About

Shu-Heng Shao is an Assistant Professor of Physics at MIT, focusing on generalizing the symmetry principle in quantum field theories and lattice models, with applications in high energy physics, condensed matter theory, and quantum gravity. His research explores the structural aspects of quantum field theories and lattice systems, particularly centered on generalized symmetries and anomalies, including a novel type of symmetry without an inverse, known as non-invertible symmetries. These symmetries have been identified in various quantum systems such as the Ising model, Yang-Mills theories, lattice gauge theories, and the Standard Model, leading to new constraints on renormalization group flows, conservation laws, and organizing principles in classifying phases of quantum matter. Born and raised in Taiwan, Shao obtained his B.S. in physics from National Taiwan University in 2010 and his Ph.D. in physics from Harvard University in 2016 under the guidance of Prof. Xi Yin. He was a long-term member at the Institute for Advanced Study in Princeton before joining the Yang Institute for Theoretical Physics at Stony Brook University as an assistant professor in 2021. In 2024, he joined MIT faculty. His notable awards include the 2026 New Horizons in Physics Prize, the 2025 Frontiers of Science Award at the International Conference on Basic Science, and recognition from the Simons Collaboration on Ultra-Quantum Matter. His work has significantly contributed to the understanding of symmetries in quantum theories, as reflected in his publications and lectures.

Research topics

• Computer Science
• Mathematics
• Quantum mechanics
• Sociology
• Theoretical physics
• Physics
• Pure mathematics
• Algorithm
• Geometry
• Mathematical physics

Selected publications

• Where non-invertible symmetries end: twist defects for electromagnetic duality

  Journal of High Energy Physics · 2026-01-19 · 2 citations

  articleOpen access1st author

  A bstract We study novel conformal twist defects in 4d Maxwell theory, around which electric and magnetic fields are exchanged. These are codimension-2 defects living at the end of topological defects for certain non-invertible global symmetries. We determine the operator spectrum of the twist defect by solving classical electromagnetic wave equations subject to a twisted boundary condition. Using techniques from defect CFT, we show that correlation functions of these defect operators factorize into two sectors: a universal generalized free-field sector, and a chiral current sector analogous to edge modes in Chern-Simons theory. In a similar setup, we also revisit the twist fields attached to non-invertible line defects in the 2d compact boson CFT. We discuss a defect ’t Hooft anomaly involving a chiral O (2) symmetry, highlighting its dynamical implications.

  PublisherOA PDFDOI

• Noninvertible symmetries: What’s done cannot be undone

  Physics Today · 2025-06-03 · 2 citations

  article1st authorCorresponding
  PublisherDOI

• Quantized Axial Charge of Staggered Fermions and the Chiral Anomaly

  Physical Review Letters · 2025-01-14 · 18 citations

  articleOpen accessSenior author

  In the 1+1D ultralocal lattice Hamiltonian for staggered fermions with a finite-dimensional Hilbert space, there are two conserved, integer-valued charges that flow in the continuum limit to the vector and axial charges of a massless Dirac fermion with a perturbative anomaly. Each of the two lattice charges generates an ordinary U(1) global symmetry that acts locally on operators and can be gauged individually. Interestingly, they do not commute on a finite lattice and generate the Onsager algebra, but their commutator goes to zero in the continuum limit. The chiral anomaly is matched by this non-Abelian algebra, which is consistent with the Nielsen-Ninomiya theorem. We further prove that the presence of these two conserved lattice charges forces the low-energy phase to be gapless, reminiscent of the consequence from perturbative anomalies of continuous global symmetries in continuum field theory. Upon bosonization, these two charges lead to two exact U(1) symmetries in the XX model that flow to the momentum and winding symmetries in the free boson conformal field theory.

  PublisherDOI

• Additivity, Haag duality, and non-invertible symmetries

  Journal of High Energy Physics · 2025-08-01 · 3 citations

  articleOpen access1st authorCorresponding

  A bstract The algebraic approach to quantum field theory focuses on the properties of local algebras, whereas the study of (possibly non-invertible) global symmetries emphasizes global aspects of the theory and spacetime. We study connections between these two perspectives by examining how either of two core algebraic properties — “additivity” or “Haag duality” — is violated in a 1+1D CFT or lattice model restricted to the symmetric sector of a general global symmetry. For the Verlinde symmetry of a bosonic diagonal RCFT, we find that additivity is violated whenever the symmetry algebra contains an invertible element, while Haag duality is violated whenever it contains a non-invertible element. We find similar phenomena for the Kramers-Wannier and Rep(D 8 ) non-invertible symmetries on spin chains.

  PublisherOA PDFDOI

• Where Non-Invertible Symmetries End: Twist Defects for Electromagnetic Duality

  ArXiv.org · 2025-09-25

  preprintOpen access1st authorCorresponding

  We study novel conformal twist defects in 4d Maxwell theory, around which electric and magnetic fields are exchanged. These are codimension-2 defects living at the end of topological defects for certain non-invertible global symmetries. We determine the operator spectrum of the twist defect by solving classical electromagnetic wave equations subject to a twisted boundary condition. Using techniques from defect CFT, we show that correlation functions of these defect operators factorize into two sectors: a universal generalized free-field sector, and a chiral current sector analogous to edge modes in Chern-Simons theory. In a similar setup, we also revisit the twist fields attached to non-invertible line defects in the 2d compact boson CFT. We discuss a defect 't Hooft anomaly involving a chiral $O(2)$ symmetry, highlighting its dynamical implications.

  PublisherOA PDFDOI

• Parity Anomaly from a Lieb-Schultz-Mattis Theorem: Exact Valley Symmetries on the Lattice

  Physical Review Letters · 2025-10-27

  preprintOpen accessSenior author

  We show that the honeycomb tight-binding model hosts an exact microscopic avatar of its low-energy SU(2) valley symmetry and parity anomaly. Specifically, the SU(2) valley symmetry arises from a collection of conserved, integer-quantized charge operators that obey the Onsager algebra. Along with lattice reflection and time-reversal symmetries, this Onsager symmetry has a Lieb-Schultz-Mattis (LSM) anomaly that matches the parity anomaly in the IR. Indeed, we show that any local Hamiltonian commuting with these symmetries cannot have a trivial unique gapped ground state. We study the phase diagram of the simplest symmetric model and survey various deformations, including Haldane's mass term, which preserves only the Onsager symmetry. Our results place the parity anomaly in $2+1\mathrm{D}$ alongside Schwinger's anomaly in $1+1\mathrm{D}$ and Witten's SU(2) anomaly in $3+1\mathrm{D}$ as 't Hooft anomalies that can arise from the Onsager symmetry on the lattice.

  PublisherOA PDFDOI

• Non-Invertible Interfaces Between Symmetry-Enriched Critical Phases

  ArXiv.org · 2025-12-29

  articleOpen access

  Gapless quantum phases can become distinct when internal symmetries are enforced, in analogy with gapped symmetry-protected topological (SPT) phases. However, this distinction does not always lead to protected edge modes, raising the question of how the bulk-boundary correspondence is generalized to gapless cases. We propose that the spatial interface between gapless phases -- rather than their boundaries -- provides a more robust fingerprint. We show that whenever two 1+1d conformal field theories (CFTs) differ in symmetry charge assignments of local operators or twisted sectors, any symmetry-preserving spatial interface between the theories must flow to a non-invertible defect. We illustrate this general result for different versions of the Ising CFT with $\mathbb{Z}_2 \times \mathbb{Z}_2^T$ symmetry, obtaining a complete classification of allowed conformal interfaces. When the Ising CFTs differ by nonlocal operator charges, the interface hosts 0+1d symmetry-breaking phases with finite-size splittings scaling as $1/L^3$, as well as continuous phase transitions between them. For general gapless phases differing by an SPT entangler, the interfaces between them can be mapped to conformal defects with a certain defect 't Hooft anomaly. This classification also gives implications for higher-dimensional examples, including symmetry-enriched variants of the 2+1d Ising CFT. Our results establish a physical indicator for symmetry-enriched criticality through symmetry-protected interfaces, giving a new handle on the interplay between topology and gapless phases.

  PublisherOA PDF

• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories

  SciPost Physics · 2025-01-09 · 28 citations

  articleOpen access

  We explore exact generalized symmetries in the standard 2+1d lattice \mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msub> </mml:math> gauge theory coupled to the Ising model, and compare them with their continuum field theory counterparts. One model has a (non-anomalous) non-invertible symmetry, and we identify two distinct non-invertible symmetry protected topological phases. The non-invertible algebra involves a lattice condensation operator, which creates a toric code ground state from a product state. Another model has a mixed anomaly between a 1-form symmetry and an ordinary symmetry. This anomaly enforces a nontrivial transition in the phase diagram, consistent with the “Higgs=SPT” proposal. Finally, we discuss how the symmetries and anomalies in these two models are related by gauging, which is a 2+1d version of the Kennedy-Tasaki transformation.

  PublisherOA PDFDOI

• Tensor Networks for Noninvertible Symmetries in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:math> and Beyond

  Physical Review X · 2025-07-16 · 8 citations

  articleOpen access

  Tensor networks provide a natural language for noninvertible symmetries in general Hamiltonian lattice models. We use ZX-diagrams, which are tensor network presentations of quantum circuits, to define a noninvertible operator implementing the Wegner duality in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>3</a:mn><a:mo>+</a:mo><a:mn>1</a:mn><a:mi mathvariant="normal">D</a:mi></a:mrow></a:math> lattice <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msub><d:mi mathvariant="double-struck">Z</d:mi><d:mn>2</d:mn></d:msub></d:math> gauge theory. The noninvertible algebra, which mixes with lattice translations, can be efficiently computed using ZX-calculus. We further deform the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi mathvariant="double-struck">Z</g:mi><g:mn>2</g:mn></g:msub></g:math> gauge theory while preserving the duality and find a model with nine exactly degenerate ground states on a torus, consistent with the Lieb-Schultz-Mattis-type constraint imposed by the symmetry. Finally, we provide a ZX-diagram presentation of the noninvertible duality operators (including noninvertible parity and reflection symmetries) of generalized Ising models based on graphs, encompassing the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mrow><j:mn>1</j:mn><j:mo>+</j:mo><j:mn>1</j:mn><j:mi mathvariant="normal">D</j:mi></j:mrow></j:math> Ising model, the three-spin Ising model, the Ashkin-Teller model, and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mn>2</m:mn><m:mo>+</m:mo><m:mn>1</m:mn><m:mi mathvariant="normal">D</m:mi></m:mrow></m:math> plaquette Ising model. The mixing (or lack thereof) with spatial symmetries is understood from a unifying perspective based on graph theory.

  PublisherDOI

• Gauging non-invertible symmetries on the lattice

  SciPost Physics · 2025-08-29 · 13 citations

  articleOpen access

  We provide a general prescription for gauging finite non-invertible symmetries in 1+1d lattice Hamiltonian systems. Our primary example is the Rep(D _8 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mn>8</mml:mn> </mml:msub> </mml:math> ) fusion category generated by the Kennedy-Tasaki transformation, which is the simplest anomaly-free non-invertible symmetry on a spin chain of qubits. We explicitly compute its lattice F-symbols and illustrate our prescription for a particular (non-maximal) gauging of this symmetry. In our gauging procedure, we introduce two qubits around each link, playing the role of “gauge fields” for the non-invertible symmetry, and impose novel Gauss’s laws. Similar to the Kramers-Wannier transformation for gauging an ordinary \mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , our gauging can be summarized by a gauging map, which is part of a larger, continuous non-invertible cosine symmetry.

  PublisherOA PDFDOI

Recent grants

• Quantum Field Theory for Topological Phases of Matter

  NSF · $225k · 2022–2024

Frequent coauthors

• Ho Tat Lam

  Massachusetts Institute of Technology

  31 shared

• Yichul Choi

  Stony Brook University

  25 shared

• Nathan Seiberg

  Institute for Advanced Study

  23 shared

• Clay Córdova

  19 shared

• Ying-Hsuan Lin

  Harvard University

  19 shared

• Yifan Wang

  New York University

  16 shared

• Pranay Gorantla

  15 shared

• Xi Yin

  Harvard University

  10 shared

Labs

• MIT Center for Theoretical Physics – a Leinweber InstitutePI

Education

• PhD, Physics

  Harvard University

  2016

Awards & honors

• 2026 // New Horizons in Physics Prize (shared with Yifan Wan…
• 2025 // Frontiers of Science Award, International Conference…
• 2023 // Simons Collaboration on Ultra-Quantum Matter
• 2023 // Frontiers of Science Award, International Conference…
• 2021 // National Science Foundation Award