Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality

Abstract

Lieb-Schultz-Mattis (LSM) theorems provide powerful constraints on the problem, i.e. whether a quantum phase or phase transition can emerge in a many-body system. We derive the topological partition functions that characterize the LSM constraints in spin systems with G_s\times G_{int} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> symmetry, where G_s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> is an arbitrary space group in one or two spatial dimensions, and G_{int} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:math> is any internal symmetry whose projective representations are classified by \mathbb{Z}_2^k <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:msubsup> </mml:math> with k <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>k</mml:mi> </mml:math> an integer. We then apply these results to study the emergibility of a class of exotic quantum critical states, including the well-known deconfined quantum critical point (DQCP), U(1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> Dirac spin liquid (DSL), and the recently proposed non-Lagrangian Stiefel liquid. These states can emerge as a consequence of the competition between a magnetic state and a non-magnetic state. We identify all possible realizations of these states on systems with SO(3)\times \mathbb{Z}_2^T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> internal symmetry and either p6m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>6</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> or p4m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mn>4</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> lattice symmetry. Many interesting examples are discovered, including a DQCP adjacent to a ferromagnet, stable DSLs on square and honeycomb lattices, and a class of quantum critical spin-quadrupolar liquids of which the most relevant spinful fluctuations carry spin- 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>2</mml:mn> </mml:math> . In particular, there is a realization of spin-quadrupolar DSL that is beyond the usual parton construction. We further use our formalism to analyze the stability of these states under symmetry-breaking perturbations, such as spin-orbit coupling. As a concrete example, we find that a DSL can be stable in a recently proposed candidate material, NaYbO _2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mn>2</mml:mn> </mml:msub> </mml:math> .

Topics & Concepts