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arXiv:2301.06493 · Physical review. B./Physical review. B

Mapping quantum geometry and quantum phase transitions to real space by a fidelity marker

Matheus S. M. de Sousa, Antonio L. Cruz, Wei Chen

Published May 17, 2023• 34 citations

Abstract

The quantum geometry in the momentum space of semiconductors and insulators, described by the quantum metric of the valence-band Bloch state, has been an intriguing issue owing to its connection to various material properties. Because the Brillouin zone is periodic, the integration of quantum metric over momentum space represents an average distance between neighboring Bloch states, which we call the fidelity number. We show that this number can further be expressed in real space as a fidelity marker, which is a local quantity that can be calculated directly from diagonalizing the lattice Hamiltonian. A linear-response theory is further introduced to generalize the fidelity number and marker to finite temperature and moreover demonstrates that they can be measured from the global and local optical absorption power against linearly polarized light. In particular, the fidelity number spectral function in two-dimensional systems can be easily measured from the opacity of the material. Based on the divergence of quantum metric, a nonlocal fidelity marker is further introduced and postulated as a universal indicator of any quantum phase transitions provided the crystalline momentum remains a good quantum number, and it may be interpreted as a Wannier state correlation function. The ubiquity of these concepts is demonstrated for a variety of topological insulators and topological phase transitions in different dimensions.

DOI: 10.1103/physrevb.107.205133

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The quantum geometry in the momentum space of semiconductors and insulators, described by the quantum metric of the valence-band Bloch state, has been an intriguing issue owing to its connection to various material properties. Because the Brillouin zone is periodic, the integration of quantum metric over momentum space represents an average distance between neighboring Bloch states, which we call the fidelity number. We show that this number can further be expressed in real space as a fidelity marker, which is a local quantity that can be calculated directly from diagonalizing the lattice Hamiltonian. A linear-response theory is further introduced to generalize the fidelity number and marker to finite temperature and moreover demonstrates that they can be measured from the global and local optical absorption power against linearly polarized light. In particular, the fidelity number spectral function in two-dimensional systems can be easily measured from the opacity of the material. Based on the divergence of quantum metric, a nonlocal fidelity marker is further introduced and postulated as a universal indicator of any quantum phase transitions provided the crystalline momentum remains a good quantum number, and it may be interpreted as a Wannier state correlation function. The ubiquity of these concepts is demonstrated for a variety of topological insulators and topological phase transitions in different dimensions.

Momentum space of a gapped quantum system is a metric space: it admits a notion of distance reflecting properties of its quantum ground state. By using this quantum metric, we investigate geometric properties of momentum space. In particular, we introduce a nonlocal operator which represents distance square in real space and show that this corresponds to the Laplacian in curved momentum space, and also derive its path-integral representation in momentum space. The quantum metric itself measures the second cumulant of the position operator in real space, much like the Berry gauge potential measures the first cumulant or the electric polarization in real space. By using the nonlocal operator and the metric, we study some aspects of topological phases such as topological invariants, the cumulants and topological phase transitions. The effect of interactions to the momentum space geometry is also discussed.

We formulate quantum geometry for non-Hermitian systems under open boundary conditions. By defining quantum-geometric quantities in both real-space and non-Bloch representations, we establish a unified framework beyond conventional Bloch band theory. Our central result is an exact equivalence between the real-space integrated quantum metric and a non-Bloch integrated quantum metric defined on the generalized Brillouin zone. We further introduce localized non-Bloch Wannier functions in the presence of the non-Hermitian skin effect and show that the non-Bloch integrated quantum metric gives the gauge-invariant part of their spread functional. These results establish quantum geometry as a natural framework for characterizing open-boundary non-Hermitian band structures and the localization properties encoded in skin modes.

We study the entanglement spectrum of noninteracting band insulators, which can be computed from the two-point correlation function, when restricted to one part of the system. In particular, we analyze a type of partitioning of the system that maintains its full translational symmetry, by tracing over a subset of local degrees of freedom, such as sublattice sites or spin orientations. The corresponding single-particle entanglement spectrum is the band structure of an entanglement Hamiltonian in the Brillouin zone. We find that the hallmark of a nontrivial topological phase is a gapless entanglement spectrum with an ``entanglement Fermi surface.'' Furthermore, we derive a relation between the entanglement spectrum and the quantum geometry of Bloch states contained in the Fubini-Study metric. The results are illustrated with lattice models of Chern insulators and

{\mathbb{Z}}_{2}

topological insulators.

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.

Spatially resolved local quantum geometric markers play a crucial role in the diagnosis of topological phases without long-range translational symmetry, including amorphous systems. Here, we focus on the nonlocality of such markers. We demonstrate that they behave as correlation functions independently of the material's structure, showing sharp variations in the vicinity of topological transitions and exhibiting a unique pattern in real space for each transition. Notably, we find that, even within the same Altland-Zirnbauer class, distinct topological transitions generate qualitatively different spatial signatures, enabling a refined, class-internal probe of topological stability. As such, nonlocal quantum geometric indicators provide a more efficient and versatile tool to understand and predict the stability of topological phase transitions.

Quantum geometry quantifies how the single-particle Bloch wavefunction changes in phase and amplitude across the Brillouin Zone. In multi-orbital systems where bands have strongly mixed orbital composition, quantum geometry plays a vital role in determining the ground state and low-energy properties of interacting electronic systems. In this work, we show that Mott metal-insulator transitions, as well as transitions between different magnetic orders within the Mott insulating phase, can be driven by the quantum geometry of the underlying Bloch band, thereby providing a mechanism complementary to conventional bandwidth-tuned Mott transitions. By studying the Kane-Mele-Hubbard model using exact diagonalization, we demonstrate that in in half-filled and topologically-trivial bands, quantum geometric properties of the Bloch states alone can act as a tuning knob for Mott metal-to-insulator and affect the competition between ferromagnetism and antiferromagnetism. We show that both transitions may be heuristically understood via non-local Coulomb scattering in a basis of exponentially localized Wannier functions. These results highlight the role of quantum geometry beyond topological settings as a governing principle for conventional Mott and magnetic physics in multi-orbital and moiré materials.