A Non-Fermi Liquid from a Charged Black Hole; A Critical Fermi Ball
aa r X i v : . [ h e p - t h ] M a r A Non-Fermi Liquid from a Charged Black Hole; A Critical Fermi Ball
Sung-Sik Lee
Department of Physics & Astronomy, McMaster University,Hamilton, Ontario L8S 4M1, Canada (Dated: October 25, 2018)
Abstract
Using the AdS/CFT correspondence, we calculate a fermionic spectral function in a 2+1 dimensionalnon-relativistic quantum field theory which is dual to a gravitational theory in the
AdS background witha charged black hole. The spectral function shows no quasiparticle peak but the Fermi surface is still welldefined. Interestingly, all momentum points inside the Fermi surface are critical and the gapless modes aredefined in a critical Fermi ball in the momentum space. S = 1 κ Z d x √− g (cid:20) R − F µν F µν + 32 (cid:21) + Z d x √− g (cid:20) ¯ ψγ µ (cid:18) ∂ µ + 12 ω bcµ Σ bc − iA µ (cid:19) ψ − m ¯ ψψ (cid:21) + Z d x √− g ǫ ¯ ψψ. (1)Here x µ = ( t, x, y, z ) is the space-time coordinate with signature ( − , , , . R is the scalar2urvature and F µν = ∂ µ A ν − ∂ ν A µ is the field strength tensor of a U(1) gauge field A µ . ψ is afour-component Dirac spinor, ω bcµ is the spin connection and Σ bc = [Γ b , Γ c ] is the generator of thelocal Lorentz transformation with Γ a , the gamma matrices. γ µ = e µa Γ a where e µa is the tetrad. Theaction describe the U(1) gauge field and the Dirac spinor coupled with gravity in the backgroundwith a negative cosmological constant which is set to be − in our unit. The last term in the actionis a boundary term defined at z = ǫ where g ǫ is the determinant of the induced metric on the2+1D space. Although the boundary term does not affect the equation of motion in the bulk, it isimportant for obtaining a non-trivial dependence of the saddle point action on the boundary valueof the spinor field[18].The above action has an AdS black hole solution given by ds = 1 z (cid:20) α ( − f ( z ) dt + dx + dy ) + dz f ( z ) (cid:21) ,A = qα ( z − , (2)where f ( z ) = p q z − (1 + q ) z . This is a special case of a more general dyonic black holesolution considered in Ref. [9]. In this coordinate system, the horizon of the black hole is at z = 1 .The metric describes the AdS space near the boundary at z = 0 . The Hawking temperature of theblack hole is T H = α π (3 − q ) [9]. For nonzero q and α , the black hole carries a nonzero charge.Non-vanishing components of the spin connection are ω ˆ t ˆ z ˆ t = z (cid:0) fz (cid:1) ′ and ω ˆ x ˆ z ˆ x = ω ˆ y ˆ z ˆ y = − f , where (ˆ t, ˆ x, ˆ y, ˆ z ) represents the local Lorentz coordinate.According to the AdS/CFT correspondence, we can view this classical gravitational theory asa strongly coupled 2+1D quantum field theory in a large N limit. The Hawking temperature cor-responds to the temperature of the boundary field theory. This theory can be motivated from theM-theory defined in AdS × S which describes the low energy physics of the 2+1D supersym-metric Yang-Mills theory with supercharges. If the infinite tower of the Kaluza-Klein modesare truncated self consistently in the M-theory, the resulting theory would contain the above theorywith particular values of the fermionic mass. Here, we will not restrict ourselves to the M-theoryand we will regard the fermionic mass as a free parameter which characterizes the corresponding2+1D quantum field theory. In particular, we will focus on the case with m = 0 where the chiralsymmetry simplifies the calculation significantly. However, the qualitative features may be similarfor other values of the mass.Recently, it has been proposed that a 2+1 dimensional U ( N ) × U ( N ) Chern-Simons-mattertheory at level k is dual to the type IIA string theory on AdS × CP in a ’t Hooft limit with a3xed λ = N/k [10]. The present gravitational theory may be related to the Chern-Simons-mattertheory in the strong coupling limit ( λ >> ) at a finite chemical potential. In this paper, instead ofattempting to establish a precise connection with a microscopic theory, we take the gravitationaltheory in Eq. (1) as our starting point and examine the dynamics of the fermion, in a hope that thegravity description may capture some universal features of strongly interacting fermions at finitedensity.The Dirac spinor is a source field which is linearly coupled with a fermionic field in the bound-ary theory. The gauge field is coupled with a conserved U(1) current which includes the current ofthe fermion. The electrostatic potential induces a nonzero density of the boundary fermions. Thechemical potential of the boundary theory is given by µ = A ( z = 0) = − qα . This is crucial inobtaining a system of fermions with a finite density. It is noted that the fermionic field that cou-ples with the Dirac spinor can be a composite field in the ultra-violet theory. In the following, wewill calculate the ‘single particle’ spectral function of the fermion which is possibly a compositeparticle.For m = 0 , the chiral symmetry enables us to focus on one chiral mode. Due to the 2+1dimensional translational symmetry, we can assume a plane wave solution for the left chiral modes ψ − and ¯ ψ − which satisfy Γ ψ − = − ψ − and ¯ ψ − Γ = − ¯ ψ − , ψ ( t, x, y, z ) = e − i ( ωt − k · r ) ψ − ( z ) , ¯ ψ ( t, x, y, z ) = e i ( ωt − k · r ) ¯ ψ − ( z ) , (3)where r = ( x, y ) and k = ( k x , k y ) . In the chiral representation with Γ ˆ0 = − II , Γ ˆ i = σ i σ i , where σ i are the Pauli matrices with i = x, y, z and I is the × identity matrix, theequation of motion for the two-component chiral spinors becomes (cid:20) iωzαf + iqz ( z − f + i k · σα z + (cid:18) zf ′ − f zf ∂ z (cid:19) σ z (cid:21) ψ − ( z ) = 0 , ¯ ψ − ( z ) (cid:20) iωzαf + iqz ( z − f − i k · σα z + (cid:18) zf ′ − f ←− ∂ z zf (cid:19) σ z (cid:21) = 0 . (4)At T = 0 ( q = 3 ), the solution near the horizon ( z → ) reads ψ − ( z ) ∼ e − iω α (1 − z ) σ z C , ¯ ψ − ( z ) ∼ ¯ Ce − iω α (1 − z ) σ z in the leading order of (1 − z ) for some fixed spinors C , ¯ C . To maintain4he causality in the boundary theory, we impose the ingoing boundary condition, ψ − ( z ) ∼ e iω α (1 − z ) , ¯ ψ T − ( z ) ∼ e − iω α (1 − z ) (5)as z → . At a finite temperature with q < , the boundary condition is modified to be ψ − ∼ e − iω ln(1 − z ) α (3 − q and ψ T − ∼ e iω ln(1 − z ) α (3 − q . Therefore there is one-parameter familyof solutions that satisfy the ingoing boundary condition near the horizon. Near the boundary ofthe AdS space ( z → ), the spinors behave as ψ − ( z ) ∼ z / χ and ¯ ψ − ( z ) ∼ z / ¯ χ , where χ and ¯ χ should be chosen so that the solution satisfies the ingoing boundary condition near the horizon.We represent the solution near the boundary as ψ − ( z ) ∼ (cid:16) zα (cid:17) / P ( ω, k )1 η, ¯ ψ T − ( z ) ∼ (cid:16) zα (cid:17) / Q ( ω, k )1 η ∗ , (6)where η and η ∗ are Grassmann numbers (not spinors) that we use to impose boundary data. Oncethe second components of the spinors are chosen to be η and η ∗ , the complex functions P ( ω, k ) and Q ( ω, k ) are uniquely determined from the boundary condition near the horizon. It is notedthat we could have chosen the boundary condition near z = 0 in different ways. This is becausethe two components of the spinors decay in the same power[19]. Different choices of boundarycondition may correspond to different field theories on the boundary[20]. However, we note thatEq. (6) is a natural choice for the following reasons. First, the vector (0 , along which we imposeboundary data is an eigenvector of σ z , the generator of the rotation in the x − y plane. Therefore,this prescription is independent of momentum direction, which guarantees that the propagator thatwe will calculate below is invariant under the rotation. Second, once the second component of ψ − is chosen as boundary data, it is natural to choose the second component of ¯ ψ − as boundary data.This can be seen by turning on a small fermion mass which mixes ψ − and ψ + , and identifying ¯ ψ − ∼ ψ † + . Although not shown here, if we choose the (1 , component as our boundary data,which is another possible choice consistent with the above conditions, we obtain the same spectralfunction.From the AdS/CFT dictionary, the Green’s function of the fermion in the boundary theoryis given by G ( ω, k ) = i ∂ S [ η ∗ ,η ] ∂η∂η ∗ , where S [ η ∗ , η ] is the gravity action evaluated for the saddle5onfiguration of the spinor fields which satisfy the boundary conditions, Eqs. (5) and (6). Thebulk spinor action in Eq. (1) vanishes at saddle points. Only the boundary term contribute to theaction and we obtain the Green’s function G ( ω, k ) = i ( P ( ω, k ) Q ( ω, k ) + 1) . (7)The quantity of physical importance is the spectral function, A ( ω, k ) = lim δ → + ImG ( ω + iδ, k ) which measures how much spectral weight a fermion with momentum k has at energy ω . - - -
20 0 20 4001020304050 Ω k FIG. 1: The contour plot of the zero temperature spectral function as a function of energy and momentumfor q = −√ and α = 10 . The darkest region represents the area with no spectral weight and the brightestregion, the highest value of the spectral function. We numerically integrate the equation of motion (4) to obtain the spectral function as a functionof ω and k = p k x + k y . Due to the rotational symmetry, A ( ω, k ) does not depend on momentumdirection. In Fig. 1, we show the spectral function at zero temperature. For a large momentum,the spectral function as a function of energy shows a broad peak which is centered at a negativeenergy ω . The broad peak does not disperse significantly as momentum changes. However, thewidth of the broad peak becomes larger as momentum increases and the edges of the broad peakdisperse as ω edge ≈ ± k + ω . There is no quasiparticle peak, which implies that the fermions arein a non-Fermi liquid state. 6 - Ω H Ω ,k = L (a) - - Ω H Ω ,k = L (b) - - Ω H Ω ,k = L (c) - - Ω H Ω ,k = L (d) - - Ω H Ω ,k = L (e) - - Ω H Ω ,k = L (f) FIG. 2: The energy distribution curves of the spectral function for q = −√ and α = 10 at momenta (a) k = 3 , (b) k = 5 , (c) k = 7 , (d) k = 9 , (e) k = 11 and (f) k = 13 . Although there is no delta function peak, the spectral function shows sharp peaks at zero energyfor momenta smaller than a critical momentum k c . To closely examine the low energy structure,we display the spectral function as a function of energy for fixed values of momentum in Fig.2. Within the numerical accuracy, the sharp peak has an algebraic singularity at ω = 0 . Weemphasize that this is not a quasiparticle peak. As is shown in Fig. 2, the zero energy peakis more pronounced at a smaller momentum and the size of the peak decreases as momentumincreases. The critical momentum k c above which the zero energy peak disappears coincides with7he momentum at which the edge of the broad peak cross the Fermi energy ω = 0 . Therefore,we interpret k c as Fermi momentum. The most striking feature is that the algebraic singularitiesat zero energy exist for all momenta below the Fermi momentum. Namely, all momentum pointsinside a two dimensional disk with | k | < k c has the singular peak at zero energy. We call the set ofthese momentum points a critical Fermi ball . Although not shown here, the Fermi momentum andthe absolute value of the energy of the broad peak increases as α increases. If we switch the signof q , the broad peak is centered at a positive energy. However, the position of the critical Fermiball does not change. - Ω- H i Ω ,k = L FIG. 3: The spectral function at k = 3 as a function of imaginary frequency with the same parameters usedin Fig. 2. To show that there is true singularity at ω = 0 , the spectral function at a momentum below k c is shown as a function of imaginary frequency in Fig. 3. Indeed, the spectral function has a strongsingularity near ω = 0 along the imaginary axis. A careful reader may note that the height of thepeak is still finite and the position of the peak is slightly away from ω = 0 . These are artifactsoriginated from the fact that Eq. (4) has been numerically integrated over a range [0 , − ǫ ] with asmall but nonzero ǫ to avoid the divergence in the equations at z = 1 . As a smaller ǫ is used, theposition of the peak moves to ω = 0 and the height of peak increases systematically. This impliesthat the singularity at ω = 0 is genuine.Unlike the Fermi liquid state or a non-Fermi liquid state with a critical Fermi surface[21] wherelow energy excitations exist only near a Fermi surface, in the present non-Fermi liquid state allmomentum points below the Fermi surface are important at low energies. Therefore we expectthat the low energy properties of this state to be drastically different from a Fermi liquid state or anon-Fermi liquid state with a critical Fermi surface. For example, low temperature thermodynamicproperties of this 2+1D non-Fermi liquid state with a critical Fermi ball will behave like a 3+1D8ritical Fermi surface. In a sense, this ‘dimensional lift’ is not surprising because the 2+1D non-Fermi liquid theory is described by the 3+1D gravity.What would be the origin of the non-Fermi liquid behavior? In the most trivial scenario, thenon-Fermi liquid behavior can be caused by a composite nature of the fermion field. If the fermionis a composite of weakly interacting fields, it will decay into multiple modes and the spectralfunction will show a broad feature without a quasiparticle peak. However, the sharp zero energypeak in the spectral function suggests that the fermion field is not a mere composite of weaklyinteracting fields. The fermion is a rather well defined excitation at low energies irrespective ofwhether it is a fundamental or composite particle. Then the non-Fermi liquid behavior can be dueto strong interactions between the fermions.Although the occurrence of the critical Fermi ball is somewhat counter-intuitive, one may un-derstand it as a consequence of strong interactions. Since the gravitational description is valid inthe strong coupling limit, where the interaction energy scale is presumably larger than the Fermienergy, even those fermions which are deep inside the Fermi surface can participate in the lowenergy physics, overcoming the kinetic energy penalty. - Ω H Ω ,k = L FIG. 4: Temperature dependence of the spectral function at k = 6 . With a fixed α = 10 , q is changed totune temperature to T = 0 (solid line), T = π (dashed line) and T = π (dotted line). Until now, we have examined the zero temperature spectral function. In Fig. 4, we comparethe spectral function at zero temperature and finite temperatures . As expected, the singular zeroenergy peak is rounded at finite temperature due to thermal fluctuations. If temperature is highenough, the sharp peak completely disappears. At finite temperatures, the Fermi momentum is notsharply defined, but the position of the broad peak is not sensitive to temperature.In summary, we solved the Dirac equation in a charged black hole background to extract a9ermionic spectral function of a 2+1 dimensional strongly coupled field theory at finite chemicalpotentials. The spectral function revealed a critical Fermi ball in the momentum space where allmomentum points inside the Fermi ball are critical.In the future, it would be interesting to study physical properties of the critical Fermi ball inmore details, such as possible instabilities and thermodynamic/transport properties. 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