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.
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 the two-dimensional $t\text{\ensuremath{-}}J$ model on a square lattice using infinite projected entangled pair states (iPEPS). At small doping, multiple orders, such as antiferromagnetic order, stripe order and superconducting order, are intertwined or compete with each other. We demonstrate the role of spin symmetry at small doping by either imposing SU(2) spin symmetry or its U(1) subgroup in the iPEPS ansatz, thereby excluding or allowing spontaneous spin-symmetry breaking, respectively, in the thermodynamic limit. From a detailed comparison of our simulations, we provide evidence that stripe order is pinned by long-range antiferromagnetic order. We also find SU(2) iPEPS, enforcing a spin-singlet state, yields a uniform charge distribution and favors d-wave singlet pairing.
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∗∼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
TBKT<T∗ due to its reduced
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
TBKT. 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.
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 cuprates.
Simulating strongly-correlated quantum many-body systems at finite temperatures is a significant challenge in computational physics. In this work, we present a scalable finite-temperature tensor network algorithm for two-dimensional quantum many-body systems. We employ the (fermionic) projected entangled pair state (PEPS) to represent the vectorization of the quantum thermal state and utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the 2D antiferromagnetic Heisenberg model, the
J1-
J2 model, and the Fermi-Hubbard model, comparing physical properties such as internal energy, specific heat, and magnetic susceptibility with results obtained from stochastic series expansion (SSE), exact diagonalization, and determinant quantum Monte Carlo (DQMC).
The emergence of superconductivity from doping strongly correlated chiral topological phases in purely repulsive two-dimensional fermionic systems is a problem of broad and fundamental interest. However, existing numerical evidence has been limited to finite-size studies, and direct thermodynamic-limit evidence for superconducting long-range order has remained lacking. Here we provide such evidence for chiral superconductivity in the triangular Hofstadter-Hubbard model by advancing a simplex tensor-network approach that simultaneously captures superconducting long-range order and chiral topological order in the presence of intrinsic charge fluctuations, a capability that has remained challenging for previous two-dimensional approaches. We show that a broad intermediate-
U chiral spin liquid is separated from the weak-
U Chern insulator by a Mott transition, together forming undoped parent chiral topological states. Upon hole doping, we identify a uniform chiral superconducting state in the infinite system, characterized by a finite complex pairing order parameter. The pairing field exhibits an almost universal phase winding over a broad interaction-doping regime, with a distinct pocket of opposite winding near the Mott criticality. In addition, the entanglement spectrum retains the chiral structure of the parent topological phases while developing additional low-energy branches upon doping. These results establish that chiral superconductivity emerges robustly from doped chiral topological phases.