TTopological invariants for holographic semimetals
Yan Liu a and Ya-Wen Sun b,c,d a Department of Space Science, and International Research Institute of MultidisciplinaryScience, Beihang University, Beijing 100191, China b School of physics & CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100049, China c Kavli Insititute for Theoretical Sciences, University of Chinese Academy of Sciences,Beijing 100049, China d CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China
Abstract
We study the behavior of fermion spectral functions for the holographic topolog-ical Weyl and nodal line semimetals. We calculate the topological invariants fromthe Green functions of both holographic semimetals using the topological Hamilto-nian method, which calculates topological invariants of strongly interacting systemsfrom an effective Hamiltonian system with the same topological structure. Non-trivial topological invariants for both systems have been obtained and the presenceof nontrivial topological invariants further supports the topological nature of theholographic semimetals. Email: [email protected] Email: [email protected] a r X i v : . [ h e p - t h ] D ec ontents M/b → M/b (cid:28) ( M/b ) c case . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.3 Spectral function at ω = ± a and k z = 0 . . . . . . . . . . . . . . 274.3 Topological invariants for holographic nodal line semimetal . . . . . . . . 31 s and e Topological states of matter are a new type of quantum states of matter that cannot bedescribed by the Landau-Ginzburg paradigm and do not possess a local order param-eter [1]. They are otherwise characterized by nontrivial topological structures in theirquantum wave functions and possess novel nontrivial properties that are stable undersmall perturbations. Many topological states of matter have been found in laboratoriesalready, e.g. topological insulators, anomalous quantum Hall effects, Weyl semimetals,etc.. As most known properties of topological states of matter have been studied in the1eakly coupled theory, an important question is if interactions, especially strong inter-actions, will change the topological properties and destroy the topological structures ofthese systems.In [2, 3] and [4], strongly coupled topological Weyl and nodal line semimetals werefound in the framework of anti-de Sitter/conformal field theory (AdS/CFT) correspon-dence, which turns a strongly coupled field theoretical problem into a weakly coupledclassical gravity problem [5, 6, 7]. The evidence that the holographic Weyl and nodalline semimetals are topological semimetals includes the anomalous Hall conductivity forWeyl semimetals [3], the induced effect of surface state [8], as well as the nodal loop fromthe dual fermion spectral functions [4]. Based on the holographic models of semimetals,many interesting observations have been made, including a prediction of nontrivial Hallviscosity in the quantum critical region due to the presence of the mixed gauge gravita-tional anomaly [9], the axial anomalous Hall effect [10], the behavior of AC conductivity[11], the disorder effect on the topological phase transition [12], and the properties ofquantum chaos in the quantum critical region [13]. Moreover it has been shown thatthere is a universal bulk topological structure for both holographic topological semimetals[4], where the near horizon behavior of the solutions determines that small perturbationscould not gap the semimetal phases. However, topological invariants could not be de-fined associated with the bulk topological structure, and for a further nontrivial pieceof evidence — the topological invariants, we have to resort to the dual Green functionsobtained from probe fermions on the bulk background.For weakly coupled topological systems, topological invariants can be defined fromthe Bloch states, i.e. the eigenstates of the weakly coupled Hamiltonians. A simpleexample is the nontrivial Berry phase associated with a closed loop in the momentumspace of many topological systems, which is calculated from the Berry connection of theeigenstates of the Hamiltonian. Equivalently, the formula for the topological invariantscould also be rewritten using Green functions, which in principle also works at the strongcoupling limit. However, the topological invariants defined from Green functions usuallyrequire an integral in the imaginary frequency axis, which is extremely time consumingwhen we only have numerical results for the Green functions. In [16, 17, 18], a methodcalled topological Hamiltonian was developed, which states that topological invariantsof a strongly coupled system could be calculated from the eigenstates of an effectiveHamiltonian in the same way as in the weakly coupled theory.As proved in [16, 17, 18], this effective topological Hamiltonian could be directlydefined from the zero frequency Green functions and it possesses the same topologicalstructure as the original strongly coupled system. Thus to calculate the topologicalinvariants in a strongly coupled holographic semimetal system, we would first need to Different holographic models for Weyl semimetal can be found in [14, 15].
In this section, we first review the basic setups and the topological structures of theholographic Weyl and nodal line semimetals as well as their phase diagrams. The basicsin this section will provide the topologically nontrivial semimetal background for thecalculation of topological invariants in Sec 4. More details could be found in [3, 4].
A Weyl semimetal breaks either time reversal or inversion symmetry [19]. For a holo-graphic Weyl semimetal, we have two important fields in the bulk: the axial gauge field A a corresponding to the time reversal symmetry breaking operator whose source intendsto separate one Dirac node into two Weyl nodes and a scalar field Φ corresponding to theDirac mass operator whose source intends to gap the system. As a mass operator in thefield theory breaks the axial symmetry, this scalar field should be axially charged in thebulk with a nonzero source at the boundary that breaks the axial symmetry explicitly.3he bulk action of the holographic Weyl semimetal system [3] is S = (cid:90) d x √− g (cid:20) κ (cid:18) R + 12 L (cid:19) − F − F + α (cid:15) abcde A a (cid:18) F bc F de + F bc F de (cid:19) − ( D a Φ) ∗ ( D a Φ) − V (Φ) (cid:21) , where D a = ∇ a − iq A a and F µν , F µν are the vector U (1) V gauge field strength and theaxial U (1) A gauge field strength separately. α is the coefficient of the Chern-Simons termwhich corresponds to the chiral anomaly and Φ is the axially charge scalar field. Thepotential term is V = m | Φ | + λ | Φ | . (2.1)We choose the mass of the scalar field to be m = − ds = u ( − dt + dx + dy ) + dr u + hdz , Φ = φ , A = A z dz . (2.2)The asymptotic AdS boundary conditions characterizing proper source terms areΦ = Mr + · · · , A z = b + · · · . (2.3)For general parameter values, there exist three kinds of near horizon solutions atzero temperature, which flow to boundary solutions at three regions of M/b . The criticalsolution corresponds to the near horizon Lifshitz solution, which flows to boundary
M/b =( M/b ) c . The Weyl semimetal phase has an AdS near horizon solution and flows to valuesof M/b < ( M/b ) c . The trivial phase has an AdS near horizon solution with a different IRAdS radius and flows to values of M/b > ( M/b ) c . The different IR AdS radius indicatesthat some degrees of freedom are gapped out along the RG flow from UV to IR. Notethat λ Φ at the horizon denotes the degrees of freedom that are not gapped out in theIR. For the reference of the following sections, we write here the near horizon geometryfor the topological phase u = r , h = r , A z = a + πa φ r e − a r , φ = √ πφ (cid:16) a r (cid:17) / e − a r , (2.4)where a is the near horizon value of the separation A z and φ is a free parameter flowingthe symmetry to different boundary values of M/b < ( M/b ) c .For the Weyl semimetal, there is a smoking gun transport coefficient: the anomalousHall conductivity σ AHE , which is only nonzero in the Weyl semimetal phase. Semi-analytic calculations showed that σ AHE = 8 αA z (cid:12)(cid:12) r = r with the horizon value of A z . Fig 1shows the anomalous Hall conductivity as a function of M/b , indicating that the phaseat
M/b < ( M/b ) c is indeed the topological Weyl semimetal phase.4 �� ��� ��� ��� ��� ��� ��������������������� �� σ ��� � α � Figure 1:
The dependance of anomalous Hall conductivity at zero temperature in the holo-graphic Weyl semimetal as a function of
M/b for m = − , q = 1 , λ = 1 / A nodal line semimetal has a nontrivial shape of Fermi surface where Fermi points connectto form a loop under certain symmetries (see [20] for a review). A topologically nontrivialnodal line semimetal cannot be gapped by small perturbations unless passing througha topological phase transition. Two important fields in the holographic setup are themassive two form field B ab whose dual source intends to deform the Dirac point to anodal loop and the axially charged scalar field whose dual source intends to gap thesystem. The action [4] is S = (cid:90) d x √− g (cid:20) κ (cid:18) R + 12 L (cid:19) − F − F + α (cid:15) abcde A a (cid:18) F bc F de + F bc F de (cid:19) − ( D a Φ) ∗ ( D a Φ) − V (Φ) − η (cid:0) D [ a B bc ] (cid:1) ∗ (cid:0) D [ a B bc ] (cid:1) − V ( B ab ) − λ | Φ | B ∗ ab B ab (cid:21) where F ab = ∂ a V b − ∂ b V a is the vector gauge field strength, F ab = ∂ a A b − ∂ b A a is the axialgauge field strength, D a = ∇ a − iq A a , D a = ∇ a − iq A a and D [ a B bc ] = ∂ a B bc + ∂ b B ca + ∂ c B ab − iq A a B bc − iq A b B ca − iq A c B ab . (2.5)The potential terms are V = m | Φ | + λ | Φ | , V = m B ∗ ab B ab , (2.6)where m is the mass of the scalar field and m is the mass of the two form field. The λ term denotes the interaction between the scalar field and the two form field. Withoutloss of generality we choose the conformal dimension for operators dual to Φ and B ab tobe 1, i.e. m = − m = 1. We also set λ = 1, λ = 0 . η = 1 for simplicity.5ote that here the real part of Φ corresponds to the operator ¯ ψψ and the imaginarypart corresponds to ¯ ψ Γ ψ as could be checked from the ward identity for J µ . However,the real and imaginary parts of B ab do not correspond to the composite operators ¯ ψ Γ µν ψ and ¯ ψ Γ µν Γ ψ . This is because ¯ ψ Γ µν Γ ψ = i (cid:15) µνρσ ¯ ψ Γ ρσ ψ , which means that the real partand imaginary part of B ab should have a self duality property in order to be dual to¯ ψ Γ µν ψ and ¯ ψ Γ µν Γ ψ . Here B ab could instead be considered to be dual to a sum of manysuch kinds of composite operators each composed of a different fermionic operator. Inthis way, B ab does not need to have the self dual property between its real and imaginaryparts. An action that could describe the two form field with the self dual property is S ∝ (cid:82) d x √− g (cid:2) i ( B ∧ H ∗ − B ∗ ∧ H + m B | B | ) (cid:3) , where H = dB − iq A ∧ B [21, 22]. The zero temperature solution can be parameterized as ds = u ( − dt + dz ) + dr u + f ( dx + dy ) , Φ = φ ( r ) , B xy = B ( r ) . (2.7)The asymptotic AdS boundary conditions with proper source terms areΦ (cid:39) Mr + · · · , B xy (cid:39) br + · · · . (2.8)We have three different kinds of near horizon geometries at zero temperature. Thecritical soluton has a Lifshitz symmetry at the horizon and flows to M/b = (
M/b ) c at theboundary. The nodal line semimetal phase has another Lifshitz near horizon solution andflows to values of M/b < ( M/b ) c . The trivial phase has an AdS near horizon solutionwith a different IR AdS radius and flows to values of M/b > ( M/b ) c . The different IRAdS radius indicates that some degrees of freedom get gapped out along the RG flowfrom UV to IR. For the reference of following sections, we list the near horzion geometryfor the topological phase u = 18 (11 + 3 √ r (cid:16) δu r α (cid:17) ,f = (cid:115) √ − b r α (cid:16) δf r α (cid:17) ,φ = φ r β ,B = b r α (cid:16) δb r α (cid:17) , where ( α, β, α ) = (0 . , . , . δf, δb ) = ( − . , − . δu for the parametervalues that we have fixed above. We thank Carlos Hoyos and Elias Kiritsis for helpful discussions on this point. .3 A universal bulk topological structure There is a universal bulk topological structure for the holographic topological semimetalsdetermined by the horizon solutions. We denote the two kinds of fields as A and φ , thefirst of which deforms the topology of the Fermi point to whatever possible configurationsand the second intends to gap the system. The conformal dimension of the two fieldsat the horizon are δ A,φ ± separately and the leading order horizon solutions of the twofields are A, φ ∼ c A,φ r δ A,φ + + · · · , where the r δ A,φ − terms are too divergent to get regularsolutions. The crucial observation is that at the horizon the two coefficients c A,φ cannotboth be nonzero due to the interaction between A and φ which leads to three differentadiabatically connected solutions. The solutions are distingushed into three categories:(1) c A (cid:54) = 0 , c φ = 0; (2) c A = 0 , c φ (cid:54) = 0, and (3) c A = 0 , c φ = 0, corresponding to threetypes of phases — the topological semimetal phase, the partially gapped phase and thecritical point. At the horizon c A,φ cannot coexist leads to the fact that at the semimetalphase, we cannot find a solution of perturbations of the gap operator that could gap thesystem. Thus small perturbations could not gap the system indicating that the semimetalphases are topological semimetals.
The existence of a universal bulk topological structure suggests that we could in principleproduce a large class of holographic zero density systems which possess a nontrivial topo-logical structure. In some cases we could obtain some specific transport behavior whichtells what is the corresponding topological state, e.g. in the Weyl semimetal case, nontriv-ial anomalous Hall conductivity shows that it corresponds to a topologically nontrivialWeyl semimetal. However, in most cases, we would not be able to tell from the bulktopological structure what would be the boundary topological structure. In condensedmatter physics, the band structure is used to characterize topological structures of weaklycoupled topological states of matter. The wave function of electrons or equivalently theHamiltonian of the system possesses a nontrivial topological structure and topological in-variants could be defined. Here for the strongly coupled topological states of matter, thereis no band theory or even no quasiparticle descriptions, however, we could still detect thetopological structure from the dual Green functions of probe fermions and calculate thetopological invariants from the Green functions. In this section we probe the holographicWeyl/nodal line semimetals with fermions and provide prescriptions for calculating the Fermion spectral function for the holographic finite density systems were first studied in [23, 24].
To probe the dual fermion spectrum of the holographic Weyl semimetal, we add a probefermion on the background geometry (2.2) and calculate the dual Green functions fromthe holographic dictionary. In five dimensions, a bulk four component spinor correspondsto a two component chiral spinor of the dual four dimensional field theory [25]. We utilizetwo spinors Ψ and Ψ with opposite masses and one standard quantization while theother alternative quantization to correspond to two opposite chiralities. For the holographic Weyl semimetal, Φ breaks the axial symmetry so that it couplesthe left chirality to the right chirality. The axial potential A z breaks the time reversalsymmetry while conserves the axial symmetry, though the two chiralities are affected indifferent ways by A z . This leads to the following action of probe fermions S = S + S + S int , (3.1) S = (cid:90) d x √− gi ¯Ψ (cid:0) Γ a D a − m f − iA a Γ a (cid:1) Ψ ,S = (cid:90) d x √− gi ¯Ψ (cid:0) Γ a D a + m f + iA a Γ a (cid:1) Ψ ,S int = − (cid:90) d x √− g (cid:0) iη Φ ¯Ψ Ψ + iη ∗ Φ ∗ ¯Ψ Ψ (cid:1) , where D a = ∂ a − i ω mn,a Γ mn , (3.2)and we choose both the axial charge and the coupling constant η to be 1. Note thatthe coupling constant in front of A z is opposite for the two spinors. We use the followingconvention of Γ-matricesΓ µ = γ µ , Γ r = γ , Γ t = (cid:18) ii (cid:19) , Γ i = (cid:18) iσ i − iσ i (cid:19) , Γ r = (cid:18) − (cid:19) . (3.3)From this form of bulk action for probe fermions, we could see that Φ corresponds tothe operators of ¯ ψψ and ¯ ψγ ψ where ψ is the boundary four component spinor operator. Equivalently one could as well choose two spinors with the same mass and the same quantizationwith the spatial Γ-matrices of one spinor having an opposite sign compared to the other spinor.
8n (3.1) Φ couples to ¯Ψ Ψ , which with Ψ , taking opposite quantizations is just theexpectation value of the dual operator of ¯ ψψ when the source of ψ is zero. Similar probefermionic action was considered in [26] to study the holographic mass effect of the fourdimensional Dirac fermions.The equations of motion are (cid:0) Γ a D a − m f − iA z Γ z (cid:1) Ψ − η φ Ψ = 0 , (cid:0) Γ a D a + m f + iA z Γ z (cid:1) Ψ − η φ Ψ = 0 , (3.4)where we have used Φ = φ ( r ) and η being a real number. We expand the bulk fermionfield as Ψ l = ( uf ) − / ψ l e − iωt + ik x x + ik y y + ik z z , l = 1 , . (3.5)Since the spacetime background is isotropic in the x -, y -plane, after substituting thebackground geometry the equations of motion for probe fermions become (cid:32) Γ r ∂ r + 1 u (cid:16) − iω Γ t + ik x Γ x + ik y Γ y (cid:17) + 1 √ uf (cid:16) i ( k z ∓ A z )Γ z (cid:17) + ( − l m f √ u (cid:33) ψ l − η φ √ u ψ ¯ l = 0(3.6)with l = (1 ,
2) and ¯ l = 3 − l . For the Weyl semimetal phase, the equations are isometricin the x - y directions and there is a ω → − ω or k z → − k z symmetry.We can solve (3.6) as a set of eight coupled functions. At the horizon the ingoingboundary condition depends on the near horizon geometry. For the topologically trivialphase, the near horizon ingoing solution for nonzero k while ω → AdS case in [25] and the imaginary part of the Green function is automaticallyzero where no Fermi surface could be found. For the topologically nontrivial and criticalphases, the near horizon ingoing boundary condition is ψ l = e i √ ∆ lr z l (cid:0) . . . (cid:1) z l (cid:0) . . . (cid:1) i √ ∆ l (cid:0) ( ω + k z + ( − l A z ) z l + ( k x − ik y ) z l (cid:1)(cid:0) . . . (cid:1) i √ ∆ l (cid:0) ( k x + ik y ) z l + ( ω − ( k z + ( − l a z )) z l (cid:1)(cid:0) . . . (cid:1) (3.7)with ∆ l = ω − k x − k y − ( k z + ( − l a z ) for ω > k x + k y + ( k z + ( − l a z ) , where · · · denotes subleading terms. We will focus on the non-negative frequency and thenear horizon boundary condition is only complex when ω > k or ω > k , where k l = (cid:113) k x + k y + ( k z + ( − l a ) with a the horizon value of A z . This is similar to the pureAdS case. 9ear the boundary r → ∞ , the Dirac fields behave as ψ = a r m f + · · · a r m f + · · · a r − m f + · · · a r − m f + · · · , ψ = a r − m f + · · · a r − m f + · · · a r m f + · · · a r m f + · · · . (3.8)Because the two chiralities couple to each other, the source of ψ , will also source ex-pectation values of ψ , . To calculate the retarded Green function, we need four differenthorizon boundary conditions and get four sets of source and expectation values. Wedenote the four boundary conditions as I, II, III, IV respectively and the source andexpectation matrices are M s = a ,I a ,II a ,III a ,IV a ,I a ,II a ,III a ,IV a ,I a ,II a ,III a ,IV a ,I a ,II a ,III a ,IV and M e = − a ,I − a ,II − a ,III − a ,IV − a ,I − a ,II − a ,III − a ,IV a ,I a ,II a ,III a ,IV a ,I a ,II a ,III a ,IV . The Green function is obtained by G = i Γ t M e M − s . After getting G we find eigenvaluesof G and read the imaginary part of the four eigenvalues. We could calculate the retardedGreen function using numerics with a very small ω for numerical convenience. The basic setup for the probe fermions on the holographic nodal line semimetal back-ground (2.7) has already been obtained in [4] and here we will elaborate on more details.The coupling of the two bulk probe spinors to the scalar field is the same as in the Weylsemimetal case while for the holographic nodal line semimetal background, there seemto be multiple consistent ways to couple the two spinors to the B ab field and it turnsout that only one way of coupling can deform the Fermi point to a circle. Expanding ψ L,R to the bulk four component spinor ψ , , we could write the action of the bulk probefermions as follows S = S + S + S int , (3.9) S = (cid:90) d x √− gi ¯Ψ (cid:16) Γ a D a − m f (cid:17) Ψ ,S = (cid:90) d x √− gi ¯Ψ (cid:16) Γ a D a + m f (cid:17) Ψ ,S int = − (cid:90) d x √− g (cid:16) i Φ ¯Ψ Ψ + i Φ ∗ ¯Ψ Ψ + L B (cid:17) , (3.10)10nd L B = − i ( η B ab ¯Ψ Γ ab γ Ψ − η ∗ B ∗ ab ¯Ψ Γ ab γ Ψ ) . (3.11)Note that the Lorentz invariance in the tangent space has been explicitly broken in thebulk and this is because we have already chosen the source and expectation to correspondto the boundary values of Γ r ψ s,e = ± ψ s,e and Γ r ψ s,e = ∓ ψ s,e . Here Γ xy γ exchangesthe position of the source and expectation spinors of Ψ , so that B ab couples to theexpectation values of both bulk spinors Ψ , . If we take two spinors of the same massand the same quantization, we would not need the γ matrix in the L B term but in theΦ term to couple the fields to the expectation values of Ψ L,R at the boundary.There are other physically consistent ways to construct the action of S B , e.g. somepossibilities are S B = − (cid:90) d x √− g ( iη B ab ¯Ψ Γ ab Ψ + iη B ∗ ab ¯Ψ Γ ab Ψ ) , (3.12) S B = − (cid:90) d x √− g ( iη B ab ¯Ψ Γ ab Ψ − iη B ∗ ab ¯Ψ Γ ab Ψ ) , (3.13) S B = − (cid:90) d x √− g ( iη B ab ¯Ψ Γ ab Ψ + iη B ∗ ab ¯Ψ Γ ab Ψ ) . (3.14)However, all these could not probe the fermion spectral functions of the nodal linesemimetal states but other systems where B ab corresponds to the source of other typesof composite operators, and only the choice of L B in (3.11) corresponds to a topologicalnodal line semimetal.The corresponding Dirac equation can be written as (cid:32) Γ r ∂ r + 1 u (cid:16) − iω Γ t + ik z Γ z (cid:17) + 1 √ uf (cid:16) ik x Γ x + ik y Γ y (cid:17) + ( − l m f √ u (cid:33) ψ l − (cid:32) η Φ √ u + ( − l η b √ uf Γ xy γ (cid:33) ψ ¯ l = 0 , (3.15)with l = (1 ,
2) and ¯ l = 3 − l .The system has an SO (2) symmetry in the k x - k y plane and only depends on k x − y = (cid:112) k x + k y . Thus without loss of generality we could work at k y = 0 in the following.For k z (cid:54) = 0 or ω (cid:54) = 0 the k z or ω terms are more important at the horizon, thus theinfalling near horizon boundary conditions are determined by k z or ω . For k z = ω = 0,the near horizon boundary conditions are determined by the k x and k y terms. Then wecould obtain the Green functions using the same formula as for the holographic Weylsemimetal phase.We could work at k z = 0 while ω → ω = 0 the imaginary part would disappear and the retarded11reen functions become real. However, here for the purpose of calculating the topologicalinvariants and also because we could still detect the imaginary poles at k z = ω = 0 whichbecomes divergences in the real part, we would focus on the k z = ω = 0 data directly.For these poles, when we introduce a very small ω high peaks of imaginary parts wouldshow up.At zero frequency and k z = 0, the four eigenvalues of the Green function are allreal and appear in pairs in the form of ( g , − g , g , − g ), where g and g are positivevalues and without loss of generality we choose g ≥ g . We denote the two branchesof eigenstates with eigenvalues g , − g as “bands I” and the two branches of eigenstateswith eigenvalues g , − g as “bands II”. An illustration of the four bands in the ω - k x planeat k y = k z = 0 is in Fig. 2. Bands crossings arise when g = g where bands I and bandsII cross at two symmetric points or when g = ∞ where bands I cross at a pole.From numerics we could tell that for background solutions in the nodal line semimetalphase, there are multiple and discrete Fermi nodal lines at k F,i = (cid:112) k x + k y in the fermionspectral functions at which a pole exists at ω = k z = 0. At the critical point, k F = 0 for ω = 0. The nodal lines at k F,i and ω = 0 are all band crossing lines of two bands. At thenodal lines the zero frequency Green functions have two infinite eigenvalues correspondingto these two crossing bands and two other finite and opposite to each other eigenvaluescorresponding to the two gapped bands.Figure 2: Illustration of “bands” I and II near a k F,i in the ω - k x plane for k y = k z = 0. Thepole k F,i is always a band crossing point of two “bands”.
One immediate question is if these poles all come from the same two bands or differentsets of two bands. For the second possibility, gapped bands at a certain k F,i might becomegapless poles at another k F,j (cid:54) = i and for this to happen, the two sets of bands have tointersect at some points in the ω - k x plane at k y = k z = 0. Fig. 3 shows the illustrationfor the spectrum in the ω - k x plane for a multiple-nodal line system where all the polesare from the same two bands (left) or from different sets of two bands (right).12igure 3: Illustration for a multiple-nodal line system in the E − k x plane where all the polesare from the same two bands (left) or come from different sets of two bands (right). To answer this question, it seems that we would need a spectral density plot of G ( ω, k )for the ω - k x plane to see which of the following possibility happens: (1) the two sets ofbands I and II would intersect at some points in the ω - k x plane and some of the poles arefrom bands I and others are from bands II; or (2) the two sets of bands do not intersectfor the whole ω - k x plane and the poles are always from bands I. However, in fact wecould distinguish these two possibilities just from the data of the zero frequency Greenfunctions. The explanation is the following. When there is a pole in the zero frequencyGreen function, i.e. at least one of the eigenvalues reaches infinity, a Fermi point wouldappear at ω = 0 in the spectral density plot for spectral weight of fermions in the ω - k x plane. The value of the zero frequency Green function eigenvalues reflects how far theband peaks are from the k x axis in the spectral density plot. When the eigenvalue issmall (large), the bands are far away from (close to) the k x axis. Thus we could use theeigenvalues of G − (0 , k x ) to denote the relative distance of the bands to the k x axis andplot a qualitative picture of spectral density plot in the ω - k x plane. In this way, to tellif all the poles come from the same bands or different bands we only need to examineif there is a band crossing point between two adjacent poles at which g = g . If g isalways larger than g when the system evoles from one pole k F,i to the next one k F,i +1 ,then we could tell that the poles are always from bands I, however, if there is a certain k F,i < k x < k F,i +1 at which g = g , the two poles should come from different sets ofbands. Fig. 4 is the qualitative behavior of the bands for
M/b (cid:39) . m f = − / The band crossing points always exist when we tune the value of
M/b in the nodal lines semimetalphase, thus the band crossing should not be accidental. ω - k x plane. We willsee in the next section that this is in fact the spectrum/band structure of the topologicalHamiltonian defined from the zero frequency Green functions and this is consistent withthe main spirit of the topological Hamiltonian approach that the zero frequency Greenfunctions capture all the topological information of the system. - - - - - Figure 4:
Eigenvalues of − G − (0 , k x ) for M/b (cid:39) . ω - k x plane.We refer to the two bands with red colour as bands I and the two bands with blue colour asbands II. The distance between adjacent poles are becoming larger as k x increases. The first observation is that the distance between adjacent poles are becoming largeras k x increases. At small k x the poles are very sharp and very close to each other and wedid not plot this area as the nodal loops are so dense that we need to run at extremelysmall intervals of k x to reveal all the poles which requires a much larger accuracy.We could see from the figures that bands I and II always intersect once and only oncein the upper ω plane between each two adjacent poles, which means that the adjacent twopoles always come from different two sets of bands. Different from the weakly couplednodal line semimetal system where the four bands are divided into two gapless bandsand two gapped bands which are always gapped, now the two gapped bands in the nodalline semimetal phase are not always gapped but soon become gapless at a larger k x andexchange the role with the other two bands. Between each adjacent two poles, there is oneand only one band crossing point in the upper ω plane. Another interesting observationis that between each two adjacent band crossing points there is always one pole and onezero of the Green function. This means that for positive m f there will also be poles.However, we will show below that different from the holographic Weyl semimetal case,these zeros do not possess nontrivial Berry phases.14hen we increase M/b to be approaching the critical value of (
M/b ) c , all the nodalloops would shrink in size and finally become a point at the critical point. Fig. 5 showsthe evolution of one k F as a function of M/b . For each of the nodal lines, we have sharpFermi surface and a linear dispersion in all the k x , k y and k z directions [4]. ��� ��� ��� ������������������������ �� � � � Figure 5:
The dependance of one branch of the nodal loop radius at zero temperature in theholographic nodal line semimetal phase as a function of
M/b . Clearly the radius of the nodalloop reaches zero when
M/b approaches the critical value. The qualitative behavior is the samefor other branches of nodal loops.
In mathematics, topological objects possess properties that are invariant under home-omorphisms, which are called topological invariants. Topological invariants could benumbers, e.g. the genus of a closed surface, or could also be groups, e.g. the funda-mental group. In the same way, topological invariants could be defined for topologicalstates of matter, which are invariant under adiabatic deformations that do not changethe topology of the underlying physical system.For weakly coupled topological systems, a simple example of a topological invariantis the Berry phase with value 0 or π , which is the phase accumulated along a closed loop γ in the momentum space for the Bloch states, i.e. eigenstates of the Hamiltonian | n k (cid:105) .The formula for Berry phase [27] is φ = (cid:73) γ A k · d k , (4.1)where the Berry connection is defined by eigenstates | n k (cid:105)A k = i (cid:88) j (cid:104) n k | ∂ k | n k (cid:105) , (4.2)15here j runs over all occupied bands and | n k (cid:105) is the eigenvector of the momentum spaceHamiltonian. Berry phase could be defined in general dimensions and here we focus on3 + 1 dimensions for our purpose. We can also write (4.1) using the Berry curvature as φ = (cid:90) S Ω · d S , (4.3)where Ω i = (cid:15) ijl (cid:0) ∂ k j A k l − ∂ k l A k j (cid:1) (4.4)and d S is the surface element of S which is a surface surrounded by the closed loop γ ,i.e. γ = ∂S .An equivalent calculation of this topological invariant is to use the Green function N ( k z ) = 124 π (cid:90) dk dk x dk y Tr (cid:104) (cid:15) µνρz G∂ µ G − G∂ ν G − G∂ ρ G − (cid:105) , (4.5)where µ, ν, ρ ∈ k , k x , k y and k = iω is the Matsubara frequency. For noninteractingsystems, the Green function G ( iω, k ) = 1 / ( iω − h ( k )) where h ( k ) is the Hamiltonianmatrix H = (cid:80) k c † k h ( k ) c k . This formula for the topological invariant is still applicablefor interacting systems, however, it involves an integration in the iω direction, which isdifficult to get in practical strongly coupled systems. This is not a problem in holographyas in principle we could get the Green function for any value of ω using numerics, which,however, is extremely time consuming.In [18, 16] it was shown that the zero frequency Green function G (0 , k ) already con-tains all the topological information. One could define an effective topological Hamilto-nian H t ( k ) = − G − (0 , k ) (4.6)and define eigenvectors using this effective topological Hamiltonian. As long as G ( iω, k )does not have a pole at nonzero ω , the topological invariants defined under the effectiveHamiltonian H t ( k ) as if the system is a weakly coupled theory with the Hamiltonian H t ( k ) would be the same as those defined in the original system. Thus we could definetopological invariants using negative valued eigenvectors of H t ( k ), i.e. effective occupiedstates n k with H t ( k ) | n k (cid:105) = − E t | n k (cid:105) and E t > ω axis in theGreen function, the topological invariant could be calculated from the weakly coupledformula defined for the effective topological Hamiltonian. Once we have obtained thetopological Hamiltonian, the procedure would be the same as the weakly coupled case.16n the following, we will first obtain the Green function at zero frequency for both theholographic Weyl and nodal line semimetal states and calculate the topological invariantsfrom occupied eigenvectors of the zero frequency Green functions. To understand thisprocedure easier, we will first start with a simple example, which is the calculation ofholographic topological invariants for the pure AdS case before going to the Weyl andnodal line cases. In the pure AdS case, the system is in fact degenerate at zero frequency, which is easyto understand as the two Weyl nodes coincide to form a Dirac node, but we could stilldistinguish the two degenerate eigenstates according to their chiralities. The retardedGreen functions for one chirality in the pure AdS case for ω > k has already been obtainedin [25]. For pure AdS, the two chiralities do not interact and we could directly get thefull Green function using two spinors of opposite masses and quantizations. In this case,the action of the two spinors are S = S + S , (4.7) S = (cid:90) d x √− gi ¯Ψ (cid:0) Γ a D a − m f (cid:1) Ψ ,S = (cid:90) d x √− gi ¯Ψ (cid:0) Γ a D a + m f (cid:1) Ψ . (4.8)To obtain the topological Hamiltonian, we focus on the ω = 0 solutions and the zerofrequency Green function. We parametrize the solution as Ψ l = ( ψ + l , ψ − l ) T with l = (1 , ω > k case, at zero frequency, the solutions as well as the Greenfunctions are real functions of k = (cid:112) k x + k y + k z . The solution of this action withinfalling boundary condition at the horizon is ψ + l = r − / K − ( − l m f + (cid:16) kr (cid:17) a + l , l = (1 ,
2) (4.9)where a +1 , are two arbitrary spinors and K ± m f + (cid:16) kr (cid:17) is the BesselK function. ψ − , couldbe obtained from the equations of motion for ψ +1 , , which in our convention of Γ-matricesis ψ − l = k µ σ µ k r (cid:0) r∂ r + ( − l m f (cid:1) ψ + l . (4.10)Four boundary conditions could be identified as four linearly independent choices of a +1 , . After expanding the solutions at the boundary we could get the two source andexpectation matrices. The final result for the retarded Green function of two chiralities17re G (0 , k ) (cid:39) N k i σ i k − mf − k i σ i k − mf , (4.11)where N = Γ[1 / − m f ]Γ[1 / m f ]4 mf is an overall normalization constant and k = ( k x , k y , k z ). Notethat when m f is negative, the Green function has poles at ω = k while when m f ispositive, the Green function has zeros instead of poles at ω = k . However, from theprocedure below, we will see that the topological structure is not affected by the valueof the scaling dimension and no matter whether the Green function has zeros or poles,topological invariants could be the same.The topological Hamiltonian H t is defined as − G − (0 , k ) from (4.6). To calculate theBerry curvature, we need to find eigenvectors of the topological Hamiltonian which areequivalent to eigenvectors of the Green function. For the pure AdS case, the eigenvaluesof the Hamiltonian are degenerate at ω = 0. Here we can treat this system as a b → | n (cid:105) = n (cid:0) k z + k, k x + ik y , , (cid:1) T , | n (cid:105) = n (cid:0) , , k z − k, k x + ik y (cid:1) T , (4.12)where n l = 1 / (cid:112) k ( k − ( − l k z ). Note that the eigenvectors for the pure AdS case arein fact the same as those in the free massless Dirac Hamiltonian. | n (cid:105) has positive chi-rality and is the eigenvector of the positive chirality Hamiltonian while | n (cid:105) has negativechirality and is the eigenvector of the negative chirality Hamiltonian.To calculate the topological invariant we define a sphere S : k = k enclosing theDirac node k = 0 where k is a constant. The system is gapped on the sphere and theformula for the topological invariant is C l = 12 π (cid:73) S Ω l · d S , (4.13)where Ω i = (cid:15) ijk F ij , with ( i , j , k ) ∈ { k x , k y , k z } (4.14)and F is the Berry curvature defined in (4.4). C defined in this way is an integer numberthat does not depend on the exact shape and radius of S as long the deformation doesnot pass through a Dirac node.On the sphere S = k (sin θ cos φ, sin θ sin φ, cos θ ) we have Ω l = ( − l e ρ / k , thusfor | n (cid:105) C = 12 π (cid:73) S Ω · d S = 12 π (cid:90) π dφ (cid:90) π dθ sin θ k − k = − | n (cid:105) C = 12 π (cid:73) S Ω · d S = 12 π (cid:90) π dφ (cid:90) π dθ sin θ k k = 1 . (4.16)The total topological invariant is then zero for pure AdS. This is clear intuitively: thedual zero density state of pure AdS only consists massless Dirac excitations. For the Weyl semimetal, the nontrivial topological invariant is defined as the Berrycurvature integrated on a closed surface S enclosing the Weyl node located at k l inthe momentum space C Weyl l = 12 π (cid:73) Ω l · d S , (4.17)and the result does not depend on the exact shape and size of S as long as there is onlyone Weyl node inside the closed surface.For the Weyl semimetal case, the zero frequency Green function, or equivalently theeffective topological Hamiltonian is also real. We will start from the easiest case: the M/b → φ is infinitely small so that could be ignored.Then we go to the more general case of small M/b . This
M/b → M/b → limit In the
M/b → φ to the background geometry andto the axial gauge field. Then the axial gauge field is a constant in the bulk with A z = a and the metric is pure AdS . As we ignore the contribution of φ , ψ and ψ do not coupletogether and could be solved independently in terms of BesselK functions at ω = 0, ψ + l = r − / K − ( − l m f + (cid:16) k l r (cid:17) a + l (4.18)where k l = ( k x , k y , k z + ( − l a ) and k l = (cid:113) k x + k y + ( k z + ( − l a ) . Compared tothe pure AdS case, the pole of the first spinor (the negative chirality one) shifts from w = k = 0 to w = k x = k y = 0 while k z = a and the pole of the second spinor shifts to k z = − a . The retarded Green function is G (0 , k ) (cid:39) N k i σ i k − mf − k i σ i k − mf , (4.19)19here the normalisation factor N takes the same form as in (4.11). The eigenvalues ofthe Green function give ±N k m f , . For negative m f the poles of the system are at k , = 0.As a is not zero, the eigenvectors are now not degenerate at zero frequency. At the twoWeyl nodes k z = ± a , i.e. one of k , is zero while the other not zero, two branches ofthe eigenvectors are gapless and the other two are gapped.Now we calculate the topological invariant at k z = a and the calculation for the otherone would be similar and give an opposite topological invariant. At k z = a , the gaplesseigenvector with a negative eigenvalue of the topological Hamiltonian is | n (cid:105) = n (cid:0) k z − a + k , k x + ik y , , (cid:1) T , (4.20)where n = 1 / (cid:112) k ( k + k z − a ) with k = (cid:113) k x + k y + ( k z − a ) . From | n (cid:105) we have Ω = − e ρ / k and the topological invariant is C Weyl1 = 12 π (cid:73) S Ω · d S = 12 π (cid:90) π dφ (cid:90) π dθ sin θ k − k = − . (4.21)It can be checked that the gapped eigenvector will only contribute a zero to the topologicalinvariant.This shows that k z = a is a Weyl node with negative chirality and the other node at k z = − a should have C Weyl2 = 1, i.e. the other node possesses an opposite chirality andtopological charge because the total topological invariant/chirality charge for the wholesystems should still be zero which is exactly the consequence of the Nielsen-Ninomiyatheorem [28].This is the simplest case that φ does not have any contribution. The next step isto calculate the topological invariant for the more general nonzero M/b case. For thiscase, the background geometry gets modified by the scalar field in the bulk and wecannot solve it analytically anymore. As the background is numerical, we do not haveanalytic solutions for the fermion Green function either. We could in principle calculatethe retarded Green functions using numerics, and then calculate the eigenstates andBerry curvature using numerics. However, this procedure requires finding eigenvectorsnumerically which usually loses a lot of accuracy. In order to avoid too much numericsand the inaccuracy, we will solve it semi-analytically by expanding near the Weyl nodesand for this to be possible we have to work at the small
M/b (cid:28) ( M/b ) c limit. M/b (cid:28) ( M/b ) c case For
M/b (cid:28) ( M/b ) c , it is expected that there would be two poles separated in the k z axis,though close to each other. When we calculate the Berry curvature we could in principle20erform the integration on any sphere that surrounds one and only one Weyl node. Whenwe reduce the size of the sphere to be smaller and smaller, we could expand the systemaround the Weyl node to solve the fermions and get the retarded Green function on thatsphere and then diagonalize it for the eigenstates. Here because of the non-analyticity inthe equations of the probe fermions near the pole, we need to take a near-far matchingmethod.We can divide the geometry into the near region and the far region, which overlapat the matching region. There are usually two scales s (cid:28) s where the near region isdefined by r (cid:28) s and far region s (cid:28) r while the matching region s (cid:28) r (cid:28) s asillustrated in Fig. 6. s is the IR expansion parameter which is important in the nearregion while not important in the far region. Here in this system s is k if we focus onthe right Weyl node at k z = a and the k x , k y and k z − a terms in the far region couldbe treated as perturbations. s is a UV parameter, e.g. s is the chemical potential µ inthe finite density case. Here s is the parameter at which the geometry starts to deviatefrom AdS . According to the background geometry (2.4) s is in fact a or equivalently b as a /b ∼ O (1). The near region is now r (cid:28) b and in this region the background is AdS .Thus we require k = (cid:113) k x + k y + ( k z − a ) (cid:28) b for the near-far matching method towork. Figure 6: Illustration of the near region r (cid:28) s and the far region s (cid:28) r . In the near region of the holographic Weyl semimetal phase, the contribution of φ almost vanishes while the leading order of A z is a constant. The near horizon geometryis still AdS . As the order of φ is extremely small at the horizon in the Weyl semimetalphase, the equations for the two spinors are decoupled in the near region. We calculatenear one of the expected poles and choose k z = a .The near region equations are (cid:32) Γ r ∂ r + 1 u (cid:16) − iω Γ t + ik x Γ x + ik y Γ y (cid:17) + 1 √ uf (cid:16) i ( k z + ( − l A z )Γ z (cid:17) + ( − l m f √ u (cid:33) ψ l = 0 , and at the near region the solutions are ψ + l = r − / K − ( − l m f + (cid:16) k l r (cid:17) a + l , (4.22)21here k l = (cid:113) k x + k y + ( k z + ( − l a ) for the upper two components, and using ψ − l = k lµ σ µ k l r (cid:0) r∂ r + ( − l m f (cid:1) ψ + l (4.23)we get ψ − l = k lµ σ µ k l r − / K − ( − l m f − (cid:16) k l r (cid:17) a + l (4.24)for the lower two components.The far region equations can be expanded in terms of k = ( k x , k y , k z − a ) around as ψ fl = ψ f l + k i ψ f il , (4.25)where k x , k y , k z − a are the small expansion parameters. The far region leading orderequations are (cid:32) Γ r ∂ r + 1 √ uf (cid:16) i ( a ∓ A z )Γ z (cid:17) + ( − l m f √ u (cid:33) ψ f l − η φ √ u ψ f l = 0 , (4.26)and to the first order in k i equations are (cid:32) Γ r ∂ r + 1 √ uf (cid:16) i ( a ∓ A z )Γ z (cid:17) + ( − l m f √ u (cid:33) ψ f jl − η φ √ u ψ f j ¯ l + 1 u (cid:16) ik j Γ j (cid:17) ψ f l = 0for k j ∈ { k x , k y } , and (cid:32) Γ r ∂ r + 1 √ uf (cid:16) i ( a ∓ A z )Γ z (cid:17) + ( − l m f √ u (cid:33) ψ f zl − η φ √ u ψ f z ¯ l + 1 √ uf (cid:16) i ( k z − a )Γ z (cid:17) ψ f l = 0for k z − a , where ¯ l = 3 − l .To solve the far region equations we need the near horizon boundary conditions whichare input determined by the matching region expansion of the near region. In the nearhorizon region of the far region, i.e. at the matching region, as φ is not important from(4.26) the leading order solutions are ψ f = (cid:32) r m f a +1 r − m f a − (cid:33) (4.27)for ψ and ψ f = r − / K − m f + (cid:16) a r (cid:17) a +2 r − / K − m f − (cid:16) a r (cid:17) σ z a +2 (4.28)22or ψ , where a ± and a +2 are constant two-component spinors. There are six independentnear horizon parameters a ± and a +2 , and to calculate the retarded Green function, we onlyneed four nontrivial linearly independent combinations of the six, which are determinedby the boundary conditions at the matching region. Note that for ψ there are only twofree parameters compared to four for ψ . This is because the expansion around k z = a is analytic for ψ and the infalling boundary conditions for ψ have already been chosenin the far region.At first order in k i of the i -th component in k = ( k x , k y , k z − a ), the solutionsare sourced by the leading order solutions and there are no new free parameters. Wesubtract all the solutions of the leading order and the near horizon solutions are onlynonzero when there are nonzero leading order sources. The first order solutions are ψ f i = (cid:32) r − − m f c +1 i r − m f c − i (cid:33) , with i ∈ { x, y, z } (4.29)where c + , − x,y,z are two component spinors that are determined by a ± and a +2 . ψ f i = r − K m f + (cid:0) a r (cid:1) c +2 i K − m f + (cid:0) a r (cid:1) c − i , with i ∈ { x, y, z } (4.30)where c + , − x,y,z are two component spinors that are determined by a ± and a +2 .Before going to the matching region to match the initial conditions of ψ , which shouldbe two sets of linearly independent combinations of (cid:18) a +1 a − (cid:19) , we could first obtain theboundary values of the fields under these six independent far region boundary conditions.For simplicity we choose these six boundary conditions to be V j ini ,i = δ ji , with i, j ∈ { , ... } , (4.31)where V ini = a +1 a − a +2 and V j ini ,i refers to the value of the i -th component of V ini under the j -th boundary condition.Now we indicate the boundary source vector as s ji and the expectation vector as e ji ,which are the i -th components of r − m f (cid:18) ψ +1 ψ − (cid:19) and r m f (cid:18) − ψ +2 ψ − (cid:19) under the j -th boundarycondition separately. We keep terms in both matrices s and e up to the first order in k x , k y and k z − a and each element of s ji or e ji would be composed of zeroth and firstorder contributions in k x , k y and k z − a . Due to the structure of the equations, some of23hese contributions would be zero, e.g. s has no O ( k x ) or O ( k y ) contributions. A fullset of nonzero elements of these two matrices s and e under the six boundary conditionscould be found in the appendix A. The exact values of these elements could be obtainedby numerically integrating the far region equations for any background in the holographicWeyl semimetal phase.In the matching region, k (cid:28) r (cid:28) b with k = (cid:113) k x + k y + ( k z − a ) we expand thenear region solutions (4.22) and (4.24) as ψ +1 = (cid:20) r m f (cid:18) m f − k − m f − Γ (cid:18) m f + 12 (cid:19)(cid:19) + r − m f − (cid:18) − m f − k m f + Γ (cid:18) − m f − (cid:19)(cid:19)(cid:21) d +1 ψ +2 = (cid:20) √ r K m f + (cid:18) h r (cid:19) + ( k − a )4 r / (cid:18) (2 m f + 1) ra K m f + (cid:18) a r (cid:19) − K m f + (cid:18) a r (cid:19)(cid:19)(cid:21) d +2 for the upper two components, and ψ − = (cid:20) m f − k − m f − r m f − Γ (cid:18) m f − (cid:19) + 2 − m f − k m f − r − m f Γ (cid:18) − m f (cid:19) k µ σ µ (cid:21) d +1 ψ − = (cid:20) a √ r K − m f − (cid:18) a r (cid:19) + ( k − a )8 a r / (cid:16) − a K − m f + (cid:18) a r (cid:19) − (2 m f + 3) rK − m f − (cid:18) a r (cid:19) (cid:17)(cid:21) d +2 for the lower two components.These expansions fix the near horizon initial boundary conditions for the far regionand we could solve the far region equations using these boundary conditions and obtainthe source and expectation matrices under infalling boundary conditions. In fact for ψ we do not need this near far matching procedure and could directly use the infallingboundary conditions at the far region and treat k as a small expansion as it is analyticalwhen expanding around k = 2 a which is not zero.Now the four infalling boundary conditions that we need are (cid:32) d + j d + j (cid:33) i = δ ji (4.32)i.e. the i -th component of (cid:18) d +1 d +2 (cid:19) under the j -th boundary condition is δ ji . These fourboundary conditions fix the six far region boundary conditions to be four and the bound-ary values of the fields under these four boundary conditions are also combinations of theboundary values of the fields under six far region boundary conditions. Let us denote s j e j as the source and expectation vectors under the far region j -th boundary condi-tion, and j runs from 1 , ...,
6. The source vector for the first matching region boundarycondition in (4.32) corresponds to (cid:32) Γ (cid:18) m f + 12 (cid:19) m f − k m f + (cid:33) ( s ) + ( s , s ) · (cid:32) Γ (cid:18) − m f (cid:19) k m f − m f + k µ σ µ (cid:18) (cid:19)(cid:33) , (4.33)and the source vector for the second matching region boundary condition in (4.32) cor-responds to (cid:32) Γ (cid:18) m f + 12 (cid:19) m f − k m f + (cid:33) ( s ) + ( s , s ) · (cid:32) Γ (cid:18) − m f (cid:19) k m f − m f + k µ σ µ (cid:18) (cid:19)(cid:33) . (4.34)The third and fourth boundary conditions correspond to s and s separately. For ex-pectations, we only need to substitute s ’s in the formulas above by e ’s.Finally we could get the source and expectation matrices at the boundary, which arecomposed of the parameters x i in the appendix A, where x i with i ∈ , ...,
36 are constantswhich are the boundary values of the solutions associated with the six far region boundaryconditions. These two matrices are very long and we do not write them out here. Thenext step is to get the Green function from the source and expectations matrices using G = i Γ t es − . The Green function obtained in this way is still quite complicated and itis difficult to obtain the eigenstates of the Green function. Now we analyze the Greenfunction more carefully to see if it could be simplified in certain limits.As the final Green function will not get modified by changing the initial boundaryconditions by a linear superposition or scaling, here we rescale both the source andexpectation matrix by a factor of √ k for simplicity. Now the determinant of the sourcematrix isdet S = 2( x x − x x ) Γ (cid:0) (cid:1) / Γ (cid:0) (cid:1) + 2 √ k z ( x x − x x )( x x − x x ) √ k Γ (cid:0) (cid:1) / Γ (cid:0) (cid:1) ++ t ( x i ) k z + t ( x i ) k , (4.35)written in orders of k , k / , k , where t and t are functions of x i which are too longto write out. As we have stated above, x i with i ∈ , ...,
36 are 36 constants that couldbe read from the boundary values of the source and expectation matrices and k = (cid:113) k x + k y + ( k z − a ) . Note that t , do not have the factor ( x x − x x ).In the pure AdS case and the M/b → ψ and ψ do not couple together itcan be checked that x x − x x = 0. In this limit det S could be simplified todet S (cid:39) ( x x − x x ) k . (4.36)25he Green function is also simplified in this limit. For M/b small enough, in prin-ciple one could find an enclosing sphere with radius k on which x x − x x (cid:28) ( x x − x x ) √ k holds and the expressions for the Green function and the topologicalinvariants could be simplified a lot as small perturbations would not change the topo-logical invariants. x i ’s are parameters that do not depend on k so it seems that if wechoose k large enough, this inequality would hold. However, in our near far matchingcalculation, the order of k at the sphere should be so small that there exists a region of r (cid:29) k where the system could still be AdS and also that 2 k at the enclosing sphereshould be smaller than 2 a so that the sphere only has one pole inside. With the largestpossible k that satisfies this constraint, we find that x x − x x (cid:28) ( x x − x x ) √ k indeed holds for small values of M/b in the holographic semimetal phase. Numericallywe have checked that for
M/b (cid:39) .
16, the ratio of the left side of the inequality over theright side could be around 7 .
7% at the sphere where we have chosen k = 10 − a or ifwe choose k = 10 − a the ratio would be around 6 .
7% for
M/b (cid:39) .
05. Here we havechosen m f = − / S = 0 and from the perturbativecalculation (4.35) the position of the pole seems also to be modified for a very smallvalue compared to a in the case of M/b → x x − x x term,which though could be ignored in the limit that we are considering. However, with anonzero ω → ω → k z (cid:54) = a . This means thatfor the imaginary part, the peak is still at a . Note that the value of anomalous Hallconductivity σ AHE (cid:39) αa should be proportional to the distance between two poles,which seems to also lead to the conclusion that the position of poles should be at ± a .Thus it is possible that there is some reason leading to the fact that summing over allperturbations at the order x x − x x would finally keep the position of the poleunchanged. We will leave this for a future study.We have found that at the sphere with a small radius k , which is away from thepole but not far away, ( x x − x x ) is very small compared to other terms as wefocus on the case that M/b is small enough. Thus the ( x x − x x ) term could beignored as the Berry curvature is a quantized number which should not be affected bysmall perturbations. In this limit, the Green function can be simplified to be G (0 , k ) (cid:39) N √ k det S N k z N ( k x − ik y ) − k z k x − ik y N ( k x + ik y ) − N k z − k x − ik y − k z N N k z N N ( k x − ik y ) − N k z N ( k x − ik y ) − N N ( k x + ik y ) N N k z N ( k x + ik y ) N k z , N = x x , N = √ (cid:0) x x − x x (cid:1)(cid:0) x x − x x (cid:1) Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) and N = x x − x x x x − x x . From numerics one can check N < N for M/b (cid:28) . We pick the negative valuednormalized eigenstate of the topological Hamiltonian, which is1 N (cid:18) − k − k z ( k x + ik y ) N , − N , − k − k z k x + ik y , (cid:19) T (4.37)with N = √ k (1+ N )( k + k z ) N √ k x + k y . With this state, we could calculate the Berry curvature,then integrate it on the small enclosing sphere and get a nontrivial topological invariant −
1. We could do a similar analysis for the pole at k z = − a and obtain the topologicalinvariant 1.Thus we have the final result for the nontrivial topological invariants for small M/b for the holographic Weyl semimetal phase. For
M/b ∼ O (( M/b ) c ) this matching methodcould not work as there is no matching region anymore. In this case we could in principlecalculate the zero frequency Green functions using the numerical method and performthe integration also numerically to get topological invariants, which we leave for futureinvestigation. ω = ± a and k z = 0We have calculated the topological invariant for the holographic Weyl semimetal usingthe ω = 0 Green function (topological Hamiltonian) in the previous subsection. In thissubsection, for completeness we will have a look at the Fermi spectrum for ω (cid:54) = 0 andto avoid tedious numerics we will also stay in the semi-analytic regime of calculation. Inthe M/b → ω > k , , instead of theformula in (4.19), we have G ( ω, k ) (cid:39) ω + k µ σ µ k − mf ω ω − k µ σ µ k − mf ω , (4.38)where k lω = (cid:113) ω − k x − k y − ( k z + ( − l a ) . The poles are at k ω = 0 or k ω = 0. At k z = 0 we could see that the two branches of k ω = 0 and k ω = 0 intersect at k z = 0while ω = ± (cid:113) k x + k y + a . This means that besides the two “band crossing” pointsat ω = 0 and k z = ± a , we have another two “band crossing” points at ω = ± a and27 - a a - a k z ω Figure 7:
The spectrum of the holographic Weyl semimetal in the limit
M/b = 0. When wehave a small nonzero
M/b , the points (0 , ± a ) will become a pseudogap and we do not havepoles at these two points any more. The points ( ± a ,
0) remain poles. k z = 0 at M/b →
0. The following figure shows the Fermi spectrum of the
M/b → M/b (cid:54) = 0, the effect of φ will change thisband intersection at ω = ± a and k z = 0 into a pseudogap. We work in the very small M/b limit and expand the system in orders of
M/b to study the leading order effectof
M/b . We could easily check from the equations of motion for the background thatthe scalar field has an order O ( M/b ) profile and backreacts to other fields to give order O (( M/b ) ) order corrections to other fields. This means that at leading order in M/b the background geometry would still be pure
AdS with A z = b = a all through the bulkand φ = φ ( r ) ∼ O ( M/b ) which could be solved from the equation of motion for φ in theAdS background.As the effect of φ at the horizon is always negligible in the holographic Weyl semimetalphase, again we take the near far matching method with ω − a → k x = k y = 0. The near region is defined by r (cid:28) b and the far region is defined by ω ∓ a (cid:28) r depending on if we want to study the up or down branch. Here we focus onthe ω → a branch and it is straightforward to generalize to the other branch.In the near region, the geometry is AdS and the near region solutions are the Hankelfunctions ψ = √ r H (1) (cid:0) , k ω r (cid:1) a +1 ω + k µ σ µ k ω √ r H (1) (cid:0) − , k ω r (cid:1) a +1 , ψ = √ r H (1) (cid:0) , k ω r (cid:1) a +2 ω + k µ σ µ k ω √ r H (1) (cid:0) − , k ω r (cid:1) a +2 , (4.39)28here k lω = (cid:113) ω − k x − k y − ( k z + ( − l a ) and we have chosen m f = − /
4. Notethat different from the ω = 0 , k z → a region where ψ has effectively zero momentumwhile ψ has an effective finite momentum, here for each of ψ l half of the componentshave effectively zero momentum while the other half nonzero. However, the solutions arestill functions of k lω /r where k lω →
0. This indicates that though for some componentsof ψ l , there is a finite ω + a = 2 a term in the equation, the ω + a = 2 a terms inthe equations are in fact first order ω − a corrections sourced by corresponding leadingorder solutions which have the ω + a coefficient in front. We define ˜ ω = ω − a and fix k x = k y = k z = 0. We expand the system at ω → a i.e. small ˜ ω . As we explained above,to study the leading order effect of M/b the background geometry is still
AdS and A z isalso a constant in the whole bulk spacetime. Then we could also expand the equations ofmotion for ψ l in orders of M/b at the far region as it is only important in the far region.The far region solutions could be written as ψ fl = ψ f l + ˜ ωψ f ωl + Mb ψ f φl + k z ψ f zl , (4.40)where M/b is small. The leading order equations are (cid:32) Γ r ∂ r + ( − l m f √ u (cid:33) ψ f l = 0 . (4.41)Before knowing how infalling boundary conditions from the near region result in the farregion, we have in the far region eight independent horizon initial boundary conditionsand linear order solutions could be determined from leading order ones. The eight initialboundary conditions for (cid:32) ψ f ψ f (cid:33) are V j ini = (cid:0) r m f δ j , r m f δ j , r − m f δ j , r − m f δ j , r − m f δ j , r − m f δ j , r m f δ j , r m f δ j (cid:1) T , j ∈ { , ..., } . The leading order solutions are just exact solutions r ± m f for the components with nonzeroboundary conditions. Note that later we will use the matching region solution to reducethese eight linearly independent boundary conditions to four by imposing infalling bound-ary conditions.The initial boundary conditions at the horizon for various components of ψ f ω,zl are r − ( − l m f − as they are sourced by leading order solutions. Thus the first order correctionsin ω and k z for far region solutions are not important as for pure AdS backgroundthe solutions r − ( − l m f − are exact solutions and do not affect the boundary source andexpectation terms. From here on we focus on ψ f φl .The initial boundary conditions for ψ f φl are all zero. We integrate the far regionequations for ψ f φl and get the following result. Under the boundary condition V , , the29nly nonzero boundary values are ψ +2 = P Mb r / (cid:18) δ j δ j (cid:19) ; for V , , only nonzero boundaryvalues are ψ − = − P Mb r − / (cid:18) δ j δ j (cid:19) ; for V , , only nonzero boundary values are ψ +1 = P Mb r − / (cid:18) δ j δ j (cid:19) ; for V , , only nonzero boundary values are ψ − = − P Mb r / (cid:18) δ j δ j (cid:19) , where P , are numbers determined by numerics and for the set of parameters that we have usedin this paper, we have P (cid:39) . P (cid:39) . (cid:18) a + j a + j (cid:19) i = δ ji , i, j ∈ { ... } . At the matching region, expanding thesolutions we get the near horizon boundary conditions for the far region solutions. Thuswe have the following far region solutions for these four infalling boundary conditionsafter matching the coefficients in the matching region. For the first boundary condition,the far region solution with infalling boundary condition obtained from the matchingregion is − i / Γ[1 / πk / ω ψ l + ( ω + a )2 / k / ω (cid:32) √ /
4) + i Γ(3 / π (cid:33) ψ l ; (4.42)for the second boundary condition, the far region solution is − i / Γ[1 / πk / ω ψ l + ( ω − a )2 / k / ω (cid:32) √ /
4) + i Γ(3 / π (cid:33) ψ l ; (4.43)for the third boundary condition, the far region solution is − i / Γ[3 / πk / ω ψ l + ( ω − a )2 − / k / ω (cid:32) / − i √ / π (cid:33) ψ l ; (4.44)and for the fourth boundary condition, the far region solution is − i / Γ[3 / πk / ω ψ l + ( ω − a )2 − / k / ω (cid:32) / − i √ / π (cid:33) ψ l , (4.45)where ψ jl corresponds to far region solutions under the j -th boundary condition (4.42).Here the value of k ω at k z = 0 is the same as k ω at k z = 0.With these four linearly independent solutions and the boundary values of ψ j , thatwe have already obtained earlier, we could now get the source and expectation matrices,which are very long and we do not write them out here. Now the determinant of thesource matrix has an extra P term compared to the M/b = 0 case meaning that thedeterminant of the source matrix is not zero anymore at ω = a , k z = 0 due to nonzero P , which makes the k ω = 0 pole vanish and becomes a pseudogap.30 .3 Topological invariants for holographic nodal line semimetal For a nodal line semimetal, there are two topological invariants as shown in [20]. The firstone is to take a circle linking the nodal loop in the momentum space of k x , k y and k z andthis loop cannot shrink to a point as it cannot deform adiabatically to unlink the nodalloop. The second topological invariant is defined on a sphere enclosing the nodal loop.In this case, the sphere also can not shrink to a point without passing singularities in theGreen function. The first topological invariant is the one responsible for the stability ofthe nodal loop under small perturbations, i.e. the nodal line semimetal does not becomegapped under small perturbations. The second topological invariant is related to whetherthe critical point is topological or not. As the second topological invariant would requiretoo much numerics, we will not consider this one in this paper.The first topological invariant is a Berry phase along the circle. We have multiplewhile discrete nodal lines in the k x - k y plane at k z = 0 in the holographic nodal linesemimetal phase and for each nodal line we could define a Berry phase. For each two oreven more nodal lines we could also define a circle linking at the same time with two ormore nodal lines, i.e. two or more nodal lines pass through the inside of the circle, whichhowever could be continuously deformed to two or more separate circles of each nodalline itself as is shown in Fig. 8. Thus in the following we will focus on the Berry phaseof each nodal line.Figure 8: Illustration of the circle to which the Berry phase is associated. From left to right: acircle with only one nodal loop passing through its inside; a circle with two nodal loops passingthrough its inside; the circle could continuously deform to two separated circles, each of whichis of the type in the first figure.
To avoid tedious numerical calculations, we choose very closely located discrete pointson the loop and calculate the Berry phase in the discrete limit. We will show that inthis case, the effective topological Hamiltonian method is still applicable and a nontrivialBerry phase of π could be obtained for the holographic nodal line semimetal phase.The procedure to calculate the Berry phase is the following. We first find the positionof the Fermi surface k F = (cid:112) k x + k y at k z = 0 and ω = 0. Then without loss of generalitywe take the circle in the k x - k z plane to be (cid:112) k z + ( k x − k F ) = k and k y = 0. Alongthis circle, we choose N points to be k z = c f cos θ and k x = k F + c f sin θ where θ = πjN , j ∈ { , ...N } and the range of θ covers 0 to 2 π as shown in Fig. 9. c f should be chosen31o be small enough such that the circle does not pass through another nodal line. Thenwe could define the Berry phase using the discrete version and calculate the total Berryphase acquired along this circle. The discrete Berry phase is defined as e − iφ i i = (cid:104) n i | n i (cid:105)|(cid:104) n i | n i (cid:105)| , (4.46)where | n i (cid:105) and | n i (cid:105) are two adjacent eigenstates along the circle. The total Berry phaseis the sum of all adjacent phases along the circle from 0 to 2 π .Figure 9: We can discrete the circle in the k x - k z plane with N points. For the nodal line semimetal, the poles are also band crossing points. At ω = 0 and k z (cid:54) = 0, the near horizon boundary condition is proportional to e −| k z | / ( u r ) which is real.The four eigenvalues of G − (0 , k ) are real and appear in ± pairs. The eigenstates of G − (0 , k ) are also real. This feature is the same as the weakly coupled theory for a nodalline semimetal and this means that the relative phase between adjacent eigenvectors couldeither be 0 or π . For k z → k z and k x contributions may be equally large and we cannot ignore k x terms anymore.Thus we first choose discrete points on the circle not very close to the k x axis.Using numerics, we choose 51 discrete points on a circle with | k − k F | = c f where c f is a small number. To see more clearly whether there is a phase change on this circle,we have the following Fig. 10 of the four components of the normalized gapless negativeeigenvalued eigenvector for k F = 931 / at M/b (cid:39) . ± − Note that we have fixed b = 1. �� ��� ��� ��� ��������������������� θπ ��� ��� ��� ��� ��������������������� θπ ��� ��� ��� ��� ��� - ��� - ��� - ��� - ��� - ������ θπ ��� ��� ��� ��� ��������������������� θπ Figure 10:
The value of the four components of the normalized gapless negative eigenval-ued eigenvector of the topological Hamiltonian, i.e. − G − (0 , k ) of the holographic nodal linesemimetal at M/b (cid:39) . k F (cid:39) . θ = π/ , / π is k z = 0. This behavior is qualitatively the same for other poles and for small deformations ofthe circle that does not pass through the nodal lines. From this figure, we could see that in the k z > k z < k z < / √ | n k z → − (cid:105) =(1 / √ , , , / √ T when k z → k z > ± / √ | n k z → + (cid:105) = (0 , / √ , − / √ , T . This shows thatat k z = 0 there is a sudden jump in the eigenvectors that the adjacent eigenvectors areorthogonal to each other, i.e. (cid:104) n k z → − | n k z → + (cid:105) = 0. According to the formula of discreteBerry phases, this gives undetermined Berry phases. However, in fact though | n k z → + (cid:105) and | n k z → − (cid:105) are orthogonal to each other, it could be that the eigenvector n k z =0 at k z = 0is not orthogonal to either of | n k z → + (cid:105) and | n k z → − (cid:105) and gives a determined result for theBerry phase. Thus the eigenvectors at the k z = 0 points play an important key role todetermine the Berry phase.Numerics could not detect small but nonzero k z regions very accurately, but we couldwork directly at k z = 0 which is easier in numerics. For each of the pole, the small circlewould intersect with the k x axis twice (i.e. k z = 0) one at k F − = k F (1 − δ ) and oneat k F + = k F (1 + δ ), where δ (cid:28) k F − and k F + . We find that for all the poles from the sametwo bands as k F (cid:39) . k F − is | n k F − (cid:105) = 1 / , , − , T while theeigenvector at k F + is | n k F + (cid:105) = 1 / , − , , T . To connect these two eigenvectors withthose of | n k z → − (cid:105) and | n k z → + (cid:105) , we find that there needs to be a π phase along the circle.When we first connect | n k z → − (cid:105) and | n k z → + (cid:105) to | n k F − (cid:105) we find that | n k z → − (cid:105) and | n k z → + (cid:105) written in this way are already continuously connected without flipping signs of either ofthe two vectors. When we connect | n k z → − (cid:105) and | n k z → + (cid:105) to | n k F + (cid:105) we find that eitherone of | n k z → − (cid:105) and | n k z → + (cid:105) has to flip the sign or there would be a π phase changeat k z = 0 and if we flip the sign of one of | n k z → − (cid:105) and | n k z → + (cid:105) , a π phase difference33ould appear in the upper or lower half plane in the k x - k z plane. Thus for these poles,we could see that there is a nontrivial Berry phase of π . An illustration on the vectorscan be found in Fig. 11 and the different vectors for different poles or zeros can be foundin Tab. 1.Figure 11: Illustration for the calculation of Berry phase for holographic nodal line semimetalphase around each pole k F,i or any zero point k ,i of the Green function. | n k z → ± (cid:105) are the samefor all these points while | n k F ± (cid:105) are different depending on the points. poles from Bands I poles from Bands II zeros of the Green function | n k F + (cid:105) / , − , , T / , − , − , T / , , − , T | n k F − (cid:105) / , , − , T / , , , − T / , , − , T | n k z → + (cid:105) (0 , / √ , − / √ , T | n k z → − (cid:105) (1 / √ , , , / √ T Table 1:
A table of | n k F ± (cid:105) and | n k z → ± (cid:105) for poles from bands I, II and zeros of the Greenfunction. The behavior of the negative valued eigenvectors for k z (cid:54) = 0 points on the small circlearound the pole is the same for all the poles and all the zeros of the Green functions,while the k z = 0 and k F, ± = k F (1 ± δ ) negative valued eigenvectors | n k F ± (cid:105) are differentdepending on whether the poles come from bands I or II. In general, for poles from bandsI, i.e. the blue colored bands in Fig. 4, | n k F ± (cid:105) are the same as above and all result ina nontrivial Berry phase of π . For the poles from bands II, e.g. for k F = 931 / k F − is | n k F − (cid:105) = 1 / , , , − T while the eigenvector at k F + is | n k F + (cid:105) =34 / , − , − , T , which are still orthogonal to both | n k z → − (cid:105) and | n k z → + (cid:105) . This meansthat for the poles from the bands II, the Berry phase for the circle around the nodal pointis still undetermined.Besides checking the Berry phase for the poles, we have also checked if there is anontrivial Berry phase at each zero of the Green functions G (0 , k ix ) = 0 and we find thatfor the zeros of the Green functions, | n k F + (cid:105) = | n k F − (cid:105) = 1 / , , − , T and results in atrivial Berry phase of 0. This is different from the Weyl semimetal case where the zerosof the Green function could still have nontrivial topological invariants and this may alsoindicate that for positive m f the poles do not have nontrivial Berry phases.Thus with the above we conclude that for the holographic nodal line semimetal phase,there is a nontrivial topological invariant associated with poles from bands I and for polesfrom bands II the Berry phase is undetermined. We have calculated the topological invariants for holographic Weyl and nodal line semimet-als. For both cases, we find that we could define a nontrivial topological invariant usingthe topological Hamiltonian method, which allows us to calculate the topological invari-ants using the zero frequency Green functions of fermionic operators. For the holographicWSM case, semi-analytic calculations allows us to get the topological invariants for verysmall
M/b , which are ± M/b we will have to use numerics. For the holographic NLSM case, differentfrom the weakly coupled models, there are multiple nodal lines which are poles at ω = 0with k F,i for the holographic model. From the zero frequency Green function we could tellthat these poles come from different sets of bands indicating that the two gapped bandsand two gapless bands exchange their roles alternatively along the k x axis. A discreteversion of Berry phase calculation shows that for half of these poles there is a nontrivial π Berry phase while for the other half coming from the other two bands, the Berry phaseis undetermined.These nontrivial topological invariants provide a further robust evidence that theholographic models are strongly coupled topologically nontrivial semimetals and theseholographic models serve as a useful arena and a useful tool for the study of variousinteresting properties of strongly topological semimetals. It would be interesting to gen-eralize these to gapped systems and provide predictions of properties of strongly coupledgapped and gapless topological states of matter.35 cknowledgments
We would like to thank Rong-Gen Cai, Chen Fang, Carlos Hoyos, Elias Kiritsis, KarlLandsteiner, Shun-Qing Shen, Sang-Jin Sin, Zhong Wang, Jan Zaanen, Long Zhang foruseful discussions. This work is supported by the National Key R&D Program of China(Grant No. 2018FYA0305800) and by the Thousand Young Talents Program of China.The work of Y.L. was also supported by the NFSC Grant No.11875083 and a grant fromBeihang University. The work of Y.W.S. has also been partly supported by startinggrants from UCAS and CAS, and by the Key Research Program of the Chinese Academyof Sciences (Grant No. XDPB08-1), the Strategic Priority Research Program of ChineseAcademy of Sciences, Grant No. XDB28000000. We are also grateful to the hospitalityof Hanyang University during the conference “Holography and Geometry of QuantumEntanglement” (APCTP) where this work was presented. A s and e In this appendix we list the elements s and e appeared in Sec. 4.2.2. Note that x i with i ∈ , ...,
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