Beyond Particles: Unparticles in Strongly Correlated Electron Matter
MMay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 1 Beyond Particles: Unparticles in Strongly Correlated Electron Matter
Philip W. Phillips
Department of Physics and Institute for Condensed Matter Theory, University of Illinois 1110 W. Green Street, Urbana, IL61801, U.S.A.
I am concerned in these lectures with the breakdown of the particle concept in strongly correlated electron matter.I first show that the standard procedure for counting particles, namely Luttinger’s theorem, breaks down anytimepole-like excitations are replaced by ones that have a divergent self-energy. Such a breakdown obtains in electronicsystems whose pole-like excitations do not extend to the edge of the Brillouin zone, as in Fermi arcs in the cuprates.Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixedpoint, unparticles are the natural candidate to explain the presence of charged degrees of freedom that have no particlecontent. The continuous mass formulation of unparticles is recast as an action in anti de Sitter space. Such an actionserves as the generating functional for the propagator. This mapping fixes the scaling dimension of the unparticle tobe d U = d/ √ d + 4 / d the spacetime dimensionof the unparticle field. The general dynamical mechanism by which bulk operators, such as the Pauli term, couple tothe scaling dimension of the boundary operator and thereby lead to a vanishing of the spectral weight at zero energyis reviewed in the context of unparticles and zeros. The analogue of the BCS gap equations with unparticles indicatesthat the transition temperature increases as the attractive interaction strength decreases, indicating that unparticlesare highly susceptible to a superconducting instability. Keywords : cuprates, unparticles, anti de Sitter
1. Introduction
In these lectures, I will focus on strong electron correlations as they play out in the normal state of thecuprates. The ultimate goal is to understand the departures from the standard theory of metals in termsof a new fixed point, namely one that describes strong coupling physics. While there are obvious featuresof the normal state that suggest that Fermi liquid theory breaks down, such as T − linear resistivity, mystarting point for approaching this problem will be the onset of Fermi arc formation as the Mott insulatoris doped. My focus on Fermi arcs as a window into the origin of non-Fermi liquid behavior in the cuprates isquite simple. As I will show, when arcs form, the standard rule which relates the total charge count to thenumber of quasiparticles fails. Consequently, Fermi arcs provide a glimpse into the breakdown of the particleconcept in the normal state of the cuprates. I will then suggest that the unparticle construct of Georgi’s provides a good description of the extra stuff that couples to the current but has no particle content. I willthen show that such scale-invariant matter has a natural mapping on to an action in anti de Sitter spaceand has a superconducting instability quite unlike the standard BCS picture.
2. Fermi arcs and the Failure of the Luttinger Count
As illustrated in Fig. (1a), lightly doping a Mott insulator results in a locus of quasiparticle excitationsin momentum space which does not form a closed surface. The quasiparticles, represented by solid dots inFig. (1a), form an arc in momentum space which does not extend to the Brillouin zone edge. Because thequasiparticle excitations correspond to poles in a single-particle Green function, G R = Z p ω − ε p − Σ( ω, p ) , (1)there are two distinct ways of viewing the termination of the arc. Either poles still exist where the would-bearc terminates but the spectral weight, Z p is too small to be detected experimentally or there is no pole atall. The first is rather traditional and requires no modification of the standard theory of metals. That is,the former is simply a limitation on experiment. However, such is not the case for the latter. To illustrate a r X i v : . [ c ond - m a t . s t r- e l ] D ec ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 2 the difficulty, consider the path shown in Fig. (1a). On either side of the Fermi arc or the extreme case ofa nodal metal in which case there is only a quasiparticle at ( π/ , π/ , the real part of the Green functionmust change sign. However, the path depicted in Fig. (1a) passes through the Fermi arc but returns withoutencountering a pole and hence has no mechanism for incurring the expected sign change. The only possibleway to end up with the correct sign of the real part of the Green function is through a line of zeros of thesingle-particle Green function as depicted in Fig. (1b). The zero surface can either be on the back-side ofthe arc or somewhere between the end of the arc and the edge of the Brillouin zone. Consequently, numerousproposals for the pseudo gap state of the cuprates have stressed that zeros of the retarded Green function(more rigorously, Det (cid:60) G R ( E = 0 , p = 0)) must be included if Fermi arcs are to be modeled. (0 ,⇡ ) ( ⇡, ,
0) ( ⇡, ⇡ ) (0 ,⇡ ) ( ⇡, ,
0) ( ⇡, ⇡ ) ReG R < ReG R > R > ReG R <
000 000 00 a ) b ) Fig. 1. (Color online) a.) Sign change of the real part of the retarded Green function in the presence of an arc of quasiparticleexcitations, blue dots. The path shown, through the Fermi arc and circumventing it on the return is problematic because thereturn path has no mechanism for picking up the sign change of the Green function. b) Solution for maintaining the correctsign of the Green function. A line of zeros must be present in one of two places: 1) either on the backside of the arc or 2)somewhere between the end of the arc and the edge of the Brillouin zone.
That zeros of the single-particle Green function represent a breakdown of the particle concept can beseen from Eq. (1). The only way for the Green function to vanish in the absence of a pole is if thedenominator diverges. This requires a divergent self energy, Σ. Such a divergence tells us that the startingpoint of particles with well-defined energy, momentum, or mass is no longer valid. Nonetheless, many haveargued that they must be included in the count, n = 2 (cid:88) p θ ( (cid:60) G R ( ω = 0 , p )) , (2)of the particle density. This seemingly innocuous expression is a consequence of what is known as Luttinger’stheorem , a prescription for counting particles. Because θ is the heavy-side step function, this equation hascontributions anytime the real part of the Green function is positive. This occurs via a pole or a zero crossing.Poles count quasiparticles and hence it is natural that they should be included in the particle count. Shouldonly poles be present, as in a Fermi liquid, then the physical content of Eq. (2) is self evident. However, thisequation also includes zero crossings where the particle concept breaks down. Nonetheless, they are presentin the Luttinger count of the particle density as pointed out explicitly first in the classic book by Abrikosov,Gorkov, and Dzyaloshinski ( Eq. (19.16) and the paragraph below it) and others .There is a simpler way of counting charges. Simply integrate the density of states n = 2 (cid:90) µ −∞ N ( ω ) (3)over all the occupied states. As a sum over all energy scales, this formula for the charge count has bothhigh-energy (UV) and low-energy (IR) physics. Hence, what Eq. (2) accomplishes is a reduction of thecharge density to a zero-energy ( ω = 0) surface. That is, the charge density has been reduced to IR physicsalone and as a consequence, all the charge content has a particle interpretation.If Eq. (2) is correct, then it is truly extraordinary because it would apply in cases where the self energyvanishes and cases in which it diverges. A classic example of a divergent self-energy is the Mott insulator a . a Dzyaloshinkskii has maintained (see discussion after Eq. (9)) that in a Mott insulator, the self energy does not diverge ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 3 Consider the Kramers-Kronig (cid:60) G R ( ω = 0 , p ) = 1 π (cid:90) ∞−∞ (cid:61) G R ( ω (cid:48) , p ) ω (cid:48) dω (cid:48) (4)relationship for the real part of the retarded Green function. A Mott insulator is a gapped system in whichthe parts of the spectral function below and above the chemical potential have the same band index. As aresult, there is no change in the band index upon integration over all frequency. Since ImG R is of a fixed sign,the integrand in Eq. (4) can change sign in the range of integration. Consequently, the integral can vanish.A zero of the single-particle Green function then is a balancing act of spectral weight above and below thechemical potential weighted with the factor of 1 /ω . As a result, the zero surface is strongly dependent onwhere the chemical potential is located. While this might seem to make sense in the context of the particledensity being determined by the location of the chemical potential, there is a subtlety here in that even for anincompressible system, in which case the chemical potential can be placed anywhere in the gap, the locationof the zero surface can change, thereby changing the particle densty . Physically, however, nothing changeswhen the chemical potential is moved in the gap. This problem was first pointed out by Rosch and canbe illustrated with the atomic limit of the Hubbard model SU (2) Hubbard model in which the Hamiltonianis simply U n ↑ n ↓ . The retarded Green function for this problem G R ( ω ) = 1 ω + µ + U + 1 ω + µ − U (5)is a sum of the two poles that constitute the lower and upper Hubbard bands. It is instructive to write bothterms over a common denominator, G R ( ω ) = 1 ω + µ + U/ − Σ loc ( ω ) , (6)which allows us to identify the self-energy,Σ( ω ) = U (cid:18) U (cid:19) ω + µ . (7)Clearly when ω = − µ , the self energy diverges, thereby leading to a zero of the Green function in contrastto the statement by Dzyaloshinskii that the self-energy does not diverge in a Mott insulator. Evaluatingat ω = 0 and substituting the result into the Luttinger theorem leads to n = 2Θ (cid:32) µµ − (cid:0) U (cid:1) (cid:33) (8)the expression that should yield the particle density. The Mott insulator corresponds to n = 1 and thisformula should yield this result for any value of the chemical potential satisfying − U/ < µ < U/
2. However,it is evident that there is a problem here: 1) for − U/ < µ <
0, n=2, 2) µ = 0, n = 1, and finally 3) for0 < µ < U/ n = 0. Hence, unlike poles, zeros are not a conserved quantity. This does not mean theyare devoid of physical content. Zeros are the key indicator that high and low energies are mixed as in theMott problem laid plain by Eq. (4). So what to make of this result? It might be argued that one couldtake care of the degree of freedom with the chemical potential by using the limiting procedure, lim T → µ ( T )which would uniquely fix the chemical potential at T = 0. For the SU(2) atomic limit of the Hubbard model,this limiting procedure places the chemical potential at the particle-hole symmetric point, µ = 0, in whichcase n = 1. Does this fix all the problems?While the importance of this limiting procedure can certainly be debated , it would be advantageous toconsider a model which even with this limiting procedure, the Luttinger count is violated. The generalization but nonetheless there are still zeros, in the sense that DetReG R ( ω = 0 , p) = 0 in a Mott insulator. These statements arecontradictory. ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 4 to a non-degenerate ground state will be considered a little later. The model we consider is a generalizationof the atomic limit of the Hubbard model, the SU(N) H = U n + n + · · · n N ) , (9)to N degenerate flavors of fermions and hence has SU ( N ) symmetry. As will be seen, it is imperative thatthis symmetry be in tact because the basic physics of Mott insulation fails when the spin rotation symmetryis broken. Exact computation of the Green function reveals that the equivalent statement of Luttinger’stheorem for the SU ( N ) model becomes n = N Θ(2 n − N ) . (10)Consider the case of n = 2 and N = 3. This equation implies that 2 = 3 and hence is false. In fact, anypartially filled band with N odd leads to a violation of Eq. (2). That Eq. (10) actually reduces to the correctresult for the SU(2) case is entirely an accident because the Θ function only takes on values of 0, 1 /
2, or 1.This problem still persists even if the degeneracy of the state in the atomic limit is lifted by a small hopping t = 0 + . Note this perturbation preserves the SU ( N ) symmetry and hence is fundamentally different fromlifting the degeneracy via a magnetic field which would break the SU ( N ) symmetry as proposed recently asa rebuttal to our work . In the limit t → µ = 0, and T = 0, the Green function becomes G ab ( ω ) = Tr (cid:18)(cid:20) c † a ω − H c b + c b ω − H c † a (cid:21) ρ (0 + ) (cid:19) with Tr the trace over the Hilbert space and ρ (0 + ) the density matrix. Here c a is the creation operator fora fermion of flavor, a . Our use of ρ (0 + ) is crucial here because ρ ( t = 0 + ) describes a pure state whereas for ρ ( t = 0) = (cid:80) u P u | u (cid:105)(cid:104) u | with probabilities satisfying (cid:80) u P u = 1 is a mixture of many degenerate groundstates | u (cid:105) . Our use of ρ (0 + ) allows us to do perturbation theory in t . For t →
0, the intermediate stateshave energy U or 0 and hence we can safely pull ω − H outside the trace. Noting that { c † a , c b } = δ ab , weobtain that G ab ( ω ) = ωδ ab − U ρ ab ω ( ω − U ) , (11)where we have introduced ρ ab = Tr (cid:0) c † a c b ρ (0 + ) (cid:1) = (cid:104) u | c † a c b | u (cid:105) , | u (cid:105) the unique ground state. Now comesthe crucial point. Consider N = 3. If the degeneracy between the iso-spin states were lifted in the startingHamiltonian, as in a recent model put forth as a possible rebuttal to our claim that Eq. (10) representsa counterexample to the Luttinger claim, only a single one of the iso-spin states would be possible and ρ ab (0 + ) = diag(1 , , (cid:54) = ρ ab ( t = 0). However, turning on a hopping matrix element places no restriction onthe permissible iso-spin states. Consider a rotationally-invariant spin singlet state on a three-site system,the unique ground state. We have that ρ ab (0 + ) = 1 / , ,
1) = ρ ab ( t = 0). For this unique groundstate, Eq. (11) has a zero at ω = U/ T → µ ( T ) = U/
2. Consequently, the Luttinger count failsto reproduce the particle density. Specifically, Eq. (11) implies that 1 = 0! The key point is that as long asthe perturbation which lifts the degeneracy does not break SU ( N ) symmetry, then ρ ab ( t = 0 + ) = ρ ab ( t = 0)and Eq. (11) survives and the recent criticism does not.The crux of the problem is that the Luttinger-Ward (LW) functional strictly does not exist when zerosof the Green function are present. Consider the LW functional, defined by δI [ G ] = (cid:90) dω Σ δG (12a) I [ G = G ] = 0 (12b)which was used by Luttinger to show that the integrand of I is a total derivative. Because Σ diverges forsome ω when G is the total Green function, it is not possible to integrate the defining differential expression inthe neighborhood of the true Green function, and therefore the LW functional does not exist. Consequently,there is no Luttinger theorem and Eq. (2) does not represent the density of a fermionic system because zerosof the Green function must be strictly excluded, a model-independent conclusion. In fact, Dzyaloshinskii ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 5 has already pointed out that when Σ diverges, the Luttinger count fails. However, he mistakenly assumesthat such a divergence is not intrinsic to a zero of the Green function and hence is irrelevant to Mott physics.As we have seen here, this is not the case.Experimentally, Fermi arc formation leads to a violation of Luttinger’s theorem. Shown in Fig. (2) is aplot of the area enclosed by the locus of k-points for which there is a maximum in the spectral function inLa − x Sr x CuO (+ plot symbol) and Bi Sr CaCu O δ ( × plotting symbol) as a function of the nominaldoping level in the pseudogap regime. Although the maxima in the spectral function form an arc as thereare zeros present on the opposite side, x FS was extracted by simply closing the arc according to a recentproposal for Bi Sr CaCu O δ (Bi-2212) and for La − x Sr x CuO (LSCO) by determining the largeFermi surface (1 − x ) defined by the k F measured directly from the momentum-distribution curves and thensubtracting unity. Hence, the key assumption that is being tested here in this definition of x FS is that eachdoped hole corresponds to a single k − state. A typical uncertainty in these experiments is ± .
02. Even whenthis uncertainty is considered, the deviation from the dashed line persists indicating that one hole does notequal one k-state and hence a fundamental breakdown of the elemental particle picture in the cuprates.0 0 . . . . . . x x F S × Yang et al. (2011)+ He et al. (2011)1
Fig. 2. Apparent doping x FS inferred from the Fermi surface reconstruction as a function of the nominal doping x in LSCOand Bi-2212.
3. Unparticles and anti de Sitter Spacetime
What I have shown so far is that there is some charged stuff which has no particle interpretation whenLuttinger’s theorem breaks down. So what is the extra stuff? The dichotomy in Fig. (3) represents thecurrent conundrum. Any excitations that arise from such a divergence of the self energy are clearly notadiabatically connected to the ones at the non-interacting or Fermi-liquid fixed point. Consequently, if anynew excitations emerge, they must arise fundamentally from a new fixed point as illustrated in Fig. 3. Allfixed points have scale invariance. To illustrate scale invariance, consider the Lagrangian L = 12 ( ∂ µ ϕ ) + m φ (13)with a mass term. Without the mass term, the scale transformation x → x/ Λ produces a Lagrangian ofthe same form just multiplied by the scale factor Λ . Hence, the Larangian without the mass term is scaleinvariant. However, with the mass term, scale invariance is lost. For a theory with a coupling constant, that ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 6 is a term of the form gϕ , scale invariance at a critical point β ( g ) ≡ dgd ln E = f ( g ) = 0 , (14)reflects an invariance of the coupling constant under a change in the energy scale. That the β − function islocal in energy implies that renormalization group (RG) procedure should be geometrizable. It is this insightthat underlies the gauge-gravity duality as we will see later in this section.While it is difficult to establish the existence of a non-trivial (non-Gaussian in the UV variables) IR fixedpoint of the Hubbard model (or any strongly coupled model for that matter), the cuprates, Mott systemsin general, display quantum critical scaling . This suggests that it is not unreasonable to assume thata strongly coupled fixed point governs the divergence of the self energy in the Fermi arc problem. Withoutknowing the details of the fixed point, what permits immediate quantitative progress here is that all criticalfixed points exhibit scale invariance and this principle anchors fundamentally the kind of single-particlepropagators that arise as pointed out by Georgi . If k is the 4-momentum, the candidate propagator musthave an algebraic form, ( k ) γ . However, such a propagator cannot describe particles because particles arenot scale invariant. Scale invariant stuff has no particular mass, hence the name unparticles. The mostgeneral form of the single-particle propagator, G U ( k ) = A d U d U π ) i ( k − i(cid:15) ) d/ − d U ,A d U = 16 π / (2 π ) d U Γ( d U + 1 / d U − d U ) , (15)involves the scaling dimension of the unparticle field, d U , as proposed by Georgi . Here k is the d -momentumwhere d is the dimension of the spacetime in which the unparticle lives. Here, we adopt the notation wherethe diagonal entries of the metric are mostly positive. Fermi liquid Σ( ω = 0 , p ) = ∞ Σ( ω = 0 , p ) = 0 new fixed point ˜ g Fig. 3. Heuristic renormalization group flow for the Fermi liquid fixed point in which the self-energy is zero or negligible in theIR and one in which zeros of the single-particle propagator appear. Since the self energy diverges in the latter, the excitationswhich appear here are not adiabatically connected to the Fermi liquid fixed point. The breakdown of the particle concept atthe new fixed point suggests an unparticle picture is valid. The horizontal axis represents the strength of the coupling constant.
Within the unparticle proposal, the spectral function should have the form A (Λ ω, Λ α k k ) = Λ α A A ( ω, k ) ,A ( ω, k ) = ω α A f A (cid:18) k ω α k (cid:19) . (16)We take α A = 2 d U − d . The scaling for a Fermi liquid corresponds to d U = ( d − /
2. Because of theconstraints on unparticles, d U always exceeds d/ − d − / d → d + 1. We term a correlated system with such scaling anun-Fermi liquid as the basic excitations are unparticles. Un-Fermi liquids are non-Fermi liquids composed ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 7 of unparticles, whose propagator is given by the unfermionic analogue of Eq. (15), S U ( k ) ∼ (cid:0) k − i(cid:15) (cid:1) d U − ( d +1) / × (cid:16) k / + cot ( d U π ) (cid:112) k − i(cid:15) (cid:17) , (17)which contains a non-local mass term. Un-Fermi liquids should not be construed as Fermi liquids with polesat the unparticle energies. Because the unparticle fields cannot be written in terms of canonical ones, thereis no sense in which a Gaussian theory can be written down from which pole-like excitations can be deduced.Note that the scaling form, if it were to satisfy any kind of sum rule, can only be a valid approximation overa finite energy range.From where do unparticles come? This can be illustrated with the continuous mass formalism . Thekey idea here is that although unparticles have no particular mass, it is still possible to use the massiveLagrangian in Eq. (13) to construct the propagator for unparticles. The trick is to view unparticles as acomposite with all possible mass . We consider the action, S φ = 12 (cid:90) ∞ f ( m ) dm (cid:20)(cid:90) d x (cid:0) ∂ µ φ∂ µ φ + m φ (cid:1)(cid:21) (18)in which the mass is explicitly integrated over with a distribution function of the form, f ( m ) ∝ ( m ) d U − d/ )and hence ϕ ≡ ϕ ( x, m ). Because of the mass integration, any scale change in the Minkowski variables, x µ , can be absorbed into a redefinition of the mass integral and hence our action is scale invariant. Thepropagator for this action, (cid:90) ∞ dm ( m ) d U − d/ p + m ∝ ( p ) d U − d/ , (19)has the algebraic form describing unparticles. An identical analysis applies to Dirac fields as well . Hence,the continuous-mass formalism lays plain that unparticles should be thought of as scalar or Dirac fields withno particular mass.While it is common b to relate the emergent unparticle field directly to the scalar field φ using a differentmass-distribution function g ( m ) through a relationship of the form φ U ( x ) = (cid:90) ∞ dm φ ( x , m ) B ( m ) , (20)this is not correct as it would imply that the unparticle field φ U has a particle interpretation in terms ofthe scalar field φ . For example, φ U would then obey a canonical commutator and the resultant unparticlepropagator could be interpreted as that of a Gaussian theory. We demonstrate this explicitly in the Appendix.In actuality, the unparticle field should not be a sum of φ ( x, m )s, which are independent functions of mass,but rather should involve some unknown product of the particle fields . Consequently, unparticle physicscannot be accounted for by a Gaussian theory. In fact, as pointed out by Georgi , unparticles should bethought of as a composite consisting of d U / . The key claim of this duality is that some strongly coupled conformally invariant fieldtheories in d-dimensions are dual to a theory of gravity in a d + 1 spacetime that is asymptotically anti deSitter ds = R z (cid:0) η µν dx µ dx ν + dz (cid:1) . (21)This spacetime is invariant under the transformation x µ → Λ x µ and z → z Λ and hence satisfies the requisitesymmetry (not the full symmetry of the conformal group) for the implementation of the gauge-gravity duality. b For example, this has been used explicitly in the derivation of the unparticle propagator by Deshpande and He . ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 8 The requirement on the AdS radius is that it exceed the Planck length, Λ P . Because the AdS radius andthe coupling constant of the boundary theory are proportional, the requirement, R (cid:29) Λ P , translates intoa boundary theory that is strongly coupled . Our current understanding of the extra dimension on thegravity side is that it represents the renormalization direction, that is the flow in the energy scale. Thescale change, x µ → Λ x µ increases the radial coordinate, z → z Λ. Consequently, moving into the bulk ofthe geometry increases the corresponding projection onto the boundary, as depicted in Fig. (4). Hence, thelimit of z = ∞ represents the full low-energy or IR limit of the strongly coupled theory, thereby providinga complete geometrization of the RG procedure. That is RG = GR. Ultimately it is the locality of the β − function with respect to energy that is responsible for this geometrical interpretation of RG. Fig. 4. Geometrical representation of the key claim of the gauge-gravity duality. A strongly coupled field theory lives at theboundary, at the UV scale. The horizontal r-direction (the extra dimension in the gauge-gravity duality) represents the runningof the renormalization scale. This is illustrated by the two projections at different values of z in the space-time. Because thespacetime is asymptotically hyperbolic, larger values of r lead to a larger projection of the boundary theory and hence fullrunning of the renormalization scale amounts to the construction of the infrared (IR) limit of the original strongly coupled UVtheory. Charging the bulk gravity theory amounts to placing a black hole at z = ∞ . Establishing a hard connection rather than a purely heuristic one between unparticles and the gauge-gravity duality is straightforward within the continuous mass formalism. The continuous mass formalisminvolves an integration over mass. However, mass is energy and hence can be replaced by an inverse length.Since there is already an extra dimension floating around in the continuous mass formalism, we can formal-ize this by making the replacement m → /z in Eq. (18). Letting f ( m ) = a δ ( m ) δ and rememberingthat dm → − /z introduces an extra factor of z δ into the continuous mass Lagrangian as can be seenfrom L = a δ (cid:90) ∞ dz R z δ (cid:20) z R η µν ( ∂ µ φ )( ∂ ν φ ) + φ R (cid:21) . (22)All the factors of z and the radius, R , can be accounted for by using the AdS metric, Eq. (21). Consequently,we have proposed that the correct starting point for the unparticle construction is the action on AdS δ S = 12 (cid:90) d δ x dz √− g (cid:18) ∂ a Φ ∂ a Φ + Φ R (cid:19) . (23)where √− g = ( R/z ) δ , and φ → ( a δ − / R / Φ. All of the factors of z δ and the z in the gradientterms appear naturally with this metric. According to the gauge-gravity duality, the on-shell action thenbecomes the generating functional for unparticle stuff which lives in an effective dimension of d = 4 + 2 δ .Therefore, we would like to have δ ≤
0. We have of course introduced extra dynamics in the z − direction.This will be eliminated by choosing the solutions which are non-normalizable at the boundary. This will fixthe scaling dimension of the unparticle field. ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 9 We have shown that the unparticle propagator falls out of this construction. To see this, we start fromthe equation of motion, z d +1 ∂ z (cid:18) ∂ z Φ z d − (cid:19) + z ∂ µ ∂ µ Φ − Φ = 0 . (24)For spacelike momenta k >
0, the solutions are identical to those of Euclidean AdS. The solution that issmooth in the interior is given byΦ( z, x ) = (cid:90) d d k (2 π ) d e i k · x z d K ν ( kz ) (cid:15) d K ν ( k(cid:15) ) ˜Φ( k ) , (25)where k is a d -momentum transverse to the radial z -direction and ν = √ d + 42 . (26)We note that this solution decays exponentially in the interior and thus, even though it is a z -dependentsolution, one can think of Φ as localized at the boundary z = (cid:15) →
0. Hence, we have eliminated the unwanteddynamics in the bulk. Here, we have explicitly cut off the AdS geometry to regularize the on-shell action S = 12 (cid:90) d d x g zz √− g Φ( z, x ) ∂ z Φ( z, x ) (cid:12)(cid:12)(cid:12) z = (cid:15) = 12 R d − (cid:15) d − (cid:90) d d p (2 π ) d d d q (2 π ) d (2 π ) d δ ( d ) ( p + q )˜Φ( p ) dd(cid:15) (cid:18) log (cid:20) (cid:15) d K ν ( p(cid:15) ) (cid:21)(cid:19) ˜Φ( q ) . (27)Interpreting this as a generating functional for the unparticle field Φ U living in a d -dimensional spacetime,we can then read the (regulated) 2-point function, which scales as p ν . We can then analytically continueto the case of timelike momenta, which corresponds to choosing the non-normalizable solution to the bulkequation of motion. This analytically continued solution will also be localized at the boundary.The correlation function for two unparticle fields in real space is then given by (cid:104) Φ U ( x )Φ U ( x (cid:48) ) (cid:105) = 1 | x − x (cid:48) | d U , (28)where d U = d √ d + 42 > d , (29)and d is the dimension of the spacetime the unparticle lives in. We note that in this construction, thereis only one possible scaling dimension for the unparticle instead of two, due to the fact that the square ofthe mass of the AdS scalar field Φ is positive c . As a result, the unparticle propagator has zeros defined by G U (0) = 0, not infinities. This is the principal result of this construction.The unparticle construction of propagators that have zeros is made possible because the mapping toanti de Sitter space fixed the scaling dimension. What this suggests is that within the general frameworkof the gauge-gravity duality, it should be possible to add various terms to the bulk gravity model whichcouple to the scaling dimension of the dual fermionic operators at the boundary to engineer propagatorsthat exhibit zeros. Indeed, such a construction is possible . Computing fermionic correlators at theboundary requires fermionic fields in the bulk described by an appropriate equation of motion. All the earlywork on fermionic correlators in AdS was based on probe fermions obeying a Dirac action S = ( ψ, ¯ ψ ) = (cid:90) d d x (cid:112) − igi ¯ ψ ( /D − m + · · · ) ψ (30) c In AdS, a stable particle is allowed to have negative mass squared as long as it satisfies the so-called Breitenlohner-Freedmanbound. ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 10 p = 8. The absence of spectral weight for a range ofenergy flanking ω = 0 results from the zeros-poles duality. in which the mass was varied. The asymptotic solutions ( r → ∞ ) to the equations of motion for ψ scale ar m and br − m . One is free to interpret either term as the source or the response. In the standard quantization,the response term vanishes at the boundary. Consequently, the associated boundary fermionic propagatoris the ratio b/a . The “ · · · ” in the action for the probe fermions reveals that bulk gravity theory is notunique because the boundary theory is not known, just the correlators. Such correlators all possess Fermisurfaces, though the excitations can even be of the marginal Fermi liquid type . However, when theboundary theory is specified, thereby fixing the gravity theory, no Fermi surfaces arise . All the spectralfunctions are of the unparticle kind with a vanishing spectral weight at the chemical potential. Whatto make then of the bottom-up constructions based on the Dirac equation? Recall, a restriction for theimplementation of the gauge-gravity duality is that the boundary theory has to have a finite couplingconstant, strictly absent from a Fermi liquid . This can be fixed by specifying extra details of the bulktheory; that is, filling in the “ · · · ” in the probe fermion action in Eq. (30). There are many terms that canbe added, all of which involve higher derivative operators. That such terms are not the leading operatorsin the bulk theory does not imply that they cannot change the boundary dynamics. A possible mechanismis that they change the scaling dimension of the boundary correlators. One such operator which does thisis the Pauli term. Consider replacing the “ · · · ” in Eq. (30) by − ip [Γ µ , Γ ν ] F µν where Γ µ is a Dirac matrix.Such a term is well known to give rise to the anomalous magnetic moment of the electron in flat space.Even if the details of the bulk spacetime vary from Reissner-Nordstrom to Schwarzschild , as long asthe spacetime at the boundary is asymptotically anti de Sitter, the fermionic propagator is gapped asshown in Fig. (5) once p exceeds a critical value. The origin of the gap is now well understood andstems from an exact duality Det (cid:60) G R ( ω = 0 , k ; p ) = 1Det (cid:60) G R ( ω = 0 , − k ; − p ) , (31)between zeros and poles as first shown by Alsup, et al. for the Reissner-Nordstrom spacetime and byVanacore and Phillips for the Schwarzschild case. For sufficiently negative values of p , only poles exist.This means that if we flip the sign of p , all the poles must be converted into zeros. Zeros imply a gapand hence a boundary theory at finite coupling. Consequently, the Pauli term appears to be a fundamentalaspect of the physics of boundary correlators in AdS as partly evidenced by the fact that it is present intop-down constructions . An example of the zeros poles duality is depicted in Fig. (6). The mechanismfor the conversion of poles to zeros is that the Pauli term couples to the scaling dimension of the probe ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 11 fermions. While this is also true for the mass term, Det (cid:60) G R is restricted in this case to be ±
1. No suchrestriction applies when the Pauli term is present. (cid:45) (cid:45) D e t R e G (cid:72) Ω (cid:61) , k (cid:76) p (cid:61) (cid:177) Fig. 6. Duality between poles and zeros simply changing the sign of the Pauli term. For negative values of p the Greenfunction exhibits poles. By duality, the poles are converted into zeros when p changes sign and becomes positive. It is theduality that underlies the suppression of the spectral weight and the opening of a gap once p exceeds a critical value.
4. Superconducting Instability
Because unparticles do not have any particular energy, they should be useful in describing physics in whichno coherent quasiparticles appear, as in the normal state of the cuprates. As a result, we have advocatedthen that unparticles provide a description of the incoherent fermionic contribution to the charge densityin the pseudo gap and strange metal regions of the cuprates . Since all formulations of superconductivitystart with well-defined quasiparticles, we explore what happens when we use a quasiparticle spectral functionwith a scaling form indicative of unparticles. Such an approach is warranted given that the cuprates exhibita color change upon a transition to the superconducting state as evidenced most strikingly by theviolation of the Ferrell-Glover-Tinkham sum rule . Some initial work along these lines has been proposedpreviously , which have all been based on the Luttinger-liquid Green function. As remarked earlier,the advantage of the unparticle approach is that it is completely general regardless of the spatial dimensionunlike the Luttinger-liquid one which must be restricted to d = 1 + 1. To obtain the general result, wework first with the scaling form of the spectral function as in Eq. (16), generalizing the procedure in .Complementary to this work is a recent paper in which the pairing itself, rather than the propagators, wastreated as unparticle-like .The equation for the existence of an instability in terms of the Green function is1 = iλ (cid:88) k | w k | G ( k + q ) G ( − k ) . (32)This gives the zero temperature result. We take the interaction strength λ as a constant of mass dimension2 − d , and w k as a filling factor. We work in the center of mass frame, such that q = ( q , q = 0. Then switch to imaginary time to work atfinite temperature, so that the new equation reads1 = λT (cid:88) n, k | w k | G ( ω n , k ) G ( − ω n , − k ) . (33)The Green function is related to the spectral function via G ( ω n , k ) = (cid:90) ∞−∞ dx A ( x, k ) x − iω n . (34) ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 12 Then we obtain, using ω n = πT (2 n + 1),1 = λ (cid:90) dxdy (cid:88) k | w k | A ( x, k ) A ( y, − k ) × tanh ( x/ T ) + tanh ( y/ T ) x + y . (35)The k -dependence is recast in terms of ξ ( k ), which is in general some function of k with units of energy,which for instance can always be done for an isotropic system. In BCS, ξ ( k ) would correspond to kineticenergy. Take (cid:80) k | w k | → (Volume) − × N (0) (cid:82) dξ so we pull out a constant density of states. The integralis now rewritten as 1 = g (cid:90) dxdy (cid:90) ω c dξA ( x, ξ ) A ( y, ξ ) × tanh ( x/ T ) + tanh ( y/ T ) x + y (36)where λN (0) × (Volume) − = g such that g is a dimensionless measure of interaction strength. Evaluatingthe integral at any temperature gives the minimal coupling to cause a pairing instability, and this equationtraces out a phase diagram for g and T .To obtain qualitative information regarding the role that scale invariance plays in the BCS instability,let us impose the scaling form at the outset. Then approximately1 = g T α A ) (cid:90) dxdy (cid:90) ω c / ˜ T dξA ( x, ξ ) A ( y, ξ ) × tanh ( x/ W ) + tanh ( y/ W ) x + y (37)where the tilde denotes the ratio of that energy to W , e.g. ˜ T ≡ TW . The scaling form of the spectral functionconfers a scaling form for g like g (cid:16) ˜ T , ˜ ω c (cid:17) = ˜ T − α A ) f g (cid:32) ˜ T ˜ ω c (cid:33) . (38)Now sequentially we take a logarithm, derivative and finally rescale the remaining integral back to obtain dgd ln ˜ T = − α A ) g + g ω c (cid:90) dxdyA ( x, ω c ) A ( y, ω c ) × tanh ( x/ T ) + tanh ( y/ T ) x + y . (39)The second term is positive-definite. This term can conceivably be small if there is relatively little spectralweight near ω c within the scaling form or, equivalently, that g is not very susceptible to changes in ω c . Inthis event, we have dgd ln ˜ T = − α A ) g + O (cid:0) g (cid:1) . (40)The right-hand side of this expression is strictly negative for our region of interest where α A >
0. Hence, wefind quite generally that the critical temperature increases as the coupling constant decreases!This stands in stark contrast to the Fermi liquid case in which just the opposite state of affairs obtains.This is illustrated clearly in Fig. (7). In the context of the cuprate superconductor problem, the opposingtrends for T c versus the pairing interaction suggests that perhaps a two-fluid model underlies the shape of the ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 13 superconducting dome assuming, of course, that a similar behaviour for T c as a function of doping persists. d Since the transition to the superconducting state breaks scale invariance, the particle picture should bereinstated. Consequently, we expect the broad spectral features dictated by the branch cut of the unfermionpropagator to vanish and sharp quasiparticle features to appear upon the transition to the superconductingstate as is seen experimentally . g unp a r t i c l e s B C S T c Fig. 7. Plot of the β -function for the superconducting transition in the ladder approximation for unfermions, Eq. (40). Wehave considered the case d U > d/
5. Closing
What I have laid out here is a way of thinking about physics when the particle concept breaks down. TheLuttinger count ensures that a purely IR limit, or particle interpretation, of the number of charges existseven if interactions are present. However, as we have shown, when zeros are present, no such theorem applies.Zeros obtain from a divergent self energy and Eq. (2) is not valid in this limit. In fact, this was pointed outexplicitly by Dzyaloshinskii but that such a state of affairs obtains in a Mott insulator was overlooked.Zeros indicate then that some charged stuff exists that has no particle interpretation. I have illustratedhere how unparticles can be used to describe such a breakdown and dynamical generation of a mass gap inbottom-up AdS constructions is consistent with the zeros picture. Ultimately, a scale invariant sector mustarise from a non-trivial fixed point. Establishing that such a non-trivial fixed point exists at strong couplingremains the outstanding problem in this field. Acknowledgements
These lectures were based on a series of papers with B. Langley, K. Dave, J. Hut-tasoit, C. Kane, R. Leigh, and M. Edalati. This work is supported by NSF DMR-1104909 which grew out ofearlier work funded by the Center for Emergent Superconductivity, a DOE Energy Frontier Research Center,Grant No. DE-AC0298CH1088.
Appendix A. Appendix
We show here that if only if a linear relationship between the unparticle and particle fields is maintained(as in Eq. (20)), then a Gaussian action for the unparticles obtains. Let us turn the action in terms of the d There is of course no connection between g and the hole-doping level, x . Certainly a generalization of this work to a dopedmodel that admits unparticles could be studied as a function of doping to see if the qualitative trends in Fig. (7) obtain. ay 9, 2018 8:19 WSPC Proceedings - 9in x 6in karpaclecturesfinal page 14 massive fields into an action in terms of unparticle fields. The original partition function is given by Z = (cid:90) D φ n e i (cid:82) d d p L [ { φ n } ] L = 12 (cid:88) n B n φ n ( p ) (cid:0) p − M n (cid:1) φ n ( − p ) . where n is to indicate a sum over the mass M n . This sum can remain a general sum over various free fields,but we will ultimately take the limit where the sum is a continuous sum over all masses. The factor B n is a weight factor that, in the continuous mass limit, will change the mass dimension of φ n . We introducea Lagrange multiplier through a factor of unity and simply integrate over the all fields that are not φ U toobtain Z = (cid:90) D φ n D φ U D λ exp (cid:40) i (cid:90) d d p (cid:32) (cid:88) n B n (cid:0) p − M n (cid:1) φ n + λ (cid:32) φ U − (cid:88) n F n φ n (cid:33)(cid:33)(cid:41) = (cid:90) D φ U D λ exp (cid:40) i (cid:90) d d p (cid:32) λφ U − λ (cid:88) n F n B n ( p − M n ) (cid:33)(cid:41) = (cid:90) D φ U exp i (cid:90) d d pφ U ( p ) (cid:32)(cid:88) n F n B n ( p − M n ) (cid:33) − φ U ( − p ) with repeated absorptions of normalization contants into the measure. The factor F n is another weightfactor, this time chosen to determine the scaling dimension of the unparticle field φ U . Bescause F n is chosento give φ U ( x ) a scaling dimension d U , in the continuous mass limit the ratio F n /B n ∼ (cid:0) M n (cid:1) d U − d . Thisis necessary because of how F n imposes the scaling dimension. Hence we identify the propagator of theunparticle field as G U ( p ) = (cid:88) n F n B n ( p − M n ) ∼ (cid:0) p (cid:1) d U − d . This argument can also be run in reverse. Namely, if we assume a Gaussian action for the unparticlesthen the Lagrange multiplier constraint in the form of Eq. (20) is implied.
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