Interacting bosons on a lattice

1
Interacting bosons on a lattice
Fabien Alet LPT, Univ. Paul Sabatier
Toulouse In collaboration with Matthias
Troyer, Guido Schmid (ETH Zürich), Stefan Wessel
(ETH Zürich ? Stuttgart), Simon Trebst (ETH
Zürich ? KITP, UCLA), George Batrouni (Nice),
Erik Sorensen (Toulouse ?McMaster), Leonid
Pryadko (UC Riverside), Pinaki Sengupta (UC
Riverside ? UCLA), Sylvain Capponi (Toulouse),
Handong Chen (Stanford ? Urbana), Soucheng Zhang
(Stanford)
2
(A short personal review on)Interacting bosons
on a lattice
Fabien Alet LPT, Univ. Paul Sabatier
Toulouse In collaboration with Matthias
Troyer, Guido Schmid (ETH Zürich), Stefan Wessel
(ETH Zürich ? Stuttgart), Simon Trebst (ETH
Zürich ? KITP, UCLA), George Batrouni (Nice),
Erik Sorensen (Toulouse ?McMaster), Leonid
Pryadko (UC Riverside), Pinaki Sengupta (UC
Riverside ? UCLA), Sylvain Capponi (Toulouse),
Handong Chen (Stanford ? Urbana), Soucheng Zhang
(Stanford)
3
Plan

• Introduction
• Phase diagrams Phase transitions
• Cold bosons on optical lattice
• Supersolids

4
Part I
Introduction

• Models
• Techniques
• Useful observables

5
Motivations for interacting bosonic models

• Natural description of Superfluid-Insulator
  transition
• Dirsordered superconducting films
• Josephson junction arrays
• 4He adsorbed on porous media

• Recent experimental achievements
• Cold atomic gases loaded on optical lattices
• Supersolidity of (bulk) solid 4He

6
Bosonic models

• Bosonic Hubbard Model
• t Hopping term
• U Onsite repulsion. Keep bosons from
  condensing
• Vk k-th neighbour repulsion
• µ Chemical potential

• Phase approximation
• , integration of fluctuations of ?i around
  ni
• Quantum phase model
• ni and Fi canonical conjugates

7
Techniques

• Analytical
• Scaling Theory (Mean Field)
• Variational approach Gutzwiller-type wave
  functions

Fisher et al. (1989)
Rokhsar, Kotliar (1991), Krauth et al. (1992)

• Numerical
• Zero temperature Series expansion
• Quantum Monte Carlo Stochastic Series
  Expansion, Worm Algorithm

Monien et al.

• Useful observables
• Structure factor , Compressibility
• BEC coherence fraction
• Superfluid density Response function
• Relation to winding-numbers in a
• path-integral

Imaginary Time
Pollock, Ceperley (1987)
Space
8
Part II
T0 phase diagrams Quantum phase transitions

• Pure case
• Longer-range interaction
• Possible application to high-Tc

9
Softcore model

• Mean-field Phase diagram

Fisher et al. (1989)
10
Nature of quantum phase transitions

• Scaling theory predictions
• Transitions at and outside the tip of the lobe
  different
• 2nd order transitions of (d1)-XY universality
  class (tip of the lobe), mean-field type
  (outside)
• Finite size scaling analysis

11
Nature of quantum phase transitions

• Scaling theory predictions
• Transitions at and outside the tip of the lobe
  different
• 2nd order transitions of (d1)-XY universality
  class (tip of the lobe), mean-field type
  (outside)
• Finite size scaling analysis

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Hardcore model

• Double-occupancy strictly forbidden
• Relation to quantum spin ½ models
• t Exchange term, V Ising term, µMagnetic
  field
• Apparition of density waves insulating phases

V2 model
Hébert et al. (2002)
13
Hardcore model

• Half-filling
• Quarter-filling
• Relation to high-Tc STM experiments ?

V1 V2 model
STRIPED SOLID
SUPER-FLUID
CHECKERBOARD SOLID
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Effective bosonic model for high-Tc
superconductors

• Low energy approach
• Starting point t-J fermionic model on a 2x2
  plaquette
• Keep only low-lying states
• Singlet, 3 Triplets, Hole pair
• CORE Method Effective action between
• these states

Altman et al. (2002)
Phys. Rev. B 70, 024516 (2004)
15
Effective bosonic model for high-Tc
superconductors

• Remarks
• 4 colours of hardcore bosons
• Each site is a 2x2 plaquette
• Competition Magnon and Cooper pairs kinetic
  terms (Js and Jc) favour uniform phases (AF and
  SC). Repulsive interactions (Coulomb V1 et V2)
  favour Pair Density Waves (PDW).
• Parameters can be estimated from microscopic
  models (CORE)
• Solve the bosonic model by Mean-Field, QMC

Phys. Rev. B 70, 024516 (2004)
16
Global phase diagram at T0
Mean-Field

• Qualitative Phase diagram

SUPRA
AF
  PDW  
QMC
 PDW 
 PDW 
MOTT INSULATOR
1/4
3/8
1/2
1/8
nboson 0
1/4
dnhole 0
3/16
1/8
1/16
Phys. Rev. B 70, 024516 (2004)
17
Relation to STM experiments

• STM experiments
• Unit cell close to 4ax4a
• Observed in different compounds
• Close to doping x1/8
• Other explanations Hole crystal stabilized by
  Coulomb, Pinned vortices, Spiral state

Hoffman et al. (2002) Vernishin et al.
(2004) Hanaguri et al. (2004)
Ca(2-x)NaxCuO2Cl2

• Hole pair crystal

• Hole crystal ?

Bi2Sr2CaCu2O8
Inconsistent period
Consistent Period 4a x 4a
Lee et al. (2003)
Sachdev et al. (2005)
Kotov et al. (2006)
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Other (more exotic) phases

• Bosons in disordered potential
• Bose Glass New compressible insulator

BOSE GLASS
Fisher et al. (1989)

• Metallic phase of bosons at T0 ?
• Probably seen in experiments on thin metal alloy
  (MoGe) films
• Bose Metal Some proposals yet no real
  demonstration

Mason, Kapiltunik
Das, Doniach (2001) Phillips, Dalidovitch (2002)
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Part III
Cold bosons on optical lattices

• Introduction and experiments with trapped
  bosonic systems
• Detecting the quantum phase transition
• Identifying the local phases
• Local quantum criticality ?

20
Cold atomic gases

• Bose-Einstein Condensation in dilute atomic
  gases
• First observed in Rubidium

(1995) Cornell, Wieman, Ketterle

• Realization of a lattice for the atoms
• Standing waves created by lasers superimpose an
  optical lattice

2002
21
Trapped atoms in optical lattice

• Lattice depth controlled by laser intensity
• Quantum phase transition as laser intensity is
  varied
• Detected by measuring momentum distribution
  function

Greiner et al. (2002)
MOTT INSULATOR
SUPERFLUID

• Hundreds of proposals
• Quantum computing, Disordered systems,
  Bose-Fermi mixtures, artificial magnetic fields,
  frustrated spin models, lattice gauge theories,
  string theory
• The cold atom Hubbard toolbox
• Not so many experimental realizations

D. Jaksch and P. Zoller, Ann. Phys. 315, 52
(2005), cond-mat/0410614
I. Bloch, Nature Physics 1, 23 (2005)
22
Trapped bosonic systems

• All realistic systems are confined by a trapping
  potential
• Site dependent chemical potential
• Inhomogeneous systems

U/t increases

• Questions
• How to detect the quantum phase transition ?
• How to quantitatively identify the local phases
  ?
• Local quantum criticality ?

Phys. Rev. A 70, 053615 (2004)
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Detecting the quantum phase transition

• In the Momentum distribution function
• Original proposition of looking for secondary
  peaks wrong !
• Typical experimental data coherence peak
  height and width
• QMC calculations
• Flatter traps needed !

Kashurnikov et al. (2002)
Stöferle et al. (2004)
TRAPPED
UNIFORM
Phys. Rev. A 70, 053615 (2004)
See Gygi et al. (2006)
24
Describing the phases
Phys. Rev. A 70, 053615 (2004)

• Qualitatively describing the local phases
• Mott-insulating plateau in the center of the
  trap
• Surrounding by a superfluid shell (1d or 2d ?)
• Local quantum criticality at the boundary ?

• Identify local phases
• Local compressibility
• Mott plateau incompressible,
• Superfluid compressible
• Increased fluctuations near the boundary

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Coherence in the superfluid ring

• Equal-time Greens function in
  the superfluid ring
• Better fitted by a 2d form
• than 1d form
• Superfluid shell has a 2d behaviour

Phys. Rev. A 70, 053615 (2004)
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Absence of quantum criticality
Phys. Rev. A 70, 053615 (2004)

• Validity of local potential approximation
• Data collapse on a single curve
• This curve varies from trap to trap
• Not reducible to homogeneous case

• Absence of singular behaviour
• A singularity emerges in 2d uniform system
• Qualitatively different in a trap
• Need to quantify this (finite size analysis)

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Effective ladder model
Phys. Rev. A 70, 053615 (2004)

• Idea Describe the critical region by an
  inhomogeneous ladder
• Linearized leg potential
• Removes structural disorder
• Quantitative agreement with
• 2d trapped system
• Allows correct finite size scaling

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Effective ladder model

• Finite-size analysis of the ladder model
• No critical chain the ladder
• No divergence
• No sign of quantum criticality

• Absence of quantum criticality
• Due to inhomogeneity (finite gradient)
• Due to coupling to the rest of the system

• Possible scenario
• Single domain formation
• No critical slowing down

Phys. Rev. A 70, 053615 (2004)
29
Part IV
Supersolids

• Definitions
• Stability of supersolids in lattice models
• Supersolidity in 4He

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What is a supersolid (SS) ?

• Coexistence of density wave order and
  superfluidity
• Density wave order at a certain ordering vector
  k DIAGONAL ORDER
• BEC for large r OFF-DIAGONAL ORDER
• In practice, look for and

• Physical picture
• Start from crystalline state
• Dope with defects (vacancies,interstitials)
• These defects might condense
• Check for stability ! (Crystal melting Phase
  separation)

Andreev-Lifshitz (1969) Chester (1970)

• SS in nature ?
• Up to recently NO
• Supersolidity in 4He (2004-)
• Suggested supersolids in cold gases in optical
  lattices
• Suggested supersolids in quantum magnets, close
  to magnetization plateaus

Kim-Chan
Chromium and dipolar interactions
SrCu2(BO3)2 ?
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Search for supersolids in lattice models

• Hardcore bosons on square lattice
• Phase diagrams (QMC)

Hébert et al. (2002)

• Interpretation Free  superfluid channels 

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Stability of Checkerboard Supersolids (1)
Phys. Rev. Lett. 94, 207202 (2005)

• Doping a checkerboard solid (Hardcore case)
• With extra-bosons
• Better create a domain wall ! SS instable
  because solid instable ? PHASE SEPARATION
• Doping with holes Same effect (hole-boson
  symmetry) ? PHASE SEPARATION

Energy gain

• Can we still have a checkerboard supersolid ?
• Add diagonal hopping t Checkerboard SS
  becomes stable
• Release hardcore constraint Softcore model

G. Schmid (unpublished)
33
Stability of Checkerboard Supersolids (2)

• Doping with holes Same as hardcore case ?
  PHASE SEPARATION
• With bosons If , extra-bosons go
  to empty sublattice ? PHASE SEPARATION
• If , extra-bosons go to occupied
  sublattice ? SUPERSOLID POSSIBLE !
• Perturbative arguments confirmed by QMC
  Agreement with Gutzwiller variational approach

Phys. Rev. Lett. 94, 207202 (2005)
D. Kovrizhin et al., EPL 2005
34
Stability of Checkerboard Supersolids (3)

• No checkerboard found at half-filling (tip of
  the Mott lobe)
• Probable reason Formation of hole domain walls
  too likely
• Solution Go to more (extreme) softcore case ?
  Quantum rotor model
• Bonus 2d Superfluid-SuperSolid quantum phase
  transition 3d Ising universality class

F.A., unpublished
35
Supersolidity to solve classical frustration

• Hardcore Bosons on triangular lattice
• Half-filling Classical degeneracy lifted by
  quantum effects t towards SS (order by disorder)

V
t
Classical limit
?2/3
3 papers on the same day ! D. Heidarian et al.,
PRL 95, 127206 (2005) R.G. Melko et al., PRL 95,
127207 (2005) S. Wessel et al., PRL 95, 127207
(2005)
Ising AF Extensive entropy
?1/3
36
Nature of triangular supersolid

• Structure of supersolid ?
• Different SS patterns from above/below
  half-filling
• Sublattices densities
• (How) Do they meet ? New SS phase with
  sublattices ?
• Answer is 1st order phase transition between
  the two SS

M. Boninsegni et al., PRL 95, 237204 (2005)
Fraction of sites With ?gt1/2
37
Hardcore Bosons on kagomé lattice

• No supersolid but partially ordered Valence Bond
  Solid (VBS)
• Unusual continuous quantum phase transition
  between SF-VBS
• Different broken symetries in SF and VBS
• Landau theory generically predicts 1st order
  phase transition
• Here second order phase transition
• Physics of deconfined quantum critical points ?

3 Bosons delocalized on hexagons
S. Isakov et al., cond-mat/0602430
Senthil et al., Science (2004)
38
Supersolidity in 4He
Torsion rod

• Kim Chan (Penn. State Univ)
• Observation of Non-Classical Rotational Inertia
  in
• Solid 4He confined in porious media (Nature,
  2004)
• Bulk solid 4He (Science, 2004)
• Recently confirmed by other groups
• Possible similar observation in solid H2 (
  )

Torsion cell
Detection
Drive
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Which kind of Supersolidity for 4He ?

• At equilibrium, SS is (most certainly) due to
  defects (vacancies,interstitials)
• General arguments of Chester (1970), Prokofev
  Svistunov (PRL, 2005)

• Bulk solid 4He is (most certainly) commensurate
• Experiments extremely low concentration of
  defects at low T
• Numerics Activation energy of vacancies (15K),
  interstitials (48K) very high

Pederiva, Ceperley

• But
• Discussion on the validity of numerics
• If bulk solid 4He would be incommensurate, low T
  properties could be modified

Galli et al.
Phenomenological theory of Anderson, Brinkmann,
Huse (Science, 2005) could explain low T data
40
Answer Bulk solid 4He is not SS

• Recent Path Integral Monte Carlo Simulations
• Ceperley, Bernu (PRL, 2004)
• Clark, Ceperley (PRL, 2006),
• Boninsegni et al. (PRL, 2006)
• No ODLRO or Superfluidity

• How can the experiments be explained ??
• Zero point motion Do not transport mass !
• Bound State of interstitial vacancy
• Polycristallite sample with SF interfaces
  But same results for different setups !
• Non-equilibrium effects Superglass ??

Ma et al.
Burovski et al.
Boninsegni et al.
41
Superglass ?

• Recent calculations of Boninsegni et al. (PRL
  96, 105301 (2006))

Initial high-T random state rapid quench
below Tc Superfluidity !
Initial low-T crystalline state

• However Simulation non-local dynamics completely
  different from reality !

• But out-of-equilibrium effects probably
  important
• Very recent experiments (Rittner and Reppy,
  cond-mat/0604528)
• Supersolidity lost for slow T annealing (and
  probably higher quality samples)

42
Conclusions

• Interacting bosonic models
• Easy to define as fermions, Easy to solve
  (analytics, numerics) from fermions !
• Yet lot of interesting physics

• Cold atomic gases
• Recent experimental breakthroughs
• Inhomogeneities due to trapping (will be removed
  in the future )
• (Too !) Many theoretical proposals

• Supersolids in lattice models
• Present but quite hard to get (phase separation
  much simpler !)
• Supersolidity can resolve classical frustration

• Supersolidity in 4He
• Kim-Chan experiments still not microscopically
  understood
• Out-of-equilibrium effects ?