The pseudo-gap phase and the duality in holographic fermionic system with dipole coupling on Q-lattice
TThe pseudo-gap phase and the duality in holographic fermionicsystem with dipole coupling on Q-lattice
Yi Ling , , ∗ Peng Liu , † Chao Niu , ‡ and Jian-Pin Wu , § Institute of High Energy Physics,Chinese Academy of Sciences, Beijing 100049, China Institute of Gravitation and Cosmology,Department of Physics, School of Mathematics and Physics,Bohai University, Jinzhou 121013, China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China
Abstract
We classify the different phases by the “pole-zero mechanism” for a holographic fermionic systemwhich contains a dipole coupling with strength p on a Q-lattice background. A complete phasestructure in p space can be depicted in terms of Fermi liquid, non-Fermi liquid, Mott phase andpseudo-gap phase. In particular, we find that in general the region of the pseudo-gap phase in p space is suppressed when the Q-lattice background is dual to a deep insulating phase, while for ananisotropic background, we have an anisotropic region for the pseudo-gap phase in p space as well.In addition, we find that the duality between zeros and poles always exists regardless of whetheror not the model is isotropic. Key Words:
Holographic Q-lattice, “pole-zero mechanism”, Fermionic system
PACS: ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] a r X i v : . [ h e p - t h ] F e b . INTRODUCTION Until now, a general theoretical framework for quantum phases and phase transitionsof strongly correlated electron systems, such as cuprate and other oxides, has not beenestablished yet. As an alternative approach, AdS/CFT correspondence [1–3] is a powerfultool to attack these strongly correlated problems and to get some possible clues on the basicprinciple behind these numerous correlated electron systems.Indeed, some exotic states of matter, including Fermi liquid, non-Fermi liquid, Mottphase and pseudo-gap phase, have been found or identified by AdS/CFT correspondence[4–11]. By adding a probe fermion on RN-AdS background, a non-linear dispersion relationis observed [5], indicating that a non-Fermi liquid phase emerges. Also, it is found thatthe low energy behavior of a fermionic system on the RN-AdS background is controlledby the
AdS near horizon geometry [6]. Later, the properties of a fermionic system onGauss-Bonnet, Lifshitz as well as hyperscaling violation geometry have been extensivelystudied [12–21]. Furthermore, a dipole coupling between the gauge field and the Dirac fieldcan be introduced to model the Mott phase [8, 9]. Many extended works on the dipolecoupling effects have been explored in more general geometries in [22–28]. Recently, interms of the “pole-zero mechanism”, a pseudo-gap phase has been detected in a holographicfermionic system with dipole coupling in RN-AdS black hole and the Schwarzschild-AdSblack hole [10, 11]. Moreover, they observed a remarkable duality between zeroes and polesin this holographic system [10], which should be interesting for experimental scientists incondensed matter physics, although this phenomenon has not been captured by experimentsyet. In this paper, we shall address the pseudo-gap phase and the duality in a holographicfermionic system with dipole coupling on Q-lattice geometry.Motivated by the idea of Q-balls [29], a Q-lattice model in a holographic frameworkwas first constructed in [30], in which the translational symmetry was broken and a metal-insulator transition observed through the study of optical conductivity. A lot of relevantwork has been investigated in this context [31–37]. In [35], we study a holographic fermionicsystem with dipole coupling on Q-lattice geometry and find that a Mott gap opens whenthe dipole coupling parameter p is beyond a certain critical value p c . An interesting result isthat the Mott gap opens much more easily when the Q-lattice background is dual to a deepinsulating phase rather than a metallic phase. Here, we shall probe the pseudo-gap phase in2his system by the “pole-zero mechanism” and study how the lattice parameters affect theformation of the pseudo-gap.Our paper is organized as follows. In Section II, a brief review of the holographic Q-lattice geometry and the Dirac equation is presented. Our main results on the classificationof quantum phases of holographic fermionic system with dipole coupling are presented forisotropic and anisotropic Q-lattice geometry in Section III and Section IV, respectively.Specifically, we will focus on the pseudo-gap phase and discuss how the lattice parametersaffect its formation. Finally, the conclusion and discussion are presented in Section V. II. HOLOGRAPHIC Q-LATTICE GEOMETRY AND THE DIRAC EQUATION
In this section, we give a brief introduction on the holographic Q-lattice model whichbreaks translational symmetry in both spatial directions and then demonstrate how to sim-plify the Dirac equation over a specific Q-lattice background. For detailed discussion, werefer to [30, 34, 35].The action with Q-lattice structure in both of the spatial directions can be written as S = (cid:90) d x √− g (cid:20) R + 6 − F − | ∂φ | − m | φ | − | ∂φ | − m | φ | (cid:21) , (1)where F = dA . φ and φ are the complex scalar fields simulating the lattices. From theabove action, the equations of motion can be deduced as R µν = g µν ( − m | φ | + m | φ | ) + ( F ρµ F νρ − g µν F ) + ∂ ( µ φ ∂ ν ) φ ∗ + ∂ ( µ φ ∂ ν ) φ ∗ , ( ∇ − m ) φ = 0 , ( ∇ − m ) φ = 0 , ∇ µ F µν = 0 . (2)To find solutions to the above equations of motion, we take the following ansatz ds = − g tt ( z ) dt + g zz ( z ) dz + g xx ( z ) dx + g yy ( z ) dy ,φ = e ik x ϕ , φ = e ik y ϕ , A = A t ( z ) dt, (3)with g tt ( z ) = (1 − z ) P ( z ) Q ( z ) z , g zz ( z ) = 1 z (1 − z ) P ( z ) Q ( z ) ,g xx ( z ) = V ( z ) z , g yy ( z ) = V ( z ) z , A t ( z ) = µ (1 − z ) a ( z ) ,P ( z ) = 1 + z + z − µ z , (4)3here k and k are two wave-numbers along x and y directions, respectively, such that the φ , is periodic in x, y direction with lattice constant 2 π/k , . In addition, in this paper, wewill set m , = − / AdS BF bound nearthe horizon.Based on the above ansatz, we obtain five second order ODEs for V , V , a , ϕ , ϕ andone first order ODE for Q . To solve the ODEs numerically, we impose a regular bound-ary condition at the horizon z = 1 and impose the following conditions on the conformalboundary: Q (0) = 1 , V (0) = V (0) = 1 , a (0) = 1 . (5)We will only focus on the standard quantisation of the scalar field where the asymptoticbehavior of ϕ , is described as ϕ , = λ , z ∆ − , − , with the UV behavior corresponding toQ-lattice deformation with lattice amplitude λ , , where ∆ ± , ± = 3 / ± (9 / m , ) / , isthe scaling dimension of the dual field of φ , . In addition, the Hawking temperature of theblack hole is given by T = P (1) Q (1)4 π . (6)As a result, each of our Q-lattice solutions can be specified by five dimensionless quantities T /µ , λ /µ ∆ − , λ /µ ∆ − , k /µ and k /µ . We shall abbreviate these quantities to T, λ , , k , respectively in what follows for simpleness.Now, we introduce the Dirac equation with dipole coupling on this Q-lattice geometry.We consider the following action, which involves the interaction between the spinor field andthe gauge field S D = i (cid:90) d x √− gζ (cid:0) Γ a D a − m ζ − ip /F (cid:1) ζ. (7)In the above action, D a = ∂ a + ( ω µν ) a Γ µν − iqA a and /F = Γ µν ( e µ ) a ( e ν ) b F ab , where ( e µ ) a form a set of orthogonal normal vector bases and ( ω µν ) a is the spin connection 1-forms.With the redefinition ζ = ( g tt g xx g yy ) − / F , and by denoting the Fourier transform of F as F ( z, k ) ≡ ( A α , B α ) T with α = 1 ,
2, and choosing the following gamma matrices,Γ = − σ − σ , Γ = iσ iσ , = − σ σ , Γ = σ σ . (8)the Dirac equation deduced from Eq.(7) can be written in a simple form as (cid:18) √ g zz ∂ z ∓ m ζ (cid:19) A B ± ( ω + qA t ) 1 √ g tt B A + p √ g zz g tt ( ∂ z A t ) B A − k x √ g xx B A + k y √ g yy B A = 0 , (9) (cid:18) √ g zz ∂ z ∓ m ζ (cid:19) A B ± ( ω + qA t ) 1 √ g tt B A + p √ g zz g tt ( ∂ z A t ) B A + k x √ g xx B A + k y √ g yy B A = 0 . (10)To solve the Dirac equation, we need to impose the following independent ingoing boundarycondition at the horizon A α ( z, k ) B α ( z, k ) = c α − i (1 − z ) − iω πT . (11)The near boundary behavior of the Dirac field will be A α B α ≈ a α z m ζ + b α z − m ζ . (12)Finally, we can read off the retarded Green function by holography a α = G αα (cid:48) b α (cid:48) . (13)To obtain the boundary Green function, we need to construct a basis of finite solutions,( A Iα , B Iα ) and ( A IIα , B IIα ) due to the four components of the Dirac fields being coupled to oneanother.
III. PSEUDO-GAP PHASES AND DUALITY ON AN ISOTROPIC Q-LATTICE
There is competition between the poles ( k = k F ) and zeros ( k = k L ) in the Green functionof a strongly coupled fermionic system in condensed matter physics. By the “pole-zero5echanism”, we can classify the different phases of a strongly coupled fermionic system [38–42]. The criterion of the phase classification is given as below.Poles ⇔ (Non-)Fermi liquid phaseZeros ⇔ Mott Insulator phaseCoexistence of poles and zeros ⇔ Pseudo-gap phaseIn a holographic framework, this “pole-zero mechanism” was first introduced in [10] tocharacterize different phases. Here, we will use this mechanism to probe the pseudo-gapphase in the holographic fermionic system with dipole coupling on Q-lattice geometry. Weexplore the case of an isotropic Q-lattice in this section, i.e., λ = λ and k = k , in whichwe can set k y = 0 without loss of generality. The case of anisotropic Q-lattice geometry willbe discussed in the next section.For definiteness, we fix m = 0 as well as q = 1 and work at an extremely low temperature T (cid:39) . p = 0, neither poles nor zeros can be observed in the determinant of theGreen function detG R because the poles (zeros) of Green function are cancelled (Fig. 1).However, once the dipole coupling is turned on, the poles or zeros emerge in detG R so thatwe can classify the phases for the fermionic system with dipole coupling in terms of the“pole-zero mechanism”. Fig. 2 shows the determinant of the Green function detG R as afunction of k x at ω = 0 for p = − . p = 4 . λ = λ = 0 . k = k = 0 .
8. One pole emerges at k x = k F (cid:39) .
222 for p = − .
5, indicating a(non-)Fermi liquid state, and a zero can be found at the same momentum k x = k L (cid:39) . p = 4 .
5, indicating a Mott state. Obviously, there is a duality between zero and poleunder p → − p , which was first revealed in [10]. When we decrease the dipole couplingto | p | = 0 .
1, we find the coexistence of pole and zero in detG R (Fig. 3), which points toa pseudo-gap phase. Again, the duality between zeros and poles remains under p → − p (Fig. 3). According to the above observations, we claim that the (non-)Fermi liquid phase,Mott phase and the pseudo-gap phase emerge in a fermionic system with dipole coupling onthe Q-lattice. By the density of state (DOS) A ( ω ), we can determine the critical point p c of the transition from (non-)Fermi liquid phase to Mott phase and find that the Mott gapopens more easily in a deep insulating phase than a metallic phase [35]. Now, we shall focuson the effect of lattice constant k , and lattice amplitude λ , on the pseudo-gap phase.6 - - k x TrG R (a) - - k x detG R (b) FIG. 1: (a): Re(Tr G R ) (solid blue line) and Im(Tr G R ) (dashed red line) for p = 0 at ω = 0. A poleis visible in the spectral function A ( k x ) (Im(Tr G R )), which indicates a Fermi surface ( k F (cid:39) . detG R ) (solid blue line) and Im( detG R ) (dashed red line) for p = 0 at ω = 0, in whichneither poles nor zeroes is observed. Here, we have fixed λ = λ = 0 . k = k = 0 . - - k x detG R (a) - - - - - k x detG R (b) FIG. 2: Re( detG R ) (solid blue line) and Im( detG R ) (dashed red line) at ω = 0. (a) ( p = − . k x = k F (cid:39) .
222 and (b) ( p = 4 .
5) shows a zero at k x = k L (cid:39) . λ = λ = 0 . k = k = 0 . By careful numerical calculation, we find that a pseudo-gap emerges when | p | (cid:46) .
605 for λ = λ = 0 . k = k = 0 .
8. When we go to a deep insulating phase with λ = λ = 2and k = k = 1 / / , the pseudo-gap phase emerges in the region of | p | (cid:46) . p space is suppressed in the deep insulating phase.For comparison, we also find that the pseudo-gap phase appears when | p | (cid:46) .
634 in the7 - - k x detG R (a) - - k x detG R (b) FIG. 3: Re( detG R ) (solid blue line) and Im( detG R ) (dashed red line) at ω = 0. Both pole andzero can be seen for p = − . k x = k L (cid:39) .
441 and k x = k F (cid:39) . p = 0 . k x = k L (cid:39) .
274 and k x = k F (cid:39) . λ = λ = 0 . k = k = 0 . RN-AdS black hole background, which is obtained simply by setting λ = λ = 0 .Before closing this section, we would like to present a brief discussion on the dualitybetween zeros and poles. When k y is set to zero, we can package the Dirac equations (9)and (10) into the following evolution equations( ∂ z − m ζ √ g zz ) ξ α + (cid:20) v − + ( − α k (cid:114) g zz g xx (cid:21) + (cid:20) v + − ( − α k (cid:114) g zz g xx (cid:21) ξ α = 0 , (14)where we have defined ξ α ≡ A α B α and v ± = (cid:113) g zz g tt ( ω + qA t ) ± p √ g tt ∂ z A t . For massless fermions,the retarder Green function is G ( ω, k ) = lim z → ξ ξ . (15)Therefore, from the evolution equation (14), one can easily find the following symmetry ofGreen’s function G ( ω, k ) = G ( ω, − k ) . (16)In addition, we can introduce the reciprocal of ξ α , /ξ α = 1 ξ α . (17) Because we have set the gauge coupling constant g F = √ g F = 2, the charge q and dipole coupling p will correspond to √ q and √ p in [10] due to theproducts g F q and g F p being the relevant quantities.
8e find that /ξ α satisfies the following evolution equations( ∂ z − m ζ √ g zz ) /ξ α + (cid:20) /v − + ( − α k (cid:114) g zz g xx (cid:21) + (cid:20) /v + − ( − α k (cid:114) g zz g xx (cid:21) /ξ α = 0 , (18)where /v ± = − (cid:113) g zz g tt ( ω + qA t ) ± p √ g tt ∂ z A t . It is easily found that there is a symmetry betweenEqs. (14) and (18) under the transformation k → − k , p → − p and ξ α → − /ξ α . CombiningEq. (16), we can easily find that there is a duality between zeros and poles when p → − p . IV. PSEUDO-GAP PHASES AND DUALITY ON AN ANISOTROPIC Q-LATTICE
Now we turn to study the pseudo-gap phase and the duality between the zeros and poleson anisotropic Q-lattices, in which we set λ = 1, λ = 0 . k = k = 0 . k x direction (setting k y = 0). As illustrated inFig. 4 and Fig. 5, we find that the (non-)Fermi liquid phase, Mott insulating phase andthe pseudo-gap phase along the k x direction emerge, depending on the values of the dipolecoupling parameter p . Furthermore, the duality between zeros and poles still holds, whichcan also be understood by a similar analysis to the last section. In a parallel manner, wealso work out the location of poles and zeros in detG R along the k y direction (setting k x = 0)for p = 6 and p = −
6. We find that k yF (cid:39) .
983 for p = − k yL (cid:39) .
983 for p = 6,which indicates that the duality remains valid in the y direction. But obviously we noticethat k xF (cid:54) = k yF ( k xL (cid:54) = k yL ), reflecting the anisotropy of the system.We now study the pseudo-gap phase. We find that the pseudo-gap phase emerges alongthe k x direction for | p | (cid:46) . k y direction for | p | (cid:46) . k x direction), the region of pseudo-gap phase is suppressed, which isconsistent with that explored on isotropic Q-lattices in the previous section. V. CONCLUSION AND DISCUSSION
By the “pole-zero mechanism” we have classified the phases which appear in a holographicfermionic system with dipole coupling on Q-lattice geometry. These phases could be Fermiliquid, non-Fermi liquid, Mott phase or pseudo-gap phase, depending on the strength of the9 - - k x detG R (a) - - k x detG R (b) FIG. 4: Re( detG R ) (solid blue line) and Im( detG R ) (dashed red line) as a function of k x (we set k y = 0) at ω = 0 with p = − p = 6 ((b)) on anisotropic Q-lattice background with λ = 1, λ = 0 . k = k = 0 .
8. (a) shows a pole at k x = k xF (cid:39) .
016 and (b) shows a zero at k x = k xL (cid:39) . - - k x detG R (a) - - k x detG R (b) FIG. 5: Re( detG R ) (solid blue line) and Im( detG R ) (dashed red line) as a function of k x (we set k y = 0) at ω = 0 with p = − . p = 0 . λ = 1, λ = 0 . k = k = 0 . p = − . k x = k xL (cid:39) . k x = k xF (cid:39) . p = 0 . k x = k xL (cid:39) .
329 and k x = k xF (cid:39) . dipole coupling. Therefore, varying the dipole coupling parameter p can induce quantumphase transitions in our holographic system. By investigating the spectral function, we canwork out the critical value p c of the phase transition from (non-)Fermi liquid to Mott phaseand see how the lattice parameters λ , and k , affect the formation of the Mott phase in[35]. Here, we have further computed the determinant of the retarded Green function andfocused on the formation of the pseudo-gap phase. We find that the region of the pseudo-gap10hase is suppressed in the deep insulating phase. For the anisotropic Q-lattice geometry,we have obtained an anisotropic pseudo-gap phase region which is also suppressed along theinsulating direction. Another interesting result is that the duality between zeros and polespreviously found in [10] still holds in either isotropic or anisotropic Q-lattice geometry andis independent of the lattice parameters. Acknowledgments
We are grateful to Xiao-Mei Kuang for valuable discussions. This work is supported by theNatural Science Foundation of China under Grant Nos.11275208, 11305018 and 11178002.Y.L. also acknowledges the support from Jiangxi young scientists (JingGang Star) programand 555 talent project of Jiangxi Province. J. P. Wu is also supported by Program forLiaoning Excellent Talents in University (No. LJQ2014123). [1] J. M. Maldacena,
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