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arXiv:2002.01312 · Physical review. B./Physical review. B

Localization anisotropy and complex geometry in two-dimensional insulators

Bruno Mera

Published Mar 17, 2020• 5 citations
arXivHTMLPDF

Abstract

The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter $\ensuremath{\tau}$. It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity $\ensuremath{\tau}$ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if, by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure $\ensuremath{\tau}$ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.

DOI
10.1103/physrevb.101.115128

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The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter \ensuremathτ\ensuremath{\tau}\ensuremathτ. It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity \ensuremathτ\ensuremath{\tau}\ensuremathτ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if, by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure \ensuremathτ\ensuremath{\tau}\ensuremathτ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.
Inspired by the duality picture between superconductivity (SC) and insulator in two spatial dimension (2D), we conjecture that the order parameter, suitable for characterizing 2D fermionic insulating state, is the disorder operator, usually known in the context of statistical transformation. Namely, the change of the phase of the disorder operator along a closed loop measures the particle density accommodating inside this loop. Thus, identifying this (doped) particle density with the dual counterpart of the magnetic induction in 2D SC, we can naturally introduce the disorder operator as the dual order parameter of 2D insulators. The disorder operator has a branch cut emitting from this ``vortex'' to the single infinitely far point. To test this conjecture against arbitrary two-dimensional lattice models, we have chosen this branch cut to be compatible with the periodic boundary condition and obtain a general form of its expectation value for noninteracting metal and/or insulator wave function, including gapped mean-field order wave function. Based on this expression, we observed analytically that it indeed vanishes for a wide class of band metals in the thermodynamic limit. On the other hand, it takes a finite value in insulating states, which is quantified by the localization length or the real-valued gauge invariant two form dubbed as the quantum metric tensor. When successively applied along a closed loop, our disorder operator plays the role of twisting the boundary condition of a periodic system. We argue this point by highlighting the Aharonov-Bohm phase associated with this nonlocal operator.
We revisit the connection between entanglement entropy and quantum metric in topological lattice systems, and provide an elegant and concise proof of this connection. In gapped two-dimensional lattice models with well-defined tight-binding Hamiltonians, we show that the entanglement entropy is intimately related to the quantum metric of electronic states, and bounded by the Chern number of (Slater-determinant) interacting topological insulators. Our results hold promising applicability to the recently discovered twisted transition metal dichalcogenides, characterized by flat topological bands at low twist angles, where these bounds can be applied at integer filling, and new pathways to enhance entanglement measures by engineering heterostructures with higher Chern numbers.
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Characterizing anisotropic correlations in quantum and statistical systems requires a coordinate-invariant framework. We introduce a geometric map based on the local informational line element, calibrated by the Euclidean benchmark scale CvacC_{\mathrm{vac}}Cvac​: ds2=Cvac/I(x,x+ε)ds^{2} = C_{\mathrm{vac}}/I(x,x+ε)ds2=Cvac​/I(x,x+ε). We prove that this map yields a smooth Riemannian structure gijg_{ij}gij​ if and only if the short-distance mutual information (MI) follows the anisotropic inverse-quadratic law (local exponent Xloc=2X_{\text{loc}}=2Xloc​=2). A key insight is that anisotropy is necessary to activate tensor geometry; isotropic MI forces conformal flatness gij∝δijg_{ij} \propto δ_{ij}gij​∝δij​, suppressing shear degrees of freedom. We employ a parameterization-invariant unimodular split gij=V2/Dγijg_{ij} = V^{2/D}γ_{ij}gij​=V2/Dγij​, which rigorously separates local density fluctuations (volume VVV) from directional anisotropy (shape/shear γijγ_{ij}γij​). We introduce ``MI Tomography,'' an operational protocol to reconstruct these geometric components from finite directional measurements. The protocol is validated using the equal-time ground state of an anisotropic 2D quantum harmonic lattice (massless relativistic scalar) on a torus, where the reconstructed shape tensor γijγ_{ij}γij​ quantitatively recovers the physical coupling anisotropy. We work strictly in the local, fixed-coarse-graining Xloc=2X_{\text{loc}}=2Xloc​=2 branch; the line element is used solely to extract the local kinematic structure (the local metric tensor), deferring global distance claims.
The quantum geometric tensor (QGT) characterizes the Hilbert space geometry of the eigenstates of a parameter-dependent Hamiltonian. In recent years, the QGT and related quantities have found extensive theoretical and experimental utility, in particular for quantifying quantum phase transitions both at and out of equilibrium. Here, we consider the symmetric part (quantum Riemannian metric) of the QGT for various multiparametric random matrix Hamiltonians and discuss the possible indication of ergodic or integrable behavior. We found for a two-dimensional parameter space that, while the ergodic phase corresponds to the smooth manifold, the integrable limit marks itself as a singular geometry with a conical defect. Our study thus provides more support for the idea that the landscape of the parameter space yields information on the ergodic-nonergodic transition in complex quantum systems, including the intermediate phase.