Fermi surface, hyperscaling violation and unified frame in effective holographic theories Bom Soo Kim Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, 69978 Tel Aviv, Israel University of Crete, Heraklion 13/12/2012 Based on 1005.4690 with C. Charmousis, B. Goutraux, E. Kiritsis, R. Meyer 1210.0540 & 1202.6062 1 / 20
Quantum phases of matter at low T (3 slides) Hyperscaling violation! (3 slides) Entanglement Entropy (EE) (4 slides) Hyperscaling violation as a unification tool (3 slides) 2 / 20
Quantum phases of matter at low T Compressible matter Fermi surfaces 3 / 20
Doiron-Leyraud et al. , Nature 447, 565 (2007) Dopong dependence and Fermi surfaces! 4 / 20
Quantum phases at low temperature Sachdev 1203.4565 and others Several physical properties of interesting quantum phases of matter do not fit into the standard Fermi liquid paradigm. So it is called non-Fermi liquid. - ρ ∼ T ( T 2 ), S ∼ T 1 / z ( T d / z ), · · · For the materials such as high T c cuprates, HF, organic insulators · · · at T = 0, the “density” of the ground states can be dialed by a quantum tuning parameter, such as doping, chemical potential, pressure · · · : “compressible” “Compressible” quantum matter can be described by a modified hamiltonian H ′ = H − µ Q , [ Q , H ] = 0 , - Q : conserved U (1) charge. - ground state is compressible if d �Q� / d µ � = 0 at T = 0 . - requires gapless modes to change the density of the ground states. - scaling argument implies d �Q� / d µ ∼ T d − 1 → d = 1 . - hard to get the compressible matter except d = 1, Luttinger liquid. 5 / 20
Compressible matter in 2+1 dimensions S.S. Lee 0905.4532 and others realized in 2+1 dimensional low energy effective theories with emergent gauge bosons and fractionalized gauge-charged fermions and bosons. c σ = ψ σ · h , σ = ↑ , ↓ , where UV gauge invariant electron split into spinon ψ σ and holon h . - an emergent gauge symmetry : ψ σ ( x ) → e i θ ( x ) ψ σ ( x ) , h ( x ) → e − i θ ( x ) h ( x ) . There exist universal, compressible non-Fermi liquid states with Fermi surface (different from FL, but same k F of free electrons). A ) 2 ψ + 1 / 4 g 2 F 2 , L = ψ † ( ∂ t − iA t − µ ) ψ − 1 / 2 m ψ † ( ∇ − i � - Fermi surface is hidden (not gauge invariant, not a physical observable) and characterized by singular, non-quasi particle low energy excitations. - propagator can be computed in a fixed gauge, z is different from UV. - compressible matter for d = 2 and θ = 1, thus d eff = 1. Luttinger liquid. 6 / 20
Hyperscaling violation exponent θ Hyperscaling violation exponent story of the gravity side 7 / 20
Hyperscaling (HS)? Hyperscaling (HS) is a property of the physical quantities based on their naive - e.g. S ∼ T d / z . scaling dimensions (power counting). HS is violated by random-field fluctuations, which dominates over thermal ones, near a quantum critical point. Specifically, the free energy F grows with modified scaling. - e.g. S ∼ T ( d − θ ) / z , thus d eff = d − θ . Fisher 1986 Holographically, HS violation is realized as a property of metric � d � i + d r 2 d s 2 = r − 2+2 θ/ d − r − 2( z − 1) f ( r ) d t 2 + � d y 2 , f ( r ) i =1 t → λ z t , ds → λ θ/ d ds . x i → λ x i , r → λ r , Pointed out first in Gouteraux-Kiritsis 1107.2116 based on an explicit solution Charmousis-Gouteraux-BSK-Kiritsis-Meyer 1005.4690, further investigated in Huijse-Sachdev-Swingle 1112.0573. 8 / 20
Holographic realization of HS violation (EMD) Charmousis-Gouteraux-BSK-Kiritsis-Meyer 1005.4690 Gouteraux-Kiritsis 1107.2116 Explicit solution is given by 2 � � i + d r 2 ds 2 = r − 2+ θ − r 2 − 2 z f ( r ) d t 2 + � d x 2 , f ( r ) i =1 � r � 2+ z − θ d 4 x √− g � 4 F 2 − 1 � � R − Z 2 ( ∂φ ) 2 + V S = , f ( r ) = 1 − r H � Z = 1 2 z − 2 e φ = r s , 4 − θ φ , z + 2 − θ r − 2 − z + θ f ( r ) , q 2 e A t = q s � V = (2 + z − θ )(1 + z − θ ) e − θ s φ , 4( z − 1) − 2 z θ + θ 2 . s = ± - Effective holographic theories(EHT) valid only for certain range of r . - Generalization of Lifshitz case with θ , or AdS with ( z , θ ). - Worked out for general p + 2 dimension. - Thermodynamic and transport properties are analyzed. - Most general IR scaling asymptotics at finite density with single φ, F . - Embedded in higher-dimensional solutions. 9 / 20
Gravity side Field theory side T µν g µν J µ A µ φ Tr (ΦΦ) , Tr (ΨΨ) ψ Tr (ΦΨ) , ˜ Tr ( F 2 ) φ ( dilaton ) ’Elementary fields’ F , Φ , Ψ ( ψ σ , h ): not gauge invariant, not measurable, ’mesonic’ operators dual to ˜ φ, φ, ψ ( c ) : measurable. Identify charge density and chemical potential in 3+1 bulk (2+1 boundary), A t ( r ) = µ + � J t � r + · · · , F tr | r =0 = � J t � . Systems with charge density need electric flux at infinity. The horizon at finite charge density is identified as deconfined phase. in fractionalized phase : flux is sourced by the horizon, Sachdev and others in the mesonic (cohesive) phases : flux is sourced by charged fields in the bulk. 10 / 20
Fermi surface identification? Entanglement Entropy How to identify Fermi surfaces in holography? Novel phases for Lifshtiz and Schr¨ odinger spaces 11 / 20
Fermi surface identification? Luttinger v.s. EE Defining fractionalization using Luttinger count for the compressible matter : the total charge density is equal to the sum over the momentum space volumes of all Fermi surfaces in the theory, weighted by the charge of the corresponding fermionic operators . � J t � = � q ℓ V ℓ . ℓ - Extremal Reissner-Nordstr¨ om BH : maximally violated, - ’electron star’ geometry : fully satisfied. * NOT able to check fractionalized Fermi surface explicitly in holography! Entanglement entropy (EE) is useful for classifying phases of matter. - FT calculation for fermionic system with fermi surface : log violation of EE Wolf, Swingle, Zhang-Grover-Vishwanath - Useful definition for systems with Fermi surface : EE show logarithmic violation of the area law . Ogawa-Takayanagi-Ugajin 1111.1023 - EMD system: concrete holographic description for Fermi surface. 12 / 20
EE for Lifshitz-type theories Dong-Harrison-Kachru-Torroda-Wang 1201.1905 For a strip geometry, − l ≤ x 1 ≤ l , 0 ≤ x i ≤ L , i = 2 , · · · , d , located at r = r F with l ≪ L , the entanglement entropy is given by �� L � d − 1 � ǫ � L � d − 1 � l � θ � � θ ( RM Pl ) d S EE = − c . 4( d − θ − 1) ǫ r F l r F EE is independent of z and modified by θ . Reduces to the AdS case for θ = 0. cf. EE of Schr¨ odinger type theories depends on z and θ. For θ = d − 1 : S EE = ( RM Pl ) d L d − 1 log 2 l ǫ , log-violation of area law! r d − 1 2 F For θ = d : S EE = ( RM Pl ) d L d − 1 l , entropy is proportional to volume! 2 r d F Entanglement entropy analysis : novel phases for d − 1 < θ < d . For finite temperature, EE can not be evaluated analytically. For T → 0, it approaches to the zero temperature result, while reproduces the thermal entropy at high temperature limit. 13 / 20
Schr¨ odinger space with hyperscaling violation BSK 1202.6062, 1210.0540 • “Codimension 2” Schr¨ odinger holography : ( D + 2)-dimensional gravity with Schr¨ odinger isometry are equivalent to D -dimensional field theory with the symmetry. ds 2 = r − 2+2 θ/ D � − r − 2( z − 1) dt 2 − 2 dtd ξ + dx 2 i + dr 2 � , D = d + 1 . • Several solutions are generated by null Melvin-twist of Schr¨ odinger solutions. Null energy condition is used to constrain ’consistent’ parameter space ( z , θ ). ( d + 1)( z − 1)( d + 2 z ) − ( d + 1) z θ + θ 2 ≥ 0 , ( z − 1)( d + 2 z − θ ) ≥ 0 . • Effects of θ : scaling dimension is shifted by θ/ 2. x | 2 θ (∆ t ) iM | ∆ � �O ( x ′ ) O ( x ) � ∼ 2 | ∆ t | , | ∆ t | ∆ − θ/ 2 e Similar to the stress-energy tensor : vacuum structure might be modified(??) 14 / 20
Proposal for minimal surfaces of “codimension 2” holography BSK 1202.6062 • EE analysis using ADM form of metric Hubeny-Rangamani-Takayanagi 0705.0016 � d � � 2 ds 2 = r − 2+2 θ/ ( d +1) − r − 2( z − 1) � + r 2( z − 1) d ξ 2 + dt + r 2( z − 1) d ξ � dx 2 i + dr 2 . i =1 - impose stationary condition involved with ξ coordinate. � - using that there is a fixed length scale associated with ξ : d ξ = L ξ . - demonstrate ( d − 1 )-d area law for EE for ( d +3 )-d Schr¨ odinger background �� ǫ � l � θ L d − 1 L ξ � θ L d − 1 L ξ � S = ( RM Pl ) ( d +1) ǫ d − z +1 − c θ , l d − z +1 4( α − 1) R θ R θ • Novel phases : d + 1 − z < θ < d + 2 − z , for θ � = 0 , d + 1 < z < d + 2 , for θ = 0 . - The latter is surprising compared to the known Lifshitz case. - What are the properties of these novel phases? Not explorered yet ... 15 / 20
Hyperscaling violation exponent as a unification tool 16 / 20
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