Transcript Putting Competing Orders in their Place near the Mott

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
Mott Transition
localized,
insulating
delocalized,
(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)
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
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).
Fluctuating Order in the Pseudo-Gap
“density” (scalar) modulations, ≈ 4 lattice spacing period
LDOS of Bi2Sr2CaCu2O8+d at 100 K.
M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science, 303, 1995
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)
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
Bose Mott Transitions
• This talk: Superfluid-Insulator QCPs of
bosons on (square) 2d lattice (connection to electronic
systems later)
Filling f=1: 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).
Is LGW all we know?
• Physics of LGW formalism is particle condensation
– Order parameter y creates particle (z=1) or
particle/antiparticle superposition (z=2) 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!
Approach from the Insulator (f=1)
Excitations:
Particles ~  †
Holes ~ 
• The particle/hole theory is LGW theory!
- But this is possible only for f=1
Approach from the Superfluid
• Focus on vortex excitations
vortex
anti-vortex
• Time-reversal exchanges vortices+antivortices
- Expect relativistic field theory for
• Worry: vortex is a non-local object, carrying
superflow
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
Dual Theory of QCP for f=1
• Two completely equivalent descriptions
- really one critical theory (fixed point)
particles=
with 2 descriptions
vortices
particles=
bosons
Mott insulator
superfluid
C. Dasgupta and B.I. Halperin,
Phys. Rev. Lett. 47, 1556 (1981);
• N.B.: vortex field is not gauge invariant
- not an order parameter in Landau sense
• Real significance: “Higgs” mass
indicates Mott charge gap
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
Vortex Degeneracy
• Non-interacting spectrum
= Hofstadter problem
• Physics: magnetic space group
and
• For f=p/q (relatively prime) all
representations are at least qdimensional
• This q-fold vortex degeneracy of vortex states is a
robust property of a superfluid (a “quantum order”)
A simple example: f=1/2
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 f=1/2
-Map bosons to spins:
=
=
spin-1/2 XY-symmetry magnet
• Suppose
=Nx+iNy
• Order in core: 2 “merons”
much more interpretation of
this case:
T. Senthil et al, Science 303, 1490 (2004).
Nz=§ 1
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
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
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
f=1/4,3/4
“Deconfined” Criticality
• Under some circumstances, these QCPs have
emergent extra U(1)q-1 symmetry
f=1/2, 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
f=1/2 T. Senthil et al, Science 303, 1490 (2004).
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
• Cuprates: model singlet formation
-Doped dimer model
-Doped staggered flux states
(generally SU(2) MF states)
need presume any exotic
phases where these are
deconfined, since gauge
fluctuations are included.
Singlet formation
g
spin
liquid
Valence
bond
solid (VBS)
La2CuO4
Neel order
x
Staggered
flux spin
liquid
• Model for doped VBS
-doped quantum dimer model
Doped dimer model
E. Fradkin and S. A. Kivelson, Mod. Phys. Lett. B 4, 225 (1990).
• Dimer model = U(1) gauge theory
i
j
• Holes carry
staggered U(1)
charge: hop only on
same sublattice
• Dual analysis allows Mott states with x>0
• x=0: 2 vortices = vortex in A/B sublattice holons
Doped dimer model: results
• dSC for x>xc with vortex PSG
identical to boson model with
pair density
g
d-wave SC
1/32 1/16
1/8 x
c
x
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
Vortex-induced LDOS of Bi2Sr2CaCu2O8+d integrated
from 1meV to 12meV at 4K
Vortices have
halos with LDOS
modulations at a
period ≈ 4 lattice
spacings
7 pA
b
0 pA
100Å
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).
Doping Other Spin Liquids
• Very general construction of spin liquid states at
X.-G. Wen and P. A. Lee (1996)
x=0 from SU(2) MFT
X.-G. Wen (2002)
• Spinons fi described by mean-field hamiltonian +
gauge fluctuations, dope b1,b2 bosons via duality
- Doped dimer model equivalent to Wen’s
“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
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
pictures (leftover)