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Solution of the Quantum Ising Spin Glass and the Superglasses

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Solution of the Quantum Ising Spin Glass and the Superglasses Solution of the quantum Ising spin glass and the superglasses Markus Müller Abdus Salam International Center of Alexei Andreanov (ICTP) Theoretical Physics Xiaoquan Yu (SISSA) Lev Ioffe (Rutgers University) The arrow of time, IHP Paris, 12th October, 2010 Arrow of time and ergodicity The arrow of time How to know the direction of increasing time? • Entropy always increases • Example: diffusion (continues forever in infinite systems) t ≠ t Interesting exception: quantum localized systems → time reversal symmetry even in infinite systems Ergodicity Fundamental postulate of thermodynamics: State of maximal entropy (=equilibrium) is reached in finite time. Eq But: NO full equilibration when ergodicity is broken t = ∞ (Eq) Occurs in particular in certain disordered systems: Glasses Interrelation between arrow of time and ergodicity? Two notions associated with time evolution and dynamics Consider two types of ergodicity breakers or « glasses »! Two types of « glasses » Frustrated disordered systems Quantum localized systems e.g. Spin glass e.g. Anderson insulator (Fermi glass) Arbitrarily large energy barriers ΔE Vanishingly small matrix elements between metastable states between distant states in Hilbert ΔE >> temperature T (classical) space ΔE >> tunneling Γ (quantum) (no energy barriers necessary) Pure states differ in global density Non-ergodic sectors differ by local profiles density matrices Number of pure states: Exponential number of non-ergodic Mean field: exponential in N sectors N ~ exp[V Ld ] Finite dimensions ? loc Destroyed by large T Robust to T (if fully localized) Robust to dephasing/coupling to Destroyed by a dephasing bath environment Non-ergodicity BUT Non-ergodicity AND no diffusion diffusion in general still possible (no clear arrow of time) Two types of « glasses » Frustrated disordered systems Quantum localized systems e.g. Spin glass e.g. Anderson insulator (Fermi glass) Arbitrarily large energy barriers ΔE Vanishingly small matrix elements between metastable states between distant states in Hilbert ΔE >> temperature T (classical) space ΔE >> tunneling Γ (quantum) (no energy barriers necessary) Pure states differ in global density Non-ergodic sectors differ by local profiles density matrices Number of pure states: Exponential number of non-ergodic Mean field: exponential in N sectors N ~ exp[V Ld ] Finite dimensions ? loc Destroyed by large T Robust to T (if fully localized) Robust to dephasing/coupling to Destroyed by a dephasing bath environment Two mechanisms of ergodicity breaking: Mutual enhancement or competition? Mean field quantum glass models • Nature and spectrum of collective excitations in quantum glasses ? Jij = 0 Transverse field Ising spin glass H = −Γ σ x − σ z J σ z (Sherrington-Kirkpatrick SK) ∑ i ∑ i ij j J 2 i i< j J 2 = → Deep quantum glass phase: ij N find Ohmic bath of collective modes; extrapolate to finite d • Interplay of glassiness and superfluidity - z delocalization of hard core bosons: σ i ↔ 2ni − 1 Meanfield model for “superglass” = amorphous, glassy supersolid t x x z z H = − ∑σ i σ j − ∑σ i Jijσ j → superglass, interesting features N i< j i< j due to competition superfluidity ↔ glass order Transverse field SK model Static approximation A. Bray, M. Moore JPC (’88); Y.Y. Goldschmidt and P.Y. Lai, PRL (‘90) Critical behavior at the Q-Glass transition Phase diagram J. Miller, D. A. Huse, Phys. Rev. Lett. (’93) classical Effective field theory (Landau expansion) PM S. Sachdev, N. Read, J. Ye (‘93,’95) K. Takahashi PRB (’07) SG QMC in the paramagnet, critical line quantum M. J. Rozenberg, D. R. Grempel, PRL (‘98); Exact diagonalization in finite N~20 SK model L. Arrachea, M. J. Rozenberg, PRL (‘01) Goldschmidt, Lai, PRL (‘90): Static approx Other mean field quantum glasses: But ?? Spherical, O(N), p-spin type models – often QPT is first order, unlike SK! T. Nieuwenhuizen; F. Ritort; G. Biroli, L. Cugliandolo et al. SK: Known properties Transverse field Ising spin glass x z z H = −Γ∑σ i − ∑σ i Jijσ j i i< j SK: mean field limit, Para- Infinite coordination z = ∞: magnet Phase transition into a glass state: Glass - Many long-lived metastable states - replica symmetry breaking Spectral gap closes (Miller, Huse PRL 1993) ? remains closed in the glass phase! Read, Sachdev, Ye, PRL (1993) Quantum TAP equations (Thouless, Anderson, Palmer ‘77: Classical SK model; Biroli, Cugliandolo ’01, MM, Ioffe ‘07) x z z H = −Γ∑σ i − ∑σ i Jijσ j i i< j How to obtain insight into excitations and dynamics in the deep quantum glass? 1. Physical, qualitative arguments à la T.A.P. (Thouless, Anderson, Palmer) 2. Rigorous solution of mean field equations Quantum TAP equations (Thouless, Anderson, Palmer ‘77: Classical SK model; Biroli, Cugliandolo ’01, MM, Ioffe ‘07) x z z H = −Γ∑σ i − ∑σ i Jijσ j i i< j Effective potential as a function of magnetizations imposed by external ex auxiliary fields h i : ∞ z 0 1 1 2 G σ i = mi = Gi mi − mi Jij m j − Jij dτχi (τ )χ j (τ ) + O 1 N ({ }) ∑ ( ) ∑ ∑ ∫0 ( ) i 2 i≠ j 2 i≠ j 0 2 Gi (mi ) = −Γ 1− mi χi (ω → 0) = dmi dhi ≈ χi (mi ) Local minima (in static approximation) (∂G ∂mi = 0) N coupled random ∂G0 Γm equations for {mi} h ≡ i = i = J m − m J 2 χ m i 2 ∑ ij j i ∑ ij j ( j ) with ~ exp[ α N] ∂mi 1− m j ≠i j ≠i i solutions! Quantum TAP equations (Thouless, Anderson, Palmer ‘77: Classical SK model; Biroli, Cugliandolo ’01, MM, Ioffe ‘07) x z z H = −Γ∑σ i − ∑σ i Jijσ j i i< j Local minima ( ∂ G ∂ m i = 0 ) ∂G0 Γm h ≡ i = i = J m − m J 2 χ m i m 2 ∑ ij j i ∑ ij j ( j ) ∂ i 1− mi j ≠i j ≠i Environment of a local minimum (potential landscape): 2 Hessian: Ηij = ∂ G ∂mi∂m j = Jij + diagonal terms Spectrum of curvatures in a minimum Gapless spectrum λΓ in the whole glass phase! Spec ⎡Hij ⎤ ≡ ρH (λ) = const × ⎣ ⎦ J 2 Ensured by marginality, see below! (at small λ) Soft collective modes Spectrum of curvatures ↔ λΓ Spec ⎡Hij ⎤ ≡ ρH (λ) = const × Distribution of “restoring forces” ⎣ ⎦ J 2 Semiclassical picture: → N collective oscillators with mass M ~ 1/Γ and frequency ω 2 Mode density ρ(ω ) = const × ΓJ 2 Continuous bath with Independent of Γ Ohmic spectral function for ω < Γ !! Generalization of known spectral function at the quantum glass transition! [Miller, Huse (SK model); Read, Ye, Sachdev (rotor models)] 1 Effective action and selfconsistency problem exp[ βF ] = Tr T exp (1) − eff Seff 1 (2) 1 1 Effective action and selfconsistencyRigorous problem confirmation β β Effective action and selfconsistency problem A. Andreanov, MM, ‘10 Effective action and selfconsistency2 problem z z 1 z z x Meaneff field= J equationsdτdτ ! Qabσa(τ)σb (τ !)+ Qaa(τ τ !)σa(τ)σa(τ !) + Γ dτσa (τ) (3) S " Zeff = Tr T exp 2eff − $ (1) !!0 a<b a a !0 # Zeff = Tr T expS e#ff # (1) S (2) exp[ βFeff] = Tr T exp eff (2)(1) − S z z (2) Qaa(τ τ !)= σa(τ)σa(τ !) eff (4) β − " #S β β β 2 z z 1 1 z z x = J 2dτdτ ! Q σ (τz)σ (τz!)+ Q (τ τ !)σ z(τ)σz (τ !) + Γ dτσx (τ) (3) eff eff =β J dτdτ ! abQaabσa(τb)σb (τ !)+ Qzaaaa(τ z τ !)σaa(τ)σaa(τ !) + Γ βdτσa (τa) (3) S Qab2 = σ (τ)σ−(τ !) q(x) (5) S "a<b" 2a a b− eff $$ a 2 !!0 !!0 a<b z z 1 a" #zS →z a !0 !0 x eff = J dτdτ ! #Q#abσa(τ)σb (τ !)+ ##Qaa(τ τ !)σa(τ)σa(τ !) + Γ## dτσa (τ) (3) S " 2 − $ !!0 a<b a a !0 # #z z z Trz T exp[ eff] # Qab = σa(τ)σb (τ !) eff (4) Qaa(τ τ !)= σa(=τ)σa(τS!)···eff S (4) (6) − ""···" # Tr# T#Sexp[ eff] Selfconsistency: z z S Qaa(τ τ !)= σa(τ)σa(τ !) eff (4) (7) z z S Qaa−(τ z τ !)="z σa(τ)σa(τ !#) eff (5) Qab = σ−a(τ)σb ("τ !) eff q#(Sx) (5) " #S → z z Qab = σa(τ)σb (τ !) eff q(x) (5) " Tr T #Sexp[→ eff] =Tr T ···exp[ S ] ω (6) To prove: z Trz T exp[ eff] Γ Γc "···=σ#a(τ)σa(···τ !) =S consteff | 2| independent of Γ! (6) (8) % "···→"# Tr T exp[# Seff] × J Tr T exp[S eff] (7) = ··· S (6)(7) Parisi flow "··· # Tr T exp[ ] Parisi flow i) Permanent gaplessness, Ohmic spectralSeff function Parisi flow ii) Constant is independent of Γ (7) 2 2 ∆ m˙ (y,∆ x)= q˙(x)[m!!(y, x)+2βxm(y, x)m!(y, x)] (9) m˙ (y, x)= q˙(x−)[m2!!(y, x)+2βxm(y, x)m!(y, x)] (8) ∆−2 2 2 2 m˙ (y, x)=z ˙ ∆z q˙(x)[∆m!!(y, x)+2ω βxm(y, x)m!(y, x)] (8) Γ < Γc P˙ (y,σ x()=τP−)σ(y,2( xτq)=˙!()x) Pconstq(˙y,(x x)) P|!!2β(|y,x ( xindependentm)(y,2 xβ)Px((y,m x()) ofy,! Γ x)!P (y, x))! (9)(8) (10) →" a a #≈2!! × J 2 − ∆2 2 − P˙ (y, x)= q˙(x) P%!!(y, x) % 2βxz (m(y,z x)Pz (y, xz))&! & (9) Parisi flow w2 (u, v)=w(u,dyP v)=(y,−1)dyPσ (u(y,)σ1)(v)σ c(u)σ (v) c (10) (11) " " # # % ! ! z z & w(u, v)= dyP (y, 1) σ β(u)σ (v)β β β (10) z z ∆2 z2 zc z x Tr T2σ (u)σ (vz) exp z " w σ∆ σ# + du[zyσz + Γσ ] z x ∆ Tr Tσ (u)σ (v2 ) exp uv u v wuvσ σ u + duu [yσ + Γσ ] ! 0 2 0 u v u u z mz˙ (y, x)= q˙(x)[m!!(y, x)+2' ββxm(y, x)0m!(y,β x)]0 ) (9) σ (u)σ z(v) z = 2 ' ) (11) σ (u)y,σΓ (v)− 2z = z ∆ (( z z ( z x (12) " # Tr Ty,σΓ (u)σ (v) exp β wuvσ σβ((+ du[yσ +(Γσ ] " # 2 ∆2 2 z βz u v z β ux u ∆ Tr T exp 2' wuv0 ∆σ2uσv + duz[y0σzu + Γσu] z ) x z z P˙ (y, x)= q˙(x) P (y,Tr xT) exp2βx (m(y,w x)uvPσ(y,σ x))+! du[yσ + Γσ ] (10) σ (u)σ (v) y,Γ = !! ' 0 ((2 0 u (v ) u u (11) " # 2 β− ' 0 β 0 ) ∆2 (( β( T ∂Z0(y, Γ) z z z x Tr T exp% 2 wuvz σuσ((vz + du[yz σu +β&(Γσu] mw(y,u,1) v)= = dyP (y, 1) σ (u=)σT (v)du σu (12)(11) Z (y,'Γ) 0∂Ty ∂Z0(y, Γ0)c " # z) m(y,01) = " !0 # = T du σu (13) ! Z0(((y, Γ) ∂y β( 0 " # T ∂Z0(y, Γβ) 2 βz ! m(y, 1) =z q(xz)= dy P∆(y,2 x)m= T(y, xz)duz σ z x (13)(12) Tr Tσ (u)σ (v) exp wuvσ σ +2 udu[yσ + Γσ ] Z0(y, Γ!)q(x)=∂y2 dy P (y,0u x)vm" (y,# x)u u (14) 0 0 z z ' 2 ! ) σ (u)σ (v) y,Γ = q1 = P (y, 1)!m (y, 1) (14)(12) β (( 2 β ( " # q(x)= dyy 2 P (y, x)m (y, x)2 (13) !∆ q1 = Pz(y,z 1)m (y, 1)z x (15) Tr T exp! 2 wuvσuσv + du[yσu + Γσu] dynamic susceptibility ' 0 !y 0 ) q = P (y, 1)m2(y, 1) (14) 1 ((
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