Optical conductivity of topological surface states with emergent supersymmetry
OOptical conductivity of topological surface stateswith emergent supersymmetry
William Witczak-Krempa and Joseph Maciejko
2, 3, 4 Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada (Dated: October 17, 2018)Topological states of electrons present new avenues to explore the rich phenomenology of corre-lated quantum matter. Topological insulators (TIs) in particular offer an experimental setting tostudy novel quantum critical points (QCPs) of massless Dirac fermions, which exist on the sample’ssurface. Here, we obtain exact results for the zero- and finite-temperature optical conductivity at thesemimetal-superconductor QCP for these topological surface states. This strongly interacting QCPis described by a scale invariant theory with emergent supersymmetry, which is a unique symmetrymixing bosons and fermions. We show that supersymmetry implies exact relations between the op-tical conductivity and two otherwise unrelated properties: the shear viscosity and the entanglemententropy. We discuss experimental considerations for the observation of these signatures in TIs.
PACS numbers: 05.30.Rt, 74.40.Kb, 71.27.+a, 11.30.Pb
Topological insulators [1, 2] allow for the experimentalstudy of new quantum states of matter. The strong spin-orbit coupling in these bulk-insulating materials leadsto unique gapless Dirac fermion surface states. Thesecan undergo quantum phase transitions forbidden in non-topological systems, and thus constitute a new platformto study the rich physics of quantum criticality [3, 4]. Aconsiderable challenge in the study of interacting QCPsis to determine their dynamical response—that is, theirresponse at finite frequency ω —both at zero and finitetemperature T , such as the optical conductivity σ ( ω, T ) .Here, we focus on the dynamical response of a novel QCPthat can appear at the surface of a three-dimensional(3D) topological insulator: it describes the interaction- FIG. 1. Phase diagram near the semimetal-superconductor(SM-SC) quantum critical point of Dirac fermions on the sur-face of a 3D topological insulator. T is the temperature and r is the nonthermal tuning parameter [see Eq. (1)]. The evo-lution of the Dirac dispersion and Cooper field potential areshown. Supersymmetry emerges at the QCP where it relatesthe Dirac fermions and the bosonic Cooper pairs. driven quantum phase transition between a single Diraccone of electrons and a gapped superconductor [5, 6](see Fig. 1). As an important step towards observingthis transition, recent experiments have reported the dis-covery of intrinsic superconductivity on the surface of a3D topological insulator, Sb Te [7]. We emphasize thatstandard 2D (or layered) systems that do not break time-reversal symmetry must have an even number of Diraccones and thus cannot host this transition. More com-plex scenarios realizing multiple copies of this QCP canoccur via f -wave pairing [8] and pair-density-wave [9]instabilities of spinless Dirac fermions on the 2D honey-comb lattice, or for interacting ultracold atomic gases inoptical lattices [10].When the chemical potential is at the Dirac point, aspecial type of symmetry emerges at the QCP [5, 6, 8,9, 11]: spacetime supersymmetry (SUSY). SUSY relatesbosons and fermions, and has been proposed to exist inextensions of the Standard Model of elementary particlephysics, but has not yet been observed. At the QCP ofFig. 1, it emerges naturally by relating the Dirac fermionsof the semimetal to the bosonic Cooper pairs of the su-perconductor. These two become degenerate at the tran-sition and in fact share a deeper relation described bySUSY. We emphasize that this is a consequence of thestrong interactions at the QCP, where long-lived exci-tations (quasiparticles) are destroyed by quantum zero-point fluctuations. We show that even in the presenceof such strong interactions, SUSY allows the exact de-termination of the zero-temperature optical conductivity σ ( ω, of the topological surface states at the QCP. Weare not aware of any known exact result for the dynamicalresponse of a realistic strongly interacting QCP in spatialdimensions higher than one. In addition, SUSY impliesthat the conductivity directly determines the shear vis- a r X i v : . [ c ond - m a t . s t r- e l ] J u l Dirac SM-SC
SC-Insulator σ ∞ π − √ π ≈ .
227 0 . η ∞ σ ∞ / ≈ . × − . × − λ corner σ ∞ / ≈ . . b − . TABLE I.
Exact results.
Comparison of the conductivity, viscosity, and entanglement entropy at two different QCPs. Leftcolumn: exact results obtained in this paper for the Dirac semimetal (SM) to superconductor (SC) QCP with emergentsupersymmetry. Right column: known approximate results for the SC to Cooper-pair-insulator QCP. The optical conductivityand dynamical shear viscosity at T = 0 are σ ( ω,
0) = σ ∞ e / ~ and η ( ω,
0) = η ∞ ω ~ . λ corner determines the entanglement entropyof nearly smooth corners [Eq. (5)]. b determines a finite- T correction to the optical conductivity of the form b ( ik B T / ~ ω ) . cosity and certain many-body entanglement properties.Our exact findings are summarized in Table I. We be-gin by describing the low-energy theory of the QCP, andthen explain how the emergent SUSY allows the exactdetermination of various properties such as the opticalconductivity. We end by discussing considerations rele-vant for the experimental observation of these signatures.The generalized Landau-Ginzburg theory for the quan-tum phase transition couples a single charge- e Diracfermion ψ to the charge- e Cooper pair bosonic field, φ , L = i ¯ ψγ µ ∂ µ ψ + 12 | ∂ µ φ | + r | φ | + λ | φ | + h ( φ ∗ ψ T iγ ψ + c . c . ) , (1)in imaginary time, where ¯ ψ = ψ † γ and γ µ , µ = 0 , , are × matrices satisfying the Pauli algebra. Wenote that time-reversal invariance forbids a fermion massterm. The QCP is obtained by tuning r to zero, andthe resulting system is strongly correlated because boththe quartic coupling λ and the fermion-boson coupling h are relevant at the noninteracting, UV fixed point λ = h = 0 . There is a single stable IR fixed point with λ = h = 0 [5, 6, 8, 9, 11], at which (1) becomes invari-ant under SUSY transformations that rotate the Diracfermion into the boson and vice-versa [12]. In line withthe requirement of SUSY, it was shown [5, 8] that thefermion and Cooper pair velocities flow to the same valueat low energies, which we henceforth set to unity. This isconsistent with the fermions and Cooper pair fields be-ing strongly coupled. As a result of the unique velocity,(1) displays emergent Lorentz invariance. By virtue ofSUSY, the fermion and boson anomalous dimensions areknown exactly [12]: η ψ = η φ = 1 / , a clear indication ofthe destruction of quasiparticles. The electric current isgiven by the sum of fermionic and bosonic contributions: J µ = ¯ ψγ µ ψ + i ( φ ∗ ∂ µ φ − c . c . ) .The QCP (1) has an important purely bosonic analogobtained by omitting the fermions, in which case it de-scribes the superconductor-to-insulator quantum phase transition obtained by localizing Cooper pairs [13]. Partof the interest in this QCP (and its optical conductivity)comes from the fact that it is believed to occur in certainthin-film superconductors [13]. The QCP that we studybelongs to a different universality class because it involvesfermions, and we shall contrast the two throughout (seeTable I). Exact charge & shear responses : As the system istuned to the QCP, the optical conductivity depends onlyon the ratio ~ ω/k B T [14]: σ ( ω, T ) = e ~ Φ (cid:18) ~ ωk B T (cid:19) , (2)where Φ( x ) is a dimensionless, universal scaling functionthat is fully determined by the universality class of thetransition. We recall that the conductivity is obtainedfrom the current-current correlator via the Kubo formula, σ = iω h J x ( ω, ~k = 0) J x ( − ω, ~k = 0) i T . An important con-sequence of the scale invariance is that the optical con-ductivity at T = 0 is a frequency-independent constant : σ ( ω,
0) = e σ ∞ / ~ , where we have defined σ ∞ = Φ( ∞ ) ,and we are working at frequencies lesser than microscopicenergy scales such that we are probing the universal re-sponse. For QCPs such as the one under consideration,this universal constant determines the charge response ofthe ground state in a system lacking quasiparticles. Wenow describe how the emergent SUSY can be used tocompute σ ∞ exactly.In supersymmetric field theories, operators are orga-nized into representations of the SUSY algebra calledsupermultiplets, the same way spin operators are orga-nized into representations of SU (2) . In our case, theelectric current J µ lies in the same supermultiplet as thestress tensor T µν , the so-called supercurrent supermul-tiplet [15]. Here, supercurrent does not refer to super-conductivity but rather to the Noether current associ-ated with SUSY. One associates to each supermultipleta so-called superfield which contains all the various com-ponents of the supermultiplet. The superfield associatedwith the supercurrent supermultiplet is denoted J µ , andis highly constrained by SUSY. Crucially, the two-pointcorrelation function of the supercurrent is entirely fixedup to an overall multiplicative constant [16–18], denoted C . Because J µ contains both the current J µ and thestress tensor T µν , this implies a relation between theirrespective two-point correlation functions. This relationin turn implies a nontrivial relation between the universalcharge and shear responses at the QCP (1).In 2D QCPs with emergent Lorentz invariance,the two-point correlation functions of the currentand the stress tensor have the power-law forms [19] h J µ ( x ) J ν (0) i = C J I µν ( x ) | x | and h T µν ( x ) T ρσ (0) i = C T I µν,ρσ ( x ) | x | , where x denotes the spacetime separation,the I ’s are dimensionless tensors without free parame-ters [20], and the constants C J,T are universal low-energyproperties related to the conductivity and viscosity, re-spectively, as we shall see below. The above discussionimplies that these are both proportional to C in our SUSYQCP, hence their ratio is fixed. We find that the partic-ular SUSY of (1) imposes C J /C T = 5 / [20]. This thenleads to a universal ratio between the zero-temperaturedynamical shear viscosity and optical conductivity at thestrongly interacting QCP (1). The dynamical shear vis-cosity η ( ω, T ) is given by the two-point function of the xy -component of the stress tensor [25, 26], and becomes η ( ω,
0) = η ∞ ω ~ at zero temperature. Fourier trans-forming from time to frequency, we find σ ∞ = π C J / and η ∞ = π C T / , and thus the universal ratio σ ∞ η ∞ = 40 , (3)which is a nontrivial fingerprint of the emergent SUSYat the QCP of (1). We emphasize that in the absenceof SUSY no relation exists in general between theconductivity and shear viscosity of QCPs. In fact, (3) isviolated at the superconductor-insulator QCP of Cooperpairs, see Table I.The emergent SUSY allows the exact calculation ofthe optical conductivity by geometric methods, and cru-cially relies on the connection between the conductivityand viscosity (3). First, the shear viscosity coefficient η ∞ , or alternatively C T , can be obtained from the secondderivative of the free energy on the squashed three-spherewith respect to the squashing parameter [15]. Remark-ably, the free energy on this spacetime geometry can becomputed exactly using the so-called SUSY localizationtechnique [27, 28], even if the theory is strongly coupled.For the QCP of interest to us, an integral expression for C T was recently obtained [29], which can be computednumerically. We were able to evaluate this integral inclosed form [20]. We then used the relation (3) to obtainan exact result for the T = 0 optical conductivity at the semimetal-superconductor QCP of 2D Dirac fermions: σ ( ω,
0) = 5(16 π − √ π e ~ ≈ . e ~ . (4)To our knowledge, this is the first exact result for the op-tical conductivity of a realistic strongly interacting QCP,and will thus serve as a benchmark for the dynamicalresponse of quantum critical systems. We note that (4)is both larger than the Dirac fermion conductivity σ ∞ = = 0 . [19] and the conductivity of the Cooper pairsuperconductor-insulator QCP σ ∞ = 0 . [30–35]. Ourresult (4) is tantalizingly close to the latter, suggestingthat even though the Dirac semimetal-superconductorQCP naively seems to have more conducting degrees offreedom, these interact more strongly. To put our ex-act result in perspective, we emphasize that σ ∞ for thesuperconductor-insulator QCP has been the subject ofnumerous studies [13, 36–38] over the past three decadesbut was reliably obtained only recently via large-scalequantum Monte Carlo simulations [30–34] and the con-formal bootstrap approach [35]. Finally, note that (4)is smaller than the conductivity of the Gaussian fixedpoint, = 0 . , in agreement with the expectationthat strong interactions reduce the charge mobility. Entanglement entropy : There is currently much in-terest in the entanglement properties of QCPs [39, 40]. Inparticular, the ground state entanglement entropy acrossa spatial region containing a sharp corner with openingangle θ contains a subleading logarithmic term whose co-efficient a ( θ ) depends only on the universality class ofthe QCP. This coefficient constitutes a new measure ofthe gapless degrees of freedom in strongly interacting sys-tems. Recent numerical work has focused on determining a ( θ ) for various interacting 2D QCPs, such as the XY andHeisenberg QCPs appearing in theories of quantum mag-netism [41–43]. For QCPs with emergent Lorentz invari-ance, the behavior of a ( θ ) near θ = π is determined bythe stress-tensor correlation coefficient C T encounteredabove [44, 45], a ( θ ) ≈ λ corner ( π − θ ) , λ corner = π C T / . (5)Using our exact result for σ ∞ , we obtain an exact resultin closed form for the corner coefficient of the semimetal-superconductor QCP occurring on the surface of a topo-logical insulator: λ corner = σ ∞ /
20 = π − √ π ≈ . .Unexpectedly, the optical conductivity at zero temper-ature entirely determines this property of the entangle-ment entropy. These two quantities are generally unre-lated in the absence of supersymmetry, as can be seen inTable I. We note that an integral expression for λ corner has been given previously [46]. In addition, our resultfor λ corner leads to an exact lower bound on a ( θ ) for all opening angles [47]: a ( θ ) ≥ (2 σ ∞ /
5) ln[1 / sin( θ/ . Optical conductivity at finite temperature : Sofar our discussion has centered on T = 0 properties. Wenow study the finite- T optical conductivity. The most re-liable statements can be made in the regime k B T (cid:28) ~ ω corresponding to the response at temperatures muchlower than the measurement frequency, where one ob-tains the nontrivial expansion [33] σ ( ω, T ) e / ~ = σ ∞ + b (cid:18) ik B T ~ ω (cid:19) − /ν + b (cid:18) ik B T ~ ω (cid:19) + · · · (6)where the dots denote higher powers of k B T / ~ ω , corre-sponding to increasingly small corrections. The dimen-sionless real coefficients b, b are universal properties ofthe QCP, and ν is the correlation-length critical expo-nent. The structure of (6) follows from simple physicalarguments, which we now briefly review. The large fre-quency expansion follows from the short time expansionof the operator product J x ( t ) J x (0) appearing in the Kuboformula for the conductivity. As t → , one can replacethe product by a series involving operators of increas-ing scaling dimensions [33], called the operator productexpansion (OPE). The operators that dominate the ex-pansion are the identity, the “mass” operator | φ | thattunes the system to the QCP in the Landau-GinzburgLagrangian (1), and the stress tensor T µν . We can thusschematically write J x J x ∼ | φ | + T µν + · · · . The co-efficients that multiply each operator in the series, omit-ted in this schematic expansion, are called OPE coeffi-cients. The parameters b, b are proportional to the OPEcoefficients multiplying | φ | and T µν , respectively. Thecorresponding powers of k B T / ~ ω in (6) are the scalingdimensions of these operators. The dimension of | φ | is ∆ r = 3 − /ν , where the correlation length exponent ν can be estimated via the (cid:15) expansion, ν ≈ . [11].A more accurate result is given by the conformal boot-strap, which predicts ∆ r = 1 . [48]. In contrast,the stress tensor is conserved and its scaling dimensionis not renormalized: ∆ T = 3 .Turning to the coefficients in Eq. (6), SUSY does notimpose any constraints on b . However, in the case of b SUSY leads to the strong result: b = 0 . (7)To understand this result, recall that b ∝ γ , where γ isan OPE coefficient multiplying the stress tensor. Thislatter coefficient can be determined from the three-pointcorrelation function h T µν J λ J ρ i at zero temperature [33].To see if SUSY constrains γ , we use a recent result forthe general form of the three-point correlation function hJ ν J λ J ρ i of the supercurrent [18]. While the preciseform of this function is fairly complicated, its crucial fea-ture is that it is characterized by a single overall constant,analogously to the two-point correlation function of thesupercurrent. By extracting the h T JJ i component of thethree-point correlation function of the supercurrent, wefind that γ and thus b vanish identically [20]. As shown in Table I, this is not the case at the superconductor-insulator QCP of Cooper pairs [33], as expected in theabsence of emergent SUSY. Sum rules : From the point of view of the frequencydependence, the finite-temperature results we have givenso far for the optical conductivity correspond to the high-frequency regime ~ ω (cid:29) k B T . In fact, we have suf-ficient information about the QCP to go even furtherand constrain the integral of the finite-temperature op-tical conductivity over all frequencies by way of a sumrule [33, 49, 50]: Z ∞ dω (cid:0) Re σ ( ω, T ) − σ ∞ e / ~ (cid:1) = 0 . (8)A dual sum rule obtained by replacing σ with /σ alsoholds [50]. The key point is that the integrand mustdecay sufficiently fast at high frequencies. This is thecase here, since in that limit the integrand scales as ( T /ω ) − /ν [Eq. (6)], and we know that ν > / [48]. Experimental realizations : Recent experimentssuggest that intrinsic (as opposed to proximity-induced)superconductivity may have been observed on the sur-face of the 3D topological insulator Sb Te [7]. Scanningtunneling microscopy data suggests an inhomogeneousdistribution of local critical temperatures T c ( r ) as highas 60 K, with global phase coherence achieved only ata much lower ∼ K. The QCP discussed here remainsstable against quenched disorder in T c , assuming it isshort-ranged, only if the Harris criterion νd > is sat-isfied, where d = 2 is the spatial dimension and ν is thecorrelation length exponent of the clean QCP [51]. Us-ing the conformal bootstrap result quoted earlier, oneobtains ν ≈ . , implying that the QCP is compro-mised by this type of disorder. Signatures of the cleanQCP will nevertheless be observable above the crossovertemperature k B T ∗ ∼ Λ W / (2 /ν − d ) ∼ Λ W . where Λ isa high-energy cutoff that can be taken as the bulk gapof the topological insulator and W is some dimensionlessmeasure of the disorder strength [52]. Given the highpower of W , one expects that the ~ ω (cid:29) k B T > k B T ∗ regime—in which the results discussed here hold—willbe reachable in the near future in samples with moder-ate amounts of disorder. Discussion & outlook : We have analyzed thedynamical response properties of a strongly interactingQCP occurring on the surface of a 3D topologicalinsulator between the gapless Dirac surface state and agapped surface superconductor. The emergence of SUSYin the low-energy limit at this QCP allowed us to deduceexact results for the dynamical response of the systemin closed form, as summarized in Table I. We found thatthe zero-temperature optical conductivity and dynamicalshear viscosity coefficient are frequency-independent,proportional to each other, and given by a simpleirrational number, Eq. (4). We further made exactstatements concerning the finite-temperature opticalconductivity, including high-frequency asymptotics andsum rules. It is natural to ask if other properties of thisQCP can be deduced from SUSY, such as the entan-glement Rényi entropies of corners [53]. More broadly,it would be worthwhile to investigate other QCPs withemergent SUSY in both two and three spatial dimensions.
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Department of Physics, Harvard University,Cambridge, Massachusetts 02138, USA Department of Physics, University of Alberta,Edmonton, Alberta T6G 2E1, Canada Theoretical Physics Institute, University of Alberta,Edmonton, Alberta T6G 2E1, Canada Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada (Dated: October 17, 2018) a r X i v : . [ c ond - m a t . s t r- e l ] J u l ONTENTS
I. Two-point functions of N = 2 SCFTs in 2+1 dimensions 2A. Two-point function of the U (1) current: C J C T σ ∞ h T J J i of N = 2 SCFTs in 2+1 dimensions 7References 12 I. TWO-POINT FUNCTIONS OF N = 2 SCFTS IN 2+1 DIMENSIONS
In supersymmetric theories, fields are grouped into different supermultiplets according to howthey transform under the supersymmetry algebra. For (2+1)-dimensional theories with N = 2supersymmetry, the R -current R µ and the stress tensor T µν are in the same supermultiplet, thesupercurrent supermultiplet. For the (2+1)D Wess-Zumino model we are considering, the R -current is simply proportional to the physical U (1) current J µ = ¯ ψγ µ ψ + i ( φ ∗ ∂ µ φ − c . c . ). In thesuperspace formalism, one associates to each supermultiplet a superfield which contains the variouscomponents of the supermultiplet. For the supercurrent supermultiplet, the superfield J µ is [1] J µ = sJ µ − ( θγ ν ¯ θ )2 T νµ + · · · , (S1)where s is the proportionality constant between R µ and J µ , ( · · · ) denotes all the components otherthan the U (1) current and the stress tensor, the γ ν ( ν = 0 , ,
2) are 2 × θ, ¯ θ are Grassmann-valued two-component spinors. In this Supplemental Material we considerMinkowski spacetime described by the metric tensor η µν = diag( − , , − + +), but results in imaginary time (Euclidean spacetime) can be obtainedsimply by replacing η µν with the Kronecker delta δ µν . Lorentz indices µ, ν, . . . are lowered (raised)with the metric tensor η µν ( η µν ), while spinorial indices α, β, . . . are lowered (raised) with theantisymmetric tensor ε αβ ( ε αβ ), where we define ε αβ ≡ −
11 0 ! = − iσ , ε αβ ≡ − ! = iσ , (S2)where σ , σ , σ are the standard Pauli matrices. Because ε αβ and ε αβ are antisymmetric tensors,one must be careful to use the second index of the pair when lowering and raising spinorial indices, θ α = ε αβ θ β , θ α = ε αβ θ β . (S3)2he gamma matrices are defined as ( γ µ ) αβ ≡ ( − iσ , σ , − σ ) , (S4)and obey the SO (1 ,
2) Clifford algebra { γ µ , γ ν } = 2 η µν . The Grassmann bilinear in Eq. (S1) isthus defined as θγ ν ¯ θ = θ α ( γ ν ) αβ ¯ θ β . One also often uses gamma matrices with two lower or twoupper spinorial indices, γ µαβ = ( − , σ , σ ) , γ αβµ = (1 , − σ , − σ ) , (S5)which can be obtained from Eq. (S4) by raising and lowering the appropriate indices. Importantly,these do not satisfy the Clifford algebra, and are real and symmetric. One can use them to writea Lorentz vector such as J µ as a symmetric bispinor J αβ , J αβ ≡ γ µαβ J µ , J µ = − γ αβµ J αβ , (S6)i.e., a second-rank symmetric tensor in spinorial indices, which has (2 × / J αβ in N = 2 superconformal field theories (SCFTs) in 2+1 dimensions [2], hJ αβ ( z ) J α β ( z ) i = c N =2 x α ( α x ββ ) ( x ) , (S7)where the definition of x will be given below. On the other hand, the general structure of the two-point functions h J µ ( x ) J ν (0) i and h T µν ( x ) T ρσ (0) i in conformal, but not necessarily superconformal,field theories was determined by Osborn and Petkou over two decades ago [3], h J µ ( x ) J ν (0) i = C J I µν ( x ) x , h T µν ( x ) T ρσ (0) i = C T I µν,ρσ ( x ) x , (S8)in 2+1 dimensions (we write x n ≡ | x | n for simplicity), where the tensors I µν ( x ) and I µν,ρσ ( x ) aregiven by I µν ( x ) ≡ η µν − x µ x ν x , I µν,ρσ ( x ) ≡
12 ( I µσ ( x ) I νρ ( x ) + I µρ ( x ) I νσ ( x )) − η µν η ρσ . (S9)Given Eq. (S1), Eq. (S7) implies that C J and C T are determined by the same universal constant c N =2 . By expanding the superspace expression (S7) in Grassmann components, we will determinehow C J and C T are related. A. Two-point function of the U (1) current: C J By spacetime translation invariance, we can set z to zero and z to z in Eq. (S7), hJ αβ ( z ) J α β (0) i = c N =2 x α ( α x ββ ) ( x ) , (S10)3here we use the notation A ( α B β ) = ( A α B β + A β B α ) for symmetrization. Equation (5) in themain text follows simply from applying Eq. (S6) to Eq. (S10). The bispinor x αβ is defined as [2] x αβ = x αβ − iε αβ θ ¯ θ, (S11)where θ ¯ θ ≡ θ α ¯ θ α and x αβ is the symmetric bispinor corresponding to x µ . To obtain the two-pointfunction of the U (1) current J µ , given Eq. (S1) one must set all θ ’s and ¯ θ ’s to zero [1] in Eq. (S10).We have s h J µ ( x ) J ν (0) i = 14 c N =2 γ αβµ ( γ ν ) α β x α ( α x ββ ) x = 14 c N =2 tr( γ µ γ λ γ ν γ ρ ) x λ x ρ x = − c N =2 I µν ( x ) x , (S12)using the identity tr( γ µ γ λ γ ν γ ρ ) = 2( η µλ η νρ + η µρ η λν − η µν η λρ ), hence we obtain C J = − s c N =2 . (S13) B. Two-point function of the stress tensor: C T For the two-point function of the stress tensor, we need to keep only terms that are quadraticin Grassmann variables (with two θ ’s and two ¯ θ ’s) on both sides of Eq. (S10). Using Eq. (S1), therelevant part of the two-point function of the supercurrent superfield J µ is thus hJ µ ( z ) J σ (0) i = 4( θγ ν ¯ θ )( θγ ρ ¯ θ ) h T νµ ( x ) T ρσ (0) i + · · · (S14)The denominator of Eq. (S10) is( x ) = (cid:18) − x αβ x αβ (cid:19) = (cid:0) x + ( θ ¯ θ ) (cid:1) = x + 3 x ( θ ¯ θ ) , (S15)observing that powers of θ ¯ θ higher than two vanish identically because of the Grassmann natureof θ and ¯ θ . The compute the numerator, we first observe that x αα = x αα − iδ α α θ ¯ θ, (S16)and we obtain γ αβµ ( γ σ ) α β x α ( α x ββ ) = − x I µσ ( x ) + 2 η µσ ( θ ¯ θ ) + · · · , (S17)where ( · · · ) denotes possible terms proportional to θ ¯ θ which do not contribute to the two-pointfunction of the stress tensor. We thus have hJ µ ( z ) J σ (0) i = c N =2 (cid:18) − x I µσ ( x ) + η µσ ( θ ¯ θ ) x + 3 x ( θ ¯ θ ) (cid:19) + · · · = c N =2 (cid:18) I µσ ( x ) + η µσ x (cid:19) ( θ ¯ θ ) + · · · , (S18)4here the dots denote all terms not proportional to ( θ ¯ θ ) . Consider now Eq. (S14). By Lorentzinvariance, we must have ( θγ ν ¯ θ )( θγ ρ ¯ θ ) = Cη νρ ( θ ¯ θ ) where C is some constant. Setting for instance ν = ρ = 1, which implies by Eq. (S4) that γ ν = γ ρ = σ , it is easily shown that ( θσ ¯ θ ) = − ( θ ¯ θ ) ,and thus ( θγ ν ¯ θ )( θγ ρ ¯ θ ) = − η νρ ( θ ¯ θ ) . (S19)Substituting this expression in Eq. (S14), and using Eq. (S18), we obtain η νρ h T νµ ( x ) T ρσ (0) i = − c N =2 (cid:18) I µσ ( x ) + η µσ x (cid:19) , (S20)by equating the coefficients of ( θ ¯ θ ) on either side of the equation. To determine the relationshipbetween C T and c N =2 , we compute the left-hand side of Eq. (S20) from the general relation Eq. (S8), η νρ h T νµ ( x ) T ρσ (0) i = 16 C T (cid:18) I µσ ( x ) + η µσ x (cid:19) , (S21)from which we obtain the relation C T = − c N =2 . (S22)We thus find that the ratio between C J and C T is a universal number: C J /C T = 2 / s . To fix theproportionality constant s for the Wess-Zumino theory, we use the fact that the UV fixed pointof Eq. (2) in the main text, the theory of a free boson and a free Dirac fermion, is also a N = 2SCFT in 2+1 dimensions. The coefficients C J and C T at this fixed point are simply the sum ofthe free boson and free Dirac fermion values, which can be computed explicitly [4], C J = C φJ + C ψJ = 10 /S , C T = C φT + C ψT = 6 /S , (S23)where S D ≡ π D/ / Γ( D/ C J C T = 53 , (S24)and thus s = p / II. EXACT EVALUATION OF THE GROUND-STATE CONDUCTIVITY σ ∞ In this section we provide the explicit calculation of the ground-state conductivity σ ∞ at thesemimetal-superconductor QCP described by the N = 2 Wess-Zumino SCFT. We closely followNishioka and Yonekura [5], who gave an integral expression for a quantity that is proportional to σ ∞ . In Ref. 5 this expression was only evaluated numerically, while we here show that this integral,and hence σ ∞ , reduces to a simple irrational number.5ishioka and Yonekura give an expression for the coefficient C T of the two-point function ofthe stress tensor in terms of a quantity called τ RR [5], C T = 3 τ RR π . (S25)At the UV fixed point of the Wess-Zumino theory one has τ RR = [5], in agreement with the valueof C T given in Eq. (S23). Using Eq. (S24), we thus have C J = 5 τ RR π . (S26)We note that the normalization of the R -current in Ref. [5] differs from the one used here [2].By Fourier transforming the two-point function h J J i in Eq. (S8), and using the standard Kuboformula for the conductivity σ ( ω ) = − iω h J x ( ω, ~k = 0) J x ( − ω, ~k = 0) i , (S27)we find σ ∞ = π C J /
2, which implies σ ∞ = 54 τ RR . (S28)In order to evaluate τ RR , one first considers the partition function of the theory on the com-pactified spacetime S b , which is a squashed three-sphere. When the squashing parameter b isset to unity, S b reduces to the regular three-sphere. The (dimensionless) free energy is given by F ( b ) = − log Z S b , where Z S b is the partition function. τ RR is then obtained by taking the secondderivative of F ( b ) with respect to b : τ RR = 2 π Re ∂ F∂b (cid:12)(cid:12)(cid:12)(cid:12) b =1 . (S29)Heuristically, each b -derivative brings down one stress tensor, so that we are left with the two-pointfunction h T T i .The crucial simplification comes because of SUSY, which leads to a powerful method calledsupersymmetric localization that allows the computation of the partition function in terms of asimple integral [6, 7]. Using Eq. (S29) then gives [5] τ RR = 2 π Z ∞ dy (cid:20) (cid:18) y − cosh(2 y/ y (cid:19) + [sinh(2 y ) − y ] sinh(2 y/ y (cid:21) , (S30)where we have used the fact that the R -charge associated with the chiral multiplet of the interactingWess-Zumino N = 2 SCFT is 2 / input theexact scaling dimension of the chiral multiplet (containing φ, ψ ).6he part proportional to 1 / π Z ∞ dy (cid:18) y − cosh(2 y/ y (cid:19) = 29 √ π . (S31)The second term is more subtle. To simplify its evaluation, we slightly deform that part of theintegrand: A ( a ) = 2 π Z ∞ dy [sinh(2 y ) − y ] sinh(2 y/ (cid:0) y + a (cid:1) , (S32)where we have introduced a shift by a/ > a → + limit: A (0 + ) = 64243 − √ π . (S33)Adding Eqs. (S31) and (S33) we obtain: τ RR = 4243 − √ π ! ≈ . , (S34)which agrees with the numerical evaluation of Eq. (S30), given in Ref. [5]. As an independentcheck of the SUSY localization calculation described above, a recent highly non-trivial conformalbootstrap calculation [9] has yielded τ RR = 0 . σ ∞ = 54 τ RR = 5243 − √ π ! ≈ . . (S35) III. THREE-POINT FUNCTION h T J J i OF N = 2 SCFTS IN 2+1 DIMENSIONS
We now consider the three-point function h T J J i of the stress tensor and two U (1) currents.The generic form of this function for (2 + 1)-dimensional CFTs is given by [3], h T µν ( x ) J λ ( x ) J ρ ( x ) i = t µνστ ( X ) η σκ η τγ I λκ ( x ) I ργ ( x ) x x x , (S36)where we define x ij = x i − x j , X = x x − x x . (S37)When the symbol x ij appears raised to an odd power, as in the denominator of Eq. (S36), itmeans | x ij | = q x ij . The second-rank tensor I µν ( x ) is defined in Eq. (S9), and the dimensionlessfourth-rank tensor t µνστ ( X ) is defined as t µνστ ( X ) = ˆ ah µν ( ˆ X ) η στ + ˆ bh µν ( ˆ X ) h στ ( ˆ X ) + ˆ ch µνστ ( ˆ X ) + ˆ eh µνστ ( ˆ X ) , (S38)7here we define h µν ( ˆ X ) = ˆ X µ ˆ X ν − η µν , (S39) h µνστ ( ˆ X ) = ˆ X µ ˆ X σ η ντ + ˆ X ν ˆ X σ η µτ + ˆ X µ ˆ X τ η νσ + ˆ X ν ˆ X τ η µσ −
43 ˆ X µ ˆ X ν η στ −
43 ˆ X σ ˆ X τ η µν + 49 η µν η στ , (S40) h µνστ ( ˆ X ) = η µσ η ντ + η µτ η νσ − η µν η στ , (S41)with ˆ X µ = X µ / | X | . The constants ˆ a, ˆ b, ˆ c, ˆ e are not linearly independent, as one has the relations3ˆ a − b + 2ˆ c = 0 , ˆ b − e = 0 , (S42)such that the three-point function h T J J i is in general specified by two independent constants.The form of the three-point function (S44) simplies tremendously if one considers a collinearframe, i.e., three spacetime points constrained to lie on a straight line: x µ = xn µ , x µ = yn µ , and x µ = zn µ , with n µ n µ = 1 [3]. We also assume for convenience that x > y > z . The three-pointfunction is then given by h T µν ( x ) J λ ( x ) J ρ ( x ) i = A T JJµνλρ ( x − y ) ( x − z ) ( y − z ) , (S43)where the fourth-rank tensor A T JJµνλρ is symmetric in both the first and second pair of Lorentzindices.As seen previously, in (2 + 1)-dimensional N = 2 SCFTs both the U (1) current and the stresstensor are part of the supercurrent supermultiplet. Therefore one can extract the three-pointfunction (S36) from the three-point function of the supercurrent superfield J αβ , whose generalform in those theories has been recently derived [2], hJ αα ( z ) J ββ ( z ) J γγ ( z ) i = x αρ x α ρ x βσ x β σ ( x ) ( x ) H ρρ , σσ γγ ( X , Θ ) , (S44)where the sixth-rank tensor H , given in Eq. (7.44) of Ref. 2, is specified by a single independentconstant d N =2 . Thus N = 2 supersymmetry reduces the number of independent constants in thethree-point function h T J J i from two to one. Our goal is to determine exactly how ˆ a, ˆ b, ˆ c, ˆ e arerelated to d N =2 .To extract Eq. (S36) from the superfield expression (S44), we need only keep the scalar compo-nent of J ββ ( z ), J γγ ( z ) and the θγ µ ¯ θ component of J αα ( z ). Switching from symmetric bispinorsto Lorentz vectors, we have hJ ν ( z ) J λ ( z ) J ρ ( z ) i = − s ( θ γ µ ¯ θ ) h T µν ( x ) J λ ( x ) J ρ ( x ) i + · · · , (S45)where θ , ¯ θ are the Grassmann coordinates associated with z and the dots represents other compo-nents of the superfield three-point function in which we are not interested. Given that θ , ¯ θ , θ , ¯ θ
8o not appear on the right-hand side of Eq. (S45), we can set them all to zero in the expansion ofEq. (S44) in components. Furthermore only terms quadratic in θ , ¯ θ need be kept.The right-hand side of Eq. (S44) is expressed in terms of two Grassmann-valued Lorentz vectors x ij and X , and the Lorentz spinor Θ , which we must expand in components. The bispinor x αβij is written as the sum of symmetric and antisymmetric parts [2], x αβij = ˜ x αβij + i ε αβ θ γijI θ ijIγ , (S46)where θ αijI ≡ θ αiI − θ αjI , I = 1 ,
2, are differences of real Grassmann coordinates. The complexcoordinates θ αi , ¯ θ αi are given in terms of the latter as θ αi = 1 √ θ αi + iθ αi ) , ¯ θ αi = 1 √ θ αi − iθ αi ) , (S47)hence θ αijI θ ijIα = 2 θ αij ¯ θ ijα = 2 θ ij ¯ θ ij , and we can write x αβij = ˜ x αβij + iε αβ θ ij ¯ θ ij . (S48)The symmetric part is defined as ˜ x αβij = x αβij + 2 iθ ( αiI θ β ) jI , (S49)where x αβij is the symmetric bispinor associated with x ij defined in Eq. (S37). Since i = j , thesecond term on the right-hand side of Eq. (S49) necessarily involves Grassmann coordinates otherthan θ , ¯ θ , and we can write ˜ x αβij = x αβij . Therefore, for our purposes x αβij = x αβij + iε αβ θ ij ¯ θ ij .Furthermore, since θ , ¯ θ , θ , ¯ θ are set to zero, we have x αβ = x αβ + iε αβ θ ¯ θ , x αβ = x αβ − iε αβ θ ¯ θ , (S50) x αβ = x αβ + iε αβ θ ¯ θ , x αβ = x αβ − iε αβ θ ¯ θ , (S51) x αβ = x αβ , x αβ = x αβ . (S52)Since x αβij , x ijαβ are symmetric in spinor indices and ε αβ , ε αβ antisymmetric, the square of x ij isgiven by x ij ≡ − x αβij x ijαβ = x ij + ( θ ¯ θ ) = x ij . The bispinor X αβ is defined as the matrixelements of [2] ˇ X = − ˇ x T ˆ x ˇ x T x x , (S53)where ˆ m denotes a matrix with two upper spinorial indices and ˇ m a matrix with two lower spinorialindices. We thus have X αβ = − x α α x α β x ββ x x = x ββ x x (cid:16) x αα x α β + ix αβ θ ¯ θ (cid:17) , (S54)9o the desired order. Finally, Θ Iα is defined as the matrix elements of [2]ˆΘ = − ˇ x T ˆ θ x + ˇ x T ˆ θ x , (S55)where ˆ θ ij is a matrix with matrix elements θ αijI = θ αiI − θ αjI . To the desired order, we obtainΘ Iα = x αβ θ β I x . (S56)We now evaluate Eq. (S44) in the collinear frame discussed earlier. In the collinear frame,Eq. (S50)-(S52) simplify to x αβ = ( x − y ) n αβ − iε αβ θ ¯ θ , (S57) x αβ = ( x − z ) n αβ − iε αβ θ ¯ θ , (S58) x αβ = ( y − z ) n αβ , (S59)where n αβ is the symmetric bispinor associated with n µ , and Eq. (S54), (S56) become X αβ = − x − y ( x − z )( y − z ) n αβ + i ( x − z ) ε αβ θ ¯ θ , (S60)Θ Iα = 1 x − z n αβ θ β I . (S61)The sixth-rank tensor H in Eq. (S44) is given by [2] H αα ,ββ ,γγ ( X , Θ ) = id N =2 (cid:20) X (cid:16) ε α ( β ε β ) α Ξ γγ + ε α ( γ ε γ ) α Ξ ββ + ε β ( γ ε γ ) β Ξ αα (cid:17) + 1 X (cid:16) X αα X γγ Ξ ββ + 3 X ββ X γγ Ξ αα − X αα X ββ Ξ γγ (cid:17) + 1 X (cid:16) ε α ( γ ε γ ) α X ββ + 5 ε β ( γ ε γ ) β X αα − ε α ( β ε β ) α X γγ (cid:17) X δδ Ξ δδ + 52 1 X X αα X ββ X γγ X δδ Ξ δδ (cid:21) , (S62)where we define Ξ αα = ε IJ Θ Iα Θ Jα . (S63)In the collinear frame, we find Ξ αα = 2 i ( x − z ) n αβ n α β θ ( β ¯ θ β )1 . (S64)Since each term in Eq. (S62) contains Ξ , which is quadratic in the Grassmann coordinates θ , ¯ θ ,we can ignore the Grassmann part of X in Eq. (S60). We therefore have X αβ = − x − y ( x − z )( y − z ) n αβ , | X | = q − X αβ X αβ = x − y ( x − z )( y − z ) . (S65)10ikewise, since H is quadratic in Grassmann coordinates we can neglect the Grassmann part ofthe x ijαβ factors in front of H in Eq. (S44). Using the latter equation, we have s ( θ γ µ ¯ θ ) h T µν ( x ) J λ ( x ) J ρ ( x ) i = 116( x − z ) ( y − z ) ∆ νλρ ( x, y, z ) , (S66)where ∆ νλρ ( x, y, z ) = γ αα ν γ ββ λ γ γγ ρ x ατ x α τ x βσ x β σ H ττ ,σσ γγ ( X , Θ )= ( x − z ) ( y − z ) ( nγ ν n ) αα ( nγ λ n ) ββ ( γ ρ ) γγ H αα ,ββ ,γγ ( X , Θ ) . (S67)For simplicity we will focus only on terms in Eq. (S66) that are proportional to θ γ λ ¯ θ , as thisturns out to be sufficient to relate ˆ a, ˆ b, ˆ c, ˆ e to d N =2 . Since on the right-hand side of Eq. (S66) θ , ¯ θ only appear in Ξ , and since the index λ appears in the combination ( nγ λ n ) ββ , the only terms inEq. (S62) that can generate θ γ λ ¯ θ are those that contain Ξ ββ . Using( nγ λ n ) ββ Ξ ββ = 2 i ( x − z ) θ γ λ ¯ θ , (S68)( nγ ν n ) αα ( γ ρ ) γγ ε α ( γ ε γ ) α = − η νρ + 4 n ν n ρ , (S69)( nγ ν n ) αα ( γ ρ ) γγ X αα X γγ = 4( x − y ) ( x − z ) ( y − z ) n ν n ρ , (S70)as well as Eq. (S66) and (S43), we obtain s ( θ γ µ ¯ θ ) A T JJµνλρ = 12 d N =2 ( η νρ − n ν n ρ ) θ γ λ ¯ θ + . . . (S71)On the other hand, we can directly calculate the left-hand side of Eq. (S71) for a general CFT in2+1 dimensions from Eq. (S36) and (S43). We obtain s ( θ γ µ ¯ θ ) A T JJµνλρ = s (cid:16) ˆ bη νρ − (2ˆ b + 3ˆ c ) n ν n ρ (cid:17) θ γ λ ¯ θ + . . . , (S72)where we have used the relations (S42) to eliminate ˆ a and ˆ e in favor of ˆ b and ˆ c . ComparingEq. 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