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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> WZW model points and whose phase transitions all have dynamical exponent z&amp;gt;1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> .