School of Physics, Beihang University, Beijing 100191, China.
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China.
Phys Rev Lett. 2023 Jun 2;130(22):226502. doi: 10.1103/PhysRevLett.130.226502.
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.
准确模拟二维（2D）Hubbard 模型是凝聚态物理和量子物理中最具挑战性的问题之一。在这里，我们开发了一种切线空间张量重整化群（tanTRG）方法，用于计算有限温度下的 2D Hubbard 模型。在 tanTRG 中，通过一种复杂度为 O(D^3)的温和演化，实现了密度算符的最优演化，其中键维数 D 控制着精度。使用 tanTRG 方法，我们在高达 8 宽度的圆柱和 10×10 正方形晶格上，提高了大规模 2D Hubbard 系统的低温计算能力。对于满带 Hubbard 模型，得到的结果与行列式量子蒙特卡罗（DQMC）的结果非常吻合。此外，tanTRG 可用于探索 DQMC 无法访问的低温、有限掺杂区域。计算得到的电荷压缩率和 Matsubara 格林函数分别反映了奇异金属和赝能隙行为。超导配对磁化率可计算到大约为跳跃能的 1/24 的低温，在这个低温下，我们发现在最佳掺杂附近，d 波配对响应最为显著。tanTRG 配备了切线空间技术，是一种在有限温度下用于强关联 2D 晶格模型的、具有良好控制、高效和精确的张量网络方法。
