Putting Competing Orders in their Place near the

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 Nax. Co. O 2¢ y. H 2 O Organics: -(ET)2 X Li. V 2 O 4 • Unusual behaviors of such materials – Power laws (transport, optics, NMR…) suggest QCP? – Anomalies nearby • Fluctuating/competing orders • Pseudogap • Heavy fermion behavior (Li. V 2 O 4)

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 Ca 2 -x. Nax. Cu. O 2 Cl 2 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 Bi 2 Sr 2 Ca. Cu 2 O 8+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 (2 nd order) QCPs – Robustness: • 1 st 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) 2 d 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 (» s 1/2) 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: • 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+i. Ny • 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 need presume any exotic • Cuprates: model singlet formation phases where these are deconfined, since gauge -Doped dimer model fluctuations are included. -Doped staggered flux states (generally SU(2) MF states)

Singlet formation g spin liquid Valence bond solid (VBS) La 2 Cu. O 4 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 • d. SC 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 Bi 2 Sr 2 Ca. Cu 2 O 8+d integrated from 1 me. V to 12 me. V at 4 K Vortices have halos with LDOS modulations at a period ≈ 4 lattice spacings 7 p. A b 0 p. A 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 b 1, b 2 bosons via duality - Doped dimer model equivalent to Wen’s “U 1 Cn 00 x” 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)