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Exploring nontrivial topology at quantum criticality on a superconducting processor

Chao Song, Ziqi Tan, Ke Wang, Sheng Yang, Fanhao Shen, Feitong Jin, Xuhao Zhu, Yujie Ji, Shibo Xu, Jiachen Chen, Yaozu Wu, Chuanyu Zhang, Yu Gao, Ning Wang, Yiren Zou, Aosai Zhang, Tingting Li, Zehang Bao, Zitian Zhu, Jiarun Zhong, Zhengyi Cui, Yihang Han, Yiyang He, Han Wang, Jianan Yang, Yanzhe Wang, Jiayuan Shen, Gongyu Liu, Zixuan Song, Jinfeng Deng, Hang Dong, Pengfei Zhang, Shao-Kai Jian, Hekang Li, Zhen Wang, Junxiang Zhang, Qiujiang Guo, Hai‐Qing Lin, Xuejia Yu, H. Wang, Fei Qing Wu

Published Apr 29, 2025• 0 citations
Publisher

Abstract

Abstract The discovery of nontrivial topology in quantum critical states has revised the classification of quantum phase transitions and opened a new direction for exploiting topological phases. However, the experimental investigation of nontrivial topology in gapless many-body systems is challenging due to the inherent complex quantum entanglement. Here, we experimentally explore the topological properties in the critical cluster Ising model by preparing its low-lying critical states on a superconducting processor with up to 100 qubits. We develop an efficient method to probe the boundary g-function, which allows us to uniquely identify the nontrivial topology of the critical systems under study. Furthermore, we probe and recognize two-fold topological degeneracy in the entanglement spectrum under periodic boundary conditions and experimentally reveal the novel bulk-boundary correspondence in topological critical systems. Our results demonstrate the low-lying critical states as useful quantum resources for investigating the interplay between topology and quantum criticality.

DOI
10.21203/rs.3.rs-6126852/v1

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Abstract The discovery of nontrivial topology in quantum critical states has revised the classification of quantum phase transitions and opened a new direction for exploiting topological phases. However, the experimental investigation of nontrivial topology in gapless many-body systems is challenging due to the inherent complex quantum entanglement. Here, we experimentally explore the topological properties in the critical cluster Ising model by preparing its low-lying critical states on a superconducting processor with up to 100 qubits. We develop an efficient method to probe the boundary g-function, which allows us to uniquely identify the nontrivial topology of the critical systems under study. Furthermore, we probe and recognize two-fold topological degeneracy in the entanglement spectrum under periodic boundary conditions and experimentally reveal the novel bulk-boundary correspondence in topological critical systems. Our results demonstrate the low-lying critical states as useful quantum resources for investigating the interplay between topology and quantum criticality.
The discovery of nontrivial topology in quantum critical states has revised the classification of quantum phase transitions and opened a new direction for exploiting topological phases. However, the experimental investigation of gapless many-body systems is challenging due to the inherent complex quantum entanglement. Here, we demonstrate an experimental study of the topological properties in the critical cluster Ising model on a superconducting processor with up to 100 qubits. By applying entanglement Hamiltonian tomography to the experimentally prepared low-lying states, we successfully reconstruct the reduced density matrices of the underlying ground states for subsystems with up to 12 qubits, based on which we extract the scaling dimension and central charge of the critical cluster Ising model. Furthermore, we recognize a robust two-fold topological degeneracy as well as the conformal-tower structure in the entanglement spectrum, experimentally revealing the bulk-boundary correspondence in topological critical systems. Our results demonstrate the low-lying states at critical points as useful quantum resources for investigating the interplay between topology and quantum criticality. Nontrivial topologies are long believed to survive only in gapped systems, yet recent progress suggests their existence even at quantum critical points. By preparing low-lying states on a superconducting processor up to 100 qubits and applying entanglement Hamiltonian tomography, the authors experimentally observe these nontrivial topological properties and suggest low-lying states as useful quantum resources.
Quantum entanglement can be an effective diagnostic tool for probing topological phases protected by global symmetries. Recently, the notion of nontrivial topology in critical systems has been proposed and is attracting growing attention. In this work, as a concrete example, we explore the quantum entanglement properties of fermionic symmetry-enriched quantum critical points by constructing exactly solvable models based on stacked multiple Kitaev chains. We first analytically establish the global phase diagram using entanglement entropy and reveal three topologically distinct gapped phases with different winding numbers, along with three topologically distinct transition lines separating them. Importantly, we unambiguously demonstrate that two transition lines exhibit fundamentally different topological properties despite sharing the same central charge. Specifically, they display nontrivial topological degeneracy in the entanglement spectrum under periodic boundary conditions, thereby generalizing the Li-Haldane bulk-boundary correspondence to a broader class of fermionic symmetry-enriched criticality. Additionally, we identify a novel Lifshitz multicritical point at the intersection of the three transition lines, which also exhibits nontrivial topological degeneracy. This work provides a valuable reference for investigating gapless topological phases of matter from the perspective of quantum entanglement.
Entanglement signatures of gapless topological phases in a ppp-wave superconductor
We explore the gapless topological phases of a ppp-wave superconductor, probing its rich topologically ordered phases and underlying quantum phenomena. The topological order of the system is characterized by studying its entanglement properties. This study confirms the bulk-boundary correspondence in the entanglement spectrum, even without a full bulk gap. For contractible bipartitions, the entanglement entropy varies non-monotonically with the chemical potential, displaying pronounced peaks at points where the bulk gap closes and reopens, signaling topological quantum phase transitions. This behavior remains robust in the thermodynamic limit. The entanglement entropy grows with system size for non-contractible bipartitions, indicating long-range entanglement in the gapless phase. These findings reveal the subtle interplay between symmetry, entanglement, and topology in gapless systems, and emphasize the role of entanglement-based diagnostics in identifying unconventional topological phases beyond the gapped paradigm.
A quantum phase transition is usually achieved by tuning physical parameters in a Hamiltonian at zero temperature. Here, we show that the ground state of a topological phase itself encodes critical properties of its transition to a trivial phase. To extract this information, we introduce an extensive partition of the system into two subsystems both of which extend throughout the bulk in all directions. The resulting bulk entanglement spectrum has a low-lying part that resembles the excitation spectrum of a bulk Hamiltonian, which allows us to probe a topological phase transition from a single wave function by tuning either the geometry of the partition or the entanglement temperature. As an example, this remarkable correspondence between the topological phase transition and the entanglement criticality is rigorously established for integer quantum Hall states.
Topological phenomena in quantum critical systems have recently attracted growing attention, as they go beyond the traditional paradigms of condensed matter and statistical physics. However, a general framework for identifying such nontrivial phenomena, particularly in higher-dimensional systems, remains insufficiently explored. In this work, we propose a universal fingerprint for detecting nontrivial topology in critical free-fermion systems protected by global on-site symmetries. Specifically, we analytically establish an exact relation between the bulk entanglement spectrum and the boundary energy spectrum at topological criticality in arbitrary dimensions, demonstrating that the degeneracy of edge modes can be extracted from the bulk entanglement spectrum. These findings, further supported by numerical simulations of lattice models, provide a universal fingerprint for identifying nontrivial topology in critical free-fermion systems.
Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Majorana zero modes at criticality. Trivial and non-trivial topological nature of criticality due to the presence of multicritical point cause an unusual topological transition along the critical line. We observe and investigate this unique transition in terms of eigenvalue spectrum. Appearance of MZMs at criticality demands integer value of topological invariant number in order to validate the concept of bulk-boundary correspondence. Hence we propose a scheme to separate the invariant number into fractional and integer contribution to establish bulk-boundary correspondence at criticality.