A collaborative research team led by Linhao Li (then a PhD student), Masaki Oshikawa (Professor), and Han Yan (Assistant Professor) from the Institute for Solid State Physics (ISSP), the University of Tokyo, with Linhao Li also affiliated with the Department of Physics and Astronomy at Ghent University, has proposed a unified theoretical framework called the Bilinear Phase Map (BPM) for generalized Kramers–Wannier dualities in quantum spin chains. By encoding duality transformations as matrices over finite fields, the team clarified how hidden symmetries, non-unitarity, boundary-condition twists, and non-invertible fusion rules emerge in lattice models, and established an exact anomaly criterion for when self-dual qudit spin chains cannot have a unique gapped ground state.

Fig.1 Three phase transitions between two different gapped phases with Z₂ × Z₂ symmetry and the dualities between them

Quantum many-body systems exhibit diverse phases characterized by symmetry, topology, and ground-state degeneracy. Understanding the distinctions between such phases, and the phase transitions connecting them, is a central issue in condensed matter physics. A key concept is duality, which relates seemingly different descriptions of the same physical system. The classic example is Kramers–Wannier duality in the Ising model, which connects paramagnetic and ferromagnetic phases and identifies the self-dual point associated with a phase transition.

Recently, Kramers–Wannier duality has attracted renewed attention as an example of non-invertible symmetry. Unlike ordinary symmetries, non-invertible symmetries do not necessarily have inverse operations and can obey generalized fusion rules. At a self-dual point, such a duality may carry an anomaly, which forbids a trivially gapped phase with a unique ground state. However, a general microscopic framework for classifying lattice dualities and diagnosing their anomalies has been lacking.

In this work, the researchers introduce the BPM framework to describe generalized Kramers–Wannier dualities systematically. In this formulation, a duality transformation is represented by a matrix over a finite field, allowing physical properties of the duality to be read from linear-algebraic data. For example, rank deficiency signals non-unitarity, the kernel encodes associated global symmetries, and boundary twists that restore unitarity can be described using the same matrix structure. The framework also provides a general way to derive non-invertible fusion rules and construct families of generalized qudit spin-chain models.

A central result is an exact anomaly criterion for a broad class of three-site dualities. The researchers study a reflection-combined BPM duality, called RBPM duality, in prime-p-level qudit chains, and prove that it is anomalous if and only if −1 is not a quadratic residue modulo p. When this condition holds, a self-dual system cannot possess a unique gapped ground state; it must instead be gapless or have ground-state degeneracy. The smallest anomalous case occurs at p = 3, revealing non-invertible symmetry constraints absent in ordinary qubit chains.

The researchers further analyze concrete p = 3 qudit models, including a staggered-dipole Ising model, and show how the BPM framework identifies self-dual points and constrains their phase structure. The framework also reveals duality relations between models that appear unrelated, demonstrating its usefulness as a systematic tool for organizing generalized quantum phases.

This work establishes a unified and algebraic framework for understanding generalized dualities in quantum spin chains, revealing deep connections between lattice symmetries, non-invertible structures, and number-theoretic conditions. By providing a clear anomaly criterion and explicit model constructions, the BPM approach opens a systematic route to classify and design exotic quantum phases beyond conventional symmetry-based descriptions. More broadly, it suggests that finite-field linear algebra can serve as a powerful organizing principle for quantum many-body systems, and it is expected to stimulate further studies on higher-dimensional generalizations, experimental realizations, and connections to quantum information and topological phases of matter.

Publication

• Journal：Physical Review Letters
• Title：Generalized Kramers-Wannier Duality from Bilinear Phase Map
• Authors： Linhao Li, Masaki Oshikawa, and Han Yan
• DOI: https://doi.org/10.1103/g1l7-bvtf