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Quantum Spin Glasses on the Bethe Lattice

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Quantum Spin Glasses on the Bethe Lattice Introduction AKLT states on the Bethe lattice A lattice model with a superglass phase Conclusions Quantum Spin Glasses 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 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 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, colloids, 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 y y 1 y y S+ = b"b# S− = b#b" Sz = 2 (b"b" − b#b#) 1 y y Total spin S = 2 (b"b" + b#b#) A general AKLT state can be written in terms of Schwinger bosons as Y “ y y y y ”M jΨ(M)i = bi"bj# − bi#bj" j 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 “ y y y y ”M jΨ(M)i = bi"bj# − bi#bj" j 0 i hiji Introduce on each site coherent states 1 y zM j~ni = (uµbµ) j 0 i p(zM) !) „ cos(θ=2) « with u = and ~n = uy~σu sin(θ=2) ei' Then „ «M=2 Y M Y 1 − ~ni · ~nj Ψ(fn^i g) = hfn^i gjΨi = (u u − u u ) = i" j# i# j" 2 hiji hiji „ « 2 X 1 − ~ni · ~nj jΨ(fn^i g)j ≡ 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 (~n0) = Z D~n1 ··· D~nz−1 T (~n0;~n1) (~n1) ··· T (~n0;~nz−1) (~nz−1) High temperature fixed point (~n) = 1: paramagnetic phase Stability of the paramagnetic fixed point against AF ordering Assume all are equal for a generation, then next generation is / (T )z−1 P1 Expand in spherical harmonics (~n) = 1 + l6=0;m clm lm(~n) P [Γ(M+1)]2 (T )(~n) = λ0 + l6=0;m λl clm lm(−~n) with λl = Γ(M−l+1)Γ(M+l+2) −1 P 2 (~n) = 1 + (z − 1)λ0 l6=0;m λl clm lm(−~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, (~n) = 1 Lower the temperature: AF instability is avoided, incompatible with odd loops Continue to lower the temperature: instability to SG phase, i (~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 = p 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 phase transition C.R.Laumann, S.
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