Monte Carlo Study of Critical Fermi Surface with Spatially Disordered Interactions

Abstract

Non-Fermi liquids are an important topic in condensed matter physics, as their characteristics challenge the framework of traditional Fermi liquid theory and reveal the complex behavior of electrons in strongly interacting systems. Both the experimentally observed smeared region and the theoretically predicted marginal Fermi liquid suggest that spatial disorder seems to be an important driver of these phenomena. By performing large-scale determinant quantum Monte Carlo (DQMC) simulations in the ferromagnetic spin-fermion model at finite $N$, beyond the large-$N$ used in previous theoretical work, we investigated the role of spatial disorder in the critical Fermi surface (FS) of this model. We proposed a corrected theory of our system, which is based on a modified Eliashberg theory and a universal theory of strange metals. This theory agrees well with the data obtained from DQMC, particularly in capturing the $ω\ln ω$ type self-energy characteristic of marginal Fermi liquid behavior, and observing the linear-in-temperature resistivity. Our findings offer strong and unbiased validation of the universal theory of strange metals, broaden the applicability of the modified Eliashberg theory, and provide insights for numerically searching for marginal Fermi liquid and linear-in-temperature resistivity.

Figures (21)

• Figure 1: Model and Phase Diagram. (a) Graph of the spin-fermion bilayer model defined in \ref{['eq:eq1']}. The top and bottom layers are fermion layers($\lambda=1,2$), and in each fermion layer, spin-1/2 fermions live on a triangular lattice. Between the fermion layers is an Ising spin layer which mediates the fermion-fermion interaction. The Ising spins have two tunable parameters (ferromagnetic interaction $J$ and transverse magnetic field $h$) to control the FM-PM phase transition. (b) $h-T$ phase diagram of model in \ref{['eq:eq1']}. In this study, we use coupling strength $\xi = 1$ and $\mu = 0.8348$ to realize a hall-filling case ($\langle n_{i\lambda} \rangle = 1.0$). The orange data points are FM-PM phase transition critical points at finite temperatures. By using the formula $T_N(h) \sim |h - h_c|^{c}$, we have fitted a phase boundary (the black line), with quantum critical point $h_c = 4.9256(3)$ and $c = 0.635(2)$. For comparison, we also plot the phase boundary of the transverse field Ising model (blue points), which corresponds to the no coupling ($\xi = 0$) case. This boundary is consistent with $(2 + 1)$D Ising universality class and can be described by $T_N(h) \sim |h - h_c|^{z \nu}$ with $\nu z \approx 0.63$ and $h_c = 4.76(7)$. The position of QCP (the black dot and the red dot on the $h$-axis) is changed due to the coupling term $\hat{H}_{sf}$. The inset is the low-energy spectral weight at QCP, which is obtained from $G(\mathbf{k}, \beta/2)$. Twisted boundary conditions are used to increase the resolution. The light red MFL region corresponds to the case when the coupling term is completely spatially random.
• Figure 1: The plot of Ising spin susceptibility $\chi(h_c, T, \mathbf{0}, 0)$ as it varies with Trotter step $\Delta \tau$. (a) $L = 12$. (b) $L = 18$. We choose $\Delta \tau = 0.02$ in simulations to ensure the convergence of observables.
• Figure 2: Fitting DQMC data by appling UT. (a) Fitting of self-energy when $\sigma_h = 0.2$. (b) Fitting of self-energy when $\sigma_h = 0.5$. (c) Fitting of Green's funtion when $\sigma_h = 1.0$. (d) Fitting of self-energy after dropping the first Matsubara frequency when $\sigma_h = 1.0$.
• Figure 2: FM-PM phase transition at finite-temperature $T = 1.0$. (a) The Ising spin susceptibilities of different system sizes at $\mathbf{q} = \mathbf{0}$ and $\Omega_n = 0$. (b) The data collapse result of Ising spin susceptibility. Critical exponents of 2D Ising universality class $\nu = 1$ and $\gamma = 7/4$ are used. The only fitting parameter is the critical field $h_N \approx 4.63$. Curves of all system sizes fall on the same line. (c) The fermion spin susceptibilities of different system sizes at $\mathbf{q} = \mathbf{0}$ and $\Omega_n = 0$. (d) The data collapse result of fermion spin susceptibility. The $h_N$ used is the one obtained from Ising susceptibility. Curves of all system sizes don't fall on the same line.
• Figure 3: Marginal Fermi liquid behavior. The fermionic self-energy shows an expected MFL scaling when a completely random coupling interaction is applied. (a) The imaginary part of the total fermionic self-energy. (b) After subtracting the thermal part, the imaginary part of the remaining quantum part can be well fitted by a MFL form $a \omega_n \ln(b/\omega_n)$. The $x$-axis values denote the fitting results, and the $y$-axis values denote the Monte Carlo data; all the points are located closely on the $y=x$ line. (c) DC resistivity proxy $\rho$ extracted from the current-current correlation function.