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Physical review. B./Physical review. B

Spin-charge separation and unconventional superconductivity in the t − J model on the honeycomb lattice

Jian-Jian Miao, Zheng-Yuan Yue, Hao Zhang, Wei-Qiang Chen, Zheng‐Cheng Gu

Published May 14, 2025• 1 citation
Publisher

Abstract

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 ($d+id$)-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 $t/J$ is large, the repulsive interaction dominates, leading to a non-Fermi liquid behavior; while in a moderate range of $t/J$, the attractive interaction dominates, leading to the SC order. Potential theoretical and experimental methods to validate the mechanism are discussed.

DOI
10.1103/physrevb.111.174518

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The physical nature of doped Mott insulators has been the subject of intensive study for decades. The single-band Hubbard model and the t\ensuremath-Jt\text{\ensuremath{-}}Jt\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\ensuremath-Jt\text{\ensuremath{-}}Jt\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.
The physical nature of doped Mott-insulator has been intensively studied for more than three decades. It is well known that the single band Hubbard model or ttt-JJJ model on the bipartite lattice is the simplest model to describe a doped Mott insulator. Unfortunately, the key mechanism of superconductivity in these toy models is still under debate. In this paper, we propose a new mechanism for the d+idd+idd+id-wave superconductivity (SC) that occurs in the small-doping region of the honeycomb lattice ttt-JJJ model based on the Grassmann tensor product state numerical simulation and spin-charge separation formulation. Moreover, in the presence of anti-ferromagnetic order, a continuum effective field theory for holons is developed near half-filling. It reveals the competition between repulsive and attractive holon interactions induced by spinon fluctuations and gauge fluctuations, respectively. At a large value of t/Jt/Jt/J, the repulsive interaction dominates, leading to the non-Fermi liquid like behavior; while in a moderate range of t/Jt/Jt/J, the attractive interaction dominates, leading to the SC order. Possible experimental detection of spin-charge separation phenomena is also discussed.
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.
Field-theoretic methods are used to investigate the large-UUU 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 and spin 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 antiferromagnetic phase (AF) as a function of hole doping. At half-filling, we deal with the underlying spin degrees of freedom of the quantum 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 $\ensuremath{\sigma}$ model with a topological Hopf term that describes the AF-VBS (valence bond solid) competition. Lastly, in the challenging doped regime, our approach allows the derivation of a $t\text{\ensuremath{-}}J$ 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.
Spin-triplet superconductivity is actively pursued in condensed matter physics due to the potential applications in topological quantum computations. The related pairing mechanism involving the interaction remains an important research topic. Here we propose a universal approach to obtain p-wave triplet superconductivity in the Hubbard models by simply changing the sign of the hopping amplitudes of the spin-down electrons, and apply it to three prototype two-dimensional lattices (honeycomb, square, and triangular). The parent Hamiltonian at half filling has long-range magnetic order, which is ferromagnetic in all three directions for the frustrated triangular lattices, and ferromagnetic (antiferromagnetic) in the xy plane (z direction) for the bipartite honeycomb and square lattices. The magnetic transitions occur at some critical interactions on honeycomb and triangular lattices, which are estimated by finite-size scalings. When the systems are doped, we find the triplet p-wave pairing is a dominating superconducting instability. We demonstrate its emergence is closely related to the strong ferromagnetic spin fluctuations induced by the doping. Our results provide an understanding of the microscopical triplet-pairing mechanism, and will be helpful in the search for spin-triplet superconducting materials.
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.
A real spin-charge separation scheme is found based on a saddle-point state of the t-J model. In the one-dimensional (1D) case, such a saddle-point reproduces the correct asymptotic correlations at the strong-coupling fixed point of the model. In the two-dimensional (2D) case, the transverse gauge field confining spinon and holon is shown to be gapped at finite doping so that a spin-charge deconfinement is obtained for its first time in 2D. The gap in the gauge fluctuation disappears at half-filling limit, where a long-range antiferromagnetic order is recovered at zero temperature and spinons become confined. The most interesting features of spin dynamics and transport are exhibited at finite doping where exotic residual couplings between spin and charge degrees of freedom lead to systematic anomalies with regard to a Fermi-liquid system. In spin dynamics, a commensurate antiferromagnetic fluctuation with a small, doping-dependent energy scale is found, which is characterized in momentum space by a Gaussian peak at (\ensuremath{\pi}/a,\ensuremath{\pi}/a) with a doping-dependent width (\ensuremath{\propto} \ensuremath{\surd}\ensuremath{\delta} , \ensuremath{\delta} is the doping concentration). This commensurate magnetic fluctuation contributes a non-Korringa behavior for the NMR spin-lattice relaxation rate. There also exists a characteristic temperature scale below thich a pseudogap behavior appears in the spin dynamics. Furthermore, an incommensurate magnetic fluctuation is also obtained at a finite energy regime. In the transport, a strong-range phase intereference leads to an effective holon Lagrangian which can give rise to a series of interesting phenomena including linear-T resistivity and a T2{\mathit{T}}^{2}T2 Hall angle. We discuss the striking similarities of these theoretical features with those found in the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprates and give a consistent picture for the latter. Electronic properties like Fermi surface and superconducting pairing in this framework are also discussed.