Yang-Lee Quantum Criticality in Various Dimensions

The Yang-Lee universality class arises when an imaginary magnetic field is tuned to its critical value in the paramagnetic phase of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>d</a:mi> <a:mo>&lt;</a:mo> <a:mn>6</a:mn> </a:math> Ising model. In <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>d</c:mi> <c:mo>=</c:mo> <c:mn>2</c:mn> </c:math> , this nonunitary conformal field theory (CFT) is exactly solvable via the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>M</e:mi> <e:mo stretchy="false">(</e:mo> <e:mn>2</e:mn> <e:mo>,</e:mo> <e:mn>5</e:mn> <e:mo stretchy="false">)</e:mo> </e:math> minimal model. As found long ago by von Gehlen using exact diagonalization, the corresponding real-time, quantum critical behavior arises in the periodic Ising spin chain when the imaginary longitudinal magnetic field is tuned to its critical value from below. Even though the Hamiltonian is not Hermitian, the energy levels are real due to the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi mathvariant="script">PT</i:mi> </i:math> symmetry. In this paper, we explore the analogous quantum critical behavior in higher-dimensional non-Hermitian Hamiltonians on regularized spheres <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:msup> <l:mi>S</l:mi> <l:mrow> <l:mi>d</l:mi> <l:mo>−</l:mo> <l:mn>1</l:mn> </l:mrow> </l:msup> </l:math> . For <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>d</n:mi> <n:mo>=</n:mo> <n:mn>3</n:mn> </n:math> , we use the recently invented, powerful fuzzy sphere method, as well as discretization by the platonic solids cube, icosahedron, and dodecahedron. The low-lying energy levels and structure constants we find are in agreement with expectations from the conformal symmetry. The energy levels are in good quantitative agreement with the high-temperature expansions and with Padé extrapolations of the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mn>6</p:mn> <p:mo>−</p:mo> <p:mi>ε</p:mi> </p:math> expansions in Fisher’s <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>i</r:mi> <r:msup> <r:mi>ϕ</r:mi> <r:mn>3</r:mn> </r:msup> </r:math> Euclidean field theory for the Yang-Lee criticality. In the course of this work, we clarify some aspects of matching between operators in this field theory and quasiprimary fields in the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>M</t:mi> <t:mo stretchy="false">(</t:mo> <t:mn>2</t:mn> <t:mo>,</t:mo> <t:mn>5</t:mn> <t:mo stretchy="false">)</t:mo> </t:math> minimal model. For <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi>d</x:mi> <x:mo>=</x:mo> <x:mn>4</x:mn> </x:math> , we obtain new results by replacing <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:msup> <z:mi>S</z:mi> <z:mn>3</z:mn> </z:msup> </z:math> with the self-dual polytope called the 24-cell.