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arXiv:2301.08827 · Physical review. B./Physical review. B

Second Euler number in four-dimensional matter

Adrien Bouhon, Yan-Qing Zhu, Robert-Jan Slager, Giandomenico Palumbo

Published Nov 22, 2024• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.110.195144

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Abstract Topological insulators are exotic material that possess conducting surface states protected by the topology of the system. They can be classified in terms of their properties under discrete symmetries and are characterized by topological invariants. The latter has been measured experimentally for several models in one, two and three dimensions in both condensed matter and quantum simulation platforms. The recent progress in quantum simulation opens the road to the simulation of higher dimensional Hamiltonians and in particular of the 4D quantum Hall effect. These systems are characterized by the second Chern number, a topological invariant that appears in the quantization of the transverse conductivity for the non-linear response to both external magnetic and electric fields. This quantity cannot always be computed analytically and there is therefore a need of an algorithm to compute it numerically. In this work, we propose an efficient algorithm to compute the second Chern number in 4D systems. We construct the algorithm with the help of lattice gauge theory and discuss the convergence to the continuous gauge theory. We benchmark the algorithm on several relevant models, including the 4D Dirac Hamiltonian and the 4D quantum Hall effect and verify numerically its rapid convergence.
Topological Euler insulators emerge in multiband systems with real Bloch Hamiltonians and wavefunctions. Their fragile topologies are characterized by the Euler class of degenerate bands and protected by the PTPTPT or C2TC_2TC2​T symmetry in two dimensions, which go beyond the tenfold KKK-theory classification of topological matter. In this work, we extend the conception of topological Euler insulators to non-Hermitian systems and propose a theoretical framework to unlock their nontrivial Euler topology. Focusing on two-dimensional, three-band non-Hermitian lattice models with symmetric Hamiltonians, we formulate a comprehensive description of their topological Euler bands, entanglement spectrum and bulk-boundary correspondence. Three typical models of non-Hermitian Euler insulators are constructed and investigated explicitly to illustrate our theory. Unique topological phase transitions and anomalous edge-band overlaps with non-Hermitian origins are further identified. Our study establishes the presence of topological Euler bands in non-Hermitian systems and unveils their intriguing physical characteristics, thereby broadening the existing territory of topological matter in non-Hermitian open systems.
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.
Four-dimensional (4D) topological insulators, which are impossible in real materials, have attracted much attention by virtue of the recent progress achieved in quantum simulations of higher-dimensional systems. In this paper, we employ the supercell approximation to investigate the disorder effects on a system that supports the 4D topological insulator phases characterized by quantized second Chern numbers and the normal insulator phase. We demonstrate that the 4D topological insulator phases are robust against weak disorders. Moreover, we reveal that disorder can transform a normal insulator to a 4D topological insulator with an emergent second Chern number, referred to as a 4D topological Anderson insulator. An effective-medium theory based on the Born approximation further confirms the numerical conclusions.
Integer and fractional Chern insulators exhibit a nonzero quantized anomalous Hall conductivity due to a spontaneous breaking of time reversal symmetry. To identify nontrivial topology in their time-reversal symmetric many-body spectra, we introduce many-body Euler numbers as a counterpart to many-body Chern numbers. Exemplarily, we perform calculations in a topological Hubbard model that can realize Chern and fractional Chern insulating phases. Furthermore, we lay out a classification scheme to realize different topological phases in interacting systems using symmetry indicators in analogy to topological band theory.
We predict the existence of a Floquet topological insulator in three-dimensional two-band systems, the Floquet Hopf insulator, which possesses two distinct topological invariants. One is the Hopf Z invariant, a linking number characterizing the (nondriven) Hopf topological insulator. The second invariant is an intrinsically Floquet Z_{2} invariant, and represents a condensed matter realization of the topology underlying the Witten anomaly in particle physics. Both invariants arise from topological defects in the system's time evolution, subject to a process in which defects at different quasienergies exchange even amounts of topological charge. Their contrasting classifications lead to a measurable physical consequence, namely, an unusual bulk-boundary correspondence where gapless edge modes are topologically protected, but may exist at either 0 or π quasienergy. Our results represent a phase of matter beyond the conventional classification of Floquet topological insulators.