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Physical Review Research

Higher-order topological biphoton corner states in two-dimensional photonic lattices

Chu Li, Meng Li, Linyu Yan, Sheng Ye, Xiaoyong Hu, Qihuang Gong, Yan Li

Published Apr 20, 2022• 20 citations
Publisher

Abstract

Higher-order topological insulators have attracted great interest for their unconventional bulk-boundary correspondence especially gapless corner states. Here, we realize topological biphoton corner states in two-dimensional lattices. In addition to the striking state localization, the correlations in topologically nontrivial cases are more than 60 times higher than those in trivial cases and the visibility of Hong-Ou-Mandel interference for output topological corner-entangled states is higher than 90.3%. We confirm that the second-order photonic topological insulators support stable biphoton states that can maintain spatial distributions in high-dimensional Hilbert space, which have potential applications in quantum algorithms, simulation, and computation.

DOI
10.1103/physrevresearch.4.023049

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Higher-order topological insulators have attracted great interest for their unconventional bulk-boundary correspondence especially gapless corner states. Here, we realize topological biphoton corner states in two-dimensional lattices. In addition to the striking state localization, the correlations in topologically nontrivial cases are more than 60 times higher than those in trivial cases and the visibility of Hong-Ou-Mandel interference for output topological corner-entangled states is higher than 90.3%. We confirm that the second-order photonic topological insulators support stable biphoton states that can maintain spatial distributions in high-dimensional Hilbert space, which have potential applications in quantum algorithms, simulation, and computation.
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Higher-order topological insulators not only exhibit the exotic bulk-boundary correspondence principle but also have an important application in quantum computing. However, it seems they have not been achieved in the discrete-time quantum walk. In this paper we construct a two-dimensional coinless discrete-time quantum walk to simulate the second-order topological insulator with zero-dimensional corner states. We show that both of the corner and edge states can be observed through the probability distribution of the walker after a multistep coinless discrete-time quantum walk. Finally, we propose a possible experimental implementation to realize this coinless discrete-time quantum walk in a three-dimensional integrated photonic circuit and verify the robustness of the corner states against the small disorder. Our work offers an alternative route to explore exotic higher-order topological matters using discrete-time quantum walks.
We propose a new correlated topological state, which we call a higher-order topological Mott insulator (HOTMI). This state exhibits a striking bulk-boundary correspondence due to electron correlations. Namely, the topological properties in the bulk, characterized by the Z_{3} spin-Berry phase, result in gapless corner modes emerging only in spin excitations (i.e., the single-particle excitations remain gapped around the corner). We demonstrate the emergence of the HOTMI in a Hubbard model on the kagome lattice, and elucidate how strong correlations change gapless corner modes at the noninteracting case.
Higher-order topological insulators have attracted significant interest in recent years. However, identifying a universal topological invariant capable of characterizing higher-order topology remains challenging. Here, we propose a entanglement topological invariant designed to characterize second-order topological systems. This entanglement topological invariant captures the entanglement of topological corner states under open boundary conditions by employing a bipartite entanglement entropy method. In several representative models, the entanglement topological invariant assumes a nonzero value exclusively in the presence of second-order topology, with its magnitude exactly matching the number of topologically protected corner states. Consequently, the proposed entanglement topological invariant not only provides a clear criterion for detecting higher-order topology, but also offers a quantitative measure for the related corner states. Our study establishes a universal and precise method for characterizing higher-order topological phases, opening avenues for their fundamental understanding and future investigations. Higher-order topological insulators have become a focus of research, but developing a universal way to characterize their unique higher-order topological properties remains a key challenge. The authors propose an entanglement topological invariant based on entanglement entropy, which not only reliably identifies second-order topological phases but also quantifies the number of topologically protected corner states.
Higher-order topological insulators are materials with a bulk energy gap featuring gapless modes at (d−n)(d-n)(d−n)-dimensional edges, where n>1n > 1n>1. Despite this modified bulk-boundary correspondence, their nontrivial topology can nevertheless be observed through the bipartite entanglement spectrum. Inspired by works suggesting fluctuations in conserved quantities could be used to study topology in one-dimensional systems, which likewise feature zero-dimensional edge modes, we extend this treatment to a two-dimensional second-order topological insulator where corner states play an analogous role. We show that the standard bipartite entanglement entropy and particle number fluctuation lack striking signals of topological phase transitions, both in the static case and in the time evolution after a quench, obscured by an area-law term from the subsystem edges. By introducing a quadripartite construction of the subsystem, we show that it is possible to isolate the topological contributions, producing sharp transition peaks in both quantities, observable by studying subsystems much smaller than the full system. As particle number fluctuations are accessible in existing experiments, these results establish a concrete and scalable path to experimental detection of the entanglement in higher-order topological insulators.
In a d -dimensional topological insulator of order d , there are zero-energy states on its corners which have close relationship with its entanglement behaviors. We studied the bipartite entanglement spectra for different subsystem shapes and found that only when the entanglement boundary has corners matching the lattice, exact zero modes exist in the entanglement spectrum corresponding to the zero-energy states caused by the same physical corners. We then considered finite-size systems in which cases these corner states are coupled together by long-range hybridizations to form multipartite entangled states. We proposed a scheme to calculate the quadripartite entanglement entropy on the square lattice, which is well described by a four-sites toy model and thus provides another way to identify the higher order topological insulators from the multipartite entanglement point of view.