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arXiv:2608.09689

Robust spin pseudogap and spin-charge separation in the $σt$-$J$ model

Zheng-Yuan Yue, Jiaxin Zhang, Zheng‐Cheng Gu

Published Aug 10, 2026• 0 citations
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Abstract

The $σt$-$J$ model, obtained from the usual $t$-$J$ model by flipping the sign of the spin-down hopping term, has been proposed to eliminate the strong interference between doped holes and the spin background, known as the phase string effect. In this work, we investigate the finite-temperature properties of the $σt$-$J$ model using infinite projected entangled-pair state (iPEPS) algorithms, in comparison with the $t$-$J$ model and its easy-plane variants. We show that these models share similar spin-pseudogap thermodynamics, with a maximum in the spin susceptibility at $T^* \sim J$ and a broad specific-heat peak. Tensor network renormalization (TNR) further provides strong evidences for a Berezinskii-Kosterlitz-Thouless (BKT) transition in the spin sector of the $σt$-$J$ model at a lower temperature $T_\text{BKT} < T^*$ due to its reduced $\mathrm{U}(1)$ spin-rotation symmetry. Crucially, in contrast to the $t$-$J$ model and to its easy-plane variants that also exhibit BKT transitions, these signatures remain robust upon doping: the doped holes preserve and even enhance the antiferromagnetic correlations in the $xy$ spin plane, leading to a weak doping dependence of both $T^*$ and $T_\text{BKT}$. Finally, we develop a slave-fermion mean-field theory for the $σt$-$J$ model, whose projective symmetry group (PSG) is selected based on numerically determined hopping and pairing correlations, and show that it explains the robust spin pseudogap upon doping.

DOI
10.48550/arxiv.2608.09689

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Robust spin pseudogap and spin-charge separation in the σtσtσt-JJJ model
The σtσtσt-JJJ model, obtained from the usual ttt-JJJ model by flipping the sign of the spin-down hopping term, has been proposed to eliminate the strong interference between doped holes and the spin background, known as the phase string effect. In this work, we investigate the finite-temperature properties of the σtσtσt-JJJ model using infinite projected entangled-pair state (iPEPS) algorithms, in comparison with the ttt-JJJ model and its easy-plane variants. We show that these models share similar spin-pseudogap thermodynamics, with a maximum in the spin susceptibility at T∗∼JT^* \sim JT∗∼J and a broad specific-heat peak. Tensor network renormalization (TNR) further provides strong evidences for a Berezinskii-Kosterlitz-Thouless (BKT) transition in the spin sector of the σtσtσt-JJJ model at a lower temperature TBKT<T∗T_\text{BKT} < T^*TBKT​<T∗ due to its reduced U(1)\mathrm{U}(1)U(1) spin-rotation symmetry. Crucially, in contrast to the ttt-JJJ model and to its easy-plane variants that also exhibit BKT transitions, these signatures remain robust upon doping: the doped holes preserve and even enhance the antiferromagnetic correlations in the xyxyxy spin plane, leading to a weak doping dependence of both T∗T^*T∗ and TBKTT_\text{BKT}TBKT​. Finally, we develop a slave-fermion mean-field theory for the σtσtσt-JJJ model, whose projective symmetry group (PSG) is selected based on numerically determined hopping and pairing correlations, and show that it explains the robust spin pseudogap upon doping.
The Hubbard model is a longstanding problem in the theory of strongly correlated electrons and a very active one in the experiments with ultracold fermionic atoms. Motivated by current and prospective quantum simulations, we apply a two-dimensional tensor network---an infinite projected entangled pair state---evolved in imaginary time by the neighborhood tensor update algorithm working directly in the thermodynamic limit. With $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry and the bond dimensions up to 29, we generate thermal states down to the temperature of 0.17 times the hopping rate. We obtain results for spin and charge correlators, unaffected by boundary effects. The spin correlators---measurable in prospective ultracold atoms experiments attempting to approach the thermodynamic limit---provide evidence of disruption of the antiferromagnetic background with mobile holes in a slightly doped Hubbard model. The charge correlators reveal the presence of hole-doublon pairs near half filling and signatures of hole-hole repulsion on doping. We also obtain specific heat in the slightly doped regime.
We study a system of doped antiferromagnet in three dimensions at finite temperatures using the ttt-JJJ model, a canonical model of strongly correlated electrons. We employ the slave-fermion representation of electrons, in which an electron is described as a composite of a charged spinless holon and a chargeless spinon. We introduce two kinds of U(1) gauge fields on links as auxiliary fields, one describing resonating valence bonds of antiferromagnetic nearest-neighbor spin pairs and the other for nearest-neighbor hopping amplitudes of holons and spinons in the ferromagnetic channel. To perform a numerical study of the system, we integrate out the fermionic holon field by using the hopping expansion in powers of the hopping amplitude, which is legitimate for the region in and near the insulating phase. The resultant effective model is described in terms of bosonic spinons, two U(1) gauge fields, and a collective field for hole pairs. We study this model by means of Monte Carlo simulations, calculating the specific heat, spin correlation functions, and instanton densities. We obtain a phase diagram in the hole concentration-temperature plane, which is in good agreement with that observed recently for clean and homogeneous underdoped samples.
We study the two-dimensional t − J model at finite temperature directly in the thermodynamic limit using purification represented by an infinite projected entangled-pair state (iPEPS). We reach temperatures down to T / t = 0.1 and hole concentrations up to 1 − n ≃ 0.25 , and provide benchmark thermodynamic-limit results for the specific heat, uniform susceptibility, and charge compressibility. We identify a susceptibility maximum T * that tracks the buildup of short-range antiferromagnetism and a shallow compressibility enhancement upon cooling in the same doping window. To expose the underlying microscopic mechanism, we introduce dopant-conditioned multipoint correlators that quantify how holes reorganize nearby exchange: Single holes weaken adjacent antiferromagnetic bonds, while nearest-neighbor hole pairs produce a cooperative response that reinforces antiferromagnetism on the parallel plaquette edge. Over the same parameter window, d -wave pairing correlations remain short-ranged. These results provide experiment-compatible thermodynamic-limit benchmarks and establish dopant-conditioned correlators as incisive probes of finite-temperature spin-texture reorganization in doped Mott insulators.
Recently, robust d-wave superconductive (SC) order has been unveiled in the ground state of the 2D t-t^{'}-J model-with both nearest-neighbor (t) and next-nearest-neighbor (t^{'}) hoppings-by density matrix renormalization group studies. However, there is currently a debate on whether the d-wave SC holds up strong on both t^{'}/t>0 and t^{'}/t<0 cases for the t-t^{'}-J model, which correspond to the electron- and hole-doped sides of the cuprate phase diagram, respectively. Here, we exploit state-of-the-art thermal tensor network approach to accurately obtain the phase diagram of the t-t^{'}-J model on cylinders with widths up to W=6 and down to low temperature as T/J≃0.06, pushing the boundaries of contemporary finite-T calculations. For t^{'}/t>0, we find a domelike SC regime with a diverging d-wave pairing susceptibility, χ_{SC}∝1/T^{α} below a characteristic temperature T_{c}^{*}. Near optimal doping, T_{c}^{*} reaches its highest value of about 0.15J. Above T_{c}^{*} yet below a higher crossover temperature T^{*}, the magnetic susceptibility becomes suppressed, which can be related to the onset of pseudogap (PG) behaviors. On the other hand, for t^{'}/t<0, we find the pairing correlations are much weaker, although there exhibits a node-antinode structure in the PG regime as observed in the hole-doped cuprates. The thermal tensor network calculations of the t-t^{'}-J model underscore both the similarities and differences in the finite-temperature phase diagram between the fundamental model and cuprates, yielding unique insights into their intricate behaviors.
This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii-Kosterlitz-Thouless (BKT) temperature TBKTT_{\text{BKT}}TBKT​ as a function of \emph{attractive} pairing strength by associating it with the onset of quasi-condensation in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established Quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS-BEC crossover theory for fermions, leads to non-monotonic behavior for TBKTT_{\text{BKT}}TBKT​ as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that TBKTT_{\text{BKT}}TBKT​ has an absolute maximum of the order of 0.1EF0.1 E_F0.1EF​. We discuss how our results compare with those derived from the Nelson Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in TBKTT_\text{BKT}TBKT​ followed by an often dramatic decrease before the system enters the BEC regime.
We apply the fermionic tensor network (TN) state method to understand the strongly correlated nature in a doped Mott insulator. We conduct a comparative study of the $\ensuremath{\sigma}t\text{\ensuremath{-}}J$ model, in which the no-double-occupancy constraint remains unchanged but the quantum phase string effect associated with doped holes is precisely switched off. Thus, the ground state of the $\ensuremath{\sigma}t\text{\ensuremath{-}}J$ model can serve as a well-controlled reference state of the standard $t\text{\ensuremath{-}}J$ model. In the absence of the phase string, the spin long-range antiferromagnetic (AFM) order is found to be essentially decoupled from the doped holes, and the latter contribute to a Fermi-liquid-like compressibility and a coherent single-particle propagation with a markedly reduced pairing tendency. In contrast, our TN calculations of the $t\text{\ensuremath{-}}J$ model indicate that the AFM order decreases much faster with doping and the single-particle propagation of doped holes gets substantially suppressed, concurrently with a much stronger charge compressibility at small doping and significantly amplified Cooper pairing tendencies. These findings demonstrate that quantum many-body interference from phase strings plays a pivotal role in the $t\text{\ensuremath{-}}J$ model, mediating long-range entanglement between spin and charge degrees of freedom, which might hold the key mechanism for high-Tc{T}_{c}Tc​ cuprates.