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

• Many interesting systems near Mott transition– Cuprates– NaxCoO2· yH2O– Organics: θ-(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)

localized, delocalized, insulating (super)conducting

Competing Orders

• Theory of Mott transition must incorporate this constraint

(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

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

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

T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).

The cuprate superconductor Ca2-xNaxCuO2Cl2

Multiple order parameters: superfluidity and density wave.Phases: Superconductors, Mott insulators, and/or supersolids

“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

Fluctuating Order in the Pseudo-Gap

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

Filling f=1: Unique Mott state w/o order, and LGW works

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

f ≠ 1: localized bosons must order

(connection to electronic systems later)

Is LGW all we know?• Physics of LGW formalism is particle condensation

– Order parameter ψ† creates particle (z=1) or particle/antiparticle superposition (z=2) with charge(s) that generate broken symmetry.

– The ψ† particles are the natural excitations of the disordered state

– Tuning s|ψ|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!

Excitations:

†Particles ~ ψ Holes ~ ψ

Approach from the Insulator (f=1)

• 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• Exact mapping from boson to vortex variables.

• Vortex carries dual U(1) gauge

charge

• Dual magnetic field

B = 2πn

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

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);

Dual Theory of QCP for f=1

superfluidMott insulator

particles=vortices

particles=bosons

C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981);

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

• 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 2π 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 q-dimensional

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

A simple example: f=1/2• A simple physical interpretation is possible for f=1/2

-Map bosons to spins:spin-1/2 XY-symmetry magnet

= =

• Suppose

ψ=Nx+iNyNz=± 1

• Order in core: 2 “merons”

much more interpretation ofthis case:T. Senthil et al, Science 303, 1490 (2004).

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)

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

• Transform as Fourier components of density with

• Gauge-invariant bilinears:

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

Critical Theory and Order

• γmn and H.O.T.s constrained by PSG

• “Unified” competing orders determined by simple MFT

-always integer number of bosons per enlarged unit cell

f=1/4,3/4• Caveat: fluctuation effects mostly unknown

“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

• charge 1/q bosons

• Can be constructed in detail directly, generalizing f=1/2 T. Senthil et al, Science 303, 1490 (2004).

(with conserved gauge flux)

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 kinematicalvariables have “spin-charge separation”, i.e. bosonic holons and fermionic spinons

N.B. This does not mean we need presume any exotic phases where these are deconfined, since gauge fluctuations are included.

- Apply dual analysis to holons• Cuprates: model singlet formation

-Doped dimer model-Doped staggered flux states

(generally SU(2) MF states)

Singlet formation

Neel order

La2CuO4

spin liquid

x

gValencebondsolid (VBS)

Staggeredflux spinliquid

• Model for doped VBS-doped quantum dimer model

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

i j

• Dimer model = U(1) gauge theory• 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

x

g

• dSC for x>xc with vortex PSG identical to boson model with pair density

d-wave SC

1/161/32 1/8 xc

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

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

100Å

b7 pA

0 pA

Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integrated from 1meV to 12meV at 4K

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).

Vortices have halos with LDOS modulations at a period ≈ 4 lattice spacings

Doping Other Spin Liquids

• Very general construction of spin liquid states at x=0 from SU(2) MFT X.-G. Wen and P. A. Lee (1996)

X.-G. Wen (2002)

• Spinons fiα 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