Research

The Watanabe Group studies the universal structure of quantum many-body systems through the unifying lens of symmetry. We work primarily on the foundations and consequences of symmetry-related phenomena in condensed matter physics — from spontaneous symmetry breaking and topological phases to the rigorous mathematical structure of quantum lattice models. Our work combines rigorous theory (proofs of general theorems, no-go results, and bounds) with broadly applicable frameworks (such as symmetry indicators usable across all 230 crystallographic space groups) and occasional predictions for real materials. Our current research spans five interconnected directions, summarized below.

1. Spontaneous symmetry breaking and Nambu–Goldstone bosons

Coset spaces G/H illustrating spontaneously broken symmetries — Coset spaces $G/H$ associated with spontaneously broken symmetries: (a) $U(1) \to$ trivial gives $S^1$; (b)(c) $SO(3) \to SO(2)$ gives $S^2$ in two distinct ways related to ferromagnetic vs. antiferromagnetic ordering.

Whenever a continuous symmetry is spontaneously broken, gapless excitations called Nambu–Goldstone (NG) bosons appear. The classic theorem of Goldstone counts one mode per broken symmetry generator. In nonrelativistic systems — where most condensed matter realizations live — the count and dispersion of NG modes can deviate dramatically from this classic picture: ferromagnetic magnons are quadratic, not linear; some symmetries break without producing the "expected" number of modes.

We established a unified counting rule that resolves these long-standing puzzles (PRL 108, 251602 (2012); PRX 4, 031057 (2014); reviewed in Annu. Rev. Condens. Matter Phys. 11, 169 (2020)). We also proved the absence of quantum time crystals in equilibrium (PRL 114, 251603 (2015)), showing that translation symmetry in time cannot be spontaneously broken in a ground state or in equilibrium states.

Recent work has uncovered surprises that test the limits of these theorems — including a one-dimensional spin model that breaks a continuous $U(1)$ symmetry at zero temperature, a remarkable counterexample to common expectations from the Mermin–Wagner theorem (PRL 133, 176001 (2024), Editors' Suggestion).

Selected press releases for this theme:

"Discovery of a one-dimensional spin model exhibiting spontaneous breaking of continuous symmetry — a new counterexample to the Mermin–Wagner theorem" (Oct 25, 2024)
"Rigorous proof of the impossibility of time crystals" (Jun 22, 2015)
"Theorem unifies superfluids and other weird materials", UC Berkeley News (Jun 8, 2012)

Related commentary and feature articles:

Nature News (2017)
"Scientists Finally Observed Time Crystals—But What the Hell Are They?", Gizmodo (2017)
"Physicists propose new definition of time crystals—then prove such things don't exist", Phys.org (Jul 2015)
Journal Club for Condensed Matter Physics (Sep 2014)
APS Physics Synopsis (2012)

2. Topological phases and symmetry indicators

Sodium chloride crystal with fractional charges at corners — Sodium chloride (NaCl) crystal: an ordinary kitchen-table material whose corners we showed carry a fractional electric charge of $\pm e/8$ — a manifestation of higher-order topology in band insulators. From Phys. Rev. X 11, 041064 (2021).

The discovery of topological insulators turned the classification of insulating band structures into a central question of condensed matter physics. We developed a symmetry-based framework that diagnoses the topology of any band insulator using only the symmetry representations of its Bloch wavefunctions. This framework covers all 230 nonmagnetic and 1651 magnetic space groups (Nat. Commun. 8, 50 (2017); Sci. Adv. 4, eaat8685 (2018)) and extends to topological superconductors in all space groups (Sci. Adv. 6, eaaz8367 (2020)). We also clarified the notion of fragile topology — phases that resist a Wannier description yet are not in the usual topological-insulator class (PRL 121, 126402 (2018), Editors' Suggestion).

The same circle of ideas led to a striking concrete prediction: the corners of a finite NaCl crystal carry a fractional charge of $\pm e/8$ (PRX 11, 041064 (2021)) — a higher-order topological phenomenon hiding in everyday salt. On the materials side, we have driven high-throughput searches for topological superconductors (PRL 129, 027001 (2022), Editors' Suggestion).

Selected press releases for this theme:

"Bulk-edge correspondence in measurement-induced topological phases" (Jun 19, 2025)
"A new database for systematic exploration of topological superconductors" (Jul 7, 2022)
"Theoretical demonstration that NaCl crystal corners carry fractional charge" (Jan 4, 2022)
"Symmetry-based diagnosis of topological superconductors" (May 1, 2020)
"Discovery of a new class of insulators between trivial and topological" (Sep 17, 2018)
"Theoretical clarification of a new type of topological insulator" (Sep 17, 2018)
"Discovery of symmetry-based indicators of band topology" (Jul 2, 2017)

Related commentary and feature articles:

"Scientists Develop Way to Identify Topological Materials", Lab Manager (Apr 17, 2019)
Journal Club for Condensed Matter Physics (Aug 2017)

3. Nonlinear electromagnetic responses in superconductors

Diagrammatic representation of optical conductivities in a multi-band superconductor — Diagrammatic representation of (a) linear and (b) second-order optical conductivities of a multi-band superconductor. From arXiv:2410.18679.

Recent terahertz experiments have made it possible to observe the nonlinear optical responses of superconductors — including third-harmonic generation, which has been interpreted as a signature of the elusive Higgs mode. Constructing a microscopic theory of these responses is, however, surprisingly subtle: naive perturbative formulas spuriously violate gauge invariance, and the contributions of collective modes (Higgs, Nambu–Goldstone) and quasiparticles must be disentangled.

We have developed a gauge-invariant formulation of nonlinear optical responses in BCS superconductors (arXiv:2410.18679; arXiv:2501.13722) and derived rigorous f-sum rules that constrain nonlinear optical conductivities at all orders (Phys. Rev. B 102, 165137 (2020)). Most recently, we are quantifying the role of quantum geometry in the competition between Higgs modes and quasiparticles in third-harmonic generation (arXiv:2512.01200).

4. Frustration-free quantum many-body systems

Rokhsar-Kivelson Markov correspondence — Rokhsar–Kivelson construction: the ground state of an RK Hamiltonian in $d+1$ dimensions corresponds to the canonical distribution of a classical Markov process in $d$ dimensions. This bridge between quantum frustration-free systems and classical stochastic dynamics underlies many of our rigorous results.

Critical fluctuations of the Ising model — Critical fluctuations of the classical Ising model on a system of linear size $L$. The horizontal axis is rescaled by $t^{1/z}$, where $z$ is the dynamical critical exponent. We rigorously proved $z \geq 2$ for the Ising model — resolving a 100-year-old open problem.

Many of the most famous models of strongly correlated physics — the AKLT chain, the toric code, Rokhsar–Kivelson dimer models, stabilizer codes — share a common mathematical structure: they are frustration-free. Each local term in the Hamiltonian is simultaneously minimized by the ground state. This special structure makes such models analytically tractable while still hosting an extraordinarily rich variety of phases (symmetry-protected, topologically ordered, fracton, …).

We are working out the general theory of gapless frustration-free systems, deriving rigorous bounds on dynamical critical exponents and finite-size gaps. Highlights include: a proof that the dynamical critical exponent must satisfy $z \geq 2$ in any gapless frustration-free system (Phys. Rev. X 15, 041050 (2025); PRB 110, 195140 (2024)), and an analogous rigorous bound for the classical Ising model that resolves a 100-year-old question about its dynamical universality class (J. Stat. Phys. 192, 76 (2025)). We have also been mapping out the landscape of frustration-free free-fermion systems (PRB 112, 115104 (2025); PRB 113, 195120 (2026); Phys. Rev. Res. 8, 023171 (2026)).

Selected press releases for this theme:

"Rigorous proof of the lower bound z ≥ 2 for the dynamical critical exponent of the Ising model — a theoretical advance on a 100-year-old open problem" (May 28, 2025)

5. Spin ice and water-ice phases

Phase diagram of high-pressure water ice on a model lattice. Phase VII (proton-disordered) and phase X (symmetric) are revealed to be the same phase, continuously connected via a crossover beyond the critical endpoint (red star). Phase VIII, with distinct symmetry, is genuinely separated from VII by a phase transition. From arXiv:2603.19620.

Liquid water and water vapor — though they look very different at atmospheric pressure — are not in fact distinct phases of matter: by going around the critical point, one can be transformed continuously into the other without ever crossing a phase transition. They are merely two faces of a single phase.

This invites a striking question for high-pressure water ice. Among its many known phases, ice VII and ice X share the same crystallographic symmetry, while ice VIII has a different one. Could VII and X then be the same phase, connected by a continuous crossover — just as liquid and vapor are?

A recent model study by our group answers this affirmatively. In a microscopic lattice model of water ice, phases VII and X are continuously connected through the screening of magnetic-monopole-like proton defects, while ice VIII, with its distinct symmetry, remains separated by a genuine phase transition (arXiv:2603.19620).

This raises a deeper, more general question that drives much of our work in this direction: within finite-temperature phases of identical symmetry, can there exist finer distinctions — analogous to the topological-phase classifications that exist at zero temperature — that survive thermal fluctuations and remain meaningful as classifying labels? We pursue this question in parallel through pyrochlore spin ice, where the ground-state "ice rules" are mathematically identical to those governing the proton arrangements in water ice and host emergent gauge fields, fractionalized excitations (magnetic monopoles), and topological order. Recent works classify topological phase transitions of pyrochlore spin ice for arbitrary spin $S$ and uncover dualities relating seemingly different models (arXiv:2603.03852; arXiv:2604.04346).

Together, these projects aim to bring the modern toolkit of topology and statistical field theory to bear on materials and questions that have been studied for over a century.