Accurate simulations of the two-dimensional (2D) Hubbard model constitute one of the most challenging problems in condensed matter and quantum physics. Here we develop a tangent space tensor renormalization group (tanTRG) approach for the calculations of the 2D Hubbard model at finite temperature. An optimal evolution of the density operator is achieved in tanTRG with a mild O(D^{3}) complexity, where the bond dimension D controls the accuracy. With the tanTRG approach we boost the low-temperature calculations of large-scale 2D Hubbard systems on up to a width-8 cylinder and 10×10 square lattice. For the half-filled Hubbard model, the obtained results are in excellent agreement with those of determinant quantum Monte Carlo (DQMC). Moreover, tanTRG can be used to explore the low-temperature, finite-doping regime inaccessible for DQMC. The calculated charge compressibility and Matsubara Green's function are found to reflect the strange metal and pseudogap behaviors, respectively. The superconductive pairing susceptibility is computed down to a low temperature of approximately 1/24 of the hopping energy, where we find d-wave pairing responses are most significant near the optimal doping. Equipped with the tangent-space technique, tanTRG constitutes a well-controlled, highly efficient and accurate tensor network method for strongly correlated 2D lattice models at finite temperature.
Accurate simulations of the two-dimensional (2D) Hubbard model constitute one of the most challenging problems in condensed matter and quantum physics. Here we develop a tangent space tensor renormalization group (tanTRG) approach for the calculations of the 2D Hubbard model at finite temperature. An optimal evolution of the density operator is achieved in tanTRG with a mild O(D^{3}) complexity, where the bond dimension D controls the accuracy. With the tanTRG approach we boost the low-temperature calculations of large-scale 2D Hubbard systems on up to a width-8 cylinder and 10×10 square lattice. For the half-filled Hubbard model, the obtained results are in excellent agreement with those of determinant quantum Monte Carlo (DQMC). Moreover, tanTRG can be used to explore the low-temperature, finite-doping regime inaccessible for DQMC. The calculated charge compressibility and Matsubara Green's function are found to reflect the strange metal and pseudogap behaviors, respectively. The superconductive pairing susceptibility is computed down to a low temperature of approximately 1/24 of the hopping energy, where we find d-wave pairing responses are most significant near the optimal doping. Equipped with the tangent-space technique, tanTRG constitutes a well-controlled, highly efficient and accurate tensor network method for strongly correlated 2D lattice models at finite temperature.
Numerical simulations of strongly correlated fermions at finite temperature are essential for studying high-temperature superconductivity and other quantum many-body phenomena. The recently developed tangent-space tensor renormalization group (tanTRG) provides an efficient and accurate framework by representing thermal density operators as matrix product operators. However, the particle number generally varies during the cooling process. The conventional strategy of fine-tuning chemical potentials to reach a target filling is computationally demanding. Here we propose a fixed-N tanTRG algorithm that stabilizes the average particle number by adaptively tuning the chemical potential within the imaginary-time evolution. We benchmark its accuracy on exactly solvable free fermions, and further apply it to the square-lattice Hubbard model. For hole-doped cases, we study the temperature evolution of charge and spin correlations, identifying several characteristic temperature scales for stripe formation. Our results establish fixed-N tanTRG as an efficient and reliable tool for finite-temperature studies of correlated fermion systems.
In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a finite thermal correlation length $\ensuremath{\xi}$, such that for system sizes $L\ensuremath{\gg}\ensuremath{\xi}$ finite-size calculations actually simulate the thermodynamic limit. In this paper, we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width W=10. We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc. are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH, we obtain an energy per site ${u}_{g}^{*}\ensuremath{\simeq}\ensuremath{-}0.6694(4)$ and a spontaneous magnetization ${m}_{S}^{*}\ensuremath{\simeq}0.30(1)$ already consistent with the ground-state properties, which is obtained from extrapolated low-T thermal data on $W\ensuremath{\le}8$ cylinders and $W\ensuremath{\le}10$ open strips, respectively. We extract an exponential divergence versus T of the structure factor S(M), as well as the correlation length $\ensuremath{\xi}$, at the ordering wave vector $M=(\ensuremath{\pi},\ensuremath{\pi})$, which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analyzing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature Tc.
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).
Understanding quantum many-body states of correlated electrons is one main theme in modern condensed-matter physics. Given that the Fermi-Hubbard model, the prototype of correlated electrons, was recently realized in ultracold optical lattices, it is highly desirable to have controlled numerical methodology to provide precise finite-temperature results upon doping to directly compare with experiments. Here, we demonstrate the exponential tensor renormalization group (XTRG) algorithm [Chen et al., Phys. Rev. X 8, 031082 (2018)], complemented by independent determinant quantum Monte Carlo, offers a powerful combination of tools for this purpose. XTRG provides full and accurate access to the density matrix and thus various spin and charge correlations, down to an unprecedented low temperature of a few percent of the tunneling energy. We observe excellent agreement with ultracold fermion measurements at both half filling and finite doping, including the sign-reversal behavior in spin correlations due to formation of magnetic polarons, and the attractive hole-doublon and repulsive hole-hole pairs that are responsible for the peculiar bunching and antibunching behaviors of the antimoments.
The two-dimensional (2D) Hubbard model, widely believed to capture the essential physics of high-T_c cuprate compounds, has attracted immense research interest ever since the discovery of superconductivity in these materials. Despite its superficial simplicity, the Hubbard model incubates abundant fascinating phenomena owing to the strong interactions between charge carriers, encompassing antiferromagnetism, the pseudogap, and unconventional superconductivity. Yet these very interactions induce strong correlations, rendering canonical perturbative approaches unreliable. With advances in modern computational hardware, numerical techniques — notably tensor network methods and Artificial Intelligence (AI) algorithms — have become indispensable to our expedition. Tensor networks encode the amplitude information of many-body quantum states in a network of interconnected tensors. Leveraging the entanglement area law, tensor networks provide efficient, systematic, and controllable representations of quantum states that would otherwise require an exponentially large number of parameters. Past decades have witnessed the development of various tensor network ansatzes, such as Matrix Product States/Operators (MPS/MPO) for one-dimensional systems and Projected Entangled-Pair States (PEPS) for two-dimensional systems, together with diverse algorithms for optimizing these ansatzes for both zero- and finite temperature scenarios. Over the past years, AI technologies have revolutionized the way we analyze scientific data. The transformer architecture, in particular, has demonstrated exceptional capabilities in the domain of natural language processing. Its attention mechanism excels at capturing long-range correlations in sequential data, making it a promising tool for studying strongly correlated many-body systems. The interpretable nature of the attention mechanism further offers valuable insights into the AI's perception of the underlying physics. In this thesis, we assemble an all-inclusive numerical toolchain for analyzing strongly correlated lattice systems. As a demonstration, we investigate the zero- and finite-temperature properties of the 2D Hubbard model on a square lattice via the infinite PEPS (iPEPS) and the eXponential Tensor Renormalization Group (XTRG), respectively. The native two-dimensional iPEPS tensor network faithfully embodies the entanglement area law and helps settle long-standing debates regarding superconducting order in the ground state of the Hubbard model. The XTRG algorithm efficiently constructs thermal density matrices across a broad temperature range via a highly effective exponential-cooling protocol, thereby enabling detailed probes of pairing and pseudogap phenomena. Afterwards, we generate a comprehensive snapshot dataset which furnishes subsequent training of an optimized encoder-only transformer model. Our novel AI architecture features a Markovian interpretation of the attention design and affords improved parallelism. The collective technological stack opens a brand new avenue for exploring the rich physics of strongly correlated many-body systems.
The performance of the density matrix renormalization group (DMRG) is strongly influenced by the choice of the local basis of the underlying physical lattice. We demonstrate that, for the two-dimensional Hubbard model, the hybrid--real-momentum-space formulation of the DMRG is computationally more efficient than the standard real-space formulation. In particular, we show that the computational cost for fixed bond dimension of the hybrid-space DMRG is approximately independent of the width of the lattice, in contrast to the real-space DMRG, for which it is proportional to the width squared. We apply the hybrid-space algorithm to calculate the ground state of the doped two-dimensional Hubbard model on cylinders of width four and six sites; at n=0.875 filling, the ground state exhibits a striped charge-density distribution with a wavelength of eight sites for both U/t=4.0 and 8.0. We find that the strength of the charge ordering depends on U/t and on the boundary conditions. Furthermore, we investigate the magnetic ordering as well as the decay of the static spin, charge, and pair-field correlation functions.