Lattice T-duality from non-invertible symmetries in quantum spin chains

TL;DR

<3-5 sentence high-level summary>The paper presents a concrete lattice realization of T-duality for the 1+1D compact boson in the XX spin chain, implementing an exact lattice duality via a non-invertible symmetry that interchanges lattice momentum and winding. By gauging the lattice Z_2 momentum symmetry and applying a lattice T-duality unitary, the authors construct a winding symmetry and a non-invertible D operator, with its action encoding T-duality at the lattice level and connecting to the Onsager algebra. They analyze the lattice ’t Hooft anomalies, showing mixed and type III anomalies that parallel continuum behavior, and prove anomaly-enforced gaplessness for any U(1)^M × U(1)^W symmetric deformation. The work further develops explicit Onsager charges, their transformation under lattice dualities, and extends the duality framework to other spin chains, revealing Spin(2k)_1 WZW points in gapless phases and guiding future explorations of lattice dualities in non-Abelian or higher-dimensional contexts.

Abstract

Dualities of quantum field theories are challenging to realize in lattice models of qubits. In this work, we explore one of the simplest dualities, T-duality of the compact boson CFT, and its realization in quantum spin chains. In the special case of the XX model, we uncover an exact lattice T-duality, which is associated with a non-invertible symmetry that exchanges two lattice U(1) symmetries. The latter symmetries flow to the momentum and winding U(1) symmetries with a mixed anomaly in the CFT. However, the charge operators of the two U(1) symmetries do not commute on the lattice and instead generate the Onsager algebra. We discuss how some of the anomalies in the CFT are nonetheless still exactly realized on the lattice and how the lattice U(1) symmetries enforce gaplessness. We further explore lattice deformations preserving both U(1) symmetries and find a rich gapless phase diagram with special $\mathrm{Spin}(2k)_1$ WZW model points and whose phase transitions all have dynamical exponent ${z>1}$.

Figures (6)

• Figure 1: T-duality in the XX model $H_\mathrm{XX}$ is implemented by the unitary operator $U_\mathrm{T}$\ref{['UTdef']}. It relates $H_\mathrm{XX}$ to its $\mathbb{Z}_2^\mathrm{M}$-gauged version $H_{\mathrm{XX}/\mathbb{Z}_2^\mathrm{M}}$. It further relates $\mathbb{Z}_2^\mathrm{M}$-gauged momentum and winding symmetries of the XX model to the winding and momentum symmetries, respectively. The lattice T-duality implemented by $U_\mathrm{T}$ precisely matches the T-duality of the compact free boson in the IR limit of the XX model.
• Figure 2: Symmetries of the XX model and their anomalies. When considering only lattice translation, the $\mathbb{Z}_2^\mathrm{C}$ symmetry generated by ${C = \prod_{j=1}^L X_j}$, and the spin rotation symmetry U(1)$^\mathrm{M}$ generated by ${Q^{\rm M} = \frac{1}{2} \sum_{j=1}^L Z_j}$, the only anomaly is the well-known LSM anomaly CGX10083745Furuya:2015coaMetlitski:2017fmdOgata2019Yao:2020xcmCS221112543. Once the winding symmetry \ref{['QWdef']} and non-invertible symmetry formed by $\mathsf{D}$\ref{['XXnoninvTra0']} are included, three new types of anomalies arise.
• Figure 3: The symmetry operators $T$, $\mathsf{D}_a$, $\mathsf{D}_b$, and $\mathsf{D}$ of the XX model act nontrivially on the conserved Onsager charges $Q_n$. The color-coded arrows in this diagram describe the action on $Q_n$ by the symmetry operators labeling them.
• Figure 4: The XYZ model \ref{['XYZhamiltonian']} with ${-1\leq \gamma \leq 1}$ and ${\Delta > 0}$ as three distinct phases, all of which are gapped, Néel ordered phases with two ground states. We color these three phases orange, green, and purple, respectively. The black solid lines separating them denote phase transitions, which are described by the compact free boson CFT at various radii ${1\leq R \leq \sqrt{2}}$. The three ${R = \sqrt{2}}$ points are described by three unitarily related XX models, all of which have their own lattice momentum and winding U(1) symmetries. These symmetries do not generally persist away from the XX model points, and we denote by dashed/dotted colored lines where such respective symmetries exist in the XYZ model.
• Figure 5: Phase diagram of the quantum spin Hamiltonian \ref{['hlambdaDef']}, which is gapless for all values of $g_2$ and $g_3$. The gapless phases are incommensurate and labeled by the number of bosonized Dirac fermions $\mathrm{C}$. Furthermore, the dark red stars denote points in the phase diagram for which the IR of \ref{['hlambdaDef']} is described by a CFT. At these CFT points, $\mathrm{C}$ corresponds to the central charge. The phase transitions are labeled by their dynamical critical exponent $z$, all of which have ${z>1}$. The ${z=2}$ critical line shown in green occurs at ${g_2 = \sqrt{1-(2g_3-1)^2}}$ for ${g_3>1/5}$. The ${z=3}$ line is shown in purple and occurs at ${g_2 = \frac{1}{2} | 3 g_3 + 1|}$. These two critical lines intersect at a multi-critical point with ${z=5}$ at ${(g_2,g_3) = (\frac{4}{5},\frac{1}{5})}$.