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Time reversal symmetry breaking superconductivity in the honeycomb t-J model

Zheng‐Cheng Gu, Hong‐Chen Jiang, D. N. Sheng, Hong Yao, Leon Balents, Xiao-Gang Wen

Published Oct 6, 2011• 0 citations
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Abstract

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Dated: October 7, 2011)We report the theoretical discovery of a novel time reversal symmetry breaking superconductingstate in the t-Jmodel on the honeycomb lattice, based on a recently developed variational method- the Grassmann tensor product state approach. As a benchmark, we use exact diagonalization(ED) and density matrix renormalization (DMRG) methods to check our results on small clusters.Remarkably, we nd systematical consistency for the ground state energy as well as other physicalquantities, such as the staggered magnetization. At low doping, the superconductivity coexists withanti-ferromagnetic ordering.

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Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Dated: October 7, 2011)We report the theoretical discovery of a novel time reversal symmetry breaking superconductingstate in the t-Jmodel on the honeycomb lattice, based on a recently developed variational method- the Grassmann tensor product state approach. As a benchmark, we use exact diagonalization(ED) and density matrix renormalization (DMRG) methods to check our results on small clusters.Remarkably, we nd systematical consistency for the ground state energy as well as other physicalquantities, such as the staggered magnetization. At low doping, the superconductivity coexists withanti-ferromagnetic ordering.
The emergence of superconductivity in doped Mott insulators has been debated for decades. In this paper, we report the theoretical discovery of a time-reversal symmetry breaking superconducting ground state in the doped Mott insulator (described by the well known ttt-JJJ model) on honeycomb lattice, based on a recently developed variational method: the Grassmann tensor product state approach. As a benchmark, we use exact diagonalization and density-matrix renormalization methods to check our results on small clusters. We find systematic consistency for the ground-state energy as well as other physical quantities, such as the staggered magnetization. At low doping, the superconductivity coexists with antiferromagnetic ordering.
In view of quite recent experimental activities on magnetic and superconducting properties of honeycomb and hexagonal lattice based materials, in this thesis we have used field-theoretic and many-body methods to investigate magnetic and superconducting properties of the large-U Hubbard model on the honeycomb lattice at half-filling and in the hole-doped regime. Within the framework of a functional-integral approach, we obtain the Lagrangian density associated with the charge (Grassmann fields) and spin [SU(2) gauge fields] degrees of freedom. The Hamiltonian related to the charge degrees of freedom is exactly diagonalized. In the strong-coupling regime, we derive a perturbative low-energy theory suitable to describe the (quantum) magnetic and superconducting phases at half-filling and in the hole-doped regime. At half-filling, we deal with the underlying spin degrees of freedom of the quantum antiferromagnetic (AF) Heisenberg model by employing a second-order spin-wave analysis, in which case we have calculated the ground-state energy and the staggered magnetization; the results are in very good agreement with previous studies. Further, in the continuum, we derive a nonlinear σ-model with a topological Hopf term that describes the AF-VBS (valence bond solid) competition. In the challenging hole-doped regime, our approach allows the derivation of a t-J Hamiltonian, and the analysis of the role played by charge and spin quantum fluctuations on the ground-state energy and, particularly, on the breakdown of the AF order at a critical hole doping; the results are benchmarked against recent Grassmann tensor product state simulations. In addition, we have performed an extensive study of the electronic structure of the doped system for each competing phase: AF, ferromagnetic (FM), and (spin-singlet pairing) s-, dx₂₋ʏ₂ – and idxʏ -wave superconducting (SC) state induced by purely electronic effects. In this context, an energetic analysis of the ground state of these phases reveal that the AF order prevails for low hole doping, while a dominantly chiral dx₂₋ʏ₂ + idxʏ superconducting state was found in the vicinity of the Van Hove singularity (high hole doping). We also stress that a thermodynamic analysis of the superconducting phase shows that the critical temperature is directly related to the exchange constant J = 4t²/U, in which t denotes the hopping amplitude and U the on-site Coulomb repulsion of the Hubbard model (purely electronic origin). Remarkably, the competition between the AF and dx₂₋ʏ₂ + idxʏ SC phases takes place by the occurrence of a first-order transition accompanied by a spatial phase separation of the referred phases.
We have numerically evaluated the energy of several kinds of wave functions considered to be candidate ground states of the two-dimensional t-J model at various hole densities. We searched a parameter space which includes d-wave and s-wave superconductivity and spin-density-wave ordering as well as the projected Fermi-liquid state. Coexistence of different orderings, such as the s+id state and d-wave spin-density-wave state, were found to be stable states. We find a phase diagram in the density-t/J plane which has coexistence of antiferromagnetism and superconductivity at very low hole concentrations and superconductivity up to rather high values of density---about 40%. At intermediate concentrations, the time-reversal symmetry-breaking s+id state is found.
We study the ground state of the doped Hubbard model on the honeycomb lattice in the small-doping and strongly interacting region. The nature of the ground state after doping holes into antiferromagnetic Mott insulating states on the honeycomb lattice remains a long-standing unsolved issue, even though tremendous effort has been spent investigating this challenging problem. In this work, we employ two complementary, state-of-the-art, many-body computational methods: constrained path (CP) auxiliary-field quantum Monte Carlo (AFQMC) with self-consistent constraint and density matrix renormalization group (DMRG) methods. Systematic and detailed cross validations are performed between these two methods for narrow systems for which DMRG can produce reliable results. AFQMC is then utilized to study wider systems to investigate the thermodynamic limit properties. The ground state is found to be a half-filled stripe state in the small-doping and strongly interacting region. The pairing correlation shows ddd-wave symmetry locally but decays exponentially with the distance between two pairs.
The physical nature of doped Mott insulators has been the subject of intensive study for decades. The single-band Hubbard model and the $t\text{\ensuremath{-}}J$ model on a bipartite lattice are widely recognized as the simplest models to describe these materials. However, the key mechanism of superconductivity in these toy models remains unclear. Here we propose a novel mechanism for the (d+idd+idd+id)-wave superconductivity (SC) that occurs in the small-doping region of the honeycomb lattice $t\text{\ensuremath{-}}J$ model based on the Grassmann tensor product state numerical simulation and spin-charge separation formulation. In the presence of anti-ferromagnetic order near half-filling, we develop a continuum effective field theory for holons, which reveals the competition between repulsive and attractive holon interactions induced by spinon fluctuations and gauge fluctuations, respectively. When t/Jt/Jt/J is large, the repulsive interaction dominates, leading to a non-Fermi liquid behavior; while in a moderate range of t/Jt/Jt/J, the attractive interaction dominates, leading to the SC order. Potential theoretical and experimental methods to validate the mechanism are discussed.
We study the honeycomb lattice $t\text{\ensuremath{-}}J$ model using the fermionic tensor network approach. By examining the ansatz with various unit cells, we discover several different stripe states with different periods that compete strongly with uniform states. At very small doping $\ensuremath{\delta}<0.05$, we find almost degenerate uniform ddd-wave superconducting ground states coexisting with antiferromagnetic order. While at larger doping $\ensuremath{\delta}>0.05$, the ground state is an approximately half-filled stripe-ordered state, where the stripe period decreases with increasing hole doping $\ensuremath{\delta}$. Furthermore, the stripe states with the lowest variational energy always display ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing symmetry. The similarity between our results and those on the square lattice contributes to a more comprehensive understanding of doped Mott insulators.