Putting Competing Orders in their Place near the Mott Transition

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Putting Competing Orders in their Place near the
Mott Transition
Leon Balents (UCSB) Lorenz Bartosch (Yale) Anton
Burkov (UCSB) Predrag Nikolic (Yale) Subir
Sachdev (Yale) Krishnendu Sengupta (Toronto)
cond-mat/0408329, cond-mat/0409470, and to appear
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Mott Transition
localized,
delocalized,
insulating
(super)conducting

• Many interesting systems near Mott transition
• Cuprates
• NaxCoO2 yH2O
• Organics ?-(ET)2X
• LiV2O4
• Unusual behaviors of such materials
• Power laws (transport, optics, NMR) suggest QCP?
• Anomalies nearby
• Fluctuating/competing orders
• Pseudogap
• Heavy fermion behavior (LiV2O4)

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Competing Orders

• Usually Mott Insulator has spin and/or
  charge/orbital order

(LSM/Oshikawa)

• Luttinger Theorem/Topological argument some
  kind of order is necessary in a Mott insulator
  (gapped state) unless there is an even number of
  electrons per unit cell

• - Charge/spin/orbital order
• - In principle, topological order (not subject of
  talk)

• Theory of Mott transition must incorporate this
  constraint

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The cuprate superconductor Ca2-xNaxCuO2Cl2
Multiple order parameters superfluidity and
density wave. Phases Superconductors, Mott
insulators, and/or supersolids
T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M.
Azuma, M. Takano, H. Takagi, and J. C.
Davis, Nature 430, 1001 (2004).
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Fluctuating Order in the Pseudo-Gap
density (scalar) modulations, 4 lattice
spacing period
LDOS of Bi2Sr2CaCu2O8d at 100 K.

M. Vershinin, S. Misra, S. Ono, Y.
Abe, Y. Ando, and A. Yazdani, Science, 303, 1995
(2004).
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Landau-Ginzburg-Wilson (LGW) Theory

• Landau expansion of effective action in order
  parameters describing broken symmetries
• Conceptual flaw need a disordered state
• Mott state cannot be disordered
• Expansion around metal problematic since large
  DOS means bad expansion and Fermi liquid locally
  stable
• Physical problem Mott physics (e.g. large U) is
  central effect, order in insulator is a
  consequence, not the reverse.
• Pragmatic difficulty too many different orders
  seen or proposed
• How to choose?
• If energetics separating these orders is so
  delicate, perhaps this is an indication that some
  description that subsumes them is needed (put
  chicken before the eggs)

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What is Needed?

• Approach should focus on Mott localization
  physics but still capture crucial order nearby
• Challenge Mott physics unrelated to symmetry
• Not an LGW theory!
• Insist upon continuous (2nd order) QCPs
• Robustness
• 1st order transitions extraordinarily sensitive
  to disorder and demand fine-tuned energetics
• Continuous QCPs have emergent universality
• Want (ultimately not today?) to explain
  experimental power-laws

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Bose Mott Transitions

• This talk Superfluid-Insulator QCPs of bosons on
  (square) 2d lattice

(connection to electronic systems later)
Filling f1 Unique Mott state w/o order, and LGW
works
f ? 1 localized bosons must order
M. Greiner, O. Mandel, T. Esslinger, T. W.
Hänsch, and I. Bloch, Nature 415, 39 (2002).
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Is LGW all we know?

• Physics of LGW formalism is particle condensation
• Order parameter ?y creates particle (z1) or
  particle/antiparticle superposition (z2) with
  charge(s) that generate broken symmetry.
• The ?y particles are the natural excitations of
  the disordered state
• Tuning s?2 tunes the particle gap ( s1/2) to
  zero
• Generally want critical Quantum Field Theory
• Theory of particles (point excitations) with
  vanishing gap (at QCP)
• Any particles will do!

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Approach from the Insulator (f1)
Excitations

• The particle/hole theory is LGW theory!
• - But this is possible only for f1

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Approach from the Superfluid

• Focus on vortex excitations

vortex
anti-vortex

• Time-reversal exchanges vorticesantivortices
• - Expect relativistic field theory for

• Worry vortex is a non-local object, carrying
  superflow

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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

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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 Mott
  charge gap

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Non-integer filling f ? 1

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

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

Aharonov-Bohm phase
2? vortex winding
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Vortex Degeneracy

• Non-interacting spectrum Hofstadter problem

• Physics magnetic space group

• For fp/q (relatively prime) all representations
  are at least q-dimensional

and

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

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A simple example f1/2
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)

• A simple physical interpretation is possible for
  f1/2
• Map bosons to spins
• spin-1/2 XY-symmetry magnet

• Suppose

?NxiNy

• Order in core 2 merons

much more interpretation of this case
T. Senthil et al, Science 303, 1490 (2004).
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Vortex PSG

• Representation of magnetic space group

• Vortices carry space group and U(1) gauge
  charges
• - PSG ties together Mott physics (gauge) and
  order (space group)
• - condensation implies both Mott SF-I transition
  and spatial order

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Order in the Mott Phase

• Gauge-invariant bilinears

• Transform as Fourier components of density with

• Vortex condensate always has some order
• - The order is a secondary consequence of Mott
  transition

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Critical Theory and Order

• ?mn and H.O.T.s constrained by PSG

• Unified competing orders determined by simple
  MFT
• always integer number of bosons per enlarged unit
  cell

• Caveat fluctuation effects mostly unknown

f1/4,3/4
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Deconfined Criticality

• Under some circumstances, these QCPs have
  emergent extra U(1)q-1 symmetry

f1/2, 1/4
f ? 1/3

• In these cases, there is a local, direct,
  formulation of the QCP in terms of fractional
  bosons interacting with q-1 U(1) gauge fields

(with conserved gauge flux)

• charge 1/q bosons

• Can be constructed in detail directly,
  generalizing f1/2

T. Senthil et al, Science 303, 1490 (2004).
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Electronic Models

• Need to model spins and electrons
• - Expect bosonic results hold if electrons are
  strongly paired (BEC limit of SC)
• General strategy
• Start with a formulation whose kinematical
  variables have spin-charge separation, i.e.
  bosonic holons and fermionic spinons

- Apply dual analysis to holons
N.B. This does not mean we need presume any
exotic phases where these are deconfined, since
gauge fluctuations are included.

• Cuprates model singlet formation
• Doped dimer model
• Doped staggered flux states
• (generally SU(2) MF states)

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Singlet formation
g
spin liquid
Valence bond solid (VBS)
La2CuO4
x
Staggered flux spin liquid

• Model for doped VBS
• doped quantum dimer model

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Doped dimer model
E. Fradkin and S. A. Kivelson, Mod. Phys. Lett. B
4, 225 (1990).

• Dimer model U(1) gauge theory

• Holes carry staggered U(1) charge hop only on
  same sublattice

• Dual analysis allows Mott states with xgt0
• x0 2 vortices vortex in A/B sublattice holons

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Doped dimer model results

• dSC for xgtxc with vortex PSG identical to boson
  model with

pair density
g
x
1/8
1/16
1/32
xc
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Application Field-Induced Vortex in
Superconductor

• In low-field limit, can study quantum mechanics
  of a single vortex localized in lattice or by
  disorder
• - Pinning potential selects some preferred
  superposition of q vortex states

locally near vortex
Each pinned vortex in the superconductor has a
halo of density wave order over a length scale
the zero-point quantum motion of the vortex. This
scale diverges upon approaching the insulator
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Vortex-induced LDOS of Bi2Sr2CaCu2O8d integrated
from 1meV to 12meV at 4K
Vortices have halos with LDOS modulations at a
period 4 lattice spacings
b
J. Hoffman E. W. Hudson, K. M. Lang,
V. Madhavan, S. H. Pan, H. Eisaki, S.
Uchida, and J. C. Davis, Science 295, 466 (2002).
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Doping Other Spin Liquids

• Very general construction of spin liquid states
  at x0 from SU(2) MFT

X.-G. Wen and P. A. Lee (1996)
X.-G. Wen (2002)

• Spinons fi? described by mean-field hamiltonian
  gauge fluctuations, dope b1,b2 bosons via
  duality
• - Doped dimer model equivalent to Wens U1Cn00x
  state with gapped spinons
• - Can similarly consider staggered flux spin
  liquid with critical magnetism

preliminary results suggest continuous Mott
transition into hole-ordered structure unlikely
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Conclusions

• Vortex field theory provides
• formulation of Mott-driven superfluid-insulator
  QCP
• consequent charge order in the Mott state
• Vortex degeneracy (PSG)
• a fundamental (?) property of SF/SC states
• natural explanation for charge order near a
  pinned vortex
• Extension to gapless states (superconductors,
  metals) to be determined

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pictures (leftover)