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arXiv:cond-mat/0609415 · Physical Review B

How to control pairing fluctuations: SU(2) slave-rotor gauge theory of the Hubbard model

Ki‐Seok Kim

Published Jun 7, 2007• 16 citations
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Abstract

We study how to incorporate Mott physics in the BCS-type superconductor, motivated by the fact that high ${T}_{c}$ superconductivity results from a Mott insulator via hole doping. The U(1) slave-rotor representation was proposed to take local-density fluctuations into account nonperturbatively, describing the Mott-Hubbard transition at half filling. Since this decomposition cannot control local pairing fluctuations, the U(1) slave-rotor representation does not give a satisfactory treatment for charge fluctuations. Extending the U(1) slave-rotor representation, we introduce an SU(2) slave-rotor representation to allow not only local-density fluctuations but also local pairing excitations. We find an SU(2) slave-rotor gauge theory of the Hubbard model in terms of two kinds of collective boson excitations associated with density and pairing fluctuations that interact with gapless fermion excitations via SU(2) gauge fluctuations. An interesting observation in this effective description is that phase fluctuations of fermion pairs arise as SU(2) gauge fluctuations. Thus, fermion-pairing excitations can be controlled by the dynamics of collective bosons in the SU(2) slave-rotor gauge theory. Performing the standard saddle-point analysis based on the SU(2) slave-rotor action, we find an interesting phase described by partial freezing of charge fluctuations near the Mott-Hubbard critical point, where local-density-fluctuation modes are condensed but local pair-excitation modes are gapped. Partial freezing of charge fluctuations causes fermion pairing to be incoherent as a result of reconciliation of superconductivity and Mott physics. The nature of this nonsuperconducting phase is identified with an anomalous metal due to the presence of incoherent pairing.

DOI
10.1103/physrevb.75.245105

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We study how to incorporate Mott physics in the BCS-type superconductor, motivated by the fact that high Tc{T}_{c}Tc​ superconductivity results from a Mott insulator via hole doping. The U(1) slave-rotor representation was proposed to take local-density fluctuations into account nonperturbatively, describing the Mott-Hubbard transition at half filling. Since this decomposition cannot control local pairing fluctuations, the U(1) slave-rotor representation does not give a satisfactory treatment for charge fluctuations. Extending the U(1) slave-rotor representation, we introduce an SU(2) slave-rotor representation to allow not only local-density fluctuations but also local pairing excitations. We find an SU(2) slave-rotor gauge theory of the Hubbard model in terms of two kinds of collective boson excitations associated with density and pairing fluctuations that interact with gapless fermion excitations via SU(2) gauge fluctuations. An interesting observation in this effective description is that phase fluctuations of fermion pairs arise as SU(2) gauge fluctuations. Thus, fermion-pairing excitations can be controlled by the dynamics of collective bosons in the SU(2) slave-rotor gauge theory. Performing the standard saddle-point analysis based on the SU(2) slave-rotor action, we find an interesting phase described by partial freezing of charge fluctuations near the Mott-Hubbard critical point, where local-density-fluctuation modes are condensed but local pair-excitation modes are gapped. Partial freezing of charge fluctuations causes fermion pairing to be incoherent as a result of reconciliation of superconductivity and Mott physics. The nature of this nonsuperconducting phase is identified with an anomalous metal due to the presence of incoherent pairing.
We propose one possible mechanism for an anomalous metallic phase appearing frequently in two spatial dimensions, that is, local pairing fluctuations. Introducing a pair-rotor representation to decompose bare electrons into collective pairing excitations and renormalized electrons, we derive an SU(2) gauge theory of the Hubbard model as an extended version of its U(1) gauge theory. Since our effective SU(2) gauge theory admits two kinds of collective bosons corresponding to pair excitations and density fluctuations, respectively, an intermediate phase appears naturally between the spin liquid Mott insulator and Fermi liquid metal of the U(1) gauge theory, characterized by softening of density-fluctuation modes as the Fermi liquid, but gapping of pair-excitation modes. We show that this intermediate phase is identified with an anomalous metallic phase because there are no electronlike quasiparticles although it is metallic.
Motivated by recent experiments on the triangular lattice Mott-Hubbard system $\ensuremath{\kappa}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}{\mathrm{Cu}}_{2}{(\mathrm{C}\mathrm{N})}_{3}$, we develop a general formalism to investigate quantum spin liquid insulators adjacent to the Mott transition in Hubbard models. This formalism, dubbed the SU(2) slave-rotor formulation, is an extension of the SU(2) gauge theory of the Heisenberg model to the case of the Hubbard model. Furthermore, we propose the honeycomb lattice Hubbard model (at half filling) as a candidate for a spin liquid ground state near the Mott transition; this is an appealing possibility, as this model can be studied via quantum Monte Carlo simulation without a sign problem. The pseudospin symmetry of Hubbard models on bipartite lattices turns out to play a crucial role in our analysis, and we develop our formalism primarily for the case of a bipartite lattice. We also sketch its development for a general Hubbard model. We develop a mean-field theory to describe spin liquids and some competing states, and apply it to the honeycomb lattice. On the insulating side of the Mott transition, we find an SU(2) algebraic spin liquid (ASL), described by gapless S=1∕2S=1∕2S=1∕2 Dirac fermions (spinons) coupled to a fluctuating SU(2) gauge field. This result contrasts with that obtained via a U(1) slave-rotor approach, which instead found a U(1) ASL. That formulation does not respect the pseudospin symmetry, and is therefore not correct on the honeycomb lattice. We construct a low-energy effective theory describing the ASL phase, the conducting semimetal phase and the Mott transition between them. This physics can be detected in numerical simulations via the simultaneous presence of substantial antiferromagnetic and valence-bond solid correlations. In the SU(2) ASL, these observables have slowly decaying fluctuations in space and time, described by power laws with the same critical exponent. We recover the results of the U(1) slave-rotor formulation in the presence of a strong breaking of pseudospin symmetry. Our analysis suggests that both a third-neighbor electron hopping, and/or pseudospin-breaking terms such as a nearest-neighbor density interaction, may help to stabilize a spin liquid phase.
We reinvestigate the bandwidth-control and doping-control Mott transitions (BCMT and DCMT) from a spin liquid Mott insulator to a Fermi liquid metal based on the slave-rotor representation of the Hubbard model [Florens and Georges, Phys. Rev. B 66, 165111 (2002); 70, 035114 (2004)] where the Mott transitions are described by softening of bosonic collective excitations. We find that the nature of the insulating phase away from half filling is different from that of half filling in the respect that a charge density wave coexists with a topological order (spin liquid) away from half filling because the condensation of vortices generically breaks translational symmetry in the presence of ``dual magnetic fields'' resulting from hole doping while the topological order remains stable owing to gapless excitations near the Fermi surface. Performing a renormalization group analysis, we discuss the role of dissipative gauge fluctuations due to the Fermi surface in both the BCMT and the DCMT.
Extending the U(1) slave-rotor representation\cite{FG_SRR} of the repulsive Hubbard model, we propose an SU(2) slave-rotor decomposition for the attractive Hubbard model, where the SU(2) slave-rotor variables represent order parameter fluctuations associated with superconductivity and charge density wave. This decomposition method allows us to modify the standard Hartree-Fock mean field theory by incorporating order parameter fluctuations on an equal footing. Deriving an effective SU(2) slave-rotor action from the attractive Hubbard model, and analyzing it at the mean field level, we demonstrate a second order phase transition driven by softening of the slave-rotor variables.
Fermionizing the charge sector and bosonizing the spin part in the SU(2) slave-boson theory, we derive an effective-field theory for dynamics of doped holes in the antiferromagnetically correlated spin background, where spin fluctuations are described by an SO(5) Wess-Zumino-Witten (WZW) theory while dynamics of doped holes is characterized by QED3{\text{QED}}_{3}QED3​ with a chemical-potential term. An important feature of our effective-field theory is the coupling term between valence-bond fluctuations and doped holes. Considering that valence-bond fluctuations are deeply related with monopole excitations of staggered U(1) gauge fields in the bosonic field theory for spin fluctuations, we demonstrate that hole dynamics helps deconfinement of bosonic spinons near the quantum critical point of the SO(5) WZW theory. We solve this effective-field theory in the Eliashberg framework and find non-Fermi-liquid physics in thermodynamics and transport, where z=3z=3z=3 criticality with dynamical exponent zzz plays an important role for hole dynamics. We discuss validity of our field theory, applying it to a doped spin chain and comparing it with the slave-fermion framework. Furthermore, we discuss instability of the anomalous metallic phase against superconductivity and density waves of doped holes, resulting from competition between gauge and valence-bond fluctuations.
We propose a mechanism of superconductivity from purely repulsive interactions in the strong-coupling regime, where the Bardeen-Cooper-Schrieffer mechanism such as the spin-fluctuation approach is difficult to apply. Based on the SU(2) slave-rotor representation of the Hubbard model, we find that the single energy scale for the amplitude formation of Cooper pairs and their phase coherence is separated into two energy scales, allowing the so-called pseudogap state where such Cooper pairs are coherent locally but not globally, interpreted as realization of the density-phase uncertainty principle. This superconducting state shows the temperature-linear decreasing ratio of superfluid weight, resulting from strong phase fluctuations.