Pomeranchuk instabilities in holographic metals
PPrepared for submission to JHEP
Pomeranchuk instabilities in holographic metals
Gast´on Giordano, Nicol´as Grandi, Adri´an Lugo,
Instituto de F´ısica La Plata IFLP-CONICET & Departamento de F´ısica, FCE-UNLPC.C. 67, 1900 La Plata, Argentina
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We develop a method to detect instabilities leading to nematic phases instrongly coupled metallic systems. We do so by adapting the well-known Pomeranchuktechnique to a weakly coupled system of fermions in a curved asymptotically AdS bulk.The resulting unstable modes are interpreted as corresponding to instabilities on thedual strongly coupled holographic metal. We apply our technique to a relativistic 3 + 1-dimensional bulk with generic quartic fermionic couplings, and explore the phase diagramat zero temperature for finite values of the fermion mass and chemical potential, varyingthe couplings. We find a wide region of parameters where the system is stable, which issimply connected and localized around the origin of coupling space. a r X i v : . [ h e p - t h ] J a n ontents A.1 The background 10A.2 The Dirac equation 11A.2.1 Separation of variables and spin components 11A.2.2 The effective Schr¨odinger equation. 13A.3 General form of the free fermionic bases 14A.3.1 Quantization of the frequencies 14A.3.2 Normalization 15
B Hamiltonian theory 16
B.1 Dynamical setup 16B.2 The free Hamiltonian 17B.3 The interaction Hamiltonian 17
C Landau description of the bulk fermions 18
C.1 Perturbative derivation 18C.2 Explicit form of the Landau interaction functions 20
D Pomeranchuk method 21
D.1 Single spinless fermion 21D.2 Multiple fermionic species with spin 23D.3 Summary and application to the holographic setup 24
E The electron star background 25F The WKB solution 29
F.1 Basics of the WKB approximation 29F.2 Application to the electron star background 31F.2.1 Fermi momenta and static wave functions 32F.2.2 Fermi velocities 34– 1 –
Introduction
Strongly correlated electron systems have been at the heart of most recent research incondensed matter theory [1]. The underlying strongly coupled dynamics is thought to beresponsible of the richness of the phase diagram of high T c superconductors. It containsnormal metallic regions, as well as an exotic phase known as “strange metal” or “non-Fermi liquid” [2] which is believed to be described by strongly coupled fermionic degreesof freedom. There are also regions in which rotational symmetry is broken, the so-called“nematic” phases, as well as inhomogeneous “smectic” and “chessboard” phases.The transition from an isotropic Fermi liquid to a nematic phase is believed to be drivenby a Pomeranchuk instability [3]. Such instability arises when an excitation of the groundstate of the Fermi liquid results in a net decrease of the total energy. In Landau theory, suchperturbation is represented by a deformation of the Fermi surface. By decomposing thedeformation onto an orthonormal basis, Pomeranchuk obtained a set of conditions underwhich the Fermi liquid is stable. This method can be generalized to lattice systems oranisotropic Fermi surfaces [4, 5] and to finite temperature and magnetic field [6–8], and itrelies on the weakness of the quasi-particle coupling, or in other words on the validity of theLandau formula. This implies that from the strange metal perspective, since the dynamicsis strongly coupled, the detection of fermionic instabilities becomes more difficult.The holographic description [9] of a homogeneous fermionic phase has been shown toaccount for some of the interesting properties of the strange metal [10–15]. In particular,the resulting spectral function is compatible with a Fermi surface with or without long-livedquasi-particles. Based on that, in this paper, we analyze Pomeranchuk instabilities of thestrange metal phase from the holographic perspective. We use a holographic background inwhich we propagate a Dirac spinor. Being weakly coupled, such spinor can be described byLandau’s theory, and its stability under Fermi surface deformations can be studied. Thisaccounts for a description of the anisotropic instabilities of the dual strange metal.The generalization of Pomeranchuk method to arbitrary curved spaces is not possible,since it relies on a momentum space representation of the fermionic system. Nevertheless,the special kind of “planar” bulk spacetimes used in holography have the additional featureof a translational symmetry in the spatial directions spanning the boundary. This allows fora labeling of the bulk states by a momentum index (cid:126)k , complemented by an additional index m labeling the oscillation mode in the holographic direction. Then the D + 1 dimensionalcurved space fermionic system can be interpreted as a ( D −
1) + 1 dimensional flat spacesystem with multiple fermion species labeled by m . Pomeranchuk method can then bestraightforwardly applied to such multi-fermion flat space weakly coupled system.In what follows, we sketch the necessary steps needed to go from a 3 + 1 dimensionalaction for spinors in AdS spacetime to a 2 + 1 dimensional Hamiltonian for a multi-fermionsystem in flat space. Then we construct the corresponding Landau theory, and apply thePomeranchuk method to it. To improve readability, only the relevant steps of the calcula-tions are shown in the bulk of the paper, leaving the details to the various appendices . It is worth to mention that Pomeranchuk instabilities were studied in the holographic context in [16]from a different perspective, focusing in the calculus of spectral functions and their distortions. – 2 –
Holographic setup
We consider a holographic background consisting of a metric and an electromagnetic fieldwith the generic “planar” form G = L (cid:18) − f dt + g dz + dx + dy z (cid:19) , A = h dt . (2.1)This is a solution of the Einstein-Maxwell equations with a negative cosmological constant,as well as possible additional matter contributions to the energy-momentum tensor. Themetric components f ( z ) and g ( z ) are functions of the holographic coordinate z , and thegeometry asymptotes AdS spacetime as z goes to zero, provided f ( z ) ∼ g ( z ) ∼ /z . Theelectric potential h ( z ) is also a function of z , and approaches a constant h ( z ) ∼ µ at theboundary, which is identified with the chemical potential on the holographic theory.Specific backgrounds with the form (2.1) are the Reisner-Nordstrom AdS black-hole[10–13], the holographic superconductor [17], and the electron star at zero [14, 15] andfinite [18, 19] temperatures. Although we keep our formalism as general as possible, inorder to get concrete results in Section 4 we apply it to the background of reference [14].In the background defined above we propagate a spinorial perturbation Ψ, whosedynamics is dictated by the free Dirac action S free = − (cid:90) d x (cid:112) | G | ¯Ψ ( / D − m ) Ψ , (2.2)where / D stands for the covariant derivative containing both curved spacetime and gaugecontributions, contracted with the curved space Dirac matrices. Since the energy-momen-tum tensor and the electric current are quadratic in the spinor perturbation, to linear orderin Ψ we can work in the probe limit in which the background (2.1) is not perturbed.The general solution to (2.2) can be decomposed asΨ( t, (cid:126)x, z ) = zf ( z ) (cid:88) αm(cid:126)k N αm(cid:126)k c αm(cid:126)k ( t ) e i(cid:126)k · (cid:126)x ψ αm(cid:126)k ( z ) , (2.3)in terms of time dependent coefficients c αm(cid:126)k ( t ) and z -dependent spinors ψ αm(cid:126)k ( z ). Here thelabel α = 1 , (cid:126)k ∈ R represents the momentum in the xy plane, while m ∈ N characterizes the number of oscillations in the z direction. Finally N αm(cid:126)k is a nor-malization constant and the factor z/f / ( z ) was introduced for calculational convenience.The spinors ψ αm(cid:126)k ( z ) are written in terms of the solution to an ordinary differential equa-tion in the variable z , with a Schr¨odinger-like form. When physically meaningful boundaryconditions are imposed at the extremes of the z range, a (possibly complex) quantizeddispersion relation is obtained ω m ( k ), with which we can write c αm(cid:126)k ( t ) ∝ exp( − iω m ( k ) t ).Expression (2.3) allows us to quantize the system by promoting the coefficients c αm(cid:126)k ( t )to operators c αm(cid:126)k and c † αm(cid:126)k satisfying fermionic anticommutation relations. Then c † αm(cid:126)k creates a fermionic perturbation with momentum (cid:126)k , spin α and mode index m , and c αm(cid:126)k annihilates it.The steps of the derivation leading to decomposition (2.3) as well as the explicit formof the components of the spinor ψ αmk ( z ) are reviewed in detail in Appendix A.– 3 – Interactions
Now we introduce interactions among the spinorial perturbations, which represent 1 /N corrections to the holographic description. We do so by supplementing the action with theadditional term S int = (cid:90) d x (cid:112) | G | T ¯ σ ¯ σ (cid:48) σσ (cid:48) ¯Ψ ¯ σ ¯Ψ ¯ σ (cid:48) Ψ σ Ψ σ (cid:48) , (3.1)where we made explicit the spinor indices σ ∈ { , , , } , and the Lorentz invariant tensor T ¯ σ ¯ σ (cid:48) σσ (cid:48) represents the most general four fermion covariant interaction in curved space [20].It is written completely in terms of Dirac matrices, and depends linearly on a set of fivecoupling constants g , . . . , g , which we assume are small.We can now obtain the Hamiltonian resulting from (2.2) and (3.1), it takes the form H = (cid:88) αm (cid:90) d k ω m ( k ) c † αm(cid:126)k c αm(cid:126)k (3.2)+ (cid:88) α α α α m m m m (cid:90) d k d k (cid:48) d q t α m ( (cid:126)k + (cid:126)q ); α m ( (cid:126)k (cid:48) − (cid:126)q ) α m (cid:126)k ; α m (cid:126)k (cid:48) c † α m ( (cid:126)k + (cid:126)q ) c † α m ( (cid:126)k (cid:48) − (cid:126)q ) c α m (cid:126)k c α m (cid:126)k (cid:48) . Here the frequencies ω m ( k ) play the role of dispersion relations for each fermionic mode.On the other hand the new tensor t α m (cid:126)k ; α m (cid:126)k α m (cid:126)k ; α m (cid:126)k contains the information about the in-teraction strengths among different modes, and it results from contracting the positionspace interaction tensor T ¯ σ ¯ σ (cid:48) σσ (cid:48) with integrals in the z direction of quartic products of thefree fermionic eigenstates ψ αm(cid:126)k ( z ).The explicit forms of the interaction tensors T ¯ σ ¯ σ (cid:48) σσ (cid:48) and t α m (cid:126)k ; α m (cid:126)k α m (cid:126)k ; α m (cid:126)k . as well as thatof the aforementioned quartic integrals, is not relevant for the moment. For the details ofthe derivation of (3.2) we refer the reader to Appendix B.With equation (3.2), we have succeded in re-writing the bulk dynamics as that of asecond quantized Hamiltonian for a two dimensional multi-fermion system in which theindex m denotes fermion species. Since the coupling in the bulk is assumed to be weak,we can safely rely on the Landau description of the Fermi liquid.Assuming that the ground state of the Hamiltonian (3.2) is characterized by a certainset of occupation numbers N αm(cid:126)k , then the excitations can be described by their variations δN αm(cid:126)k . The grand canonical energy of an excitation is then written as the Landau formula δ Ω( T, µ ) = (cid:90) d (cid:126)k (cid:88) αm (cid:15) m ( k ) δN αm(cid:126)k + 12 (cid:90) d (cid:126)k d (cid:126)k (cid:48) (cid:88) αmα (cid:48) m (cid:48) f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) δN αm(cid:126)k δN α (cid:48) m (cid:48) (cid:126)k (cid:48) , (3.3) where f αm ¯ α ¯ m ( (cid:126)k, (cid:126) ¯ k ) is the so-called “interaction function” which is obtained from the tensor t α m (cid:126)k ; α m (cid:126)k α m (cid:126)k ; α m (cid:126)k , and the quasiparticle dispersion relation (cid:15) m ( k ) takes the form (cid:15) m ( k ) = ω m ( k ) + (cid:88) α (cid:48) m (cid:48) (cid:90) d (cid:126)k (cid:48) f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) N α (cid:48) m (cid:48) (cid:126)k (cid:48) . (3.4)For future use, notice that the quasiparticle dispersion relation equals the frequency pluscorrections of first order in the coupling constants.– 4 –or the most general covariant quartic perturbation (3.1), the interaction functiontakes a particularly simple angular dependence in the momentum plane. Indeed, if wewrite (cid:126)k = k (cos θ, sin θ ) then it can be decomposed in only three Fourier modes, as f αm ; α (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) = f αmk ; α (cid:48) m (cid:48) k (cid:48) + cos( θ − θ (cid:48) ) f cαmk ; α (cid:48) m (cid:48) k (cid:48) + sin( θ − θ (cid:48) ) f sαmk ; α (cid:48) m (cid:48) k (cid:48) , (3.5)where the constant, sine and cosine coefficients depend on a reduced subset of the com-ponents of the tensor t α m (cid:126)k ; α m (cid:126)k α m (cid:126)k ; α m (cid:126)k , or in other words on the position space interactiontensor T ¯ σ ¯ σ (cid:48) σσ (cid:48) contracted with integrals of the fermionic states ψ αm(cid:126)k ( z ).The details of the construction of the Landau description starting with the underlyingHamiltonian (3.2), as well as the explicit form of the interaction function (3.5) in terms ofthe interaction tensor and wavefuntion integrals, are presented in Appendix C.We can now make use of the Pomeranchuk technique for the above defined two dimen-sional multicomponent Landau Fermi liquid, in order to diagnose instabilities arising froman anisotropic deformation of its Fermi surface.We start by decomposing the deformation on the occupation numbers, that charac-therize the excitations of the Fermi liquid, in the form δN αm(cid:126)k = H (cid:0) − (cid:15) m ( k ) + δg αm ( (cid:126)k ) (cid:1) − H ( − (cid:15) m ( k )) = δ (cid:0) − (cid:15) m ( k ) (cid:1) δg αm ( (cid:126)k ) + 12 δ (cid:48) (cid:0) − (cid:15) m ( k ) (cid:1) δg αm ( (cid:126)k ) + . . . (3.6) where H ( · ) is the Heavyside unit-step function, and δg αm ( (cid:126)k ) are arbitrary functions ofthe momentum characterizing the excitation. When plugged back into (3.3) the deltafunctions force the evaluation of the expression on the Fermi momentum k mF defined by (cid:15) m ( k mF ) = 0. The remaining angular integrals result on a grand canonical energy that isa quadratic form in the Fourier components δg αm ( c,s ) n of the parameters δg αm ( (cid:126)k ) | k = k mF = (cid:80) ∞ n =0 ( δg αmcn cos( nθ ) + δg αmsn sin( nθ )).If the grand canonical energy of an excitation is negative, the system decreases itsenergy by creating more such excitations. To avoid such instability, we need to imposethat the aforementioned quadratic form is positive definite. This is guaranteed providedall the minors of the quadratic kernel are positive, or in other words (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k mF v mF (cid:32) δ αmα (cid:48) m (cid:48) + 2 π k m (cid:48) F v m (cid:48) F f αmα (cid:48) m (cid:48) (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M × M > , ∀ M ∈ N (3.7) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k mF v mF (cid:32) δ αmα (cid:48) m (cid:48) + π k m (cid:48) F v m (cid:48) F ( f cαmα (cid:48) m (cid:48) + if sαmα (cid:48) m (cid:48) ) (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M × M > , ∀ M ∈ N (3.8)where v mF = d(cid:15) m ( k ) /dk | k = k mF are the Fermi velocities and f (0 ,c,s ) αmα (cid:48) m (cid:48) are the Landau pa-rameters, given by the Fourier components in (3.5) evaluated at the corresponding Fermimomenta. Notice that, provided the Fermi velocities are positive, the quotient k mF /v mF canbe removed from the prefactors. Moreover, we can perturbatively replace k mF /v mF in theparenthesis by its free version, defined by ω m ( k m free f ) = 0 and v m free F = dω m ( k ) /dk | k = k m free F .A review of Pomeranchuk method, including the details of the calculations resultingin formulas (3.7)-(3.8), can be seen in Appendix D.– 5 – Summary and application example
In summary, in order to test the anisotropic Pomeranchuk instabilities of an holographicFermi liquid, we need to1. Obtain the fermionic modes ψ αmk ( z ) and their frequencies ω m ( k ). This is done bysolving a Schr¨odinger-like equation in the holographic direction. Notice that we onlyneed the modes with very small frequency, since we will evaluate them at the freeFermi momentum k m free F where the frequency vanishes.2. Calculate the free Fermi velocities as v m free F = dω mF /dk | k = k m free F . Incidentally, thisis why we need modes with non-vanishing small frequency, even if just those withvanishing frequency contribute to the quantities evaluated at the free Fermi momenta.3. Integrate a particular set of quartic products of fermionic modes in the holographicdirection, in order to go from T ¯ σ ¯ σ (cid:48) σσ (cid:48) to t α m (cid:126)k ; α m (cid:126)k α m (cid:126)k ; α m (cid:126)k and then to the Landau param-eters f ( c,s ) αmα (cid:48) m (cid:48) .4. Check whether all the minors in (3.7)-(3.8) are positive. Notice that each minor of M × M has degree M on the interaction function, which is linear in the couplingconstants. This implies that each condition imposes a polynomial restriction of order M in coupling space.At any point in coupling space at which condition 4 is not satisfied, the quadratic formhas a negative eigenmode, implying that the grand canonical energy is decreased by thecorresponding excitation. This results in an instability of the system.We applied the above steps to the specific example of the zero temperature electronstar background of [14]. The electron star represents the holographic dual to the groundstate of a highly degenerate system of fermions at zero temperature. It is constructed bycoupling the gravitational and electromagnetic degrees of freedom in the bulk to a perfectfluid representing the fermions, whose equation of state is obtained in the Thomas-Fermiapproximation. A summary of the electron star solution is given in Appendix E.In order to obtain the fermionic modes and calculate the Fermi momenta and velocitiesas required by points 1 and 2, we used a Wentzel-Kramers-Brillouin (WKB) approxima-tion on the spinorial perturbation. This is consistent with the use of the Thomas-Fermiapproximation for the bulk fluid. The relevant details of the WKB method are presentedin Appendix F.The resulting unstable regions of the phase diagram can be seen in Fig. 1, in whichdifferent coupling planes are presented. We see that as M grows (and so does the degree ofthe polynomial in point 4 above) the region in coupling space where the system is stable getssmaller, converging to an isolated wide island around the origin, which is simply connected.– 6 – tableM = = = = = = = = Figure 1 . Phase diagrams showing the unstable regions in the five-dimensional coupling space g , g , g , g , g . Each plot corresponds to one of the ten two-dimensional coupling planes. We seethat there is an island of stability around the origin. The lighter regions become unstable at higher M . The plots correspond to ˆ m = 0 . λ = 2 and γ = 100 (see Appendix E for details on theseparameters). – 7 – Conclusions and outlook
We developed a general method to study anisotropic fermionic instabilities on a stronglycoupled Fermi liquid via the AdS/CFT duality. It entails to perform a Pomeranchukanalysis on the dual bulk fermion, adapting the formalism to a curved space-time witha planar slice. The key step is to rewrite the single fermion in the bulk in terms ofits modes in the holographic direction, resulting on a multi-component systems of lowerdimensional fermions which move in the planar slice. By analyzing the minors of a quadraticform, we are able to detect when the fermionic modes become unstable under anisotropicdeformations of their Fermi surface. The resulting phase transition breaks the rotationalsymmetry in the planar slice, and consequently in the boundary theory, potentially leadingto a nematic phase.We kept the formalism as general as possible, being suitable to be applied to anyEuclidean-invariant boundary theory. This includes the holographic metals [10–13], thezero temperature electron star [14, 15], the finite temperature electron star [18, 19] and theholographic superconductor [17], as well as many other examples. We applied the methodto the zero-temperature electron star background, being able to identify the unstable regionon a five dimensional coupling space.The method can be extended to more general situations relevant to condensed mattersystems. Some examples of possible future directions are: • Finite doping: the inclusion of a doping axis should be straitforward, following thelines described in [21]. • Finite magnetic field: the inclusion of a magnetic field would require a magneticallycharged background [22], and it would modify the bulk fermion coupling. Also, thePomeranchuk analysis needs to be adapted [7]. • Finite temperature fluid: as it stands, the method can be applied to the backgroundof [18] in which the temperature is included via the presence of a horizon, whilethe bulk fluid is approximated as a finite temperature Fermi liquid. Including theeffect of temperature in the bulk matter would modify the background, as well as thePomeranchuk analysis [6]. • Lifshitz scaling: considering Lifshitz geometries [23] would require a rewriting of theasymptotic conditions on the near boundary fermion, but the rest of the analysisremains mostly unchanged. • Lattice fermions: describing an underlying lattice in the holographic setup wouldimply the inclusion of a lattice-like metric [24] or momentum relaxation [25], and amodification of the Pomeranchuk analysis according to the lines of [4].These are only some of the situations to which the method can be extended, we think thatthey deserve further investigation. – 8 – knowledgements
The authors thank Pablo Rodr´ıguez Ponte and Ignacio Salazar Landea for relevant contri-butions during the early stages of this work. This work has been funded by the CONICETgrants PIP-2017-1109, PIP-2015-0688 and PUE 084 “B´usqueda de Nueva F´ısica”, andUNLP grants PID-X791, PID-11/X910. – 9 –
Fermionic states in a holographic background
In this Appendix, we work out a basis of free fermionic modes in a charged planar holo-graphic background. We do so by writing the background in terms of vielbein and spinconnection (section A.1), deriving the form of the corresponding Dirac equation (sectionA.2), separating it into spin components and Fourier modes in the planar and time direc-tions (section A.2.1), and obtaining an effective Schr¨odinger equation that describes thefermion profile in the holographic direction (section A.2.2). We write the solution (sectionA.3) and notice that boundary conditions would generically imply a quantization of thefrequencies as functions of the momenta (section A.3.1), resulting in a bunch of 2+1 dimen-sional dispersion relations. We finally obtain the normalization constants, and check thatwith the so defined basis the resulting second quantized theory satisfy correctly normalizedcommutation relations (section A.3.1).
A.1 The background
We work with a generic planar asymptotically AdS charged background, with the form G = L (cid:18) − f dt + g dz + d(cid:126)x z (cid:19) = η AB ω A ω B ,A = h dt = A A ω A . (A.1)Here f , g and h , and consequently ω A and A A , are functions of z . Since we want to couplefermions to the present background, we need the explicit form of the vielbein and dualvector basis { ω A , e A } that we wrote in the second equality. In terms of the planar indices A = t, x, y, z , they read ω t ≡ L √ f dt , e t ≡ L √ f ∂ t ,ω x ≡ Lz dx , e x ≡ zL ∂ x ,ω y ≡ Lz dy , e y ≡ zL ∂ y ,ω z ≡ L √ g dz , e z ≡ L √ g ∂ z . (A.2)This allows us to calculate the non-zero components of the spin connection { ω AB } byimposing metricity ω AB = − ω BA , and torsionless dω A + ω AB ∧ ω B = 0 conditions. Theytake the form ω tz = − ω tz = + ω zt = (ln f ) (cid:48) √ g ω t ,ω xz = + ω xz = − ω zx = − z √ g ω x ,ω yz = + ω yz = − ω zy = − z √ g ω y . (A.3)This allows us to write the Dirac equation for a charged fermion in the present background,as we do in the next subsection. – 10 – .2 The Dirac equationA.2.1 Separation of variables and spin components We consider a four-component Dirac spinor Ψ with charge q under the U (1) gauge fieldcoupled to gravity through the covariant derivative / D Ψ ≡ Γ A D A Ψ = Γ A (cid:18) e A (Ψ) + i ω BC A Σ BC Ψ − i q A A Ψ (cid:19) , (A.4)where the Dirac gamma-matrices obey { Γ A , Γ B } = 2 η AB , and Σ AB ≡ [Γ A , Γ B ] / i are thegenerators in the spinorial representation of the local Lorentz group in 3 + 1 dimensions.We use along the paper the following representation for the gamma matricesΓ t ≡ (cid:32) i σ i σ (cid:33) , Γ x ≡ (cid:32) − σ σ (cid:33) , Γ y ≡ (cid:32) σ σ (cid:33) , Γ z ≡ (cid:32) σ σ (cid:33) , (A.5)where { σ , σ , σ } are the Pauli matrices.The free fermionic modes satisfy the Dirac equation( / D − m ) Ψ = 0 . (A.6)To solve it, we find convenient to work in momentum spaceΨ ω(cid:126)k ( t, (cid:126)x, z ) = Ψ ω(cid:126)k ( (cid:126)x, z ) e − iωt = 1 N ω(cid:126)k zf ( z ) e i ( (cid:126)k · (cid:126)x − ωt ) ψ ω(cid:126)k ( z ) , (A.7)where ω , (cid:126)k are the energy and momentum along the xy plane, N ω(cid:126)k is a normalizationconstant, and the factor z/f ( z ) / has been included for later convenience. We can userotational invariance to refer the momentum to the x -axis, as R [ θ ] (cid:32) k x k y (cid:33) = (cid:32) k (cid:33) , with R [ θ ] = e i θ σ = (cid:32) cos θ sin θ − sin θ cos θ (cid:33) . (A.8)This allows us to write ψ ω(cid:126)k ( z ) = S [ θ ] ψ ωk ( z ) where S [ θ ] = e i θ Σ xy = cos θ xy sin θ (cid:32) cos θ × − sin θ × sin θ × cos θ × (cid:33) , (A.9)is the spinor representation of the rotation matrix (A.8), where 1 × is the 2 × xy ≡ i [Γ x ; Γ y ] = i Γ xy . Theresulting form of the equation (A.6) for ψ ωk ( z ) now reads (cid:32) (cid:112) g ( z ) Γ z ∂ z − i (cid:112) f ( z ) Γ t ( ω + q h ( z )) + i k z Γ x − m L (cid:33) ψ ωk ( z ) = 0 . (A.10)Notice that this equation is real due to the choice of gamma matrices (A.5).– 11 –et us now introduce the projectorsΠ α ≡ (cid:0) − α +1 Γ z Γ t Γ x (cid:1) = (cid:32) ×
00 0 (cid:33) , α = 1 (cid:32) × (cid:33) , α = 2 (A.11)With their help we can decompose the Dirac field into “spin” states as ψ ωk ( z ) = (Π +Π ) ψ ωk ( z ) = (cid:80) α =1 ψ αωk ( z ), where the projected fields are ψ αωk ( z ) ≡ Π α ψ ωk ( z ) = (cid:32) ψ (1) ωk ( z )0 (cid:33) , α = 1 (cid:32) ψ (2) ωk ( z ) (cid:33) , α = 2 (A.12)where each of the ψ ( α ) ωk have two components. Inserting the resulting decomposition in(A.7), we obtain a solution with energy ω momentum (cid:126)k and “spin” α of the formΨ αω(cid:126)k ( t, (cid:126)x, z ) = Ψ αω(cid:126)k ( (cid:126)x, z ) e − iωt = 1 N αω(cid:126)k zf ( z ) e i ( (cid:126)k · (cid:126)x − ωt ) S [ θ ] ψ αωk ( z ) . (A.13)Plugging this back into the field equation (A.10) we get a decoupled system for the bi-spinors ψ ( α ) ωk ( z ), as ψ ( α ) (cid:48) ωk ( z ) + (cid:112) g ( z ) (cid:32) ω + q h ( z ) (cid:112) f ( z ) i σ + ( − ) α k z σ − m L σ (cid:33) ψ ( α ) ωk ( z ) = 0 , α = 1 , . It is worth to notice that from the representation (A.5) the generators of the Lorentz subgroup in 2 + 1dimensions areΣ tx = 12 i (cid:32) − σ σ (cid:33) , Σ ty = 12 i (cid:32) σ σ (cid:33) , Σ xy = 12 i (cid:32) − × × (cid:33) , (A.15)From here it should be clear that ψ ( α ) ωk ( z ) are not Dirac spinors in 2 + 1 dimensions, since they mix underLorentz transformations. – 12 – .2.2 The effective Schr¨odinger equation. We want to transform (A.14) into a second order Schr¨odinger equation to which we canapply our intuitions regarding wave functions. In order to do that, let us introduce thefunctions f ± k ( z ) ≡ (cid:112) g ( z ) (cid:32) z k ± ω + q h ( z ) (cid:112) f ( z ) (cid:33) . (A.16)Now, if we consider first α = 2, we can parameterize the bi-spinor ψ (2) ωk ( z ) in terms of twofunctions ψ (2) − ωk ( z ) and φ (2) ωk ( z ), in the form ψ (2) ωk ( z ) ≡ (cid:32) (cid:113) f + k ( z ) φ (2) ω,k ( z ) ψ (2) − ωk ( z ) (cid:33) , (A.17)then substituting in (A.14) we get from its components the pair of coupled first orderequations φ (2) ωk (cid:48) ( z ) + (cid:18) f + k (cid:48) ( z )2 f + k ( z ) − m L (cid:112) g ( z ) (cid:19) φ (2) ωk ( z ) + (cid:113) f + k ( z ) ψ (2) − ωk ( z ) = 0 , (A.18) ψ (2) − ωk (cid:48) ( z ) + m L (cid:112) g ( z ) ψ (2) − ωk ( z ) + f − k ( z ) (cid:113) f + k ( z ) φ (2) ωk ( z ) = 0 , (A.19)from the first of which we can rewrite ψ (2) − ωk ( z ) = − (cid:113) f + k ( z ) (cid:18) φ (2) ωk (cid:48) ( z ) + (cid:18) f + k (cid:48) ( z )2 f + k ( z ) − m L (cid:112) g ( z ) (cid:19) φ (2) ωk ( z ) (cid:19) , (A.20)and now plugging this into the second equation, we get a second order Schr¨odinger-likeequation for φ (2) ωk − φ (2) ωk (cid:48)(cid:48) ( z ) + U ( z ) φ (2) ωk ( z ) = 0 , (A.21)where the potential is given by U ( z ) = g ( z ) (cid:18) k z − ( ω + qh ( z )) f ( z ) + m L (cid:19) − f + k (cid:48)(cid:48) ( z )2 f + k ( z ) + 34 (cid:18) f + k (cid:48) ( z ) f + k ( z ) (cid:19) − mL (cid:112) g ( z ) (cid:32) ln (cid:32) f + k ( z ) (cid:112) g ( z ) (cid:33)(cid:33) (cid:48) , (A.22) If instead we consider α = 1, we can parametrize the bi-spinor ψ (1) ωk ( z ) in terms of thefunctions ψ (1)+ ωk ( z ) and φ (1) ωk ( z ), as ψ (1) ωk ( z ) ≡ (cid:32) ψ (1)+ ωk ( z ) (cid:113) f + k ( z ) φ (1) ω,k ( z ) (cid:33) . (A.23)After introducing it in (A.14) we get as before a pair of coupled first order equations φ (1) ωk (cid:48) ( z ) + (cid:18) f + k (cid:48) ( z )2 f + k ( z ) + m L (cid:112) g ( z ) (cid:19) φ (1) ωk ( z ) − (cid:113) f + k ( z ) ψ (1)+ ωk ( z ) = 0 , (A.24) ψ (1)+ ωk (cid:48) ( z ) − m L (cid:112) g ( z ) ψ (1)+ ωk ( z ) − f − k ( z ) (cid:113) f + k ( z ) φ (1) ωk ( z ) = 0 . (A.25)– 13 –rom the first equation we can obtain ψ (1)+ ωk ( z ) = 1 (cid:113) f + k ( z ) (cid:18) φ (1) ωk (cid:48) ( z ) + (cid:18) f + k (cid:48) ( z )2 f + k ( z ) + m L (cid:112) g ( z ) (cid:19) φ (1) ωk ( z ) (cid:19) , (A.26)and by plugging it into the second equation of (A.24) we get a second order Schr¨odinger-likeequation for the function φ (1) ωk , with the form − φ (1) ωk (cid:48)(cid:48) ( z ) + U ( z ) φ (1) ωk ( z ) = 0 , (A.27)with exactly the same potential (A.22) as before.A last remark: we could try a solution of equations (A.14) such that: ψ (1) ωk ( z ) = ψ (2) ω ( − k ) ( z ) in (A.17) or (A.23). However, that would not be a smart choice since it wouldintroduce a vanishing denominator at (A.20) or (A.26) whenever f + k ( z ) vanishes at negativevalue of the momenta. A better definition is to write (A.17) and (A.23) with φ (1) ωk ( z ) = φ (2) ωk ( z ) ≡ φ ωk ( z ) the unique solution of (A.27), as we will show nextly. A.3 General form of the free fermionic basesA.3.1 Quantization of the frequencies
We must now impose boundary conditions on our solutions, in order to ensure smoothnesseverywhere. In particular, we impose regular boundary conditions in the ultraviolet, andwe can chose either regular or in-going boundary conditions in the infrared.The boundary conditions have an important consequence: the frequency ω is not freebut is related to the modulus k of the two-momentum through a dispersion relation thatis in general labelled by some integer index m , resulting in ω = ω m ( k ) (see section F.2for explicit forms of dispersion relations). This allows us to replace the indices kω in ourfunctions of the previous sections by km .Taking into account these facts, and collecting the results given in (A.14)-(A.27) wecan write for the fermionic mode in (A.13), the general formΨ αm(cid:126)k ( (cid:126)x, z ) = 1 N αmk zf ( z ) e i(cid:126)k · (cid:126)x e i θ Σ xy − ( − α ψ (1)+ mk ( z ) − ( − α (cid:113) f + k ( z ) φ mk ( z ) − α (cid:113) f + k ( z ) φ mk ( z ) − α ψ (2) − mk ( z ) , (A.28)where ψ (2) − mk ( z ) and ψ (1)+ mk ( z ) are given in (A.20) and (A.26) respectively, in terms of thesolution φ mk ( z ) of (A.21) (or equivalently of (A.27)). Notice that the 4-tuple in parenthesison the right corresponds to what was called ψ αωk ( z ) in equation (A.13), and that will bereferred bellow as ψ αmk ( z ) in attention to the quantization of the frequencies.– 14 – .3.2 Normalization To completely define the modes (A.28) we need to fix their normalization. To this end weintroduce in the space of spinors the following scalar product [26](Ψ ; Ψ )( t ) ≡ (cid:90) Σ t dz d x (cid:112) | H | Ψ ( t, (cid:126)x, z ) † Ψ ( t, (cid:126)x, z ) , (A.29)where Σ t is the space of constant t and H = L (cid:18) g ( z ) dz + d(cid:126)x z (cid:19) ; (cid:112) | H | = L (cid:112) g ( z ) z , (A.30)is the induced metric on it. It is not difficult to prove that in the space of solutions to theDirac equation (A.6) the operator i∂ t is hermitian, i.e. (Ψ ; i∂ t Ψ )( t ) = ( i∂ t Ψ ; Ψ )( t ) . (A.31)Standard arguments then show that eigenspinors of i∂ t with different eigenvalues are or-thogonal with respect to (A.29). By applying this result to the spinors (A.28) we knowthat they result orthogonal for different m ’s. By using this fact we straightforwardly getthe orthonormality relation (cid:0) Ψ αm(cid:126)k ; Ψ α (cid:48) m (cid:48) (cid:126)k (cid:48) (cid:1) ≡ (cid:90) Σ t dz d (cid:126)x √ H Ψ † αm(cid:126)k ( (cid:126)x, z ) Ψ α (cid:48) m (cid:48) (cid:126)k (cid:48) ( (cid:126)x, z ) = δ αα (cid:48) δ mm (cid:48) δ ( (cid:126)k − (cid:126)k (cid:48) ) , (A.32)if the normalization constant is fixed such that |N αmk | = (2 π ) L (cid:90) ∞ dz (cid:115) g ( z ) f ( z ) ψ αmk ( z ) † ψ αmk ( z ) . (A.33)Finally, after the calculations above, the general solution of (A.6) can be written incoordinates space as Ψ( t, (cid:126)x, z ) = (cid:90) d (cid:126)k (cid:88) αm c αm(cid:126)k ( t ) Ψ αm(cid:126)k ( (cid:126)x, z ) . (A.34)For later use, it is worth to remind that (A.32) yields for the above solutions the complete-ness relation (cid:90) d (cid:126)k (cid:88) αm Ψ αm(cid:126)k ( (cid:126)x, z ) Ψ † αm(cid:126)k ( (cid:126)x (cid:48) , z (cid:48) ) = 1 (cid:112) | H | δ ( z − z (cid:48) ) δ ( (cid:126)x − (cid:126)x (cid:48) ) . (A.35)As we will see in the next section, the above defined normalization provides canonicalanti-commutation relations for the corresponding Hamiltonian variables. This allows us todefine canonical creation and annihilation operators for the fermionic modes in the bulk.With this, we will be able to establish a standard Landau description for the bulk Fermiliquid. – 15 – Hamiltonian theory
In this Appendix we develop the Hamiltonian theory for the fermions in the bulk, using theorthonormal basis obtained in the previous section. In section B.1, we define our dynamicsfor the fermionic degrees of freedom, and write the generic form of the correspondingHamiltonian. Then in section B.2 we deal with its free part, while the interacting part isworked out in section B.3.
B.1 Dynamical setup
The action for our spinor field that propagates in the asymptotically AdS bulk reads S Ψ = (cid:90) d x (cid:112) −| G | L Ψ = − (cid:90) d x (cid:112) −| G | (cid:0) ¯Ψ ( / D − m )Ψ + T σ σ σ σ ¯Ψ σ ¯Ψ σ Ψ σ Ψ σ (cid:1) , (B.1)where the σ ’s are spin indices running from 1 to 4, and we have defined the conjugatespinor ¯Ψ ≡ Ψ † i Γ t . We included a four-fermion interaction term, which takes the mostgeneral form that respects covariance in four-dimensional curved space and fermion numberconservation. It is written in terms of an invariant tensor T σ σ σ σ given by T σ σ σ σ = g δ σ σ δ σ σ + g (Γ ) σ σ (Γ ) σ σ + g (Γ a ) σ σ (Γ a ) σ σ + g (Γ a Γ ) σ σ (Γ a Γ ) σ σ − g (cid:16) [Γ a , Γ b ] Γ (cid:17) σ σ (cid:0) [Γ a , Γ b ] Γ (cid:1) σ σ , (B.2)where we defined Γ ≡ − i Γ t Γ x Γ y Γ z . This tensor satisfies the condition T σ σ σ σ = T σ σ σ σ ,as well as the Lorentz invariance property T σ σ σ σ = S [Λ] σ δ S [Λ] σ δ T δ δ δ δ S [Λ] − δ σ S [Λ] − δ σ , (B.3)for any Lorentz transformation Λ, where S [Λ] denotes its spinorial representation. Thisproperty will be useful in what follows, in particular when we apply it to the rotation onthe xy plane represented by the unitary matrix S [ θ ] defined in (A.9).From the action (B.1) the momentum conjugate to the spinor field is π = i (cid:112) | H | Ψ † , (B.4)allowing us to write the Hamiltonian as H = (cid:90) d (cid:126)x dz (cid:16) π ∂ t Ψ − (cid:112) −| G | L Ψ (cid:17) , (B.5)where L Ψ can be read from (B.1).Inserting the momentum (B.4) into the canonical anti-commutation relations for co-ordinates and momenta, we get that the spinor field satisfies { Ψ( t, (cid:126)x, z ) , Ψ † ( t, (cid:126)x (cid:48) , z (cid:48) ) } = 1 (cid:112) | H | δ ( z − z (cid:48) ) δ ( (cid:126)x − (cid:126)x (cid:48) ) . (B.6)Then from the decomposition (A.34) we get the standard anti-commutation relations { c αm(cid:126)k ( t ) , c † α (cid:48) m (cid:48) (cid:126)k (cid:48) ( t ) } = δ αα (cid:48) δ mm (cid:48) δ ( (cid:126)k − (cid:126)k (cid:48) ) , (B.7)identifying c † αm(cid:126)k and c αm(cid:126)k as the creation and annihilation operators respectively for thefermionic modes in the bulk. In the rest of this section we rewrite the bulk dynamics givenby (B.5) in terms of them. – 16 – .2 The free Hamiltonian Let us first take the free part of the Hamiltonian (B.5), which reads H free = − (cid:90) d (cid:126)x (cid:90) dz (cid:112) | H | (cid:18) i Ψ † (cid:18) i ω BCt Σ BC − i q h ( z ) (cid:19) Ψ − (cid:112) | G tt | ¯Ψ (Γ z D z + Γ i D i − m )Ψ (cid:17) (B.8) now using the Dirac equation we get H free = 12 (cid:90) d (cid:126)x (cid:90) dz (cid:112) | H | i Ψ † ∂ t Ψ + h.c. . (B.9)Then inserting the decomposition (A.34) and using the orthogonality relation (A.35) weget the free Hamiltonian in terms of creation and annihilation operators, as H free = (cid:88) αm (cid:90) d (cid:126)k ω m ( k ) c † αm(cid:126)k c αm(cid:126)k . (B.10)In other words, the bulk degrees of freedom are described by a set of independent fermionicspecies labelled with an integer index m corresponding to the mode on the z direction. Eachspecies has a dispersion relation given by ω m ( k ), with a two-fold degeneracy given by thespin α . B.3 The interaction Hamiltonian
Regarding the interacting part of the Hamiltonian, we have H int = (cid:90) dz d (cid:126)x (cid:112) | H | T σ σ σ σ ¯Ψ σ ¯Ψ σ Ψ σ Ψ σ . (B.11)Plugging the decomposition (A.34) into (B.11) we get H int = (cid:88) α α α α m m m m (cid:90) d k . . . d k δ (2) ( (cid:126)k + (cid:126)k − (cid:126)k − (cid:126)k ) t α m (cid:126)k ; α m (cid:126)k α m (cid:126)k ; α m (cid:126)k c † α m (cid:126)k c † α m (cid:126)k c α m (cid:126)k c α m (cid:126)k , (B.12) where the (cid:126)x integral has been performed explicitly giving origin to the momentum conser-vation δ -function, while the z integral is contained in the definition of the momentum-spaceinteraction tensor t α m (cid:126)k ; α m (cid:126)k α m (cid:126)k ; α m (cid:126)k = L (2 π ) S [ θ ] † δ σ S [ θ ] † δ σ T σ σ σ σ S [ θ ] σ δ S [ θ ] σ δ I α m k ; α m k ; δ δ α m k ; α m k ; δ δ , (B.13)in terms of the integrals I α m k ; α m k ; δ δ α m k ; α m k ; δ δ = (cid:90) dz z (cid:112) g ( z ) f ( z ) ¯ ψ α m k ; δ ( z ) N α m k ¯ ψ α m k ; δ ( z ) N α m k ψ δ α m k ( z ) N α m k ψ δ α m k ( z ) N α m k , (B.14)where functions ψ αωk ( z ) are defined as in equation (A.13) and correspond to the 4-tuplesin parenthesis on the right of equation (A.28). As we show below, only a subset of theintegrals (B.14) is needed for a Landau description of the bulk fluid.– 17 – Landau description of the bulk fermions
In the previous section we rewrite the dynamics of the Dirac spinor in the bulk as that ofa bunch of fermionic species in one less dimension, indexed by the energy mode m and thespin α . In the present appendix, we develop a description in terms of the Landau theoryfor a multi-component Fermi liquid. We do that by using a perturbative approach. C.1 Perturbative derivation
We work in the grand canonical ensemble at chemical potential µ and temperature T = 1 /β (that we will take to zero at the end of the calculations). We define the expectation valueof an operator O by (cid:104)O(cid:105) = 1 Z tr (cid:16) e − β ( H − µ N ) O (cid:17) , where Z = tr (cid:16) e − β ( H − µ N ) (cid:17) = e − β Ω( T,µ ) , (C.1)where N is the number of particles and Ω( T, µ ) is the grand canonical potential. The lastformula can be inverted according toΩ(
T, µ ) = − β log Z . (C.2)To perform a perturbative approximation, we write the Hamiltonian as H = H free + H int , (C.3)and assume that the constants g i in (B.2) are small. With the help of the free Hamiltonian H free we define the zeroth order quantities (cid:104)O(cid:105) free = 1 Z free tr (cid:16) e − β ( H free − µ N ) O (cid:17) where Z free = tr (cid:16) e − β ( H free − µ N ) (cid:17) . (C.4)Then expanding (C.1) to first order in g i , we obtain Z ≈ Z free (1 − β (cid:104) H int (cid:105) free ) , (C.5)which implies for the grand canonical potentialΩ( T, µ ) ≈ − β log Z free + (cid:104) H int (cid:105) free . (C.6)For the first term in (C.6) we can write Z free = e − β (cid:82) d (cid:126)k (cid:80) αm ω m ( k ) N αm(cid:126)k , (C.7)where N αm(cid:126)k are the occupation numbers of the one-particle states N αm(cid:126)k = (cid:104) c † αm(cid:126)k c αm(cid:126)k (cid:105) .This implies for the free part of the grand canonical potential − β log Z free = (cid:90) d (cid:126)k (cid:88) αm ω m ( k ) N αm(cid:126)k . (C.8)– 18 –n the other hand, to write the second term in (C.6), we use (B.12) to have (cid:104) H int (cid:105) free = (cid:88) α α α α m m m m (cid:90) d k . . . d k δ (2) ( (cid:126)k + (cid:126)k − (cid:126)k − (cid:126)k ) t α m (cid:126)k ; α m (cid:126)k α m (cid:126)k ; α m (cid:126)k (cid:68) c † α m (cid:126)k c † α m (cid:126)k c α m (cid:126)k c α m (cid:126)k (cid:69) free (C.9) Calculating the expectation value in the right hand side (cid:68) c † α m (cid:126)k c † α m (cid:126)k c α m (cid:126)k c α m (cid:126)k (cid:69) free = N α m (cid:126)k N α m (cid:126)k × (C.10) × (cid:16) δ α α δ α α δ m m δ m m δ ( (cid:126)k − (cid:126)k (cid:48) ) δ ( (cid:126)k − (cid:126)k ) − δ α α δ α α δ m m δ m m δ ( (cid:126)k − (cid:126)k ) δ ( (cid:126)k − (cid:126)k ) (cid:17) , we replace back in (C.9) to get (cid:104) H int (cid:105) free = 12 (cid:90) d (cid:126)k d (cid:126)k (cid:48) (cid:88) αmα (cid:48) m (cid:48) f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) N αm(cid:126)k N α (cid:48) m (cid:48) (cid:126)k (cid:48) , (C.11)where the “Landau interaction functions” f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) are defined according to f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) ≡ t αm(cid:126)k ; α (cid:48) m (cid:48) (cid:126)k (cid:48) α (cid:48) m (cid:48) (cid:126)k (cid:48) ; αm(cid:126)k − t αm(cid:126)k ; α (cid:48) m (cid:48) (cid:126)k (cid:48) αm(cid:126)k ; α (cid:48) m (cid:48) (cid:126)k (cid:48) + t α (cid:48) m (cid:48) (cid:126)k (cid:48) ; αm(cid:126)kαm(cid:126)k ; α (cid:48) m (cid:48) (cid:126)k (cid:48) − t α (cid:48) m (cid:48) (cid:126)k (cid:48) ; αm(cid:126)kα (cid:48) m (cid:48) (cid:126)k (cid:48) ; αm(cid:126)k . (C.12)Here we used the definition (B.13), and re-absorbed into the couplings an infinite constant δ ( (cid:126)
0) coming from collapsing the momentum integrals . The Landau interaction functionscontain all the information about the interactions on the multi-component Fermi liquid.They verify the following relations f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) = f α (cid:48) m (cid:48) αm ( (cid:126)k (cid:48) , (cid:126)k ) ; f αmαm ( (cid:126)k, (cid:126)k ) = 0 . (C.13)Now we can use (C.11) and (C.12) to write the grand canonical potential asΩ( T, µ ) = (cid:90) d (cid:126)k (cid:88) αm ω m ( k ) N αm(cid:126)k + 12 (cid:90) d (cid:126)k d (cid:126)k (cid:48) (cid:88) αmα (cid:48) m (cid:48) f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) N αm(cid:126)k N α (cid:48) m (cid:48) (cid:126)k (cid:48) . (C.14)A perturbation δN αm(cid:126)k of the ground state occupation numbers N αm(cid:126)k gives us a vari-ation of the grand canonical potential with the form δ Ω( T, µ ) = (cid:90) d (cid:126)k (cid:88) αm (cid:15) m ( k ) δN αm(cid:126)k + 12 (cid:90) d (cid:126)k d (cid:126)k (cid:48) (cid:88) αmα (cid:48) m (cid:48) f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) δN αm(cid:126)k δN α (cid:48) m (cid:48) (cid:126)k (cid:48) , (C.15)where the quasiparticle dispersion relation (cid:15) m ( k ) has been defined as (cid:15) m ( k ) = ω m ( k ) + (cid:88) α (cid:48) m (cid:48) (cid:90) d (cid:126)k (cid:48) f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) N α (cid:48) m (cid:48) (cid:126)k (cid:48) . (C.16)For any perturbation δN αm(cid:126)k the quantity δ Ω( T, µ ) must be positive to have a stable groundstate. Whenever it becomes negative, an instability is triggered. Formally this is accomplished by regularizing through the introduction of a finite volume in coordinatespace, and then taking the infinite volume limit at the end of the calculations. – 19 – .2 Explicit form of the Landau interaction functions
We want to obtain a more explicit form of the Landau interaction functions f αm α (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) )suitable of being implemented into a numerical code. The integrals (B.14) we need are I αmk ; δ δ α (cid:48) m (cid:48) k (cid:48) ; δ δ = I αm(cid:126)k ; α (cid:48) m (cid:48) (cid:126)k (cid:48) ; δ δ α (cid:48) m (cid:48) (cid:126)k (cid:48) ; αm(cid:126)k ; δ δ , (C.17)and they satify I αmk ; δ δ α (cid:48) m (cid:48) k (cid:48) ; δ δ = I α (cid:48) m (cid:48) k (cid:48) ; δ δ αmk ; δ δ what can be used to rewrite (B.13) as t αm(cid:126)k ; α (cid:48) m (cid:48) (cid:126)k (cid:48) α (cid:48) m (cid:48) (cid:126)k (cid:48) ; αm(cid:126)k = S [ θ − θ (cid:48) ] † δ σ T σ δ δ σ S [ θ − θ (cid:48) ] σ δ I αmk ; δ δ α (cid:48) m (cid:48) k (cid:48) ; δ δ , (C.18) t αm(cid:126)k ; α (cid:48) m (cid:48) (cid:126)k (cid:48) αm(cid:126)k ; α (cid:48) m (cid:48) (cid:126)k (cid:48) = S [ θ − θ (cid:48) ] † δ σ T σ δ σ δ S [ θ − θ (cid:48) ] σ δ I αmk ; δ δ α (cid:48) m (cid:48) k (cid:48) ; δ δ . (C.19)To further disentangle the angle dependence, we find convenient to use the explicit formof the spinorial rotations matrices (A.9) to write S [ θ − θ (cid:48) ] † δ σ S [ θ − θ (cid:48) ] σ δ = P δ σ σ δ + cos( θ − θ (cid:48) ) P cδ σ σ δ + sin( θ − θ (cid:48) ) P sδ σ σ δ , (C.20)where the constant, cosine and sine auxiliary tensors are given respectively as P δ σ σ δ ≡ (cid:16) δ δ σ δ σ δ − Γ xyδ σ Γ xyσ δ (cid:17) = + P σ δ δ σ ,P cδ σ σ δ ≡ (cid:16) δ δ σ δ σ δ + Γ xyδ σ Γ xyσ δ (cid:17) = + P cσ δ δ σ ,P sδ σ σ δ ≡ (cid:16) δ δ σ Γ xyσ δ − Γ xyδ σ δ σ δ (cid:17) = − P sσ δ δ σ . (C.21)By inserting this expressions into (C.18)-(C.19) we finally obtain from (C.12) a completelyfactorized form for the Landau interaction functions, as f αm ; α (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) = f αmk ; α (cid:48) m (cid:48) k (cid:48) + cos( θ − θ (cid:48) ) f cαmk ; α (cid:48) m (cid:48) k (cid:48) + sin( θ − θ (cid:48) ) f sαmk ; α (cid:48) m (cid:48) k (cid:48) , (C.22)written in terms of three independent constant, cosine and sine components, which aregiven according to f αmk ; α (cid:48) m (cid:48) k (cid:48) = I αmk ; δ δ α (cid:48) m (cid:48) k (cid:48) ; δ δ (cid:16) T δ δ δ δ − T δ δ δ δ + Γ xyδ σ (cid:16) T σ δ σ δ − T σ δ δ σ (cid:17) Γ xyσ δ (cid:17) = + f α (cid:48) m (cid:48) k (cid:48) ; αmk ,f cαmk ; α (cid:48) m (cid:48) k (cid:48) = I αmk ; δ δ α (cid:48) m (cid:48) k (cid:48) ; δ δ (cid:16) T δ δ δ δ − T δ δ δ δ − Γ xyδ σ (cid:16) T σ δ σ δ − T σ δ δ σ (cid:17) Γ xyσ δ (cid:17) = + f cα (cid:48) m (cid:48) k (cid:48) ; αmk ,f sαmk ; α (cid:48) m (cid:48) k (cid:48) = I αmk ; δ δ α (cid:48) m (cid:48) k (cid:48) ; δ δ (cid:16)(cid:16) T δ δ δ σ − T δ δ σδ (cid:17) Γ xyσδ − Γ xyδ σ (cid:16) T σδ δ δ − T σδ δ δ (cid:17)(cid:17) = − f sα (cid:48) m (cid:48) k (cid:48) ; αmk . (C.23)Notice that we dropped any auxiliary quantity, having obtained a formula for the Landauinteraction functions which is completely written in terms of the interaction tensor (B.2)and the integrals (C.17) of the form (B.14), where the radial functions were defined inequations (A.13)-(A.28) in terms of the solutions of equation (A.27).– 20 – Pomeranchuk method
We give in this section a brief introduction to Pomeranchuk’s method to detect instabilitiesin a Landau fermi liquid. We focus on the isotropic, two dimensional case, since it iswhat concern us in this paper. We begin by explaining the method for a single spinlessfermion (section D.1), the generalization for multiple species of spinful fermions is givenlater (section D.2).
D.1 Single spinless fermion
Given a two-dimensional system of spinless fermion with quasiparticle dispersion relation (cid:15) ( k ), its Fermi surface is defined in momentum space by the relation (cid:15) ( k F ) = 0. In theground state all the single quasiparticle states with momenta k ≤ k F are occupied, whilethose with momenta k > k F are empty. In other words, we can write the occupationnumber as N (cid:126)k = H ( − (cid:15) ( k )), where H is the Heavyside step function.A fermionic excitation can then be represented by a small deformation of the Fermisurface, charactherized by a variation of the occupation numbers δN (cid:126)k . The excitationenergy at weak coupling is then given by Landau’s formula (C.15) δ Ω = (cid:90) d (cid:126)k (cid:15) ( k ) δN (cid:126)k + 12 (cid:90) d (cid:126)k d (cid:126)k (cid:48) f ( (cid:126)k, (cid:126)k (cid:48) ) δN (cid:126)k δN (cid:126)k (cid:48) , (D.1)where f ( (cid:126)k, (cid:126)k (cid:48) ) is the Landau interaction function. The variation δN (cid:126)k on the occupationnumbers of the state take the values δN (cid:126)k = 0 , ± (cid:126)k of momentum space.They can be parametrized as δN (cid:126)k = H (cid:0) − (cid:15) ( k ) + δg ( (cid:126)k ) (cid:1) − H ( − (cid:15) ( k )) , (D.2)where δg ( (cid:126)k ) is an auxiliary function that characterizes the deformation of the Fermi surface.By using the expansion of the Heavyside function in terms of Dirac delta function and itsderivatives, we get δN (cid:126)k = δ (cid:0) − (cid:15) ( k ) (cid:1) δg ( (cid:126)k ) + 12 δ (cid:48) (cid:0) − (cid:15) ( k ) (cid:1) δg ( (cid:126)k ) + . . . . (D.3)When replacing (D.3) into (D.1), the Dirac delta functions enforce the integrands to beevaluated at (cid:15) ( k ) = 0, namely at the Fermi surface. Using polar coordinates in momentumspace, this implies that the integrand must be evaluated at k = k F and then the momentumintegrals reduce to angular ones δ Ω = k F v F (cid:90) dθ δg ( θ ) + k F v F (cid:90) dθ dθ (cid:48) f ( θ − θ (cid:48) ) δg ( θ ) δg ( θ (cid:48) ) , (D.4)where by using rotational invariance we wrote f ( θ − θ (cid:48) ) = f ( k F , θ ; k F , θ (cid:48) ) and δg ( θ ) = δg ( k F , θ ). Here the Fermi velocity is defined as v F = d(cid:15) ( k ) /dk | k = k F . This is a shorthand calculation, if the reader is uncomfortable with it, (D.3) can be smoothed by makingthe replacement H ( x ) → / (1 + exp( − b x )) and then taking the b → ∞ limit. – 21 – Fourier expansion now allows us to write f ( θ − θ (cid:48) ) = ∞ (cid:88) n =0 f cn cos (cid:0) n ( θ − θ (cid:48) ) (cid:1) + ∞ (cid:88) n =1 f sn sin (cid:0) n ( θ − θ (cid:48) ) (cid:1) , (D.5)and similarly, to parameterize the deformations by the amplitudes { δg cn ; δg sn } in the de-composition δg ( θ ) = ∞ (cid:88) n =0 δg cn cos ( n θ ) + ∞ (cid:88) n =1 δg sn sin ( n θ ) . (D.6)Replacing (D.5) and (D.6) in (D.4) we obtain: δ Ω = π k F v F (cid:18) π k F v F f c (cid:19) δg c + π k F v F ∞ (cid:88) n =1 (cid:18) π k F v F f cn (cid:19) (cid:0) δg cn + δg sn (cid:1) . (D.7)We see that the excitation energy ends up written as a quadratic form in the deformationamplitudes { δg cn ; δg sn } . In order to have a stable system, δ Ω has to be positive for anypossible excitation, or in other words for any possible amplitudes. This implies that theabove defined quadratic form must be positive definite, leading to the stability conditions:1 + 2 π k F v F f c > , π k F v F f cn > , ∀ n ∈ N . (D.8)If any of these conditions is violated, the system becomes unstable. Notice that the pa-rameters f sn completely disappear from the calculation. The parameters f cn are called the“Landau parameters” of the Fermi liquid.It is important to stress that, when working in perturbation theory, the dispersionrelation is defined by (C.16) as (cid:15) ( k ) = ω ( k ) + (cid:90) d (cid:126)k (cid:48) f ( (cid:126)k, (cid:126)k (cid:48) ) N (cid:126)k (cid:48) . (D.9)This implies that only the zeroth order forms of the Fermi momentum and Fermi velocityhave to be kept in equations (D.8). Indeed, since the second term in those inequalitiescontains a Landau parameter f cn , it is already first order in the coupling constants. Thuswe can replace (D.8) by 1 + 2 π k free F v free F f c > , π k free F v free F f cn > , ∀ n ∈ N (D.10)where k free F is obtained from ω ( k free F ) = 0 and we defined v free F = dω ( k ) /dk | k = k free F .– 22 – .2 Multiple fermionic species with spin The above procedure can be easily generalized to many species of spinful fermions, theresulting quadratic form being in general non-diagonal in the indices denoting spin α andspecies m . As we know, the excitation energy of such fermionic system is given by (C.15) δ Ω = (cid:88) αm (cid:90) d k (cid:15) m ( k ) δN αm ( (cid:126)k ) + 12 (cid:88) αmα (cid:48) m (cid:48) (cid:90) d k d k (cid:48) f αmα (cid:48) m (cid:48) ( (cid:126)k, (cid:126)k (cid:48) ) δN αm ( (cid:126)k ) δN α (cid:48) m (cid:48) ( (cid:126)k (cid:48) ) . (D.11)When the deformations of the occupation numbers get spin and species indices δN αm ( (cid:126)k ),so do the functions δg αm ( (cid:126)k ) parameterizing the deformations δN αm ( (cid:126)k ) = H (cid:0) − (cid:15) m ( k ) + δg αm ( (cid:126)k ) (cid:1) − H ( − (cid:15) m ( k )) . (D.12)Notice that each species and spin component has its own Fermi momentum k mF at whichit dispersion relation vanishes (cid:15) m ( k mF ) = 0. Going through the same steps as before we getfor the energy fluctuation δ Ω = (cid:88) αm k mF v mF (cid:90) dθ δg αm ( θ ) + 12 (cid:88) αm (cid:48) αm (cid:48) k mF v mF k (cid:48) m (cid:48) F v (cid:48) m (cid:48) F (cid:90) dθ dθ (cid:48) f αmα (cid:48) m (cid:48) ( θ − θ (cid:48) ) δg αm ( θ ) δg α (cid:48) m (cid:48) ( θ (cid:48) ) , (D.13)where f αmα (cid:48) m (cid:48) ( θ − θ (cid:48) ) = f αmα (cid:48) m (cid:48) ( k αmF , θ ; k α (cid:48) m (cid:48) F , θ (cid:48) ) and v mF = d(cid:15) m ( k ) /dk | k = k mF .Now we must decompose the interaction function as in (D.5) f αmα (cid:48) m (cid:48) ( θ − θ (cid:48) ) = ∞ (cid:88) n =0 f αmα (cid:48) m (cid:48) cn cos (cid:0) n ( θ − θ (cid:48) ) (cid:1) + ∞ (cid:88) n =1 f αmα (cid:48) m (cid:48) sn sin (cid:0) n ( θ − θ (cid:48) ) (cid:1) . (D.14)Although from (C.22) we can already see that in our case only the modes n = 0 , δg αm ( θ ) = ∞ (cid:88) n =0 δg αmcn cos ( n θ ) + ∞ (cid:88) n =1 δg αmsn sin ( n θ ) , (D.15)in terms of the deformation amplitudes { δg αmcn , δg αmsn } .Going ahead to plug the decompositions (D.14) and (D.15) into (D.13), we get thegeneralization of (D.7) to be δ Ω = (cid:88) αmα (cid:48) m (cid:48) π k mF v mF (cid:32) δ αmα (cid:48) m (cid:48) + 2 π k m (cid:48) F v m (cid:48) F f αmα (cid:48) m (cid:48) c (cid:33) δg αmc δg α (cid:48) m (cid:48) c ++ ∞ (cid:88) n =1 (cid:88) αmα (cid:48) m (cid:48) π k mF v mF (cid:32) δ αmα (cid:48) m (cid:48) + π k m (cid:48) F v m (cid:48) F f αmα (cid:48) m (cid:48) cn (cid:33) ( δg αmcn δg α (cid:48) m (cid:48) cn + δg αmsn δg α (cid:48) m (cid:48) sn ) ++ ∞ (cid:88) n =1 (cid:88) αmα (cid:48) m (cid:48) π k mF v mF k m (cid:48) F v m (cid:48) F f αmα (cid:48) m (cid:48) sn ( δg αmsn δg α (cid:48) m (cid:48) cn − δg αmcn δg α (cid:48) m (cid:48) sn ) . (D.16)On a stable state, this quadratic form has to be positive definite. Neccesary and sufficientconditions for stability can be obtained by considering the following two disantangled cases.– 23 – Putting δg αmc (cid:54) = 0 and δg αmcn = δg αmsn = 0 for n ≥
1, we have that the first line in(D.16) must be positive definite, that applying Sylvester’s criterion is equivalent to, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k mF v mF (cid:32) δ αm ¯ α ¯ m + 2 π k m (cid:48) F v m (cid:48) F f αmα (cid:48) m (cid:48) c (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M × M > , ∀ M ∈ N , (D.17)where | · · · | M × M stands for the M -th minor, that is the determinant of the M × M upper-left submatrix. • Considering now the opposite case: δg αmc = 0 and δg αmcn (cid:54) = 0 , δg αmsn (cid:54) = 0 for n ≥ δ Ω = (cid:80) ∞ n =1 δ Ω n defined bythe second and third lines in (D.16) for each n separetely, since they do not couple.To simplify the analysis, we find convenient to introduce the perturbation vectors inindices ( αm ): (cid:126)u n ≡ ( δg αmcn ) and (cid:126)v n ≡ ( δg αmsn ). Then we can write, δ Ω n = (cid:126)u nt S n (cid:126)u n + (cid:126)v nt S n (cid:126)v n − (cid:126)u nt A n (cid:126)v n + (cid:126)v nt A n (cid:126)u n , (D.18)where the matrices S n = S n t and A n = − A n t are read from (D.16). By furtherdefining the complex vectors (cid:126)z n ≡ (cid:126)u n + i (cid:126)v n , (D.18) takes the simple form δ Ω n = (cid:126)z n † H n (cid:126)z n ; H n = S n + i A n . (D.19)From here it is clear that the positivity condition of δ Ω is just that the hermitianmatrices H n = H n † must be positive, for any n ∈ N . By using the explicit expressionsof S n and A n , this is equivalent to ask that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k mF v mF (cid:32) δ αmα (cid:48) m (cid:48) + π k m (cid:48) F v m (cid:48) F ( f αmα (cid:48) m (cid:48) cn + if αmα (cid:48) m (cid:48) sn ) (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M × M > , ∀ M ∈ N . (D.20)In conclusion, if any of the minors in (D.17) and (D.20) is negative, the quadratic formhas a negative mode that would lead to an excitation with negative energy: δ Ω <
0, thustriggering an instability on the fermionic system.
D.3 Summary and application to the holographic setup
In summary, in order to perform the stability analysis, all we need are • The value of the free Fermi momenta k m free F , obtained from ω m ( k m free F ) = 0. Indeed,notice that, since k mF is always positive, we can remove it from the pre-factor in (D.17)and (D.20). Regarding the second term, since k mF is multiplied by the interactionfunction, we can replace it by k m free F . • The value of the free Fermi velocities v m free F = dω m ( k ) /dk | k = k m free F , as long as weassume that v mF is always positive in other to remove it from the overall pre-factor in(D.17) and (D.20). • The Fourier components of the Landau parameters, evaluated at the free Fermi mo-menta k m free F . Notice that equation (C.22) implies that only the modes with n = 0 , ω m ( k m free F ) = 0, we need only static wave functions in (C.17).– 24 – The electron star background
In this appendix, we summarize the construction of the electron star solution. We do so inorder to fix notation and to have a fully self-contained discussion. However, all the resultspresented in this section were obtained in the original reference [14] to which the interestreader is referred.We work with the previously presented Ansatz (A.1) ds = L (cid:18) − f ( z ) dt + g ( z ) dz + d(cid:126)x z (cid:19) ,A = h ( z ) dt . (E.1)The field equations of (3 + 1)-dimensional Einstein-Maxwell theory in the presence of anegative cosmological constant − /L plus matter in the form of a charged perfect fluidare R MN − g MN R − L g MN = κ (cid:16) T Maxwell MN + T Fluid MN (cid:17) , ∇ N F MN = e J M , (E.2)with κ and e the gravitational and electromagnetic couplings respectively. Here the con-tributions to the energy-momentum tensor and current density read T Maxwell MN = 1 e (cid:18) F MP F PN − g MN F P Q F P Q (cid:19) ,T Fluid MN = ( ρ + p ) u M u N + p g MN ,J M = σ u M , (E.3)The functions p , ρ and σ are the pressure, energy density and charge density of the fluid,respectively, and u M its four-velocity. Furthermore, the conservation equations must hold ∇ N T MN = 0 ; ∇ M J M = 0 . (E.4)Since we aim to describe a relativistic system of fermions of mass m , we can try a descriptionof the energy and charge densities in terms of the density of states in flat space of a freefermion gas g ( E ) = π − E ( E − m ) / . This description would be accurate as long as thewavelength of the fermions is much shorter than the local curvature radius. Furthermore,we must consider the equation of state of the system in terms of a grand canonical potentialΩ = − p V . At zero temperature the fluid functions read σ = (cid:90) ε F m dE g ( E ) = 1 π (cid:90) (cid:15) F m dE E ( E − m ) / ,ρ = (cid:90) ε F m dE g ( E ) E = 1 π (cid:90) (cid:15) F m dE E ( E − m ) / ,p = − ρ + ε F σ . (E.5)where ε F is the bulk Fermi energy. Consistency with Ansatz (E.1) imposes that all thefunctions be dependent only on z . Moreover, the velocity of the fluid must be unitary,– 25 –hich identifies it with the vierbein temporal vector, i.e. u M ∂ M ≡ e t = 1 / ( L √ f ),see (A.2). This last fact implies that the bulk Fermi energy coincides with the local ( i.e. measured by comoving observers) chemical potential ε F ≡ h/L √ f .It is convenient to work with the dimensionless quantitiesˆ p = κ L p , ˆ ρ = κ L ρ , ˆ σ = e κ L σ , ˆ h = h/γ . (E.6)Here we introduced γ ≡ e L/κ . In terms of these scaled variables (E.5) can be integratedto obtain the explicit expressionsˆ σ = ˆ β (cid:32) ˆ h f − ˆ m (cid:33) , ˆ ρ = ˆ β (cid:115) ˆ h f − ˆ m (cid:32) − ˆ m ˆ h √ f + 2 ˆ h f (cid:33) − ˆ m ln ˆ h √ f + (cid:115) ˆ h f − ˆ m + ˆ m ln ˆ m , ˆ p = − ˆ ρ + ˆ h √ f ˆ σ , (E.7)where we introduced the two independent parameters ˆ β and ˆ m that can be used to definethe electron star ˆ β ≡ e γπ , ˆ m ≡ m Lγ . (E.8)By plugging (E.1) and (E.7) in (E.2) we get the field equations z f (cid:48) f + z g (cid:48) g + ˆ σ z g ˆ h √ f + 4 = 0 ,z f (cid:48) f − z ˆ h (cid:48) f + (3 + ˆ p ) z g − ,z ˆ h (cid:48)(cid:48) h + ˆ σ z g ˆ h √ f (cid:32) z ˆ h (cid:48) h − f ˆ h (cid:33) = 0 . (E.9)These equations have to be solved numerically for each value of ˆ m and ˆ β to obtain themetric and gauge functions ( f, g, ˆ h ).Regarding the boundary conditions in the IR, we observe from (E.7) that the electronstar extends from infinity to the radius z s such thatˆ p ( z s ) = ˆ ρ ( z s ) = ˆ σ ( z s ) = 0 , ⇐⇒ T Fluid MN ( z s ) = J M ( z s ) = 0 , (E.10)whose position is determined by the equationˆ h ( z s ) − ˆ m f ( z s ) = 0 . (E.11)Outside the electron star, i.e. < z < z s , we have that σ = p = ρ = 0, implying that theequations of motion (E.9) reduce to f (cid:48) f + g (cid:48) g + 4 z = 0 ; 1 z f (cid:48) f − ˆ h (cid:48) f + 3 g − z = 0 ; ˆ h (cid:48)(cid:48) = 0 , (E.12)– 26 –hose general solution is the AdS Reissner-N¨ordstrom black hole f ( z ) = c z − ˆ M z + ˆ Q z , g ( z ) = c z f ( z ) , ˆ h ( z ) = ˆ µ − ˆ Q z . (E.13)To specify this solution we have to give the four integration constants. ( c, ˆ M , ˆ µ, ˆ Q ). In theelectron star context they must be obtained by matching ( f, g, ˆ h, ˆ h (cid:48) ) at the radius z = z s with the solution inside the star.On the other hand, in the IR limit z → ∞ the solution acquires the form of a Lifshitzmetric. In fact, an Ansatz for large z of the form: f = 1 z λ (1 + f z α + . . . ) , g = g ∞ z (1 + g z α + . . . ) , ˆ h = h ∞ z λ (1 + h z α + . . . ) , (E.14)solves (E.9) if α takes some of the following three values α = 2 + λ , α ± = 1 + 2 λ ± (cid:115) λ − λ + 40 λ − − ˆ m λ (4 − λ ) (1 − ˆ m ) λ − . (E.15)The parameter λ is called the dynamical critical exponent. One can show that an IRnon-singular, asymptotically exact Lifshitz solution exists only if1 ≤ − ˆ m ≤ λ , (E.16)and the root selected is the negative one: α = α − , see [14] for details. From the leadingorder of (E.9) we get, g ∞ = 36 λ ( λ − β ( λ (1 − ˆ m ) − ,h ∞ = 1 − λ ,g ∞ (3 + ˆ p ( ∞ )) = 1 + 2 λ + h ∞ λ . (E.17)The first two equations determine the coefficients that define the Lifshitz metric in termsof λ and ˆ m , while that the last equation relates implicitly ˆ β in terms of λ and ˆ m also. Soit is possible, and convenient to make numerics, to think the Lifshitz parameter and themass as the free parameters of the electron star. The expansions (E.14) result completelydetermined by ( λ, ˆ m ), and by f that remains free. However, the scalings ( t, x, y, z ) → ( b t, a x, a y, a z ) leave the Ansatz (E.1) invariant and induce the invariance of the system(E.9) under z → a z , g → a − g ,f → b − f , ˆ h → b − ˆ h . (E.18)These scalings allow to fix the leading order term of f in (E.14) together with the absolutevalue of f , while the sign to get the solution with the right behaviour in the UV results tobe the negative one. In the paper we work out the electron star solution by fixing f = − z/z s (cid:28)
1) and IR ( z/z s (cid:29)
1) respectively. / z s f ( z ) / z s g ( z ) / z h ( z ) s Figure 2 . We plot the electron star background functions f , g and h vs. z/z s (light blue line),compared to the Lifshitz solution (orange) and AdS Reissner-N¨ordstrom black hole (lihgt green)respectively. The parameters are chosen as ˆ m = 0 . λ = 2. An important point to be stressed is that, in the weak gravity regime both κ (cid:28) L and κ (cid:28) /m conditions should hold, or what is the same, e (cid:28) π ˆ β and e (cid:28) / ˆ m . On theother hand, in [14] was shown that the flat treatment (E.7) is consistent only if we restrictthe electron star parameters to the region ˆ β ∼ , ˆ m ∼
1. Together with the weak gravityconditions, they imply that γ ∼ e ∼ m L (cid:29) . (E.19)We will see in appendix F that these conditions are closely linked to the validity of theWKB approximation for the fermionic perturbations.– 28 – The WKB solution
In the electron star background, we are assuming that we have a large number of particlesinside one AdS radius. This is equivalent to the statement that the particle wavelengthis much shorter than the characteristic length of the background in which it is moving.This implies that we can solve the effective Schr¨odinger equation (A.21) (or equivalently(A.27)) in the Wentzel-Kramers-Brillouin (WKB) approximation, as we do in the rest ofthis section.
F.1 Basics of the WKB approximation
Let us remember basic facts about the WKB approximation (see for example [27]). Let usconsider the one-dimensional Schr¨odinger equation written in the form − φ (cid:48)(cid:48) ( z ) + U ( z ) φ ( z ) = 0 . (F.1)We will assume that the potential is repulsive and positive at the boundary, implying thatat z close enough to z = 0 we have U ( z ) > U (cid:48) ( z ) <
0. As we move into positivevalues of z we have a first turning point z at which U ( z ) = 0 and U (cid:48) ( z ) <
0. At largervalues of z additional turning points z r appear at which U ( z r ) = 0.Away from any given turning point, we introduce the functions u ± ( z ; z r ) = | U ( z ) | − exp (cid:18) ± (cid:90) zz r dz (cid:112) U ( z ) (cid:19) . (F.2)As can be checked by direct substitution, they are good approximate solutions of (F.1) aslong as U ( z ) is large enough, which means in particular that we can use them away fromthe turning points. On the other hand, close to any turning point we can linearize thepotential, and the solution is then written in terms of Airy functions A i and B i . Then theWKB approximate solution around z r takes the form φ WKB ( z ) = L ( r )+ u + ( z ; z r ) + L ( r ) − u − ( z ; z r ) , z r − z (cid:29) | U (cid:48) ( z r ) | − A ( r )+ A i ( w ) + A ( r ) − B i ( w ) (cid:12)(cid:12) w = U (cid:48) ( z r ) − ( z − z r ) , | z − z r | (cid:28) (cid:12)(cid:12)(cid:12) U (cid:48) ( z r ) U (cid:48)(cid:48) ( z r ) (cid:12)(cid:12)(cid:12) R ( r )+ u + ( z ; z r ) + R ( r ) − u − ( z ; z r ) , z − z r (cid:29) | U (cid:48) ( z r ) | − (F.3)where L ( r ) ± , R ( r ) ± and A ( r ) ± are numerical constants to be determined in order to ensure theboundary conditions and the continuity around each turning point.The coefficients of the functions u ± ( x, x r ) at left and right of a given turning point arelinearly related, as in (cid:32) R ( r )+ R ( r ) − (cid:33) = M ( z r ) (cid:32) L ( r )+ L ( r ) − (cid:33) , (F.4)where the explicit form of the matrix M ( z r ) is obtained by matching the functions u ± ( z ; z r )across the turning point using the intermediate Airy form of the solution. It has the– 29 –xpression M ( z r ) = M when U (cid:48) ( z r ) >
0, and M ( z r ) = M † when U (cid:48) ( z r ) <
0, with M ≡ e + i π (cid:32) − i − i (cid:33) . (F.5)The relation between the approximate WKB solutions around any pair of successiveturning points can be found by shifting the limits of integration in (F.2) from z r to r r +1 u ± ( z ; z r ) = ϕ ± r u ± ( z ; z r +1 ) , (F.6)in terms of the connection coefficients ϕ ± r , which read ϕ ± r = exp (cid:18) ± (cid:90) z r +1 z r dz (cid:112) U ( z ) (cid:19) . (F.7)Compatibility with the form (F.3) for the solution around each turning point thus impliesthe additional linear relation (cid:32) L ( r +1)+ L ( r +1) − (cid:33) = W ( z r ) (cid:32) R ( r )+ R ( r ) − (cid:33) , (F.8)in terms of the connection matrix W ( z r ) = (cid:32) ϕ + r ϕ − r (cid:33) . (F.9)With all the above, we can express the whole set of coefficients of the solution aroundeach of the turning points, in terms of L (0)+ and L (0) − , as follows (cid:32) L ( r )+ L ( r ) − (cid:33) = W ( z r − ) M ( z r − ) . . . W ( z ) M ( z ) (cid:32) L (0)+ L (0) − (cid:33) , (cid:32) R ( r − R ( r − − (cid:33) = M ( z r − ) W ( z r − ) M ( z r − ) . . . W ( z ) M ( z ) (cid:32) L (0)+ L (0) − (cid:33) . (F.10)From this, we can write the relation between the initial (leftmost) and final (rightmost)coefficients for a system with a total of R turning points. It takes the form (cid:32) R ( R − R ( R − − (cid:33) = V (cid:32) L (0)+ L (0) − (cid:33) , (F.11)where V = (cid:32) V V V V (cid:33) = M ( z R − ) W ( z R − ) M ( z R − ) . . . W ( z ) M ( z ) . (F.12)In the present context, Bohr-Sommerfeld quantization relations arise when we imposeboundary conditions at both extremes of the z axis. This leads to constraints on the matrixelements of V and consequently on the parameters of the theory.– 30 – .2 Application to the electron star background After the re-scalings ω = γ ˆ ω, k = γ ˆ k , together with (E.6) and (E.8), the potential (A.22)entering into the effective Schr¨odinger equation acquires the following form U ( z ) = γ g ( z ) (cid:32) ˆ k z − (ˆ ω + q ˆ h ( z )) f ( z ) + ˆ m (cid:33) + γ ˆ m (cid:112) g ( z ) (cid:16) ln (cid:16) f + k ( z ) (cid:112) g ( z ) (cid:17)(cid:17) (cid:48) + (cid:18)(cid:113) f + k ( z ) (cid:19) (cid:48)(cid:48) (cid:113) f + k ( z ) . (F.13) As we will be working in an electron star background, we have to consider the large γ limit.From (F.13) we then see that this regime coincides with the semiclassical limit where theWKB approximation is reliable. In such limit we can neglect the last two terms in (F.13)and consider U ( z ) ≈ γ g ( z ) (cid:32) ˆ k z − (ˆ ω + q ˆ h ( z )) f ( z ) + ˆ m (cid:33) . (F.14)Close to the AdS boundary z = 0, we can replace the functions f, g and ˆ h by theircorresponding UV expansions (E.13), obtaining U ( z ) ∼ γ ˆ m /z . This implies that thepotential diverges at the boundary. Since the resulting u − ( z, z ) function in (F.2) divergesas we move into the boundary z = 0, in order to have a normalizable solution, we need toimpose L (0) − = 0.In the deep IR, the functions take their Lifshitz form (E.14), and we get U ( z ) ∼ γ g ∞ (ˆ k − ˆ ω z λ − ). In consequence, for zero frequency the potential goes to a positiveconstant at infinity. The corresponding function u + ( z, z R − ) diverging, we need to impose R ( R − = 0. Combined with the conditions at the boundary we obtain R ( R − = V L (0)+ =0 or in other words V = 0 , (F.15)where the matrix element V was defined in (F.12) in terms of integrals involving thepotential U ( z ). For finite frequency on the other hand, the potential diverges into negativevalues. This results in an u − ( z, z R − ) with the form of a wave moving into smaller z values, i.e. entering the bulk from infinity, which implies that we must impose R ( R − − = 0 in orderto avoid causality issues. Since R ( R − − = V L (0)+ = 0 this implies V = 0 . (F.16)Equations (F.15) and (F.16) impose a constraint between the parameters entering into U ( z ). It often has a discrete set of solutions, being then understood as a quantizationcondition (see below).According to the summary on section D.3, using this information we need to obtainthe free Fermi momentum k m free F and Fermi velocity v m free F = dω m ( k ) /dk | k = k m free F for eachmode m in the z direction, as well as its static wave function ψ δαmk m free F ( z ) or in other wordsthe solution φ (2) kω of the effective Schr¨odinger equation (A.21) with k = k F and ω = 0.– 31 – .2.1 Fermi momenta and static wave functions Of particular interest to our problem are the Fermi momenta, i.e. the values k mF free ofthe momentum k satisfying ˆ ω m ( k mF free ) = 0. This implies that the corresponding potentialin (F.14) has a vanishing frequency, and then goes to a positive constant at infinity. Inconsequence it has none or an even number of turning points.The two last terms in the parenthesis in (F.14) are positive outside the star, andnegative inside it, according to the rule (E.11). This implies that for k large enough, theparenthesis remains always positive and there are no turning points. On the other hand,for k smaller than some critical value k ∗ , the potential becomes negative somewhere insidethe star, giving rise to a pair of turning points z , z . Only in this last case we are able tofind values of the free Fermi momenta, all of them inside a “Fermi ball” [10] of radius k ∗ .The connection matrix (F.12) has only three factors V = M W ( z ) M † and therelevant matrix element in equation (F.15) takes the form V = 2 cos (cid:18)(cid:90) z z dz (cid:112) − U ( z ) (cid:19) = 0 , (F.17)implying that the Bohr-Sommerfeld condition (F.15) reads γ (cid:90) z z dz (cid:118)(cid:117)(cid:117)(cid:116) g ( z ) (cid:32) q ˆ h ( z ) f ( z ) − ( k mF free ) z − ˆ m (cid:33) = (cid:18) m + 12 (cid:19) π , m = 0 , , . . . . (F.18)In terms of an integer m , this equation determines the free Fermi momentum k mF free of the m -th fermionic mode. The results of these calculations are shown in Fig. 3 k F Figure 3 . The Fermi momenta are plotted in terms of m for different values of γ = 100 , , ,
250 from bottom to top. For each value of γ there exists a maximum value of the Fermi momentum. Next, we can replace the obtained values of the free Fermi momenta k mF free into equa-tions (F.2)-(F.9) to obtain the corresponding solutions of (A.21) φ (2) kω = φ WKB m ( z ) neededto build the static wave-functions. In our calculations we found useful to replace the Airyfunctions interpolating around the turning points in (F.3) by a quartic polynomial. Resultsare shown in Fig. 4. – 32 – Φ W K B ( z ) - - - ϕ W K B ( z ) - ϕ W K B ( z ) - - - - ϕ W K B ( z ) - ϕ W K B ( z ) - - - ϕ W K B ( z ) - ϕ W K B ( z ) - - - - ϕ W K B ( z ) Figure 4 . Profiles of the function φ WKB m ( z ) are shown for the values m ∈ { , . . . , } respectively.We see the smoothly interpolated solutions (light blue curves), and purely the WKB solutions(orange curves) that in the vicinity of the turning points were replaced by a quartic polynomial. – 33 – .2.2 Fermi velocities To calculate the Fermi velocities v m free F = dω m ( k ) /dk | k = k m free F , we need to concentrate inlow energy excitations, with momenta very near to the Fermi momentum k = k free F + dk .This would result into small frequencies ˆ ω = 0 + d ˆ ω . As discussed previously, for anynon-zero frequency the potential has an odd number of turning points. As our frequencyis very small, along with the two turning points of the zero frequency case z , z , only athird one z appears. As d ˆ ω goes to zero, the third turning point z moves to infinity.In this case the resulting connection matrix V in (F.12) has the five factors V = M † W ( z ) M W ( z ) M † , and (F.16) readstan (cid:18) (cid:90) z z dz (cid:112) − U ( z ) − π (cid:19) = − i e − (cid:82) z ω ) z dz √ U ( z ) . (F.19)The solution for the dispersion relation is necessarily complex, defining a problem of quasi-normal modes. Since the integral in the exponent of the right hand side of (F.19) divergesas d ˆ ω goes to zero, the right hand side is very small. This implies that we can write (F.19)as (cid:90) z z dz (cid:112) − U ( z ) = (cid:18) m + 12 (cid:19) π . (F.20)To the first order in d ˆ k and d ˆ ω , we getˆ k free F d ˆ k (cid:90) z z dz g ( z ) z (cid:112) − U ( z ) − d ˆ ω (cid:90) z z dz qh ( z ) g ( z ) f ( z ) (cid:112) − U ( z ) = 0 , (F.21)implying for the Fermi velocity v free F = ˆ k free F (cid:82) z z dz z g ( z ) √ − U ( z ) (cid:82) z z dz g ( z ) h ( z ) f ( z ) √ − U ( z ) , (F.22)where the potential and the turning points are evaluated at zero frequency and ˆ k = ˆ k F .In is worth to mention that a possible obstacle to get straight the dispersion relationat leading order from (F.20) arises from the non-analyticity of the complex right hand sideof (F.19). However, as shown in [15], it only affects (in fact, determines!) the imaginarypart of the dispersion relation, while the real part is given as usual by the analytical lefthand side. – 34 – eferences [1] S. Sachdev, Quantum Phase Transitions , Cambridge University Press (2000),10.1017/CBO9780511622540.[2] G.R. Stewart,
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