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arXiv:1112.4562 · Physical Review B

Time-reversal symmetry breaking phase in the Hubbard model: A variational cluster approach study

Xiancong Lu, L. Chioncel, Enrico Arrigoni

Published Mar 19, 2012• 8 citations
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Abstract

We study the stability of the time-reversal symmetry breaking staggered-flux phase of a single band Hubbard model, within the variational cluster approach. For intermediate and small values of the interaction $U$, we find metastable solutions for the staggered-flux phase, with a maximum current per bond at $U\ensuremath{\approx}3.2$. However, allowing for antiferromagnetic and superconducting long-range order it turns out that in the region at and close to half filling the antiferromagnetic phase is the most favorable energetically. The effect of nearest-neighbor interaction is also considered. Our results show that a negative nearest-neighbor interaction and finite doping favors the stability of the staggered-flux phase. We also present preliminary results for the three-band Hubbard model obtained with a restricted set of variational parameters. For this case, no spontaneous time-reversal symmetry breaking phase is found in our calculations.

DOI
10.1103/physrevb.85.125117

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We study the stability of the time-reversal symmetry breaking staggered-flux phase of a single band Hubbard model, within the variational cluster approach. For intermediate and small values of the interaction UUU, we find metastable solutions for the staggered-flux phase, with a maximum current per bond at U\ensuremath≈3.2U\ensuremath{\approx}3.2U\ensuremath≈3.2. However, allowing for antiferromagnetic and superconducting long-range order it turns out that in the region at and close to half filling the antiferromagnetic phase is the most favorable energetically. The effect of nearest-neighbor interaction is also considered. Our results show that a negative nearest-neighbor interaction and finite doping favors the stability of the staggered-flux phase. We also present preliminary results for the three-band Hubbard model obtained with a restricted set of variational parameters. For this case, no spontaneous time-reversal symmetry breaking phase is found in our calculations.
The results of a semiclassical variational calculation of the t-J model are presented. The generalized staggered flux phases are introduced. It is found that these flux phases are stable for a large parameter range of the phase diagram. An interesting feature of these flux phases is that for small doping they have the best potential energy and for larger doping they have the best kinetic energy. It is also found that only those flux phases that do not break the local time reversal and parity symmetries are energetically favored. For sufficiently small J/t and doping the spiral phase is stable.
The results of a semiclassical variational calculation of the t-J model are presented. The generalized staggered flux phases are introduced. It is found that these flux phases are stable for a large parameter range of the phase diagram. An interesting feature of these flux phases is that for small doping they have the best potential energy and for larger doping they have the best kinetic energy. It is also found that only those flux phases that do not break the local time reversal and parity symmetries are energetically favored. For sufficiently small J/t and doping the spiral phase is stable.
The discovery of high temperature superconductivity inspired a number of novel proposals, one of which, put forward by C.M.Varma, involves the breaking of time-reversal symmetry to explain the physics of the underdoped pseudogap phase. It was proposed that time-reversal symmetry is spontaneously broken as a result of strong repulsion between the Cu-O electrons to form loop-currents in the system. In this work, we developed a general theory to study the quantum phase transitions in the 2 dimensional strongly interacting electronic systems in which time-reversal symmetry is spontaneously broken in the ground state. We first applied the theory of magnetic groups to identify electronic current-loop patterns in two physically relevant systems: (i) 2-band model involving spinless electrons on a honeycomb lattice with next-nearest-neighbor interactions; (ii) 3-band CuO2CuO_{2}CuO2​ model with and without lattice distortions. Next, by examining the correlation function within the standard ring and ladder Dyson series approximations, we identify the effective Hamiltonian with the relevant interactions responsible for creating low-energy fluctuations near the quantum critical point. The mean-field analysis of this effective Hamiltonian elucidated the fact that time-reversal symmetry breaking in a 2-band model is in the same universality class as the interband particle-hole pair condensation instability which occurs in the semi-conductors under large enough particle-hole attraction. Using Hubbard-Stratonovich transformation and functional integral method, we are able to investigate this instability and the static susceptibility in the condensed phase both in half-filling and doped case. Away from half-filling, because the condensates are metallic and couple to the gapless collective intraband particle-hole excitations, we find that the static susceptibility is generically negative as a result of this coupling, which implies that the condensates are unstable.
The variational cluster approximation is used to study the ground-state properties and single-particle spectra of the three-component fermionic Hubbard model defined on the two-dimensional square lattice at half filling. First, we show that either a paired Mott state or color-selective Mott state is realized in the paramagnetic system, depending on the anisotropy in the interaction strengths, except around the SU(3) symmetric point, where a paramagnetic metallic state is maintained. Then, by introducing Weiss fields to observe spontaneous symmetry breakings, we show that either a color-density-wave state or color-selective antiferromagnetic state is realized depending on the interaction anisotropy and that the first-order phase transition between these two states occurs at the SU(3) point. We moreover show that these staggered orders originate from the gain in potential energy (or Slater mechanism) near the SU(3) point but originate from the gain in kinetic energy (or Mott mechanism) when the interaction anisotropy is strong. The staggered orders near the SU(3) point disappear when the next-nearest-neighbor hopping parameters are introduced, indicating that these orders are fragile, protected only by the Fermi surface nesting.
Based on the self-energy-functional approach proposed recently [M. Potthoff, Eur. Phys. J. B 32, 429 (2003)], we present an extension of the cluster-perturbation theory to systems with spontaneously broken symmetry. Our method applies to models with local interactions and accounts for both short-range correlations and long-range order. Short-range correlations are accurately taken into account via exact diagonalization of finite clusters. Long-range order is described by variational optimization of a ficticious symmetry-breaking field. In comparison with related cluster methods, our approach is more flexible and, for a given cluster size, less demanding numerically, especially at zero temperature. An application of the method to the antiferromagnetic phase of the Hubbard model at half-filling shows good agreement with results from quantum Monte Carlo calculations. We demonstrate that the variational extension of the cluster-perturbation theory is crucial to reproduce salient features of the single-particle spectrum.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.