Rotons and Vortices near a Continuous SuperfluidtoSupersolid transition

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Rotons and Vortices near a Continuous
Superfluid-to-Supersolid transition
Arun Paramekanti (University of Toronto)
CAP2006 (Brock University, June 13)
Funding NSERC, University of Toronto, A.P.Sloan
Foundation
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• Quantum Condensed Matter Physics
• Interplay of quantum mechanics and interactions
  with 1023 electrons/atoms
• Stable phases (eg superconductors, non-Fermi
  liquids, Mott insulators), their collective
  fluctuations, their response to perturbations
• Quantum phase transitions
• Out of equilibrium dynamics

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• Research interests
• High temperature superconductivity
• A. P., M. Randeria and N. Trivedi, PRB (2005)
• BCS-BEC crossover physics
• E. Zhao and A. P. (in preparation)
• Supersolids
• R. Melko, A. P., A. Burkov, D.N. Sheng, A.
  Vishwanath, L. Balents, PRL 95, 127207 (2005)
• E. Zhao and A. P., PRL 96, 105303 (2006)
• Exploring these issues also in cold atomic gases

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Bose condensed superfluid Delocalized bosons,
broken U(1) symmetry, persistent flow,
Crystal/Solid Density order, localized bosons,
broken translational symmetry,
Supersolid Density order (broken translation
symmetry) coexisting with superfluidity
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Crystal defect condensation and supersolidity in
4He
Vacancy
Interstitial
Cold
Warm
Cool
e
k
Fewer phonons, defect is more coherent, acquires
dispersion
Strongly interacting with phonons, localized
Quantum degeneracy Can defects Bose condense?
Old A.F. Andreev, I.M. Lifshitz, Sov. Phys. JETP
(1969) G. V. Chester, PRA (1970) A.J. Leggett,
PRL (1970) Others Recent B. Clark, D. Ceperley
(PRL 2006) P.W. Anderson, W.F. Brinkman, D. A.
Huse, Science (2005) Others
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Superfluidity in He4 in the high pressure solid
phase?
Pressurized He4
200 mK
E. Kim and M.Chan, Science (2004)
Also K. Shirahama, et al (APS 2006)
A.S.C. Rittner, J. Reppy, cond-mat/0604528 (SS
disappears with annealing)
Supersolid should show nonclassical rotational
inertia due to superfluid component remaining at
rest (Leggett, 1970)

• Supersolid?
• Microcrystallites? Superglass? (N. Prokofiev
  and coworkers)

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A simpler problem Lattice bosons
Bose condensed superfluid Delocalized bosons,
broken U(1) symmetry, persistent flow,
Crystal/Solid Broken lattice symmetries (eg
site-charge order, bond-charge order)
Supersolid Broken lattice symmetries coexisting
with superfluidity
Hard-core lattice bosons and quantum magnets
n(r) SZ(r) b(r) S(r)
Analogy between quantum fluids/crystals and
phases of magnetic systems T. Matsubara H.
Matsuda, Prog. Theor. Phys. (1956)
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Lattice models of supersolids Connection to
quantum magnets

• Borrow calculational tools from magnetism
  studies e.g., mean field theory, spin waves
• Visualize states e.g., superfluids and
  supersolids

Crystal SZ ,n order
Superfluid SX ,ltbgt order
Supersolid Both order
Breaks spin rotation (phase rotation) symmetry
Breaks both symmetries
Breaks lattice symmetries
K.S. Liu M.E. Fisher, J. Low Temp. Phys. (1973)
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Low temperature STM studies of cuprate
superconductors
Ca2-xNaxCuO2Cl2 (Tc8-15K)
4a0 x 4a0 unit-cell pattern from strong
interaction between holes. Poor screening at low
doping? Is there a nearby supersolid phase?
Nondispersive pattern over 10-100 meV range
T. Hanaguri, et al, Nature (2004), M. Franz (NV)
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Engineering quantum Hamiltonians Cold atoms in
optical lattices
Coherent Superfluid
Incoherent Mott insulator
Increasing lattice depth
M.Greiner, et al, Nature (2002)
Can one realize other interesting phases?
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Theoretical questions to try and address

• Is it useful to try and approach from the
  superfluid rather than from the crystal? What
  drives the transition?
• What is the nature of excitations in a superfluid
  close to the transition into a supersolid?
• Can we use any of this to understand,
  qualitatively, the STM experiments on the cuprate
  superconductors?

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Model Bosons on the Triangular Lattice
Boson model
Quantum spin model
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Bosons on the Triangular Lattice Superfluid
Superfluid

• How does the superfluid evolve with increasing
  interactions?

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Bosons on the Triangular Lattice Crystal and
Frustrated Crystal
3
3
1
1
Frustrated at ltngt1/2
Crystal at ltngt1/3

• How does the Ising frustration get resolved by
  quantum perturbations?

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Bosons on the Triangular Lattice Phase Diagram at
half-filling
Superfluid order
Crystal order
R.G. Melko, et al, PRL (2005)
D. Heidarian, K. Damle, PRL (2005)
(2m, -m, -m)
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Bosons on the Triangular Lattice Approaching from
the superfluid phase Spin wave theory
How do interactions affect the excitation
spectrum in the superfluid?
BZ
Q
-Q
Roton minimum hits zero energy, signalling
instability of superfluid towards a supersolid
phase
G. Murthy, D. Arovas and A. Auerbach, PRB (1997)
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Bosons on the Triangular Lattice Landau theory of
the transition and going beyond the instability

• Focus on low energy modes Q,-Q, 0
• Construct Landau theory with complex r, r

Pinning and selection of density pattern
R. Melko, et al (PRL 2005)
w lt 0 2m,-m,-m
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Bosons on the Triangular Lattice Full Phase
Diagram
S. Wessel, M. Troyer, PRL (2005) M. Boninsegni,
N. Prokofiev, PRL (2005)
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Superflow and rotons near the SF-SS phase
transition

• How does a uniform superflow affect the
  superfluid?
• Idea View as a toy problem to gain an
  understanding of the superfluid vortex

• Assume we are in the superfluid phase and study
  the excitation spectrum in the moving frame using
  spin wave theory

Phase twist wavevector K
Result Superflow can cause the roton gap to
collapse above a Jc
K.a 0
Why? Josephson coupling gets renormalized by
cos(K.a) in the moving frame, so interactions
effectively stronger
K.a 0.3
K.a 0.4
Related work Superflow decay near a
Superfluid-Mott transition, E. Altman, et al (PRL
2005)
E. Zhao and A. P., PRL (2006)
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Structure of a superfluid vortex near the SF-SS
transition

• Uniform superflow causes the collapse of the
  roton gap beyond a Jc Erot
• Superflow around a vortex Js 1/r, so a
  critical radius Rv 1 / Erot within which we
  expect supersolid order to emerge

Numerical solution of the classical XXZ model
with a vortex
SF
SS
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Making contact with superconductors A slave
boson approach
Write the electron as a product of spinon and
chargon
Uniform superconductor Spinons are in a BCS-type
paired state and the bosons are condensed at
zero momentum.
Supersolid Spinons are in a BCS-type paired
state and the bosons form a supersolid phase
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Checkerboard pattern in the vortex core in
Bi2Sr2CaCu2O8d (Tc89K)
Perhaps this signals proximity to a supersolid
phase? Going from a SF to a SS to an (ab)normal
core, so coherence peak can be well-defined
except at very short distances
J.E. Hoffman, et al, Science (2002)
Consequences

• The roton should be seen in inelastic X-ray
  scattering measurements if charge modulation
  significant
• Uniform superflow should also induce the
  supersolid phase close to the transition. Also
  suggested in SDWSC proposal E. Demler, S.
  Sachdev, Y. Zhang (PRL 2001)

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Tunneling density of states in the uniform
superfluid

• Within slave boson mean field theory

• Using the BCS form for the spinon Green function
  and the boson Green function for XXZ model at
  large-S

Elastic Electron spinon uniform condensate
Inelastic Electron spinon condensate
fluctuations
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Tunneling density of states in the uniform
superfluid
Elastic peak
Inelastic peak
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Tunneling density of states in the supersolid

• Slave boson decomposition and constraint

• Slave boson Hamiltonian

• Condensing bosons into a nonuniform superfluid
  state with site modulation will lead to site and
  bond modulations of the spinon Hamiltonian
• May be possible to understand bias reversal of
  the STM experiments discussed in D. Podolsky et
  al, PRB (2003)

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Summary

• Approaching the supersolid from the superfluid
  phase via roton condensation
• Superflow induced collapse of the roton gap
• Supersolid pattern around vortices in the
  uniform superfluid
• Formulation of electronic supersolids in a slave
  boson language
• Tunneling into the uniform superconductor
  (elastic/inelastic) and modulated superconductor
  (site/bond)

Future Directions

• Superflow in the supersolid phase and induced
  anisotropies
• Numerical calculations of tunneling in the
  supersolid using slave boson language
• Tunneling near a vortex state in the slave boson
  picture

Open issues

• Spin physics? (To some extent E. Demler, et al
  (2001)) Spinful rotons?
• Mott insulator physics? (L. Balents, et al, PRB
  (2005) L. Bartosch, S. Sachdev, cond-mat/0604105)

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Supersolids appear to exist in nature,
E. Kim and M.H.Chan (Nature,2004) T. Hanaguri, et
al (Nature,2004)