Vison crystal in quantum spin ice on the breathing pyrochlore lattice

Recent excitement in the quantum spin ice (QSI) community has come from the experimental discovery of pseudospin-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mfrac><a:mn>1</a:mn><a:mn>2</a:mn></a:mfrac></a:math> breathing pyrochlores, including <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>Ba</b:mi><b:mn>3</b:mn></b:msub><b:msub><b:mi>Yb</b:mi><b:mn>2</b:mn></b:msub><b:msub><b:mi>Zn</b:mi><b:mn>5</b:mn></b:msub><b:msub><b:mi mathvariant="normal">O</b:mi><b:mn>11</b:mn></b:msub></b:mrow></b:math>, in which inversion symmetry is broken by the “up” and “down” tetrahedra taking different physical sizes. We show here that the often-neglected <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>J</d:mi><d:mrow><d:mi>z</d:mi><d:mo>±</d:mo></d:mrow></d:msub></d:math> coupling between Kramers ions, in combination with the breathing nature of the lattice, can produce an imaginary ring flip term. This can lead to an unconventional “<e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>U</e:mi><e:msub><e:mrow><e:mo>(</e:mo><e:mn>1</e:mn><e:mo>)</e:mo></e:mrow><e:mrow><e:mi>π</e:mi><e:mo>/</e:mo><e:mn>2</e:mn></e:mrow></e:msub></e:mrow></e:math> phase”, corresponding to a maximally dense packing of visons on the lattice. The coherent, emergent QED dynamics of conventional QSI persist in this phase, in a manner reminiscent of fragmentation in spinon crystals. We characterize the excitations of the system within the enlarged QSI phase diagram, showing that the imaginary ring flip acts both as a chemical potential for visons and as an effective three-photon vertex akin to strong light-matter coupling. The coupling causes a structured high-energy continuum to emerge above the photon dispersion, which is naturally interpreted as three photon up-conversion in a nonlinear optical crystal.