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Physical review. B, Condensed matter

SU(2) gauge invariance and order parameters in strongly coupled electronic systems

Elbio R. Dagotto, Eduardo Fradkin, Adriana Moreo

Published Aug 1, 1988• 147 citations
Publisher

Abstract

We show that the spin-\textonehalf{} Heisenberg model (the strong-coupling limit of the Hubbard model) is also the strong-coupling limit of an SU(2) lattice gauge theory with fermions. The local SU(2) gauge symmetry is manifest. The role of this gauge invariance is investigated in both the Hamiltonian and path-integral formulations. Off half-filling, our results reveal the existence of a lattice SU(2) bosonic matrix field which is a natural candidate for a condensate order parameter.

DOI
10.1103/physrevb.38.2926

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We show that the spin-\textonehalf{} Heisenberg model (the strong-coupling limit of the Hubbard model) is also the strong-coupling limit of an SU(2) lattice gauge theory with fermions. The local SU(2) gauge symmetry is manifest. The role of this gauge invariance is investigated in both the Hamiltonian and path-integral formulations. Off half-filling, our results reveal the existence of a lattice SU(2) bosonic matrix field which is a natural candidate for a condensate order parameter.
We study the symmetry properties of the Hubbard model with spin-orbit interactions of Rashba and Dresselhaus type. These interactions break the rotational symmetry in spin space, so that the magnetic order cannot be excluded by using the Bogoliubov inequality method. Nevertheless, we rigorously show that the existence of the magnetic long-range orders may be ruled out when the Rashba and Dresselhaus coupling constants are equal in modulus, whereas the $\ensuremath{\eta}$ pairing can be always ruled out, regardless of the microscopic parameters of the model. These results are obtained by imposing locally the SU(2) gauge symmetry on the lattice, and rewriting the spin-orbit interactions in such a way that they are included in the path order of the gauge field on the lattice.
Superconductivity in the strongly correlated electronic systems described by the Hubbard model for slightly less than half-filled band is shown to arise from the condensation of charge + e holons which are bosons. The holon effective mass is found to be very close to that of the band electrons and they have a strong nearest neighbour repulsion induced by the spin fluctuations. They interact with an U(1) part of the collective SU(2) gauge field (named S-gauge field) of the spin system. This results in spontaneous symmetry breaking of the holon field and the appearance of the Anderson-Higgs phenomenon. The interesting possibility of the SU(2) gauge field-spinon coupled system retaining some gapless excitation spectrum inspite of the spontaneous symmetry breaking in the holon field is pointed out.
We demonstrate explicitly the SU(2) gauge invariance of the large-UUU limit of the Hubbard model, i.e., the Heisenberg model written in terms of electron operators with a constraint of one particle per site. We use this result to demonstrate the equivalence of two apparently very different mean-field theories.
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.
The spin-rotationally invariant SU(2)×U(1) approach to the Hubbard model is extended to accommodate the charge degrees of freedom.Both U(1) and SU(2) gauge transformation are used to factorize the charge and spin contribution to the original electron operator in terms of the emergent gauge elds.By tracing out gauge bosons the form of paired states is established and the role of antiferromagnetic correlations is explicated.We argue that in strongly correlated electron system collective instanton excitations of the phase eld (dual to the charge) arise with a great degree of stability, governed by gauge ux changes by an integer multiple of 2π.Furthermore, it is shown that U(1) and SU(2) gauge elds play a similar role as phonons in the BCS theory: they act as the the "glue" for fermion pairing.
The antiferromagnetic phase of two-dimensional (2D) and three-dimensional (3D) Hubbard model with nearest neighbor hopping is studied on a bipartite cubic lattice by means of the quantum $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ rotor approach that yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of the model and satisfy the Mermin-Wagner theorem. The collective variables for charge and spin are isolated in the form of the space-time fluctuating U(1) phase field and rotating spin-quantization axis governed by the SU(2) symmetry, respectively. As a result interacting electrons appear as composite objects consisting of bare fermions with attached U(1) and SU(2) gauge fields. An effective action consisting of a spin-charge rotor and a fermionic field is derived as a function of the Coulomb repulsion UUU and hopping parameter ttt. At zero temperature, our theory describes the evolution from a Slater (U⪡t)(U⪡t)(U⪡t) to a Mott-Heisenberg (U⪢t)(U⪢t)(U⪢t) antiferromagnet. The results for zero-temperature sublatice magnetization (2D) and finite temperature (3D) phase diagram of the antiferromagnetic Hubbard model as a function of the crossover parameter U∕tU∕tU∕t are presented and the role of the spin Berry phase in the interaction driven crossover is analyzed.