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arXiv:1202.2397 · Europhysics Letters (EPL)

The Euler number of Bloch states manifold and the quantum phases in gapped fermionic systems

Yu-Quan Ma, Shi-Jian Gu, Shu Chen, Heng Fan, Wu-Ming Liu

Published Jul 1, 2013• 47 citations
arXivHTMLPDF

Abstract

We propose a topological Euler number to characterize nontrivial topological phases of gapped fermionic systems, which originates from the Gauss-Bonnet theorem on the Riemannian structure of Bloch states established by the real part of the quantum geometric tensor in momentum space. Meanwhile, the imaginary part of the geometric tensor corresponds to the Berry curvature which leads to the Chern number characterization. We discuss the topological numbers induced by the geometric tensor analytically in a general two-band model. As an example, we show that the zero-temperature phase diagram of a transverse field XY spin chain can be distinguished by the Euler characteristic number of the Bloch states manifold in a -dimensional Bloch momentum space.

DOI
10.1209/0295-5075/103/10008

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We propose a topological Euler number to characterize nontrivial topological phases of gapped fermionic systems, which originates from the Gauss-Bonnet theorem on the Riemannian structure of Bloch states established by the real part of the quantum geometric tensor in momentum space. Meanwhile, the imaginary part of the geometric tensor corresponds to the Berry curvature which leads to the Chern number characterization. We discuss the topological numbers induced by the geometric tensor analytically in a general two-band model. As an example, we show that the zero-temperature phase diagram of a transverse field XY spin chain can be distinguished by the Euler characteristic number of the Bloch states manifold in a -dimensional Bloch momentum space.
We study the Riemannian metric and the Euler characteristic number of the Bloch band in a one-dimensional spin model with multisite spins exchange interactions. The Euler number of the Bloch band originates from the Gauss-Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone. We study this approach analytically in a transverse field XY spin chain with three-site spin coupled interactions. We define a class of cyclic quantum distance on the Bloch band and on the ground state, respectively, as a local characterization for quantum phase transitions. Specifically, we give a general formula for the Euler number by means of the Berry curvature in the case of two-band models, which reveals its essential relation to the first Chern number of the band insulators. Finally, we show that the ferromagnetic-paramagnetic phase transition in zero temperature can be distinguished by the Euler number of the Bloch band.
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.
Two-dimensional Euler insulators are novel kinds of systems that host multigap topological phases, quantified by a quantized first Euler number in their bulk. Recently, these phases have been experimentally realized in suitable two-dimensional synthetic matter setups. Here, we introduce the second Euler invariant, a familiar invariant in both differential topology (Chern-Gauss-Bonnet theorem) and in four-dimensional Euclidean gravity, whose existence still needs to be explored in condensed matter systems. Specifically, we first define two specific models in four dimensions that support a nonzero second Euler number in the bulk together with peculiar gapless boundary states. Second, we discuss its robustness in general spacetime-inversion invariant phases and its role in the classification of topological degenerate real bands through real Grassmannians. Our results naturally generalize the second Chern and spin Chern numbers to the case of four-dimensional phases that are characterized by real Hamiltonians and open doors for implementing such higher-dimensional phases in artificial engineered systems, ranging from ultracold atoms to photonics and electric circuits.
We generalize the concept of topological invariants for mixed states based on the ensemble geometric phase (EGP) to two-dimensional band structures. In contrast to the geometric Uhlmann phase for density matrices, the EGP leads to a proper Chern number for Gaussian, finite-temperature, or nonequilibrium steady states. The Chern number can be expressed as an integral of the ground-state Berry curvature of a fictitious lattice Hamiltonian, constructed from single-particle correlations. For the Chern number to be nonzero the fictitious Hamiltonian has to break time-reversal symmetry.
Mapping a many-body state on a loop in parameter space is a simple way to characterize a quantum state. The connections of such a geometrical representation to the concepts of Chern number and Majorana zero mode are investigated based on a generalized quantum spin system with short and long-range interactions. We show that the topological invariants, the Chern numbers of corresponding Bloch band, is equivalent to the winding number in the auxiliary plane, which can be utilized to characterize the phase diagram. We introduce the concept of Majorana charge, the magnitude of which is defined by the distribution of Majorana fermion probability in zero-mode states, and the sign is defined by the type of Majorana fermion. By direct calculations of the Majorana modes we analytically and numerically verify that the Majorana charge is equal to Chern numbers and winding numbers.
Understanding the relationship between quantum geometry and topological invariants is a central problem in the study of topological states. In this work, we establish the relationship between the quantum metric and the Euler curvature in two-dimensional systems with space-time inversion IST{I}_{ST}IST​ symmetry satisfying IST2=+1{I}_{ST}^{2}=+1IST2​=+1. As IST{I}_{ST}IST​ symmetry imposes the reality of the wave function with vanishing Berry curvature, the well-known inequality between the quantum metric and the Berry curvature is not meaningful in this class of systems. We find that the non-Abelian quantum geometric tensor of two real bands exhibits an intriguing inequality between the off-diagonal Berry curvature and the quantum metric, which in turn gives the inequality between the quantum volume and the Euler invariant. Moreover, we show that the saturation condition of the inequality is deeply related to the ideal condition for Euler bands, which provides a criterion for the stability of fractional topological phases in interacting Euler bands. Our findings demonstrate the potential of the quantum geometry as a powerful tool for characterizing symmetry-protected topological states and their interaction effect.