Disorder operators in two-dimensional Fermi and non-Fermi liquids through multidimensional bosonization

TL;DR

The paper develops a multidimensional bosonization framework to study U(1) disorder operators in two-dimensional fermionic systems, linking the disorder parameter to density fluctuations. It shows that in a 2D Fermi liquid the disorder parameter scales as $l_A \ln l_A$, while at a quantum critical point with a gapless boson it scales as $l_A \ln^2 l_A$, and in a composite Fermi liquid it scales as $l_A$, revealing how global charge fluctuations encode universal FL/NFL physics. Through explicit diagonalization and finite-patch analyses, the authors connect the disorder scaling to Landau parameters and structure factors, and they predict distinct NFL signatures, such as $R \ln^2 R$ scaling for critical scalar coupling. The results provide a non-local diagnostic that complements entanglement measures and align with numerical observations, offering a path to explore other NFLs and gauge-field coupled systems with disorder operators.

Abstract

Disorder operators are a type of non-local observables for quantum many-body systems, measuring the fluctuations of symmetry charges inside a region. It has been shown that disorder operators can reveal global aspects of many-body states that are otherwise difficult to access through local measurements. We study the disorder operator for U(1) (charge or spin) symmetry in two-dimensional Fermi and non-Fermi liquid states, using the multidimensional bosonization formalism. For a region $A$, the logarithm of the charge disorder parameter in a Fermi liquid with isotropic interactions scales asymptotically as $l_A\ln l_A$, with $l_A$ being the linear size of the region $A$. We calculate the proportionality coefficient in terms of Landau parameters of the Fermi liquid theory. We then study models of the Fermi surface coupled to gapless bosonic fields realizing non-Fermi liquid states. In a simple spinless model, where the fermion density is coupled to a critical scalar, we find that at the quantum critical point the scaling behavior of the charge disorder operators is drastically modified to $l_A \ln^2 l_A$. We also consider the composite Fermi liquid state and argue that the charge disorder operator scales as $l_A$.

Figures (6)

• Figure 1: $-\ln Z(\theta)$ of a $50\times 50$ free fermion lattice model with a fixed $20\times 20$ region. The numerical results are fitted with a quadratic function $a\theta^2$ in the region $\theta\in (-\pi,\pi)$. $-\ln Z(\theta) \propto \theta^2$ is valid inside the region $\theta\in (-\pi,\pi)$ to a very good precision, and there is a smooth transition linking different intervals at $\theta=-\pi,\pi$.
• Figure 2: Illustration of the modes defined by Eq. \ref{['MB_Eq_ab']}.
• Figure 3: The ratio $\delta_1(g)$ vs contact interaction strength $g$. Numerical results with $N=320$ are shown in red dots, while the solid black line is the field theory prediction Eq. \ref{['BC_Eq_deltag']}.
• Figure 4: The ratio $\delta_2(g)$ vs contact interaction strength $g$. Numerical results with $N=320$ are shown in red dots and the solid black line is the function $\frac{1}{1+2g}$.
• Figure 5: $G(q)/\frac{k_F}{2\pi^2}$ vs $q$ for $k_c=1$ and three different values of $k_{\Phi}$. The critical case is depicted by the solid black line, while the free fermion case is shown with a dashed line for comparison.