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arXiv:0706.2657 · Physical Review B

Self-consistent slave rotor mean-field theory for strongly correlated systems

Erhai Zhao, Arun Paramekanti

Published Nov 1, 2007• 63 citations
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Abstract

Building on the work by Florens and Georges [Phys. Rev. B 70, 035114 (2004)], we formulate and study a self-consistent slave rotor mean-field theory for strongly correlated systems. This approach views the electron, in the strong correlation regime, as a composite of a neutral spinon and a charged rotor field. We solve the coupled spinon-rotor model self-consistently using a cluster mean-field theory for the rotors and various Ans\"atze for the spinon ground state. We illustrate this approach with a number of examples relevant to ongoing experiments in strongly correlated electronic systems such as (i) the phase diagram of the isotropic triangular lattice organic Mott insulators, (ii) quasiparticle excitations and tunneling asymmetry in the weakly doped cuprate superconductors, and (iii) the cyclotron mass of carriers in commensurate spin-density wave and U(1) staggered flux (or $d$-density wave) normal states of the underdoped cuprates. We compare the estimated cyclotron mass with results from recent quantum oscillation experiments on ortho-II $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6.5}$ by Doiron-Leyraud et al. [Nature (London) 447, 565 (2007)] which appear to find Fermi pockets in the magnetic field induced normal state. We comment on the relation of this normal ground state to Fermi arcs seen in photoemission experiments above ${T}_{c}$. This slave rotor mean-field theory can be generalized to study inhomogeneous states and strongly interacting models relevant to ultracold atoms in optical lattices.

DOI
10.1103/physrevb.76.195101

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Building on the work by Florens and Georges [Phys. Rev. B 70, 035114 (2004)], we formulate and study a self-consistent slave rotor mean-field theory for strongly correlated systems. This approach views the electron, in the strong correlation regime, as a composite of a neutral spinon and a charged rotor field. We solve the coupled spinon-rotor model self-consistently using a cluster mean-field theory for the rotors and various Ans"atze for the spinon ground state. We illustrate this approach with a number of examples relevant to ongoing experiments in strongly correlated electronic systems such as (i) the phase diagram of the isotropic triangular lattice organic Mott insulators, (ii) quasiparticle excitations and tunneling asymmetry in the weakly doped cuprate superconductors, and (iii) the cyclotron mass of carriers in commensurate spin-density wave and U(1) staggered flux (or ddd-density wave) normal states of the underdoped cuprates. We compare the estimated cyclotron mass with results from recent quantum oscillation experiments on ortho-II YBa2Cu3O6.5\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6.5}YBa2​Cu3​O6.5​ by Doiron-Leyraud et al. [Nature (London) 447, 565 (2007)] which appear to find Fermi pockets in the magnetic field induced normal state. We comment on the relation of this normal ground state to Fermi arcs seen in photoemission experiments above Tc{T}_{c}Tc​. This slave rotor mean-field theory can be generalized to study inhomogeneous states and strongly interacting models relevant to ultracold atoms in optical lattices.
The cluster slave-spin method is introduced to systematically investigate the solutions of the Hubbard model including the symmetry-broken phases. In this method, the electron operator is factorized into a fermionic spinon describing the physical spin and a slave-spin describing the charge fluctuations. Following the U(1)U(1)U(1) formalism derived by Yu and Si [Phys. Rev. B 86, 085104 (2012)], it is shown that the self-consistent equations to explore various symmetry-broken density wave states can be constructed in general with a cluster of multiple slave-spin sites. We employ this method to study the antiferromagnetic (AFM) state in the single band Hubbard model with the two- and four-site clusters of slave spins. While the Hubbard gap, the charge gap due to the doubly occupied states, scales with the Hubbard interaction UUU as expected, the AFM gap $\mathrm{\ensuremath{\Delta}}$, the gap in the spinon dispersion in the AFM state, exhibits a crossover from the weak- to strong-coupling behaviors as UUU increases. Our cluster slave-spin method reproduces not only the traditional mean-field behavior of $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}U$ in the weak-coupling limit, but also the behavior of $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{t}^{2}/U$ predicted by the superexchange mechanism in the strong-coupling limit. In addition, the holon-doublon correlator as functions of UUU and doping xxx is also computed, which exhibits a strong tendency toward the holon-doublon binding in the strong coupling regime. We further show that the quasiparticle weight obtained by the cluster slave-spin method is in a good agreement with the generalized Gutzwiller approximation in both AFM and paramagnetic states, and the results can be improved beyond the generalized Gutzwiller approximation as the cluster is enlarged from a single site to four sites. Our results demonstrate that the cluster slave-spin method can be a powerful tool to systematically investigate the strongly correlated system.
A slave-spin representation of fermion operators has recently been proposed for the half-filled single and multiband Hubbard model. We show that with the addition of a gauge variable, the formalism can be extended to finite doping. We solve the resulting spin problem using the cluster mean-field approximation. This approximation takes short-range correlations into account by exact diagonalization on the cluster, whereas long-range correlations beyond the size of clusters are treated at the mean-field level. In the limit where the cluster has only one site and the interaction strength UUU is infinite, this approach reduces to the Gutzwiller approximation. There are some qualitative differences when the size of the cluster is finite. We first compute the critical UUU for the Mott transition as a function of a frustrating nearest-neighbor interaction on lattices relevant for various correlated systems, namely, the cobaltates, the layered organic superconductors and the high-temperature superconductors. For the triangular lattice, we also study the extended Hubbard model with nearest-neighbor repulsion. In addition to a uniform metallic state, we find a $\sqrt{(3)}\ifmmode\times\else\texttimes\fi{}\sqrt{(3)}$ charge density wave in a broad doping regime, including commensurate ones. We find that in the large UUU limit, intersite Coulomb repulsion VVV strongly suppresses the single-particle weight of the metallic state.
The multiorbital Hubbard model is expressed in terms of quantum phase variables (``slave rotors'') conjugate to the local charge, and of auxiliary fermions, providing an economical representation of the Hilbert space of strongly correlated systems. When the phase variables are treated in a local mean-field manner, similar results to the dynamical mean-field theory are obtained, namely a Brinkman-Rice transition at commensurate fillings together with a ``preformed'' Mott gap in the single-particle density of states. The slave-rotor formalism allows to go beyond the local description and take into account spatial correlations, following an analogy to the superfluid-insulator transition of bosonic systems. We find that the divergence of the effective mass at the metal-insulator transition is suppressed by short range magnetic correlations in finite-dimensional systems. Furthermore, the strict separation of energy scales between the Fermi-liquid coherence scale and the Mott gap, found in the local picture, holds only approximately in finite dimensions, due to the existence of low-energy collective modes related to zero-sound.
In the generalized-tJtJtJ model the effect of a large local Coulomb repulsion is accounted for by restricting the Hilbert space to states with at most one electron per site. In this case the electronic system can be viewed in terms of holes hopping in a lattice of correlated spins, where holes are the carriers doped into the half-filled Mott insulator. To explicitly capture the interplay between the hole dynamics and local spin correlations we derive a formulation of the generalized-tJtJtJ model where doped carrier operators are used instead of the original electron operators. This ``doped carrier'' formulation provides a starting point to address doped spin systems, and we use it to develop a fully fermionic, mean-field description of doped Mott insulators. This mean-field approach reveals a mechanism for superconductivity---namely, spinon-dopon mixing---and we apply it to the $t{t}^{\ensuremath{'}}{t}^{\ensuremath{''}}J$ model as of interest to high-temperature superconductors. In particular, we use model parameters borrowed from band calculations and from fitting angle-resolved photoemission spectroscopy data to obtain a mean-field phase diagram that reproduces semiquantitatively that of hole- and electron-doped cuprates. The mean-field approach hereby presented accounts for the local antiferromagnetic and ddd-wave superconducting correlations which, we show, provide a rational for the role of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$ in strengthening superconductivity as expected by experiments and other theoretical approaches. As we discuss how ttt, ${t}^{\ensuremath{'}}$, and ${t}^{\ensuremath{''}}$ affect the phase diagram, we also comment on possible scenarios to understand the differences between as-grown and oxygen-reduced electron-doped samples.
A new mean field theory is proposed for the Mott transition in the Hubbard model based on the slave rotor representation of the electron operator. This theory provides a better description of the role of the long range charge correlation in the Mott insulating state and offers a good estimation of the critical correlation strength for the Mott transition. We have constructed a new variational wave function for the Mott insulating state based on this new slave rotor mean field theory. We find this new variational wave function outperforms the conventional Jastrow type wave function with long range charge correlator in the Mott insulating state. It predicts a continuous Mott transition with non-divergent quasiparticle mass at the transition point. We also show that the commonly used on-site mean field decoupling for the slave rotor corresponds to the Gutzwiller approximation for the Gutzwiller projected wave function with only on-site charge correlator, which can not describe the Mott transition in any finite dimensional system.
We generalize the dynamical-mean field (DMFT) approximation by including into the DMFT equations some length scale $\ensuremath{\xi}$ via a momentum dependent external self-energy ${\ensuremath{\Sigma}}_{\mathbf{k}}$. This external self-energy describes nonlocal dynamical correlations induced by the short-ranged collective spin density wave--like antiferromagnetic spin (or the charge density wave--like charge) fluctuations. At high enough temperatures these fluctuations can be viewed as a quenched Gaussian random field with a finite correlation length. This generalized $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ approach is used for the numerical solution of the weakly doped one-band Hubbard model with repulsive Coulomb interaction on a square lattice with the nearest and the next nearest neighbor hopping. The effective single impurity problem in this generalized $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ is solved by the numerical renormalization group. Both types of the strongly correlated metals, namely: (i) The doped Mott insulator and (ii) the case of the bandwidth $W\ensuremath{\lesssim}U$ (UUU---value of the local Coulomb interaction) are considered. The densities of states, the spectral functions, and the angle resolved photoemission spectra calculated within the $\mathrm{DMFT}+{\ensuremath{\Sigma}}_{\mathbf{k}}$ show a pseudogap formation near the Fermi level of the quasiparticle band.