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.

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 »

• 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

w(ω)=Πω(dyPy) 1(y)χω(y) *Π (y) χ (y)= σz σz != ω self-consistency ω ω ω y,c ∆2 " − # * 1 w(ω)Πω(y) − 2 * with the proper polarizability in the presencew of(ω)= a ﬁeld dyPy 1(y)χω(y) * ! Πω(y)

* 1 1 Effective action and selfconsistency problem Effective action and selfconsistency problem 1 exp[ βF ] = Tr T exp (1) − eff Seff Effective action and selfconsistency problem exp[ βF ] = Tr T exp (2)(1) − eff Seff (2) β β exp[ βFeff] = Tr T exp eff (1) − 2 S z z 1 z z x eff = J βdτdτ ! Qabσ (τ)σ (τ !)+ Qaa(τ τ !)σ (τ)σ ((2)τ !) + Γ βdτσ (τ) (3) S a b 2 − a a a 2 "a

w(ω)= dyP1(y)χω(y) ! 1

Eﬀective action and selfconsistency problem

exp[ βF ] = Tr T exp (1) − eﬀ Seﬀ (2)

β β 2 z z 1 z z x = J dτdτ ! Q σ (τ)σ (τ !)+ Q (τ τ !)σ (τ)σ (τ !) + Γ dτσ (τ) (3) Seﬀ ab a b 2 aa − a a a " a $ a !!0 #a

z z Qaa(τ τ !)= σa(τ)σa(τ !) eff (4) − " #S 1 z z Qab = σa(τ)σb (τ !) eff q(x) (5) Eﬀective action and selfconsistency problem " #S →

exp[ βF ] = TrTrTTexp exp[ eff] (1) − eff = ···Seff S (6) "··· # Tr T exp[ ] (2) Seff (7) β β 2 z z 1 z z x = J dτdτ ! Q σ (τ)σ (τ !)+ Q (τ τ !)σ (τ)σ (τ !) + Γ dτσ (τ) (3) Seff ab a b 2 aa − a a a " a $ a !!0 #a

z z Qab = σa(τ)σb (τ !) eff q(x) (5) R(τ)=Q "(τ) Q #(S )=→ Q (τ) q(x 1) (9) aa − aa ∞ aa − → Tr T exp[ ] = ··· Seff (6) "··· # Tr T exp[ ] Seff (7) z z Πω(y) χω(y)= σ σ y,c = (10) ω ω 2 " − # 1 J R(ω)Πω(y) − % z z ω Γ Γ σ (τ)σ (τ !) = const | | independent of Γ! (8) with the proper polarizability% inc →" thea presencea # of a field× Jy2 %

Πω(y) R(τ)=Q (τ) Q ( )=Q (τ) q(x 1) (9) aa − aa ∞ aa − → %

z z Πω(y) χω(y)= σ Rσ(ωy,c)== dyP (y)χω(y) (10) (11) ω ω 2 " − # 1 J R(ω)Πω(y) ! − % with the proper polarizability in the presence of a ﬁeld y %

Πω(y) 2 dyP (y)χ0(y)=1 (12) ! %

z z σa(τ)σa(τ !) eff,c R(ω)= dyP (y)χω(y) (11) " #S ≡ 2 2 Πω 0(y)=Π0 !A(y)ω + o(ω ) (13) → −

% %2 dyP (y)χ0(y)=1 (12) R(ω)=R0 + δRω (14) Rigorous! confirmation A. Andreanov, MM, ‘10 2 2 2 Πω 0(y)=δΠR0 =A(cy)ωω ++ oO(ω(ω) ) (13) (15) → ω− | | % % Parisi flow R(ω)=R0 + δRω (14) 2) Proof that coefficient c is independent of Γ: δR = c ω + O(ω2) (15) ω | | • Assume scaling functions, such as: 2 Γ2 y J y Γ Linear pseudogap in This follows if c(t) is dropping relatively quicklyP (y from) the fixed| | point* * value to zero when x T/Γ. (16) ∼ J 2 Γ y Γ frozen field≈ distribution! & % (Palmer, Thouless; Pankov ’06) 1 χ (y)= χˆ(y/Γ, ω/Γ) (17) ω Γ Parisi flow • Obtain a selfconsistency problem which is independent of Γ, its solution∆ 2yields c = O(1) ! m˙ (y, x)= q˙(x)[m!!(y, x)+2βxm(y, x)m!(y, x)] (18) − 2 ∆2 P˙ (y, x)= q˙(x) P !!(y, x) 2βx (m(y, x)P (y, x))! (19) 2 − ! " w(u, v)= dyP (y, 1) σz(u)σz(v) (20) # $c # β β z z ∆2 z z z x Tr Tσ (u)σ (v) exp 2 wuvσuσv + du[yσu + Γσu] z z $ 0 0 & σ (u)σ (v) y,Γ = (21) # $ β %% β % ∆2 z z z x Tr T exp 2 wuvσuσv + du[yσu + Γσu] $ 0 0 & %% % T ∂Z (y, Γ) β m(y, 1) = 0 = T du σz (22) Z (y, Γ) ∂y # u$ 0 #0 q(x)= dy P (y, x)m2(y, x) (23) # 2 q1 = P (y, 1)m (y, 1) (24) #y dynamic susceptibility

Π (y) χ (y)= σz σz = ω ω ω ω y,c ∆2 # − $ 1 w(ω)Πω(y) − 2 ' with the proper polarizability in the presence of a ﬁeld y '

Πω(y) self-consistency '

w(ω)= dyP1(y)χω(y) #

Π (y) w(ω)= dyP (y) ω 1 ∆2 1 w(ω)Πω(y) # − 2 ' static susceptibility '

β ∂m χ (y) χ (y)= dv( σz σz m2(y)) = 0 ≡ ω=0 # u v $y − ∂y #0 magnetization

y m (y) m(y)= dy!χ (y!) 1 ≡ 0 #0 What survives of the phenomenology beyond mean field?

• Criticality ?

• Gapless collective modes ?

Expect: Large connectivity → MF describes well the spectrum, except at the lowest energies Beyond mean field Quantum spin glass with

• Exchange matrix Jij random, |Jij| ~ J • Large connectivity z

Repeat Quantum-TAP / semi-classical analysis! Beyond mean field Quantum spin glass with

• Exchange matrix Jij random, |Jij| ~ J • Large connectivity z

Eigenvalue and eigenvector spectrum of random matrix Jij (D>2) Feigelman, Ioffe, Dotsenko (95)

z = ∞

z < ∞ Semicircle + Corrections in regime ~1/z5/6: • localization • spectral deviations Beyond mean field Quantum spin glass with

• Exchange matrix Jij random, |Jij| ~ J • Large connectivity z

Eigenvalue spectrum of the TAP Hessian Hij (d>3)

∞ > z  1 + semiclassical treatment of oscillators

~ Jz−1 6 J htyp Delocalized low-energy excitations, at least down to ω min   loc Beyond mean field?

Preliminary conclusions

• Criticality of quantum glass → large density of soft collective modes delocalized to very low energies (provided the connectivity of the exchange matrix is large enough) Similar to boson peak in structural gasses!

• Marginality reflects the competition of many metastable minima:

→ Spin glass-type ergodicity breaking counteracts quantum localization.

→ Instead it may enhance transport of energy and charge, as well as decoherence (as compared to a pure, ordered, but gapped phase).

Open questions: How does the low energy spectrum really look like? Is there a mobility gap for many body excitations? Ordering transitions in random

Hertz, Fleishman, Anderson, PRL (79) systems Bray, Moore, J. Phys. C (80) Scenario for the ordering transitions in Hamiltonian • Disordered magnets • Spin glasses • Dirty superfluids (SI transition) Susceptibility matrix ( ≥ 0!)

Hartree-Fock Hamiltonian

→ Transition when extended Hartree Fock state condenses → HF modes at finite energy are delocalized → Does a mobility edge re- emerge after the SG transition?? Ordering transitions in random

Hartree-Fock Hamiltonian

Remark: Dirty bosons: transition when single particle → Transition when extended density matrix acquires delocalized zero-mode. – Expect Hartree Fock state condenses a finite mobility edge of (many-body) excitations and → HF modes at finite energy are Arrhenius transport before the SI transition! (d ≥ 3) delocalized MM ’09; Ioffe, Mézard ’09, Feigelman, Ioffe, Mézard ‘10 → Does a mobility edge re- emerge after the SG transition?? What about glassiness AND superfluidity?

“superglasses” = amorphous, glassy supersolids

Motivation: supersolidity observed in defectful (glassy?) quantum solids (Chan, Dalibard) Superglasses ?! Xiaoquan Yu, MM ‘10

x z z H = −Γ∑σ i − ∑σ i Jijσ j i i< j

t x x z z H = − ∑σ i σ j − ∑σ i Jijσ j N i< j i< j

“Self-generated transverse field” 4

1 4 (Π˜ (y) Π˜ (y))/ω2 A(y) aˆ(y/Γ), aˆ(x) min(1, 1/x5). Superglasses0 − ?!ω ≡ ∼ Γ3 ∼ With the above assumptions one gets aXiaoquan set of self-consistency Yu, MM ‘10 equations which are completely independent of Γ! In particular the above assumptions immediately lead to the coeﬃcient B being independent of Γ. ˜ ˜ x2 z 1z 5 If they(Π0 have(yH) = aΠ− solution,ωΓ(y))σ/ω this− A proves(σy)J σ thataˆ( they/Γ limit), aˆΓ(x) Γmin(1is indeed, 1/x given). by the above scaling form. − ∑ i ≡∑ i ∼ij Γj3 $∼ c Order parameters superglass:i i< j With the above assumptions one gets a set of self-consistency equations which are completely independent of Γ! In particular the above assumptions immediately lead to the coeﬃcient B being independent of Γ. t x x z z 1 x If they have a solution, this proves that the limit Γ Γc is indeed givenM = byσ the above scaling form. (29) H = − ∑σ i σ j − ∑σ$i Jijσ j N % i & Order parameters superglass: N i< j i< j

1 x 1 z 2 Competing order parameters: M = σ qEA = σi (29) (30) N % i & N % & M signals ferromagnetism or 1 z superfluidity of hard core bosonsq = σ σ↔z 22n − 1 (30) EA N %i i & i

If M and qEA exist simultaneously → Meanfield model for superglasses « Superglass »

Gingras, Melko et al PRL (10) Carleo, Tarzia, Zamponi PRL (09)

QMC (3d) “Mean field”

QMC and cavity Quenched randomness Random, frustrated lattice ? Low T behavior – QPT ? ? Local structure of the superglass ? 4

1 4 (Π˜ (y) Π˜ (y))/ω2 A(y) aˆ(y/Γ), aˆ(x) min(1, 1/x5). 0 − ω ≡ ∼ Γ3 ∼ With the above assumptions one gets a set of self-consistency equations which are completely independent of Γ! In 2 particular1 the above assumptions immediately5 lead to the coeﬃcient B being independent of Γ. (Π˜ 0(y) Π˜ ω(y))/ω A(y) aˆ(y/Γ), aˆ(x) min(1, 1/x ). − ≡ ∼IfΓ3 they have a solution,∼ this proves that the limit Γ Γc is indeed given by the above scaling form. Order parameters superglass: $ With the above assumptions one gets a set of self-consistency equations which are completely independent of Γ! In particular the above assumptions immediately lead to the coeﬃcient B being independent of Γ. Mean field superglass 1 If they have a solution, this proves that the limit Γ Γc is indeed given by the above scaling form.XiaoquanM =Yu, MMσ ‘10x (29) Order parameters superglass: $ N % i & t x x z z H = − ∑σ i σ j − ∑σ i Jijσ j N i< j i< j 1 x 1 z 2 M = σi q = (29)σ (30) N % & EA N % i &

1 •z Static2 approximation is exact when M = 0! qEA = σi (30) N % →& obtain T = 0 phase transition glass-to- superglass exactly!

• For superfluid-to-superglass transition: Use static approximation 4

1 (Π˜ (y) Π˜ (y))/ω2 A(y) aˆ(y/Γ), aˆ(x) min(1, 1/x5). 0 − ω ≡ ∼ Γ3 ∼ With the above assumptions one gets a setPhase of self-consistency diagram equations which are completely independent of Γ! In particular the above assumptions immediately lead to the coeﬃcient B being independent of Γ. 4 If they have a solution, this proves that the limit Γ Γc is indeed given by the above scaling form. Order parameters superglass: $ PM ˜ ˜ 2 1 PM Classical5 (Π0(y) Πω(y))/ω A(y) 1 aˆ(y/Γ), aˆ(x) min(1, 1/x ). − ≡ M =∼ Γ3σx ∼ (29) N % i & SK transition With the above assumptions one gets a set of self-consistency equations which are completely independent of Γ! In particular the above assumptions immediately lead to the coeﬃcient B being independent of Γ. If they have a solution, this proves that the limit Γ 1Γc is indeed given by the above scaling form. Order parameters superglass: q =$ σz 2 (30) EA N % i &

1 x M = σix (29) xx N∂% s & y χy = % & . (31) ∂hx 1 q = σz 2 (30) EA N % i & xx Suppression of superfluidity t dyP (y)χy =1. (32) by the linear pseudogap! ! x xx ∂ s y T 0 1 χy = % & → Transition at finite J/t ≅ 0.97 (31) ∂hx → y Analogue with LR interactions in finite d?

xx t dyP (y)χy =1. (32) ! 4

1 (Π˜ (y) Π˜ (y))/ω2 A(y) aˆ(y/Γ), aˆ(x) min(1, 1/x5). 0 − ω ≡ ∼ Γ3 ∼ With the above assumptions one gets a set of self-consistency equations which are completely independent of Γ! In particular the above assumptions immediately lead to the coeﬃcient B being independent of Γ. If they have a solution, this proves that the limit Γ Γc is indeed given by the above scaling form. Order parameters superglass: $

1 M = σx (29) N % i &

1 q = σz 2 (30) EA N % i &

x xx ∂ s y T 0 1 χy = % & → (31) ∂hx → y

Structure of the superglassxx t dyP (y)χy =1. (32) ! • Superfluid and glass try to avoid each other: sx and sz 2 are anticorrelated (33) % i & % i & • Superfluid order parameter M is non-monotonous with T

Superfluid enhanced by weakening of the glass (= disorder) ! Conclusions • Quantum spin glass: • Low energy excitations of the MF quantum glass: collective oscillators → Ohmic bath • Extrapolation to finite dimensions: spin glass order favors soft collective excitations, tendentially counteracting quantum localization Ergodicity broken – Arrow of time intact

• Superglass: • Coexistence of superfluid and glassy density order • Long range interactions suppress density of states and thus superfluidity down to T = 0 • Non-monotonous superfluid order in the superglass