Putting Competing Orders in their Place near the Mott Transition

1. 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

2. Mott Transition localized, delocalized, • 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) insulating (super)conducting

3. Theory of Mott transition must incorporate this constraint 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)

4. 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, Nature430, 1001 (2004).

5. 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 (2004).

6. 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)

7. 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

8. 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, Nature415, 39 (2002).

9. 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!

10. Approach from the Insulator (f=1) Excitations: • The particle/hole theory is LGW theory! - But this is possible only for f=1

11. 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

12. 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

13. N.B.: vortex field is not gauge invariant - not an order parameter in Landau sense Dual Theory of QCP for f=1 • Two completely equivalent descriptions - really one critical theory (fixed point) with 2 descriptions particles= bosons particles= vortices Mott insulator superfluid C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); Real significance: “Higgs” mass indicates Mott charge gap

14. 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

15. Vortex Degeneracy Non-interacting spectrum = Hofstadter problem Physics: magnetic space group For f=p/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”)

16. Nz=§ 1 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).

17. 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

18. 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

19. 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

20. “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).

21. 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)

22. Singlet formation g spin liquid Valence bond solid (VBS) La2CuO4 x Staggered flux spin liquid • Model for doped VBS • doped quantum dimer model

23. i j 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 x>0 x=0: 2 vortices = vortex in A/B sublattice holons

24. d-wave SC Doped dimer model: results dSC for x>xc with vortex PSG identical to boson model with pair density g x 1/8 1/32 1/16 xc

25. 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

26. 7 pA 0 pA 100Å Vortex-induced LDOS of Bi2Sr2CaCu2O8+dintegrated from 1meV to 12meV at 4K Vortices have halos with LDOS modulations at a period ≈ 4 lattice spacings b J. HoffmanE. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).

27. 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 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

28. 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

29. pictures (leftover)