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Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions

Bose-Einstein in Quantum

Giuseppe Carleo, Marco Tarzia, and Francesco Zamponi∗ Phys. Rev. Lett. 103, 215302 (2009) Collaborators: Florent Krzakala, Laura Foini, Alberto Rosso, Guilhem Semerjian

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

March 29, 2010 Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Outline

1 Motivations Supersolidity of He4 Helium 4: Monte Carlo results

2 The quantum cavity method Regular lattices, Bethe lattices and random graphs Recursion relations Bose-Hubbard models on the Bethe lattice

3 A lattice model for the superglass Extended Hubbard model on a random graph Results A variational argument

4 Discussion Disordered Bose-Hubbard model: the Bose 3D glass model with quenched disorder Quantum Biroli-M´ezard model: a lattice glass model

5 Conclusions Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Outline

1 Motivations Supersolidity of He4 Helium 4: Monte Carlo results

2 The quantum cavity method Regular lattices, Bethe lattices and random graphs Recursion relations Bose-Hubbard models on the Bethe lattice

3 A lattice model for the superglass Extended Hubbard model on a random graph Results A variational argument

4 Discussion Disordered Bose-Hubbard model: the Bose glass 3D spin glass model with quenched disorder Quantum Biroli-M´ezard model: a lattice glass model

5 Conclusions Experimental Results

Discovery by Kim & Chan in He4 (Nature & Science 2004). Motivations• The quantumExperimental cavity method A lattice modelResults for the superglass Discussion Conclusions Reproduced after by many4 other groups. Motivations:•• Discovery supersolidity by Kim & Chan of Hein He4 (Nature & Science 2004). Non-classicalReproduced rotational inertia after observed by many in other groups. •4 He (Kim and Chan)

P=51 bars

P=51 bars

Possible interpretation: supersolidity

4 Supersolidity excluded in perfect He (Boninsegni, Ceperley et al.) Supersolidity strongly enhanced by fast quenches (Rittner and Reppy) History dependent response and some evidence for aging (Davis et al.)

Are BEC and superfluidity possible in disordered ? Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Helium 4: Monte Carlo results

Quantum Monte Carlo of He4 at high pressure P > 32 bar Quench from the down in the solid phase 2 hcp 0 glass superglass 1 -2 o -3 0.0359 A 0 -4

g(r) hcp glass 10 -6 1 log (n(r)) o -3 hcp 0.0292 A -8 0 0 2 4 6 8 10 12 14 o -10 0 5 10 r(A) o r (A) ODLRO observed in the one-particle Density-density correlations similar to the density matrix → BEC, superfluidity liquid (large Lindemann ratio) At P = 32 bar, n0 = 0.5% and ρs /ρ = 0.6 Boninsegni et al., PRL 96, 105301 (2006) Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Helium 4: Monte Carlo results

0 0 Amorphous condensate wavefunction: n(r − r ) ∼ n0φ(r)φ(r )

Plot of φ(x, y, z) on slices at fixed z Boninsegni et al., PRL 96, 105301 (2006) Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Many open problems

Difficulties of QMC Strong problems in the glass phase No access to real time dynamics: is this phase really metastable? Difficulty in determining the phase boundary: what is the phase diagram?

Open questions What are the physical ingredients (disorder, frustration)? What is the nature of the transition? Is it accompanied by slow dynamics in the liquid phase? Where does superfluidity come from? Strong interaction and disorder: need for a nonperturbative analysis

Our result – G.Carleo, M.Tarzia, FZ, PRL 103, 215302 (2009) A solvable model displaying a thermodynamic superglass phase Disorder is self-induced by frustration in absence of external potentials Realistic mechanism for He4 Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Outline

1 Motivations Supersolidity of He4 Helium 4: Monte Carlo results

2 The quantum cavity method Regular lattices, Bethe lattices and random graphs Recursion relations Bose-Hubbard models on the Bethe lattice

3 A lattice model for the superglass Extended Hubbard model on a random graph Results A variational argument

4 Discussion Disordered Bose-Hubbard model: the Bose glass 3D spin glass model with quenched disorder Quantum Biroli-M´ezard model: a lattice glass model

5 Conclusions Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Regular lattices, Bethe lattices and random graphs

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

The tree allows for a simple recursive solution but the frustration is lost Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Regular lattices, Bethe lattices and random graphs

Cavity approximation Replace the lattice by a random graph of the same connectivity:

The exact solution on the random graph can still be obtained (cavity method) Key property: loops have size log L, locally tree-like Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Regular lattices, Bethe lattices and random graphs

Cavity approximation Replace the lattice by a random graph of the same connectivity:

The exact solution on the random graph can still be obtained (cavity method) Key property: loops have size log L, locally tree-like

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 Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Regular lattices, Bethe lattices and random graphs

Cavity approximation Replace the lattice by a random graph of the same connectivity:

The exact solution on the random graph can still be obtained (cavity method) Key property: loops have size log L, locally tree-like

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 Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Classical ferromagnet on the Bethe lattice

P H = −J hi,ji σi σj

P βJσ(σ +···+σ ) Zg+1(σ) = Zg (σ1) ... Zg (σz−1) e 1 z−1 σ1,...,σz−1

Zg (σ) eβhg σ Normalized probability : ηg (σ) = = Zg (+)+Zg (−) 2 cosh(βhg )

Recursion on the effective magnetic field : z−1 hg+1 = β atanh (tanh(βJ) tanh(βhg ))

Fixed point when g → ∞ : (infinitesimal field to break the symmetry)

h = 0 at high temperature h 6= 0 at low temperature

True magnetic field can be recovered with z instead of z − 1 neighbors on the root Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Quantum lattice models

P † † H = −J hi,ji(ai aj + aj ai ) + V (n) = K + V (n) −βH P Z = Tr e |ni = |n1,..., nN i 11 = |nihn| n

Quantum model ⇔ Classical model with one additional dimension (imaginary time)

" M # M β D − β K E Z = lim P exp − P V (nα) Q nα|e M |nα+1 M→∞ M n1,...,nM α=1 α=1 becomes a path integral :

N R Q h R β i Z = Dni (τ) exp − 0 dτV (n(τ) W[n(τ)] i=1

where the ni (τ) are piecewise constant integer functions (periodic)

For a hopping h at time τh, ni → ni + 1, and one of the neighbors j has nj → nj − 1 p p Wh = J ni (τh) + 1 nj (τh). Q W[n(τ)] = h Wh Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Quantum lattice model on the Bethe lattice

Path integral construction valid for any graph

Classical degree of freedom = trajectories ni (τ) Recursive computation on trees, formally similar, but n(τ) (function) instead of σ ∈ {+1, −1} P βJσ(σ +···+σ ) Zg+1(σ) = Zg (σ1) ... Zg (σz−1) e 1 z−1 σ1,...,σz−1 becomes R Zg+1[n(τ)] = Dn1(τ)Zg [n1(τ)] ... Dnz−1(τ)Zg [nz−1(τ)] w[n, n1,..., nz−1]

The quantum cavity method

Functional recurrence equations for the local action Zg [n(τ)] = exp{−S[n(τ)]} A quadratic action gives back DMFT (correct for z → ∞) Laumann, Scardicchio, Sondhi (2008), Byczuk, Vollhardt (2008)

Numerical resolution for bosons and spins – Krzakala, Rosso, Semerjian, FZ (2008) Represent Z[n(τ)] by a weighted sample of trajectories Use the iteration equation to construct a new sample Works only if Z[n(τ)] is a probability Numerical method similar to QMC or GFMC, but here L = ∞ (no FSS) Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Bose-Hubbard models on the Bethe lattice

Test: study of the Mott transition in the Bose-Hubbard model G. Semerjian, M. Tarzia, FZ, PRB 80, 014524 (2009)

P † † U P P H = −J hi,ji(ai aj + aj ai ) + 2 i ni (ni − 1) − i µni

3 3 Cavity B-DMFT 2.5 Mean Field 2.5 Monte Carlo Monte Carlo Mean Field Cavity 2 2 µ/U Z=4 µ/U Z=6 1.5 1.5

1 1

0.5 0.5

0 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 J/U J/U

We can compute many observables

Local density hni i and condensate wavefunction hai i D † E Local green function ai (τ)ai (0) and correlation hni (τ)ni (0)i D † E Spatial correlations ai aj Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Outline

1 Motivations Supersolidity of He4 Helium 4: Monte Carlo results

2 The quantum cavity method Regular lattices, Bethe lattices and random graphs Recursion relations Bose-Hubbard models on the Bethe lattice

3 A lattice model for the superglass Extended Hubbard model on a random graph Results A variational argument

4 Discussion Disordered Bose-Hubbard model: the Bose glass 3D spin glass model with quenched disorder Quantum Biroli-M´ezard model: a lattice glass model

5 Conclusions Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions A lattice model for the superglass

Extended Hubbard model on a regular random graph at half-filling and U = ∞: P † † P P H = −J hi,ji(ai aj + aj ai ) + V hi,ji ni nj − i µni G. Carleo, M. Tarzia, FZ, PRL 103, 215302 (2009)

We study the model on a regular random graph of L sites and connectivity z = 3 No disorder in the interactions Frustration: loops of even and odd size Large loops, locally tree-like graph: Bethe lattice without boundary Solution for L → ∞ possible via the cavity method Classical model (J = 0): spin glass transition (like Sherrington-Kirkpatrick model) Glass phase is thermodynamically stable RSB, many degenerate glassy states, slow dynamics

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) Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Results at half-filling and J = 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 Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Phase diagram

Phase diagram at half-filling and J = 1

0.8 0.7 Liquid 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 Motivations The quantum cavity method A lattice model for the superglass Discussion 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 Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Outline

1 Motivations Supersolidity of He4 Helium 4: Monte Carlo results

2 The quantum cavity method Regular lattices, Bethe lattices and random graphs Recursion relations Bose-Hubbard models on the Bethe lattice

3 A lattice model for the superglass Extended Hubbard model on a random graph Results A variational argument

4 Discussion Disordered Bose-Hubbard model: the Bose glass 3D spin glass model with quenched disorder Quantum Biroli-M´ezard model: a lattice glass model

5 Conclusions Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Disordered Bose-Hubbard model: the Bose glass

P † † U P P H = −J hi,ji(ai aj + aj ai ) + 2 i ni (ni − 1) − i (µ + εi )ni

εi ∈ [−∆, ∆] quenched external disorder

Mott : one particle/site Strong localization ⇒ no BEC, ρc = 0 Zero compressibility Bose glass: additional defects Anderson localization, ρc = 0 Finite compressibility

No frustration, no RSB No slow dynamics

Fisher, Weichman, Grinstein, Fisher, PRB 40, 546 (1989)

Open question

Does the Bose Glass phase exist on the Bethe lattice? Ioffe and Mezard,´ arXiv:0909.2263 Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions 3D spin glass model with quenched disorder

P † † P H = −J hi,ji(ai aj + aj ai ) + hi,ji Vij (ni − 1/2)(nj − 1/2)

Tam, Geraedts, Inglis, Gingras, Melko, arXiv:0909.1845

QMC for the 3D cubic lattice with couplings Vij = ±V

0.9 a 0.8 0.7

0.6 qEA

0.5 3ρs 0.4 χSG/20 0.3

0.2 0.1 0 0.6 0.8 1 1.2 1.4 Same phase diagram V/t 0.8 But frustration is induced by b c 0.7 χSG/20 ρs quenched disorder qEA 0.6 qEA 3ρ χSG 0.5 s

0.4 0.3

0.2 0.1 0 0 0.002 0.004 0 0.5 1 1.5 2 1/N

Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Quantum Biroli-M´ezard model: a lattice glass model

Towards a more realistic model of structural glasses: the Biroli-M´ezard model P † † P P H = −J hi,ji(ai aj + aj ai ) + V i ni qi θ(qi ) − i µni P qi = j∈∂i nj − `

Classical model (J = 0): glass transition similarly to Hard Spheres

Random First Order Transition: discontinuous qEA, 2nd order Glassy phase both on 3D cubic and Bethe lattices (quantitatively similar) Self-generated disorder and RSB Very slow dynamics (divergence stronger than power-law) Believed (by some) to be in the same universality class of particle systems (e.g. Lennard-Jones, Hard Spheres): RFOT theory of the glass transition

Add quantum fluctuations (J 6= 0) A quantum glass transition? Slow dynamics? Aging? Nature of the transition (first or second order)? Preliminary indications of a quite complex phase diagram Work in progress – L. Foini, G. Semerjian, FZ Many body interaction could be realized in experiment with cold molecules Buchler, Micheli, Zoller, Nature 3, 726 (2007) Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Outline

1 Motivations Supersolidity of He4 Helium 4: Monte Carlo results

2 The quantum cavity method Regular lattices, Bethe lattices and random graphs Recursion relations Bose-Hubbard models on the Bethe lattice

3 A lattice model for the superglass Extended Hubbard model on a random graph Results A variational argument

4 Discussion Disordered Bose-Hubbard model: the Bose glass 3D spin glass model with quenched disorder Quantum Biroli-M´ezard model: a lattice glass model

5 Conclusions Motivations The quantum cavity method A lattice model for the superglass Discussion Conclusions Conclusions

Our results Quantum cavity method: a general method to deal with frustrated boson and spin systems Exact solution for Bethe lattice models for L → ∞ Semi-realistic model for interacting bosons displays a frustration-induced superglass phase Second order spin glass like transition Thermodynamically stable glass phase

Related works

Quantum Mode Coupling Theory – Reichmann, Miyazaki B-DMFT – Vollhardt, Hofstetter, et al. Monte Carlo – Boninsegni, Prokof’ev, Svistunov, et al. Variational calculations – Biroli, Chamon, FZ

Perspectives Experiments: He4 and cold molecules Lattice glass models for structural glasses – Foini, Semerjian, FZ Simulations of binary mixtures – Biroli, Carleo, Tarzia, FZ