k , and N l . s . the numberof lattice sites. Then we find that for finite coupling g , after summing a geometric series oftree-level mixing diagrams, G g ( k , ω ) ∼ ω − v | k − k F ( k ) | − g G ( k , ω ) . (13)In particular, for G ( k , ω ) = cω − with ∆ ≤
1, one finds a dominant low-frequency behaviorcharacteristic of a non-Fermi liquid which has vanishing quasiparticle residue [with marginalFermi liquid behavior precisely at ∆ = 1, when the naive ω − is modified to have ω log( ω )behaviour]. For ∆ >
1, the residue does not vanish, but the theory is still novel in thatthe quasiparticle width does not agree with that of standard Fermi liquid theory. As wedescribed above, these results are true in a regime where k F (cid:29) ω (cid:29) a defect , where thezero-temperature Green’s functions used above should be a good approximation to the true(finite- but low-temperature) answers.Now, we are in a position to add one simple observation on top of the basic pictureadvocated in [8]: in holographic models which undergo a dimerization transition as inSec. II, the phase transition also drives an interesting transition in the structure of theFermi surface. The main point is that the low frequency behavior of the Green’s function (cid:82) dte iωt (cid:104)O FJ ( t ) O F † J (cid:48) (0) (cid:105) changes drastically in the dimerization transition. In the undimerizedstate, we will have non-Fermi liquid behavior just as in [8]. However, in the dimerized phase,the spectrum in the dimer sector is gapped. This means that at low frequencies, insteadof exhibiting power-law behavior, lim ω → G ( k , ω ) = A for some nonzero constant A . Thusin this phase, we have a conventional Fermi liquid whose Fermi surface is shifted from theoriginal k F .Therefore, in this semi-holographic setting, the dimerization transition of Sec. II becomesa transition between a conventional Fermi liquid phase (dimerized) and a non-Fermi liquidphase (undimerized). These transitions are somewhat reminiscent of the phase transitionsin Kondo lattice models discussed in [14] and references therein.Finally, we note that if one is purely interested in finding realizations of the non-Fermiliquid phase, without studying phase transitions of the Fermi surface, one can also simplystudy the BPS lattice model made only of D5-branes and generalizations thereof. In thiscase, the Schwarzschild AdS black brane (with probe D5-branes wrapping AdS subspacesin the AdS ) correctly captures the physics down to temperatures of order T ∼ λ / N / a defect .1 FIG. 2: Dimerized configuration of interest. Note that we made the distance between J = (2 j +1)thsite and J = (2 j + 2)th site smaller than that between J = (2 j )th site and J = (2 j + 1)th site, sothat we have one unique dimerized configuration below the critical temperature. After incorporating semi-holographic fermions, now without the constraint ω (cid:29) a defect asthere is no dimerization transition, even very low frequency excitations above the Fermisurface are governed by (8) and (13) at large N . IV. DOUBLE TRACE DEFORMATION: BAND STRUCTURE OF DIMERVIBRATIONS
Next, we visit the landscape of holographic dimer models with certain double trace defor-mations added to the boundary Lagrangian. For simplicity, let us consider the 1-dimensionalarray of dimers (see Fig.2). Let us label sites so that the J = (2 j + 1)th site is paired upwith the J = (2 j + 2)th site with j ∈ Z . Our inputs are:1. There are bosonic Hermitian operators O BJ which corresponds to a bosonic fluctuationof a probe brane originating from the J th site. The fluctuation take the asymptoticform (for frequency ω ) e − iωt (cid:2) α J ( ω ) (cid:8) z − ∆ J + ... (cid:9) + β J ( ω ) (cid:8) z ∆ J + ... (cid:9)(cid:3) . (14)2. We stay in the dimerized phase where the coefficients α j +1 ( ω ), β j +1 ( ω ), α j +2 ( ω ),and β j +2 ( ω ) are related as follows: α j +2 ( ω ) β j +2 ( ω ) = t ( ω ) t ( ω ) t ( ω ) t ( ω ) α j +1 ( ω ) β j +1 ( ω ) . (15)2Originally, the dimers are basically decoupled from each other and each has a discretevibration spectrum at ω = ω n where t ( ω n ) = 0. We now deform the theory by doubletrace operators, and determine the resulting band structure. A. Double trace deformation
We add a double trace deformation of the form∆ L d . t . = h (cid:48) (cid:88) j ∈ Z O B j O B j +1 (16)to the Lagrangian. The effect of double trace deformations on the dual gravitational de-scription is well known [23–27]. In our context, the standard recipe leads to α j +1 ( ω ) β j +1 ( ω ) = h h α j ( ω ) β j ( ω ) (17)where h ≡ (2∆ − h (cid:48) . Note that this is a relation between (2 j )th site and (2 j + 1)th sitebelonging to different dimers, not a relation between the (2 j + 1)th site and the (2 j + 2)thsite which together form a dimer.Instead of the totally reflecting boundary conditions implied by the undeformed condition α J = 0, the deformed boundary condition (17) implies that when a fluctuation associatedwith (the bulk field dual to) O B j hits the (2 j )th AdS boundary, part of the wave getsreflected back towards the (2 j − h ) instead getstransmitted to the (2 j + 1)th site. B. Band structure
In summary, the equation of motion on the probe brane (top-down or bottom-up) givesthe relation (15) whereas the double trace deformation yields the relation (17). We now lookfor most general (spatially normalizable) time-dependent, but nondissipative, modes withthese relations. For simplicity, we use the periodic boundary condition with a number oflattice sites N l . s . = 2 N dimer and then take the thermodynamic limit N dimer → ∞ at the end. These “transparent” boundary conditions modify the propagator of the scalar fields with interestingconsequences for loop corrections, as explored in [28–31]. α , β ) and evolving through the chain, we get: α β = t ( ω ) t ( ω ) t ( ω ) t ( ω ) α β = T ( ω ) α β , (18) α β = h h t ( ω ) t ( ω ) t ( ω ) t ( ω ) α β , ... (19)Continuing this way, and getting back to the original site, our periodic boundary conditionimposes α β = ht ( ω ) ht ( ω ) t ( ω ) h t ( ω ) h N dimer α β . (20)Taking the thermodynamic limit N dimer → ∞ , we conclude that there exists a nondissipativesolution with frequency ω ( k ) and with Bloch momentum k if and only ifdet ht ( ω ( k )) ht ( ω ( k )) t ( ω ( k )) h t ( ω ( k )) h − e ika e ika = 0 with k ∈ (cid:104) − πa , + πa (cid:105) . (21)The band structure is encoded in ω ( k ). C. A simple toy model
If we know T ( ω ), it is a simple matter to map out the band structure numerically. Asan illustration, let us perform this exercise for the caricature toy model described fully inAppendix A. For the special case of ∆ = 1 we display the first few bands for several valuesof h in Fig. 3. Without the double trace deformation, the full spectrum is just N dimer copies of the spectrum of a single dimer, independent of k . We see that for small h we stillhave a band structure with very narrow bands centered around the mode spectrum of theuncoupled dimers, ω n = nπa for nonzero integer n . Around h = 0 . h is the coupling constant of a double trace deformation, such a spatiallyinhomogeneous condensate would not be visible at the leading order N classical action, butonly in the order 1 free energy induced from loops. For higher values of h the lowest two Note that, as the equations of motion only depend on ω , the bands have symmetry around ω = 0. ! ! ! ! ! Ω a ! ! ! ! ! Ω a ! ! ! ! ! Ω a ! ! ! ! ! Ω a ! ! ! ! ! Ω a FIG. 3: Band structure of the global AdS toy model for ∆ = 1 and h = 0 .
05 (top left panel), h = 0 .
82 (top middle panel), h = 1 (top right panel), h = 5 (bottom left panel) and h = 50 (bottomright panel). bands undergo an interesting reorganization, changing the topology of the band structure.For very large h we once more approach degenerate k -independent bands (but now with ω ≈ α J has to vanish at each site for h = 0while β J has to vanish at each site for h = ∞ , giving t ( ω n ) = 0 rather than t ( ω n ) = 0]. V. TOWARDS A HOLOGRAPHIC HUBBARD MODEL
While the double trace deformations we introduced allow dimer vibrations, or “mesons,”to propagate through the whole crystal, they do not lead to transport of defect fermions.The double trace deformation we have introduced is the product of two U (1) J invariantoperators, so it preserves all the U (1) J global symmetries. In particular, the difference innumber of defect fermions at the two ends of each dimer, or “baryon number,” is conservedand thus the defect fermions cannot move. Incidentally, the analysis of this section goes through in the same way for fermionic operators O FJ . Itwould be interesting to explore implications of this topology change in band structure for such fermionicexcitations. FIG. 4: (a)The high-temperature phase with two branes at each site, with lattice sites equallyspaced. (b)A possible low-temperature configuration. (c)Another possible low-temperature con-figuration on which we will focus.
The only way to actually introduce a moving charge carrier seems to be to introduce acharged field or object in the bulk that can effectively carry baryon number. One potentialsuch object would be the W -boson of a U (2) non-Abelian gauge field living on the stacks ofbranes. So we imagine doing something like:1. At each odd J = (2 j + 1)th site we put two anti-D5-brane, while we put two D5-braneat each even J = (2 j )th site, and this time we equally space lattice sites (see Fig.4).At each site, we label two species of defect fermions by χ L,J and χ R,J .2. In order to make charge carriers move around, we consider deforming the Lagrangianby a conventional hopping term∆ L hop = t h (cid:88) J χ † L,J χ R,J + Hermitian conjugate . (22)Here t h can be a complex coupling constant.There is a two geometrically distinct configurations with the same energy (see Fig.4). De-formations of the theory as well as 1 /N corrections can lift this degeneracy. Here we will6exclusively focus on the state depicted in Fig.4(c).The scalar partner of the W -boson is dual to an operator of the form χ † L,J χ R,J on a givensite, so turning on a nontrivial source for such a field activates a conventional hopping term inthe system, rather than the terms quartic in defect fermions we have been implicitly dealingwith by resorting to double trace deformations. This hopping operator affects the dynamicsat leading order and is not suppressed like the double trace deformations we consideredabove. Of course, one can always choose t h to be a small parameter, and treat the problemperturbatively in t h for | t h | (cid:28) U (2) J current living onthe D-branes on the J -th site. Note that in the full D3/D5 system the coupled scalar/vectorsector on the J -th site is dual to the operators χ † a,J χ b,J , the chiral condensate, and χ † a,J γ χ b,J ,the dual current. As γ = i is just a number, these two operators are really just real andimaginary parts of the hopping term deformation on each site. The labels a , b run over L and R . There are 4 real operators worth of terms we can add to the Lagrangian, and thereare 4 real operators worth of conserved currents dual to the massless gauge field mode inthe bulk. As discussed before, in the full D3/D5 system there is a second operator withthe same global charge assignments and dimension 3. For minimalistic bottom-up modelswithout such an operator presumably no bulk scalar field mixing with the vector is required.In order to analyze the effect of turning on the off-diagonal components it is convenientto treat the problem as a simple U (2) N l . s . gauge theory living on a segment of AdS , where N l . s . denotes the number of lattice sites. Let us parametrize the gauge field on each site as A µ,J = A Lµ,J ( z, t ) h µ,J ( z, t ) h ∗ µ,J ( z, t ) A Rµ,J ( z, t ) . (23)A convenient gauge choice is A z,J = 0. For a minimalistic model one can take the intrinsicmetric to be a segment of pure AdS ds = 1 z ( − dt + dz ) (24)running from the UV boundary at z = 0 to some “hard wall” at z = z ∗ . The hard wall at z ∗ simply reflects the fact that we are studying a state of the system in which the branesare reconnected, and do not reach all the way to the horizon. The detailed state of the fieldtheory is captured by the boundary conditions imposed at the hard wall.7For the purpose of phenomenological model building, the boundary conditions imposedin the IR are part of the input. But there is a particular “geometric” set of IR boundaryconditions that corresponds to the connected bridge configurations we studied before. There,the boundary conditions on the diagonal components of the gauge fields follow by continuityof the fields and their radial derivatives in the true bridged configuration. If we let σ denote asingle valued radial coordinate along the brane, with z ( σ ) an increasing function on the rightbrane and a decreasing function on the left brane, then the natural boundary conditions are: A Lt,J +1 − A Rt,J = 0 , ∂ σ ( A Lt,J +1 − A Rt,J ) = ∂ z A Lt,J +1 + ∂ z A Rt,J = 0 at z = z ∗ . (25)Note that these boundary condition gives the desired breaking of the U (2) N l . s . to the U (1) N l . s . associated with the U (1) gauge fields living on the bridges: A Lt,J +1 = A Rt,J ≡ A t, ( J,J +1) . (26)Having different boundary conditions on the left and right fields directly breaks each U (2)to U (1) × U (1) and then, as in the brane setups studied earlier, the boundary conditionsensure that the left U (1) field of the ( J + 1)th site is identified with the right U (1) field ofthe J th site.Last but not least, we have to determine the boundary conditions on the off-diagonalcomponents of the gauge field, h t,J and h ∗ t,J . From the bottom-up point of view the bestway to think about the IR boundary conditions is to introduce an “IR-brane-localized Higgsfield.” As in the cases of interest the boundary conditions always connect U (2) gauge fieldsliving on neighboring sites, one can introduce complex, bi-fundamental scalars connectingneighboring sites. That is, for every bridge with label ( J, J + 1), one adds an IR branelocalized Lagrangian for a 2 by 2 matrix of scalar fields φ ( J,J +1)
T r | D µ φ ( J,J +1) | = T r | ∂ µ φ ( J,J +1) − iA µ,J φ ( J,J +1) + iφ ( J,J +1) A µ,J +1 | . (27)One particularly interesting form of the vacuum expectation value (vev) of the IR-branelocalized Higgs field is φ ( J,J +1) = m . (28)Eq.(28) is the unique choice if we want a vev that only gives quadratic terms mixing A Rt,J with A Lt,J +1 but no other quadratic terms involving the diagonal components of the8gauge fields. This form of the vev is “geometric” in the sense that it can describe branesreconnecting as we saw above. This form of a scalar field expectation value gives rise to IRbrane localized mass terms for the gauge fields of the form L IR = (cid:88) J | m | (cid:0) ( A Lt,J +1 − A Rt,J ) + | h t,J | + | h t,J +1 | (cid:1) . (29)The boundary conditions on a gauge field with a finite boundary mass matrix ( m ) ab (dueto the boundary Higgs) are in general [32, 33] ∂ z A at = ( m ) ab A bt at z = z ∗ . (30)We see that the above Higgs vev and the resulting IR Lagrangian (29) in the limit of large m give exactly the boundary condition (25) that we know encode the correct geometricconditions on the bridges, together with a Dirichlet boundary condition on the off-diagonalcomponents of the gauge field.In this language it is now straightforward to turn on the actual fermion hopping interac-tion of Eq. (22). Asymptotically, we have A R,Lt,J = α R,LJ + β R,LJ /z, h t,J = γ J + δ J /z. (31)Turning on a nontrivial hopping interaction simply tells us that we are studying bulk gaugefield configurations in which we impose the UV boundary condition that γ J = t h .In the full D3/D5 system it is impossible to study this deformation reliably. For onething, turning on the off-diagonal gauge field components requires one to know the fullnon-Abelian DBI action governing the dynamics of these fields. However, this action is notknown beyond the few lowest dimension terms in powers of F µν . Even worse, to reliablystudy the bridged configurations we really need the analogue of the DBI action that governsa brane/anti-brane system including the tachyon field. This is certainly beyond the scopeof present-day D-brane technology.On the other hand, at the level of bottom-up model building, we have assembled all ofthe ingredients we need to study a holographic realization of a generic lattice model withhopping fermions. Thus, one can take one’s favorite unsolved lattice model (for examplethe Hubbard model), and parametrize one’s ignorance by writing down a correspondinghigher dimensional brane system with an effective action for both the gauge field and the IRHiggs field. This action will have free parameters, which should be matched against known9properties of the boundary lattice model. While it is not obvious that such a rewritingwill be advantageous, it may offer some new approaches to this class of problems, just asbottom-up models of hadron physics have done for the study of QCD. Acknowledgements
We are grateful to J. Polchinski and S. Sachdev for very helpful conversations. We alsothank S. Shenker and E. Silverstein for interesting discussions about related subjects. S.K.and S.Y. thank the theory group at the University of Washington at Seattle for hospitalityduring the completion of this work. S.K. is also happy to acknowledge the warm hospitality ofthe string theory group at the Kavli Institute for Theoretical Physics and the UCSB PhysicsDepartment while the bulk of this work was completed. He is supported by the NSF undergrant no. 0756174, by the DOE under contract DE-AC03-76SF00515, and by the StanfordInstitute for Theoretical Physics. S.Y. thanks the Kavli Institute for Theoretical Physics forunofficial hospitality and the MIT Center for Theoretical Physics for official hospitality whilethis work was vaguely in progress. He is supported by the Stanford Institute for TheoreticalPhysics and NSF Grant No. 0756174.
Appendix A: Hypergeometric-ology
In this section we present a simple toy model of probe brane fluctuations in the dimerizedphase. On the probe brane, let us suppose that there is a scalar field governed by thefollowing effective action: S toy = (cid:90) dtdx √− g (cid:2) − g µν ( ∂ µ φ ∗ )( ∂ ν φ ) − m | φ | (cid:3) , (A1)where g µν is a caricature “two-AdS ” induced metric given by g (caricature) µν dx µ dx ν = 1 (cid:2) cos (cid:8) π (cid:0) xa − (cid:1)(cid:9)(cid:3) (cid:0) − dt + dx (cid:1) , x ∈ [0 , a ] . (A2)This describes a global AdS on the bridge.Making a coordinate transformation to ρ ≡ π ( xa − ), we get: (cid:20) − ∂ ρ + ( amπ ) cos ρ (cid:21) φ ω ( ρ ) = (cid:16) aωπ (cid:17) φ ω ( ρ ) , ρ ∈ (cid:104) − π , + π (cid:105) . (A3)0By a further transformation v ≡ ρ , v ∈ [0 , , (A4) φ ω ( v ) ≡ (cid:18) v (1 − v ) (cid:19) ψ ω ( v ) , (A5)the equation of motion can be brought into hypergeometric form: ∂ ψ ω ∂v + 14 v (1 − v ) (cid:20)(cid:26) − (cid:16) aωπ (cid:17) (cid:27) v − (cid:26) − (cid:16) aωπ (cid:17) (cid:27) v + (cid:26) − (cid:16) amπ (cid:17)(cid:27)(cid:21) ψ ω = 0 . (A6)We can now bring this equation into the standard Gauss’ hypergeometric form: a hyper ≡ aωπ + 12 , b hyper ≡ − aωπ + 12 , c hyper ( c hyper − ≡ (cid:16) amπ (cid:17) −
34 (A7) ψ ω ( v ) ≡ (cid:18) v (cid:19) − c hyper2 (cid:18) − v (cid:19) c hyper − a hyper − b hyper − F ω ( v ) , (A8)0 = (cid:20) v (1 − v ) d dv + ( c hyper − ( a hyper + b hyper + 1) v ) ddv − a hyper b hyper (cid:21) F ω ( v ) . (A9)After reorganizing a bit we get ∆(∆ − ≡ (cid:16) maπ (cid:17) [choose the positive root so that (∆ − ≥ − ∆] .φ ( v ) = (cid:18) v (cid:19) − ∆2 (cid:18) − v (cid:19) ∆ − [ α ( ω ) (cid:18) v (cid:19) − (cid:26) F (cid:18) − ∆ + (cid:16) aωπ (cid:17) , − ∆ − (cid:16) aωπ (cid:17) ; 32 − ∆; v (cid:19)(cid:27) + β ( ω ) (cid:26) F (cid:18)
12 + (cid:16) aωπ (cid:17) , − (cid:16) aωπ (cid:17) ; 12 + ∆; v (cid:19)(cid:27) ]= (cid:18) v (cid:19) − ∆2 (cid:18) − v (cid:19) ∆ − [ α (cid:48) ( ω ) (cid:26) F (cid:18)
12 + (cid:16) aωπ (cid:17) , − (cid:16) aωπ (cid:17) ; 32 − ∆; 1 − v (cid:19)(cid:27) + β (cid:48) ( ω ) (cid:18) − v (cid:19) − (cid:26) F (cid:18) ∆ + (cid:16) aωπ (cid:17) , ∆ − (cid:16) aωπ (cid:17) ; 12 + ∆; 1 − v (cid:19)(cid:27) ] . The hypergeometric functions appearing above are connected through frequency-dependent matrices: F (cid:0) + (cid:0) aωπ (cid:1) , − (cid:0) aωπ (cid:1) ; + ∆; v (cid:1)(cid:0) v (cid:1) − F (cid:0) − ∆ + (cid:0) aωπ (cid:1) , − ∆ − (cid:0) aωπ (cid:1) ; − ∆; v (cid:1) = t ( ω ) t ( ω ) t ( ω ) t ( ω ) F (cid:0) + (cid:0) aωπ (cid:1) , − (cid:0) aωπ (cid:1) ; − ∆; 1 − v (cid:1)(cid:0) − v (cid:1) − F (cid:0) ∆ + (cid:0) aωπ (cid:1) , ∆ − (cid:0) aωπ (cid:1) ; + ∆; 1 − v (cid:1) , Note that v ∼ (cid:0) ρ + π (cid:1) as ρ → − π whereas (1 − v ) ∼ (cid:0) ρ − π (cid:1) as ρ → π , giving rise to in the exponents. They are also ∆-dependent, where ∆ is a parameter of the theory. t ij ( ω ) can be expressed in terms of Gamma functions as t ( ω ) = Γ (cid:0) − ∆ (cid:1) Γ (cid:0) ∆ − (cid:1) Γ (cid:0) + aωπ (cid:1) Γ (cid:0) − aωπ (cid:1) , (A10) t ( ω ) = Γ (cid:0) ∆ + (cid:1) Γ (cid:0) ∆ − (cid:1) Γ (cid:0) ∆ + aωπ (cid:1) Γ (cid:0) ∆ − aωπ (cid:1) , (A11) t ( ω ) = Γ (cid:0) − ∆ (cid:1) Γ (cid:0) − ∆ (cid:1) Γ (cid:0) − ∆ + aωπ (cid:1) Γ (cid:0) − ∆ − aωπ (cid:1) , (A12)and t ( ω ) = Γ (cid:0) ∆ + (cid:1) Γ (cid:0) − ∆ (cid:1) Γ (cid:0) + aωπ (cid:1) Γ (cid:0) − aωπ (cid:1) . (A13) [1] S. A. Hartnoll and C. P. Herzog, Phys. Rev. D77 , 106009 (2008), 0801.1693.[2] M. Fujita, Y. Hikida, S. Ryu, and T. Takayanagi, JHEP , 065 (2008), 0810.5394.[3] A. Karch and A. O’Bannon, JHEP , 024 (2007), 0705.3870.[4] S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Phys. Rev. Lett. , 031601 (2008),0803.3295.[5] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. Vegh (2010), 1003.1728.[6] S. Kachru, A. Karch, and S. Yaida, Phys. Rev. D81 , 026007 (2010), 0909.2639.[7] S. Hellerman (2002), hep-th/0207226.[8] T. Faulkner and J. Polchinski (2010), 1001.5049.[9] J. Polchinski and E. Silverstein, to appear (2010).[10] S.-S. Lee, Phys. Rev.
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