Fermionic spectral functions in backreacting p-wave superconductors at finite temperature

Journal of High Energy Physics, Apr 2017

We investigate the spectral function of fermions in a p-wave superconducting state, at finite both temperature and gravitational coupling, using the AdS/CF T correspondence and extending previous research. We found that, for any coupling below a critical value, the system behaves as its zero temperature limit. By increasing the coupling, the “peak-dip-hump” structure that characterizes the spectral function at fixed momenta disappears. In the region where the normal/superconductor phase transition is first order, the presence of a non-zero order parameter is reflected in the absence of rotational symmetry in the fermionic spectral function at the critical temperature.

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Fermionic spectral functions in backreacting p-wave superconductors at finite temperature

Received: October Fermionic spectral functions in backreacting p-wave G.L. Giordano N.E. Grandi A.R. Lugo Open Access c The Authors. We investigate the spectral function of fermions in a p-wave superconducting nite both temperature and gravitational coupling, using the AdS=CF T correspondence and extending previous research. We found that, for any coupling below a critical value, the system behaves as its zero temperature limit. By increasing the coupling, the \peak-dip-hump" structure that characterizes the spectral function at xed momenta disappears. In the region where the normal/superconductor phase transition is rst order, the presence of a non-zero order parameter is re ected in the absence of rotational symmetry in the fermionic spectral function at the critical temperature. superconductors; at; AdS-CFT Correspondence; Gauge-gravity correspondence; Holography and Contents 1 Introduction 2 Bosonic sector: the holographic p-wave superconductor 2.1 The bulk gravitational solution Vanishing Beckenstein-Hawking entropy Non-vanishing Beckenstein-Hawking entropy The boundary QFT interpretation Dirac spinors in the gravitational background Ingoing boundary conditions for vanishing Beckenstein-Hawking en Ingoing boundary conditions for non-vanishing Beckenstein-Hawking The ultraviolet behavior Normal modes and the Dirac cones Fermionic operators in the QFT Numerical results A About the positivity of the spectral function B On the spectral function in conformally at backgrounds Introduction eld theory the \large N limit". systems. Roughly, it describes a d-dimensional quantum eld theory in the strong coupling In this setup, to model superconductors at nite temperature and chemical potential, a dynamical charged eld describing the condensate needs to be added to the bulk superconductor models. symmetry due to the scalar condensate [4, 5, 11{13]. condensate is some component of a vector eld. In order for it to be charged, a nonnon-zero gravitational coupling and nite temperature [7]. regarding the spectral function are added at the end. Bosonic sector: the holographic p-wave superconductor The bulk gravitational solution 0; 1; 2; 3 and signature ( + ++), and gauge group SU(2) with Hermitean generators a preserve the diagonal subgroup of U(1)gauge SO(2)rotation can be obtained. However they always have free cosmological constant through strength is de ned by F MaN derived from this action read S(bos) = @N AaM + abc AbM AcN . The equations of motion GMN + GMN = rSF MaN + abc AbS F McN = 0 ; FSaM FNaS where \r" stands for the usual geometric covariant derivative. We are interested in solutions of the form [6, 14, 17, 34] G = A = f (z) s(z)2 dt2 + (z) dt 0 + W (z) dx 1 : To understand this ansatz, we rst observe that a non-vanishing (z) breaks the SU(2) the equations (2.2) we get s f g2 W 0 0 = s0 = 2 W 2 where we have de ned the dimensionless coupling2 scaling symmetries 2The coupling constant gYM in [15] and ours are related by, = 1=(p2 gYM). some of the elds. (x; y ; g; W ) (x ; z ; ; W ) (z) = z + : : : ; W (z) = WUV + W U0V z + : : : ; f (z) = 1 + f3 z3 + : : : ; s(z) = 1 g(z) = 1 + g3 z3 : : : ; 2 W 02UV z4 + : : : ; nite Beckenstein-Hawking entropy cases in the forthcoming sections. Now, by taking T = s(zIR) jf 0(zIR)j Vanishing Beckenstein-Hawking entropy (z) = W (z) = wIR f (z) = 1 s(z) = sIR + g(z) = gIR 4 s20 gI2R wIR 2 I2R gIR wIR z3 e 2 gIR wIR z + : : : ; 2 I2R gIR wIR z3 e 2 gIR wIR z + : : : ; values of the coupling are displayed. We see that rotational and gauge symmetries are signals the breaking of the U(1) 0 remaining gauge symmetry. (corresponding to gYM 0:71) a horizon at zh the extremal AdS-Reissner-Nordstrom black hole3 2:462 forms and the solution tends to W (z) = 0 ; (z) = 1 f (z) = g(z) = s(z) = 1 ; For couplings above this quantum critical value only the AdS-Reissner-Nordstrom solution is present. Non-vanishing Beckenstein-Hawking entropy 0. The (z) = zh) + : : : ; W (z) = WIR + WI0R (z zh) + : : : ; f (z) = fI0R (z s(z) = sIR + s0IR (z g(z) = gIR + gI0R (z zh) + : : : ; zh) + : : : ; zh) + : : : : W (z) = 0 ; g(z) = s(z) = 1 ; (z) = f (z) = 1 black hole solution 3Strictly speaking, the AdS-Reissner-Nordstrom black hole is de ned for 0 zh, see next subsection. Reissner-Nordstrom black hole solution. where Q2 condition implies that the relation numerically found, 0:995 and zh 6 should hold, consistent with the values T = zh = electrostatic forces. The boundary QFT interpretation When 0 critical line). When c < interpreted as a nite chemical potential for the particles charged under such symmetry, three regions of the gravitational coupling, namely 0:62 the system has a second order phase transitions from the 0:995 the phase transitions become rst order (red critical line). of the unstable branch (T2) respectively. the symmetry broken phase ceases to exist. respectively, for three di erent typical couplings = 0:62 = c (where Tc = 0:0252852 ) and = 0:80 > c (where Tc = 0:0103462 ). function changes over the whole phase diagram. T plane. The black point indicates the critical coupling = 0:40 < c = 0:62 = c (green), = 0:80 > c (blue). (1) (2) (3) The free energy for the values of results from the QFT point of view. Dirac spinors in the gravitational background generators f g = f 3=2; 1=2; 2=2g where 1; 2 ; 3 are the Pauli matrices. To ABg, and a spin connection f!ABg, !03 = !03 = +!30 = !13 = +!13 = !31 = S(fer) = and to introduce the projectors on eigenstates of 3 given by P 3) with eigenvalprinciple [37, 38]. If we x the boundary value of the Dirac elds to be the left (right)handed part ), the action to be considered is y i 0 and h = h (x) dx dx resulting equations of motion read is the induced metric on the boundary. The = 0 : the spinors as ( detG Gzz) 1=4 ei k x k(z) = z3=2f 4 s 21 ei k x 1 k(z) where as usual !23 = +!23 = Local gamma-matrices obey f A DA where the 2+1 dimensional two-dimensional spinors The equations of motion (3.4) then reads 4The generators of the Lorentz subgroup in 2 + 1 dimensions are, k = k+ = i 1=2 the boosts generators and y)-plane. The left/right colored spinors k (z) are thus Dirac spinors of the boundary theory. Now we can make explicit the color indices by writing follows to introduce the four-tuples ~uk(z) and ~vk(z) as k = ( k i) where and the 4 4 real matrix g W=2 g W=2 g W=2 g W=2 In terms of this notation, equations (3.7) are compactly written as +~u0k(z) + i U(z) ~vk(z) = 0 ; ~vk0(z) + i U(z) ~uk(z) = 0 ; + at the ~uk(z) = C(z; z0) ~u(0) k ~vk(z) = i S(z; z0) ~u(0) + C(z; z0) ~vk(0) ; k C(z; z0) = P cosh S(z; z0) = P sinh dz0 U(z0) dz0 U(z0) P eRzz0 dz0 U(z0) + P e Rzz0 dz0 U(z0) ; P e Rzz0 dz0 U(z0) : (3.12) to analyze the behavior of the solutions in the IR and UV limits. at zIR ! 1. The matrix U(z) is constant there U(1) = BB sIR ) satisfying 2 = gI2R kx Let us take z0 = zir operators read 1, and let us consider the region z > zir 1. There the evolution C(z; zir) ' P S(z; zir) ' P cosh( +(z zir)) 12 2 sinh( +(z zir)) 12 2 cosh( (z zir)) 12 2 sinh( (z zir)) 12 2 D P t UIR ; (3.16) i D~ P t UIR ~vk(ir) + P t ~u(ir) + i D~ P t UIR ~vk(ir) ; k P t ~v(ir) + i D~ P t UIR ~u(kir) + k P t ~v(ir) i D~ P t UIR ~u(kir) : (3.17) k 2. The condition that the solution z limit we have Let us rst consider 2 > 0, and then yielding the solution implying the behaviors ~u(ir) and ~u C(z; zir) ' 1 ; S(z; zir) ' 0 ; ~uk(z) ' ~u(kir) ; ~vk(z) ' ~vk(ir) : ~uk(z) ' P ~vk(z) ' P where the initial values of the elds satisfy the linear relation ~u(ir) = i P D~ P t UIR ~vk(ir) : k the case Similar result is obtained when boundary conditions, with the replacement 2 < 0 and/or 2 < 0 under the imposition of ingoing i sign(!) p 2 in (3.18). Regarding 4 T (1 z) C ; C horizon region zir < z 1. The evolution operators (3.12) at leading order then read 1, and let us look at the near C(z; zir) ' cos 4 T ln 1 1 From (3.11) they yield the solution ~uk(z) ' cos ~vk(z) ' 4 T ln ! 4 T S(z; zir) ' i sin 4 T ln 1 1 ~u(ir) + sin C ~u(kir) + cos 4 T 1 1 ln ~v(ir), as After imposing this condition the near horizon solution reads, ~uk(z) ' e i 4 !T ln 11 zzir ~u ~vk(z) ' e i 4 !T ln 11 zzir ~v The ultraviolet behavior boundary z ! 0. There the matrix U(z) goes to a constant U(0) = BBB and the evolution operators (3.12) at leading order are Plugging back into (3.11), the leading behavior results C(z; 0) ' 1 ; S(z; 0) ' UUV z : ~vk(z) ' ~vk(UV) + i z UUV ~u(kUV) : Normal modes and the Dirac cones to de ne fermionic normal modes as regular solutions with ~u(UV) ~uk(zUV) = ~0. Due to ~u(UV) = C(zUV; zir) ~u(ir) k k i S(zUV; zir) ~v ~v(UV) = i S(zUV; zir) ~u(ir) + C(zUV; zir) ~v k k i L ~u(kir) holds where, S(zUV; zir) L) ~u(kir) ; C(zUV; zir) L) ~u(kir) ; exist a non trivial solution i C(!; kx; ky) det (C(zUV; zir) S(zUV; zir) L) = 0 : of this equation: S(zir; 0) ' UUV zir. In such case, equation (3.32) reads that the eigenvalues of UUV, given by ( +; vanish. The equations They de ne the two UV Dirac cones =2; 0; 0), and that satisfy being essentially blind to the UV part of the geometry. Then for 2 = 0 we can this time becomes left with the case 2 = 0, where 2 = gI2R kx 2 = 0. This de nes the that rotational symmetry in the momentum plane is broken in the IR. representation of SU(2). The two point correlation functions of such operator can be the retarded fermionic correlator closely following references [15]. So(fner)shell = (2 )3 ~u(kUV)y C ~v(UV) k of the elds; from (3.31) we get ~v(UV) = Mk ~u(UV) k k Then the on-shell action is i (S(zUV; zir) C(zUV; zir) L) (C(zUV; zir) So(fner)shell = (2 )3 ~u(kUV)y C Mk ~u(kUV) : S = i hO~~(k) O~~y(0)icjret 3 3(k) GR(k) = C From here, using equation (3.39), we can read the correlation function GR(k) = Mk C : fermionic dynamics is the spectral function (k), de ned as Im Tr GR(k) = Im Tr (Mk C) : is singular for them. (b) (kx; ky) = (0:2; 0:1) . (c) (kx; ky) = (0:2; 0:4) . 0:20 Tc and UV (in violet) Dirac cones. Numerical results contained in the gures displayed in this section. We plotted the spectral function as a xed values of (kx; ky), for di erent values of and T . (kx; ky) and low c, and a low temperature T < Tc. It is seen that the intersections perature. Each gure corresponds to a xed and !. In all cases, as the temperature with gure 5 sub gure (f), or gure 7 sub gure (b) with gure 6 sub gure (f), we see that correspond to three xed values of , smaller, equal and bigger than c . Each gure in The value of ! decreases from gure to gure inside each set. It is evident by comparing the spectral function is milder for > c than in the cases c. Notice also that even (kxF ; kyF ) = lifetime. In it matches with the value of the normal phase. Discussion extending previous work in the literature. (d) (kx; ky) = (k ; 0). (e) (kx; ky) = (0:2; 0:1) . (f ) (kx; ky) = (0:2; 0:4) . (a) (kx; ky) = (k ; 0). (b) (kx; ky) = (0:2; 0:1) . (c) (kx; ky) = (0:2; 0:4) . (g) (kx; ky) = (k ; 0). (h) (kx; ky) = (0:2; 0:1) . (i) (kx; ky) = (0:2; 0:4) . ing (purple) phases, at xed (kx; ky), for three di erent temperatures and couplings: (b) (c) T = 0:20 Tc ; = 0:40 < c ; (d) (e) (f) T = 0:28 Tc ; = 0:62 = c ; (g) (h) (i) T = 0:40 Tc ; = 0:80 > c . the normal modes. gravitational coupling is increased. the critical temperature of the superconducting transition. (a) T = Tc. (b) T = 0:85 Tc. (c) T = 0:60 Tc. (d) T = 0:43 Tc. (e) T = 0:20 Tc. (f ) T = 0. = 0:40 < c, ! = 0:25 . It (a) T = Tc. (b) T = 0:85 Tc. (c) T = 0:60 Tc. (d) T = 0:43 Tc. (e) T = 0:20 Tc. (f ) T = 0. = 0:40 < c, ! = 0:18 . It (a) T = Tc. (b) T = 0:85 Tc. (c) T = 0:60 Tc. (d) T = 0:43 Tc. (e) T = 0:20 Tc. (f ) T = 0. = 0:40 < c, ! = 0:12 . It (a) T = Tc. (b) T = 0:79 Tc. (c) T = 0:60 Tc. (d) T = 0:38 Tc. (e) T = 0:28 Tc. (f ) T = 0. = 0:62 = c, ! = 0:25 . It (a) T = Tc. (b) T = 0:79 Tc. (c) T = 0:60 Tc. (d) T = 0:38 Tc. (e) T = 0:28 Tc. (f ) T = 0. = 0:62 = c, ! = 0:18 . It (a) T = Tc. (b) T = 0:79 Tc. (c) T = 0:60 Tc. (d) T = 0:38 Tc. (e) T = 0:28 Tc. (f ) T = 0. = 0:62 = c, ! = 0:12 . (a) T = 0:80 Tc. (b) T = 0:40 Tc. (c) T = 0. = 0:80 > c, ! = 0:25 . represent the UV and IR Dirac cones respectively. (a) T = 0:80 Tc. (b) T = 0:40 Tc. (c) T = 0. = 0:80 > c, ! = 0:18 . represent the UV and IR Dirac cones respectively. Acknowledgments Salam ICTP for kind hospitality and nancial support during some stages of the work. CONICET, Argentina. About the positivity of the spectral function (a) T = 0:80 Tc. (b) T = 0:40 Tc. (c) T = 0. = 0:80 > c, ! = 0:12 . represent the UV and IR Dirac cones respectively. The starting point is the (on-shell) conserved current J A(x; z) J A(z; k) = i s pf DAJ A(z; k) = Q0(z; k) = 0 2 Im k+(z) k (z) = 2 Im ~uk(z)y C ~vk(z) 2 Im ~uk(z)y S(z) 1 ~u0k(z) Here the matrix S(z) last equality we used the equations of motion (3.10). can evaluate it in known limits. the relation (3.38) and the de nition (3.41) we obtain QUV(k) Q(0; k) = i C ~u(kUV) y GR(k) GR(k)y If we assume that QUV(k) we conclude that, that the hermitian matrix i GR(k) GR(k)y is non-negative. From the de nition (3.42) 0, and being the vector ~u(UV) arbitrary, we must conclude (!; kx; ky) GR(k) GR(k)y (a) ! = 0, T = Tc. (b) ! = 0, T = 0:95 Tc. (c) ! = 0, T = 0:85 Tc. (d) ! = 0, T = 0:20 Tc. Figure 14. The spectral function at xed = 0:40 < di erent temperatures. Right: (1) gure; (2) (a) ! = 0, T = Tc. (b) ! = 0, T = 0:97 Tc. (c) ! = 0, T = 0:79 Tc. (d) ! = 0, T = 0:41 Tc. di erent temperatures. Right: (1) gure; (2) (a) ! = 0, T = Tc. (b) ! = 0, T = 0:80 Tc. (c) ! = 0, T = 0:40 Tc. = 0:80 > di erent temperatures. Right: (1) a function of ! at (kx; ky)= = (0:20; 0:10); (3) as a function of ! at (kx; ky)= = (0:20; 0:40). = 0:4. ~uk(z) ' e z BB b4 CC B@ b3 AC not relevant here. From (A.2) we get QIR(k) Q(z; k)jz!1 = lim BB2 Im( +) e + z 2 BB b2 CC SIR1BBB b2 CC B z 2 BB b4 CC SIR1BBB bb43 CCAC B@ b3 AC @ ) z BB b4 CC B@ b3 AC SIR1 BBB@ bb43 CCACCCCCACCCCA SIR Sjz!1 = 3 (A.7) S2 are real and given respectively by, except when we are outside both IR Dirac cones, i.e. 2 > 0, in which case QIR(k) = 0. e ectively (k) 0, being zero outside both IR Dirac cones or in other words, is strictly positive inside some IR Dirac cone and null outside both of them. where ~b is read from (3.26) in terms of ~u(kir). From (A.2) we get, ~uk(z) ' exp i 4 !T ln(1 z) ~b QIR(k) = 2~by ~b > 0 where in space of momenta. 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Fermionic spectral functions in backreacting p-wave superconductors at finite temperature, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP04(2017)087