Nonpertubative Many-Body Theory for the Two-Dimensional Hubbard Model at Low Temperature: From Weak to Strong Coupling Regimes

TL;DR

The paper tackles the challenge of finite-temperature symmetry breaking in the 2D Hubbard model, where the Mermin-Wagner theorem forbids true long-range order. It introduces a general symmetrization scheme and combines it with a covariance-based GW framework (GW-covariance) to compute one- and two-body observables in the half-filled 2D Hubbard model, benchmarking against DQMC. By enforcing FDR and WTIs via covariance and using the $\chi$-sum rule as a reliability diagnostic, the authors demonstrate good agreement for spin correlations and Green's functions at low temperature and intermediate-to-strong coupling, and show that the approach remains reliable away from the critical region. The work provides a non-perturbative pathway to study strong-coupling and doped regimes relevant to cuprate superconductors, offering a principled balance between conservation-law constraints and Pauli-principle consistency.

Abstract

In theoretical studies of two-dimensional (2D) systems, the Mermin-Wagner theorem prevents continuous symmetry breaking at any finite temperature, thus forbidding a Landau phase transition at a critical temperature $T_c$. The difficulty arises when many-body theoretical studies predict a Landau phase transition at finite temperatures, which contradicts the Mermin-Wagner theorem and is termed a pseudo phase transition. To tackle this problem, we systematically develop a symmetrization scheme, defined as averaging physical quantities over all symmetry-breaking states, thus ensuring that it preserves the Mermin-Wagner theorem. We apply the symmetrization scheme to the GW-covariance calculation for the 2D repulsive Hubbard model at half-filling in the intermediate-to-strong coupling regime and at low temperatures, obtaining the one-body Green's function and spin-spin correlation function, and benchmark them against Determinant Quantum Monte Carlo (DQMC) with good agreement.The spin-spin correlation functions are approached within the covariance theory, a general method for calculating two-body correlation functions from a one-particle starting point, such as the GW formalism used here, which ensures the preservation of the fundamental fluctuation-dissipation relation (FDR) and Ward-Takahashi identities (WTI). With the FDR and WTI satisfied, we conjecture that the $χ$-sum rule, a fundamental relation from the Pauli exclusion principle, can be used to probe the reliability of many-body methods, and demonstrate this by comparing the GW-covariance and mean-field-covariance approaches. This work provides a novel framework to investigate the strong-coupling and doped regime of the 2D Hubbard model, which is believed to be applicable to real high-$T_c$ cuprate superconductors.

Figures (11)

• Figure 1: Real-space lattice configurations for paramagnetic and antiferromagnetic solutions of the 2D Hubbard model. (a) Lattice configuration of the paramagnetic solution, with lattice vectors $a_x=(a,0)$ and $a_y=(0,a)$. All sites are equivalent. (b) Lattice structure with AB sublattices for the antiferromagnetic solution with Néel order (A sublattice: blue; B sublattice: red), spanned by $a_1 = (2a,0)$ and $a_2 = (a,a)$.
• Figure 2: Momentum dependence of the inverse spin susceptibility for the half-filled Hubbard model on a $12\times12$ lattice with $U=8$. Results are shown for decreasing temperatures, corresponding to inverse temperatures (a) $\beta=2$, (b) $\beta=3$, (c) $\beta=4$, (d) $\beta=6$, (e) $\beta=8$, and (f) $\beta=16$. The momentum path follows $(0,0)\to(0,\pi)\to(\pi,\pi)\to(0,0)$. Blue curves denote the symmetrized GW-covariance approximation, while red symbols with error bars show DQMC benchmarks. A systematic improvement in the agreement between GW-covariance and DQMC is observed as temperature is lowered.
• Figure 3: Temperature dependence of the static spin correlation function in spatial space for the half-filled Hubbard model ($U=8$) on a $12\times12$ lattice. Panels (a)-(d) display results for lattice separations $\mathbf{r} = (0,0)$, $(1,0)$, $(1,1)$, and $(2,0)$, respectively. The DQMC benchmarks (red points with error bars) are compared with the GW-covariance result $\chi_{\mathrm{sp}}$ (blue dashed curve) and the $\chi$-sum rule estimate $\chi_{\mathrm{sp}}^{\mathrm{(sr)}}$ (blue solid curve).
• Figure 4: Spin correlation function as a function of interaction strength $U$ for a $16 \times 16$ lattice at low temperature ($\beta = 8$). The DQMC results (red discrete points with error bars) exhibit a continuous evolution. In contrast, the GW-covariance approximation (blue curve) reveals a two-branch structure: a paramagnetic (Slater) branch for $U < U_c$ and a pseudo-antiferromagnetic (Mott-Heisenberg) branch for $U > U_c$, with a critical $U_c \simeq 2.8$ leading to a pronounced discontinuity. The GW-covariance results on the antiferromagnetic branch are obtained via the $\chi$-sum rule estimate.
• Figure 5: Imaginary-time Green's function $G(\mathbf{k},\tau)$ for the half-filled Hubbard model on a $12\times12$ lattice with $U=8$, comparing the symmetrized GW approximation (blue solid lines) and DQMC results (red dot-dashed lines). Panels (a)-(d) correspond to the nodal point $\mathbf{k} = (\pi/2, \pi/2)$, while panels (e)-(h) show the antinodal point $\mathbf{k} = (\pi, 0)$. The columns represent increasing inverse temperatures, i.e., (a,e) $\beta = 2$, (b,f) $\beta =3$, (c,g) $\beta =4$, and (d,h) $\beta =6$.