Bosonic Mott Transitions on the Triangular Lattice

1
Two studies of frustration on the triangular
lattice
1. Bose Mott transitions on the Triangular Lattice
2. Is there room for exotica in Cs2CuCl4?
Investigating the 1d-2d crossover
KIAS Workshop on Emergent Quantum Phases in
Strongly Correlated Electronic Systems, October
2005.
2
Frustrating Mott Transitions on the Triangular
Lattice

• Leon Balents
• Anton Burkov

• Roger Melko
• Arun Paramekanti
• Ashvin Vishwanath
• Dong-ning Sheng

ORNL
cond-mat/0506457
cond-mat/0505258
3
Outline (1)

• XXZ Model
• persistent superfluidity at strong interactions
• supersolid
• Dual vortex theory of Mott transition
• Field theory
• Mott phases in (dual) mean field theory
• Supersolid as melted Mott state, and a candidate
  for deconfined Mott criticality

4
Bose Mott Transitions

• Superfluid-Insulator transition of bosons in a
  periodic lattice now probed in atomic traps

Filling f1 Unique Mott state w/o order, and LGW
works
f ? 1 localized bosons must order
Interesting interplay between superfluidity and
charge order!
M. Greiner, O. Mandel, T. Esslinger, T. W.
Hänsch, and I. Bloch, Nature 415, 39 (2002).
5
Triangular Lattice

• Hard-core no double occupancy

• S1/2 XXZ model with FM XY and AF Ising exchange

Ising particle-hole symmetric

• Frustration Cannot satisfy all Jz interactions
• - no simple crystalline states near
  half-filling

any solid order determined by kinetic energy
6
Supersolid Phase

• Recent papers on XXZ model find supersolid phase
  near ½-filling

• D. Heidarian, K. Damle, cond-mat/0505257
• R. G. Melko et al, cond-mat/0505258
• M. Troyer and S. Wessel, cond-mat/0505298

T0
ODLRO
½ filling

DLRO
from M. Troyer and S. Wessel
from Melko et al
7
Supersolid Phases
0
antiferromagnetic
ferrimagnetic
spontaneous magnetization phase separation
superfluid on ¼ ¼-filled honeycomb interstitial
lattice of 1/3-triangular solid
superfluid on 1/2 -filled triangular
interstitial lattice of honeycomb
antiferromagnetic solid
particle-hole transform not identical
expect stabilized by 2nd neighbor hopping
8
Surprises

• Superfluidity survives even when VJz ! 1 !

Symptomatic of frustration superfluid exists
within extensively degenerate classical
antiferromagnetic ground state Hilbert space
topology of this space leads to proof of
diagonal LRO at Jz 1

• Persistent superfluidity is exceedingly weak

close to Mott insulator

• Energy difference between 2 supersolid states is
  nearly unobservable

9
Mott Transition

• Goal continuum quantum field theory
• - describes particles condensing at QCP

• Conventional approach use extra/missing bosons
• Leads to LGW theory of bose condensation
• Built in diagonal order, the same in both Mott
  and SF state

vortex
anti-vortex

• Dual approach use vortices/antivortices of
  superfluid
• non-LGW theory, since vortices are non-local
  objects
• focuses on Mott physics, diagonal order is
  secondary
• theory predicts set of possible diagonal orders

10
Duality
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett.
47, 1556 (1981) D.R. Nelson, Phys. Rev. Lett.
60, 1973 (1988) M.P.A. Fisher and D.-H. Lee,
Phys. Rev. B 39, 2756 (1989)

• Exact mapping from boson to vortex variables

• Dual magnetic field B 2?n

• Vortex carries dual U(1) gauge charge

• All non-locality is accounted for by dual U(1)
  gauge force

11
Dual Theory of QCP for f1

• Two completely equivalent descriptions
• - really one critical theory (fixed point) with 2
  descriptions

particles bosons
particles vortices
superfluid
Mott insulator
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett.
47, 1556 (1981)

• Real significance Higgs mass indicates
  insulating dielectric constant

12
Non-integer filling f ? 1

• Vortex approach now superior to Landau one
• need not postulate unphysical disordered phase

C. Lannert, M.P.A. Fisher, and T. Senthil, Phys.
Rev. B 63, 134510 (2001) S. Sachdev and K.
Park, Annals of Physics, 298, 58 (2002)

• Vortices experience average dual magnetic field
• - physics phase winding

Aharonov-Bohm phase in vortex wavefunction
encircling dual flux
2? winding of boson wavefunction on encircling
vortex

• Vortex field operator transforms under a
  projective representation of lattice space group

13
Vortex Degeneracy

• Non-interacting spectrum honeycomb Hofstadter
  problem

• Physics magnetic space group

and other PSG operations

• For fp/q (relatively prime) and q even (odd),
  all representations are at least 2q
  (q)-dimensional

• This degeneracy of vortex states is a robust
  property of a superfluid (a quantum order)

14
1/3 Filling

• There are 3 vortex flavors ?1,?2,?3 with the
  Lagrangian

• Dual mean-field analysis predicts 3 possible
  Mott phases

vlt0
vgt0
1/3 solid of XXZ model
Expect deconfined Mott QCP with fluctuations
included
15
½-Filling

• 2 2 4 vortex flavors with pseudo-spinor
  structure z?

- Space group operations appear as rotations
T2
T3
R2?/3
T1
T3
T2
T1
R2?/3
ordering wavevectors

• Order parameters

dz
XXZ supersolid diagonal order parameter
dx
dy
16
Dual ½-Filling Lagrangian
quartic
8th and 12th order

• Emergent symmetry
• Quartic Lagrangian has SU(2)U(1)U(1)g
  invariance
• SU(2)U(1) symmetry is approximate near Mott
  transition
• Leads to skyrmion and vortex excitations of
  SU(2) and U(1) order parameters

• Mean field analysis predicts 10 Mott phases

- e.g. v,w1lt0
note similarity to XXZ supersolids
17
Hard-Spin Limit Beyond MF analysis

• Example v,w1lt0

• Solution

- Z2 gauge redundancy

• Hard-spin (space-time) lattice model

• Z2 gauge field

• U(1) gauge field

• CP1 field

• XY field

18
Phase Diagram
tz
Z2
2-sublattice supersolid
Mott
Jz1 XXZ model
SS2
3-sublattice supersolid
SF
SS3
t?

• Blue lines LGW roton condensation transitions

• Red lines non-LGW transitions
• - Diagonal order parameters change simultaneously
  with the superfluid-insulator transition

• Should be able to understand supersolids as
  partially melted Mott insulators

19
Physical Picture
SS3 ferrimagnetic supersolid
ferrimagnetic columnar solid

• Superfluid to columnar VBS transition of
  ¼-filled honeycomb lattice!

20
Skyrmion

• VBS Order parameter pseudo-spin vector

(100)
(-100)
(010)
(0-10)
(001)
(00-1)

• Skyrmion
• -integer topological index
• finite size set by irrelevant cubic anisotropy

• Boson charge is bound to skyrmion!

NbQ
21
Mott-SS3 Criticality

• SS3-Mott transition is deconfined quantum
  critical point
• Non-compact CP1 universality class
• Equivalent to hedgehog-free O(3) transition

MotrunichVishwanath
skyrmions condense superfluid

• Disordering of pseudospin

• Hedgehogs skyrmion number changing events

22
Conclusions (1)

• Frustration in strongly interacting bose systems
  seems to open up a window through to observe a
  variety of exotic phenomena
• The simplest XXZ model exhibits a robust
  supersolid, and seems already quite close to
  non-trivial Mott state
• It will be interesting to try to observe Mott
  states and deconfined transitions by perturbing
  the XXZ model slightly (Chromium condensate?)
• Cartoon pictures of the supersolid and Mott
  phases may be useful in suggesting how this
  should be done

23
Is there room for exotica in Cs2CuCl4? Checking
the consistency of a prosaic 1d-2d crossover.

• L.B.
• O. Starykh, University of Utah

24
Cs2CuCl4 magnetic structure

• (Very good) approximate conservation of total Sa

25
2d Spin Liquid Physics?
R. Coldea et al, 2003.

• Broad inelastic neutron spectra have been
  interpreted as evidence for exotic physics
• - Scenario some exotic effective field theory
  governs intermediate energy behavior

E TN
E J, D?
E J
ordered
Decoupled chains
exotic

• Is there room?
• investigate possibility of direct crossover
• i.e. assume most relevant perturbations of
  decoupled chains drive ordering, and study
  resulting phase diagram (can be done by RGchain
  mean field theory)

26
Measurement of Couplings
R. Coldea et al, 2002.

• Single-magnon energies of fully-polarized state
  (in a-direction) exactly related to Hamiltonian
  parameters

J ¼ 0.37 meV

• Fit gives

J ¼ 0.3 J D ¼ 0.05 J
quasi-1d?

• Spatially anisotropic S1/2 antiferromagnet with
  non-negligible DM interaction

27
Low-T phase diagram
R. Coldea et al, 2001.

• Very different behavior for two field
  orientations indicates importance of DM
  interaction

transverse
longitudinal

• Phase diagram in transverse field roughly agrees
  with classical analysis

spiral (cone) observed here

• How well can we understand this phase diagram
  from a quasi-1d approach?

28
S1/2 AF Chain a primer

• Exact solution
• - Power-law spin (and dimerization) correlations

c.f. Affleck and Oshikawa, 1999
1
1/2
1
0
h/hsat
1/2
M

• XY AF correlations grow with h and remain
  commensurate
• Ising SDW correlations decrease with h and
  shift in k

• Even all amplitudes of these correlations are
  known (HikiharaFurusaki, 2004)

29
An Academic Problem

• Dh0, J J
• problem J is frustrated S? doesnt couple on
  neighboring chains
• naïve answer spiral state with exponentially
  small gap due to twist term
• True answer effective 2nd neighbor chain
  couplings generated (J)4/J3

Spatially anisotropic triangular lattice AF

• Probable GS four-fold degenerate diagonal
  dimer state

30
Why its academic

• Even D0.05J À (J)4/J3 (with constants)
• DM allows relevant coupling of Sb? and Sc? on
  neighboring chains
• immediately stabilizes spiral state
• small J perturbatively makes spiral weakly
  incommensurate

marginal dim 2
relevant dim 1
31
Transverse Field

• DM term becomes more relevant
• b-c spin components remain commensurate XY
  coupling of staggered magnetizations still
  cancels by frustration (reflection symmetry)
• Spiral (cone) state just persists for all fields.

Experiment
Order decreases with h here due to vanishing
amplitude as hsat is approached
Order increases with h here due to increasing
relevance of DM term
h
32
Longitudinal Field

• DM term Sb Sc Sz S
• wavevector mis-match for hgt0 DM irrelevant for
  
• With DM killed, sub-dominant instabilities take
  hold
• Two important couplings for hgt0

dim 12?R2
dim 1/2?R2
spiral cone state
collinear SDW

• Critical point

1
Predicts spiral state for hgthc ¼ 0.9 hsat ¼ 7.2 T
observed for hgt7.1T
1/2
1
0
h/hsat
33
Naïve Phase Diagram
T
collinear SDW
cone
(DM) cycloid
polarized
?
D/J 0.1
1
0.9
h/hsat
0
Experiment (on same scale)
spin liquid ?
no order observed (yet)
cycloid
S
h
break in scale

• Guess spin liquid region is really SDW with
  low ordering temperature
• - expected since amplitude of SDW interaction
  vanishes at h0, and relevance (in RG sense)
  decreases with h.

34
Beyond the naïve

• Collinear state is not truly collinear
• irrelevant DM involves
• effective oscillating field in c-direction with
  h Sb i ? 0 result is very elongated cycloid

• Collinear SDW state locks to the lattice at
  low-T
• irrelevant (1d) umklapp terms become relevant
  once SDW order is present (when commensurate)
• strongest locking is at M1/3 Msat

• Same uud state predicted by large-S expansion
  (Chubukov)

T
collinear SDW
cone
(DM) cycloid

• coincidentally uud state seems to occur near
  maximum Tc of collinear region

polarized
?
1
0.9
uud
h/hsat
0
0.1
35
Cs2CuBr4

• Isostructural to Cs2CuCl4 but believed to be
  less quasi-1d

• Magnetization plateau at M1/3 Msat observed for
  longitudinal but not transverse fields

T. Ono et al, 2004
(additional feature at 2/3 Msat)

• Commensurate Collinear order of some sort has
  apparently been observed in Cs2CuCl4 recently
  (Coldea, private communication)

36
Conclusions (Cs2CuCl4)

• A quasi-1d approach based on direct decoupled
  chain ! ordered crossover is quite successful in
  explaining low-energy behavior
• Work in progress to calculate ordering
  temperature, wavevector, spin stiffness, etc.
  quantitatively
• Appears likely the spin liquid state is just
  another ordered (quasi-collinear) phase with low
  Tc
• perhaps can observe uud commensurate state?
• Exotic scenario with intervening non-trivial
  fixed point seems rather unlikely

• A proper theoretical calculation (open problem!)
  of the inelastic spectrum in a 1d-2d crossover is
  sorely needed.