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arXiv:1007.2200 · Physical Review B

Momentum space metric, nonlocal operator, and topological insulators

Shunji Matsuura, Shinsei Ryu

Published Dec 13, 2010• 62 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.82.245113

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Europhysics Letters (EPL)•Dec 1, 2012•7 citations
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.
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.
By acting the projector to the filled lattice eigenstates on a specific position, or applying local electron annihilation operator on the many-body ground state, one can construct a quantum state localized around a specific position in a solid. The overlap of two such local states at slightly different positions defines a quantum metric in real space, which manifests even in systems as simple as particles in a box. For continuous systems like electron gas, this metric weighted by the density gives the momentum variance of electrons, which is readily measurable by ARPES. The presence of disorder curves the real space manifold and gives rise to various differential geometrical quantities like Riemann tensor and Ricci scalar, indicating the possibility of engineering differential geometrical properties by disorder, as demonstrated by lattice models of 2D metals and topological insulators.
By applying the projector to the filled lattice eigenstates on a specific position, or applying the local electron annihilation operator on the many-body ground state, one can construct a quantum state localized around a specific position in a solid. The overlap of two such local states at slightly different positions defines a quantum metric in real space, which manifests even in systems as simple as particles in a box. For continuous systems like electron gas, this metric weighted by the density gives the momentum variance of electrons, which is readily measurable by ARPES. The presence of disorder curves the real space manifold and gives rise to various differential geometrical quantities like Riemann tensor and Ricci scalar, indicating the possibility of engineering differential geometrical properties by disorder, as demonstrated by lattice models of 2D metals and topological insulators.
Quantum geometry plays a fundamental role across many branches of modern physics, yet its full characterization in nonequilibrium systems remains a challenge. Here, we propose a framework for quantum geometry in Floquet topological insulators by introducing a time-resolved quantum metric tensor, defined via the trace distance between micromotion operators in momentum-time space. For class A in two spatial dimensions, we find a general inequality linking the Floquet quantum metric tensor and the Floquet topology: the associated quantum volume is bounded below by the Floquet topological invariant. This relation is found to also hold in class AIII in one dimension, where the Floquet geometric tensor may be notably reduced due to time-reflection symmetry. This work will be useful in digesting the general aspects of quantum geometry in periodically driven systems in connection with their topological characterization.
In this paper, we stress the importance of momentum-space geometry in the understanding of two-dimensional topological phases of matter. We focus, for simplicity, on the gapped boundary of three-dimensional topological insulators in class AII, which are described by a massive Dirac Hamiltonian and characterized by an half-integer Chern number. The gap is induced by introducing a magnetic perturbation, such as an external Zeeman field or a ferromagnet on the surface. The quantum Bures metric acquires a central role in our discussion and identifies a cigar geometry. We first derive the Chern number from the cigar geometry and we then show that the quantum metric can be seen as a solution of two-dimensional non-Abelian BF theory in momentum space. The gauge connection for this model is associated to the Maxwell algebra, which takes into account the Lorentz symmetries related to the Dirac theory and the momentum-space magnetic translations connected to the magnetic perturbation. The Witten black-hole metric is a solution of this gauge theory and coincides with the Bures metric. This allows us to calculate the corresponding momentum-space entanglement entropy that surprisingly carries information about the real-space conformal field theory describing the defect lines that can be created on the gapped boundary.
We characterize the quantum phase transitions of gapped time-reversal invariant spin chain systems by nontrivial Z 2 numbers derived from a quantized Berry phase in a (1 + 1)-dimensional Bloch momentum space. We study this approach analytically in a general two-band model and give a concrete example in a transverse field XY spin chain system. In our scheme, an extra local gauge transformation is performed to the spin system by a time-dependent twist operator, which endows the Hamiltonian of the system with the topology of a torus without changing its energy spectrum. The Z 2 number is obtained by a loop integral of the Berry gauge potential along a quarter of the Brillouin zone. We show that the different phases of the XY spin chain system are distinguished by the Z 2 topological order in momentum space.