Introduction AKLT states on the Bethe lattice A lattice model with a superglass Conclusions

Quantum Spin on the Bethe lattice

Francesco Zamponi∗ with C.R.Laumann, S. A. Parameswaran, S.L.Sondhi, PRB 81, 174204 (2010) with G.Carleo and M.Tarzia, PRL 103, 215302 (2009)

∗Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure, 24 Rue Lhomond, 75231 Paris Cedex 05

June 2, 2010 Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Outline

1 Introduction Classical spin glasses Random lattices and the cavity method Ising antiferromagnet on a random graph

2 AKLT states on the Bethe lattice AKLT states: mapping on a classical model AKLT models on the tree AKLT models on the random graph Phase diagram Spectrum of the quantum spin phase

3 A lattice model with a superglass phase Extended Hubbard model on a random graph Results A variational argument

4 Conclusions Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Outline

1 Introduction Classical spin glasses Random lattices and the cavity method Ising antiferromagnet on a random graph

2 AKLT states on the Bethe lattice AKLT states: mapping on a classical model AKLT models on the tree AKLT models on the random graph Phase diagram Spectrum of the quantum spin glass phase

3 A lattice model with a superglass phase Extended Hubbard model on a random graph Results A variational argument

4 Conclusions Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Classical spin glasses

Frustration in many-body problems: spin glasses A classical example: the Edwards-Anderson model on a square lattice: P H = hi,ji Jij Si Sj Jij = ±1 with probability 1/2 Its mean field version: the Sherrington-Kirkpatrick model: P 2 H = i,j Jij Si Sj Jij Gaussian with zero mean and hJij i = 1/N High temperature: paramagnetic, m = 0 Low temperature: amorphous magnetization profile, mi = hSi i 6= 0 −1 P −1/2 Average magnetization m = N i mi ∼ N → 0 −1 P 2 Edwards-Anderson order parameter qEA = N i mi −1 P ˙ ¸ −1 P ˙ ¸2 Linear susceptibility χ = N ij Si Sj finite; χSG = N ij Si Sj Mean field theory: Huge degeneracy of classical ground states Many metastable states, slow relaxation Frustration appears as a key ingredient in many different systems: structural glasses, , spin glasses, electron glasses, optimization problems ... What happens to the spin glass phase in presence of quantum fluctuations? Tunnelling between different SG states? Speed up the relaxation? Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Regular lattices, Bethe lattices and random graphs

A useful tool: Bethe approximation Discard the small loops of the lattice, graph becomes a tree:

Provides an “improved” mean field theory since the local connectivity remains finite The tree allows for a simple recursive solution but the frustration is lost: no loops Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Regular lattices, Bethe lattices and random graphs

A more useful tool for frustrated systems: Cavity approximation Replace the lattice by a random graph of the same connectivity: Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Regular lattices, Bethe lattices and random graphs

A more useful tool for frustrated systems: Cavity approximation Replace the lattice by a random graph of the same connectivity:

Regular random graphs have many good properties: Equal probability to all graphs of N sites such that each site has z neighbors Free-energy is self-averaging (a single sample is representative of the average) No boundary, all sites play statistically the same role (“periodic boundary conditions”) Key property: typical graphs are locally tree-like; loops have a length diverging logarithmically with the size N of the system. The exact solution on the random graph can still be obtained (cavity method, M´ezard-Parisi 2001) Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Regular lattices, Bethe lattices and random graphs

A more useful tool for frustrated systems: Cavity approximation Replace the lattice by a random graph of the same connectivity:

Unfrustrated phase (RS cavity method) Correlations decay fast enough Long loops can be neglected → back to a tree Cavity approximation is equivalent to Bethe approximation Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Regular lattices, Bethe lattices and random graphs

A more useful tool for frustrated systems: Cavity approximation Replace the lattice by a random graph of the same connectivity:

Frustrated phase (1RSB cavity method) Correlations do not decay fast enough The recursion relation on a tree is initiated from a given boundary condition For each boundary condition, a different fixed point is obtained One has to sum over boundary conditions in a consistent way Cavity approximation describes the glassy phase Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Ising antiferromagnet on a random graph

Ising antiferromagnet: P P H = J hi,ji Si Sj + h i Si Equivalent to a model of particles with repulsive interaction: P P H = V hi,ji ni nj − µ i ni

On the regular random graph: Typical graphs characterized by many (large) loops of even and odd length Graph is not bipartite Frustrates antiferromagnetic ordering: spin glass phase, without disorder in the couplings Glassy phase for large J and small h (large V , close to half-filling) Same phase diagram of the SK model Krzakala and Zdeborova, EPL 81, 57005 (2008) Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Outline

1 Introduction Classical spin glasses Random lattices and the cavity method Ising antiferromagnet on a random graph

2 AKLT states on the Bethe lattice AKLT states: mapping on a classical model AKLT models on the tree AKLT models on the random graph Phase diagram Spectrum of the quantum spin glass phase

3 A lattice model with a superglass phase Extended Hubbard model on a random graph Results A variational argument

4 Conclusions Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions AKLT states

Schwinger bosons: a representation of a spin S

† † 1 † † S+ = b↑b↓ S− = b↓b↑ Sz = 2 (b↑b↑ − b↓b↓) 1 † † Total spin S = 2 (b↑b↑ + b↓b↓)

A general AKLT state can be written in terms of Schwinger bosons as

Y “ † † † † ”M |Ψ(M)i = bi↑bj↓ − bi↓bj↑ | 0 i hiji

where hiji are the links of a given graph.

Fixed connectivity z: then zM bosons per site, spin S = zM/2 Then the maximum possible spin on each link is 2S − M: the state is annihilated by any Hamiltonian H constructed out of projectors on spin S0 > 2S − M on each link (which are polynomials of ~Si · ~Sj ) Affleck, Kennedy, Lieb, Tasaki, 1987 Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions AKLT states: mapping on a classical model

Y “ † † † † ”M |Ψ(M)i = bi↑bj↓ − bi↓bj↑ | 0 i hiji Introduce on each site coherent states

1 † zM |~ni = (uµbµ) | 0 i p(zM) !)

„ cos(θ/2) « with u = and ~n = u†~σu sin(θ/2) eiϕ Then

„ «M/2 Y M Y 1 − ~ni · ~nj Ψ({nˆi }) = h{nˆi }|Ψi = (u u − u u ) = i↑ j↓ i↓ j↑ 2 hiji hiji „ « 2 X 1 − ~ni · ~nj |Ψ({nˆi })| ≡ exp(−MH ) H = − ln cl cl 2 hiji

A classical antiferromagnetic model at temperature T = 1/M Arovas et al. 1988 Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions

AKLT models on the tree (AKLT, 1987)

„ « P 1−~ni ·~nj Hcl = − hiji ln 2

M “ 1−~n0·~n1 ” T (~n0,~n1) = 2

0 1 R 1 z−1 ψ (~n0) = Z D~n1 ··· D~nz−1 T (~n0,~n1)ψ (~n1) ··· T (~n0,~nz−1)ψ (~nz−1)

High temperature fixed point ψ(~n) = 1: paramagnetic phase

Stability of the paramagnetic fixed point against AF ordering Assume all ψ are equal for a generation, then next generation is ψ ∝ (T ψ)z−1 P∞ Expand in spherical harmonics ψ(~n) = 1 +  l6=0,m clmψlm(~n) P [Γ(M+1)]2 (T ψ)(~n) = λ0 +  l6=0,m λl clmψlm(−~n) with λl = Γ(M−l+1)Γ(M+l+2) −1 P 2 ψ(~n) = 1 + (z − 1)λ0 l6=0,m λl clmψlm(−~n) + O( )

λl 1 2 Instability if = : critical point for l = 1, Mc = λ0 z−1 z−2 d d “ λ1 ” “ M ” Decay of correlations: h~n0 · ~nd i ∝ = λ0 M+2 d 1 P d(i,j) P d−1 “ M ” χAF = N ij (−1) h~ni · ~nj i ∝ d z(z − 1) M+2 diverges at Mc Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions AKLT models on the random graph

High temperature: paramagnetic phase, ψ(~n) = 1 Lower the temperature: AF instability is avoided, incompatible with odd loops

Continue to lower the temperature: instability to SG phase, ψi (~ni ) is a random variable Spin frozen in random direction; degenerate states corresponding to different patterns

„ «2d 1 X 2 X d−1 M χ = [h~ni · ~nj i] ∝ z(z − 1) SG N M + 2 ij d

“ ”2 Finite only if 1 M 1, or M M = √ 2 z−1 M+2 < < SG z−1−1 Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Phase diagram

Blue line: AF instability on the tree Green line: SG instability on the random graph Dots: critical states z = 2 corresponds to one dimensional chain: no

C.R.Laumann, S. A. Parameswaran, S.L.Sondhi, FZ, PRB 81, 174204 (2010) Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Spectrum of the quantum spin glass phase

AKLT mapping provides a classical vector model with a Spin Glass phase Classical Gibbs measure decomposes into a collection of clustering pure states PN α = 1 ... N with essentially disjoint support: P({nˆ}) = α=1 wαPα({nˆ}) Corrections are exponentially small in N i In each of these states, the ψα(ˆni ) – and thus the local magnetizations – are macroscopically different: qEA 6= 0 We want to show that the quantum AKLT ground state |Ψi itself is a superposition over a collection of degenerate ground states |Ψαi each of which PN √ corresponds to one of the classical clustering states: |Ψi = α=1 wα|Ψαi

Argument 2 1 Define quantum states |Ψαi by |h{nˆi }|Ψαi| = Pα({nˆi }) 2 Since different states have different support (exponentially in N). Therefore −N hΨα|O|Ψβ i ∼ e for any operator. Different states are almost orthogonal. √ 3 PN Define |Φi = α=1 wα|Ψαi. Interference terms are negligible: PN hΦ|O|Φi ∼ α=1 wαhΨα|O|Ψαi = hΨ|O|Ψi ⇒ |Φi = |Ψi 4 PN −N −N 0 = hΨ|H|Ψi = α=1 wαhΨα|H|Ψαi + O(e ) ⇒ hΨα|H|Ψαi ∼ e

G.Biroli, C.Chamon, FZ, PRB 78, 224306 (2008) Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Outline

1 Introduction Classical spin glasses Random lattices and the cavity method Ising antiferromagnet on a random graph

2 AKLT states on the Bethe lattice AKLT states: mapping on a classical model AKLT models on the tree AKLT models on the random graph Phase diagram Spectrum of the quantum spin glass phase

3 A lattice model with a superglass phase Extended Hubbard model on a random graph Results A variational argument

4 Conclusions Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions A lattice model with a superglass phase

Extended Hubbard model on a regular random graph at half-filling and U = ∞: P † † P P H = −t hi,ji(ai aj + aj ai ) + V hi,ji ni nj − i µni

We study the model on a regular random graph of L sites and connectivity z = 3

Frustrated antiferromagnetic interaction Solution for L → ∞ possible via the cavity method Classical model (t = 0): spin glass transition (like Sherrington-Kirkpatrick model) RSB, many degenerate glassy states

Methods Quantum cavity method: solution for L → ∞ Canonical Worm Monte Carlo: limited to L < 240 by ergodicity problems Variational calculation + Green Function Monte Carlo (for illustration)

G. Carleo, M. Tarzia, FZ, PRL 103, 215302 (2009) Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Results at half-filling and t = 1

0.4

0.3

/ρ Condensate fraction: c 0.2 SGF ρ + 2 ρc = lim|i−j|→∞ ai aj = a 0.1 | h i |

0 0 1 2 3 4 5 0.2

0.15 β = 1.5 β = 2 β = 3 EA 0.1 Edwards-Anderson order parameter: q β = 4 β = 5 1 P 2 qEA = i (δni ) δni = ni ni 0.05 L − h i

0 012345 0.12 L = 240 L = 160 0.09 L = 80

SG 0.06 Spin-Glass susceptibility: χ = 1 R β d P n ( ) n (0) 2 0.03 χSG L 0 τ i,j δ i τ δ j 2.5 2.75 3 h i 0 0 0.5 1 1.5 2 2.5 3 3.5 V Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Phase diagram

Phase diagram at half-filling and t = 1

0.8 0.7 0.6 0.5 T 0.4 Superfluid Glass 0.3 0.2 Superglass 0.1 0 0.5 1 1.5 2 2.5 3 3.5 4 V Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions A variational argument

Many spin glass states Each spin glass state breaks translation invariance P Simplest variational wavefunction: hn|Ψi = exp( i αi ni ) A different set of parameters for each spin glass state Optimization of the parameters αi depends on the initial condition

0.8 0.6 0.4 0.2 /ρ i 0 δn 0.2 − 0.4 − 0.6 − 0.8 − 0 10 20 30 40 50 60 70 80 i

Stability of the glass state Green Function Monte Carlo: |Ψ(τ)i = e−τH |Ψi The time τ needed to escape from the initial state |Ψi increases with system size Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Outline

1 Introduction Classical spin glasses Random lattices and the cavity method Ising antiferromagnet on a random graph

2 AKLT states on the Bethe lattice AKLT states: mapping on a classical model AKLT models on the tree AKLT models on the random graph Phase diagram Spectrum of the quantum spin glass phase

3 A lattice model with a superglass phase Extended Hubbard model on a random graph Results A variational argument

4 Conclusions Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Conclusions

1 AKLT states on regular random graphs provide a model for quantum spin glasses 2 Spin glass phase is characterized by many almost degenerate ground states 3 No tunnelling between SG states, matrix element is exponentially small in N 4 Extended Hubbard model on random graph shows a super-spin glass phase

Future work: identify a more realistic model!