Z(2) Excitations and Disorder in SU(2) Spin Models and Gauge Theories.

Noncollinear magnetic order is typically characterized by a tetrad ground state manifold (GSM) of three perpendicular vectors or nematic directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S(3)/Z(2), S(3)/Z(4), and S(3)/Q(8), respectively. Q(8) denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z(2), Z(4), and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This non-Abelian topological phase possesses a 22-fold ground state degeneracy on the torus arising from the 22 representations of the Drinfeld double of Q(8).Noncollinear magnetic order is typically characterized by a tetrad ground state manifold (GSM) of three perpendicular vectors or nematic directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S(3)/Z(2), S(3)/Z(4), and S(3)/Q(8), respectively. Q(8) denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z(2), Z(4), and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This non-Abelian topological phase possesses a 22-fold ground state degeneracy on the torus arising from the 22 representations of the Drinfeld double of Q(8).

arXiv:1012.5671Physical Review LettersJan 25, 201215 citations

Quantum Error Correction and Z(2) Lattice Gauge Theories

Seyong Kim

$Z(2)$ lattice gauge theory plays an important role in the study of the threshold probability of Quantum Error Correction (QEC) for a quantum code. For certain QEC codes, such as the well-known Kitaev's toric/surface code, one can find a mapping of the QEC decoding problem onto a statistical mechanics model for a given noise model. The investigation of the threshold probability then corresponds to that of the phase diagram of the mapped statistical mechanics model. This can be studied by Monte Carlo simulation of the statistical mechanics model. In~\cite{Rispler}, we investigate the effects of realistic noise models on the toric/surface code in two dimensions together with syndrome measurement noise and introduce the random coupled-plaquette gauge model, 3-dimensional $Z(2) \times Z(2)$ lattice gauge theory. This new Z(2) gauge theory model captures main aspects of toric/surface code under depolarizing and syndrome noise. In these proceedings, we mainly focus on the aspects of Mont Carlo simulation and discuss preliminary results from Monte Carlo simulations of mapped classes of Z(2) lattice theories.$Z(2)$ lattice gauge theory plays an important role in the study of the threshold probability of Quantum Error Correction (QEC) for a quantum code. For certain QEC codes, such as the well-known Kitaev's toric/surface code, one can find a mapping of the QEC decoding problem onto a statistical mechanics model for a given noise model. The investigation of the threshold probability then corresponds to that of the phase diagram of the mapped statistical mechanics model. This can be studied by Monte Carlo simulation of the statistical mechanics model. In~\cite{Rispler}, we investigate the effects of realistic noise models on the toric/surface code in two dimensions together with syndrome measurement noise and introduce the random coupled-plaquette gauge model, 3-dimensional $Z(2) \times Z(2)$ lattice gauge theory. This new Z(2) gauge theory model captures main aspects of toric/surface code under depolarizing and syndrome noise. In these proceedings, we mainly focus on the aspects of Mont Carlo simulation and discuss preliminary results from Monte Carlo simulations of mapped classes of Z(2) lattice theories.

arXiv:2501.13611International Symposium on Lattice Field Theory (LATTICE)Jan 30, 20250 citations

The Hidden Fermions in Z(2) Theories

Mark Srednicki

Springer eBooksJan 1, 19830 citations

The Z(2) lattice gauge vacuum and the transverse ising model: Two sides of a coin

MANFRED L. RISTIG, J. W. Kim, J. W. Clark

Springer eBooksApr 7, 20083 citations

Theory of Spin Lattices and Lattice Gauge Models

J. W. Clark, MANFRED L. RISTIG

Springer eBooksJan 1, 19979 citations

Z(2)

lattice gauge theory plays an important role in the study of the threshold probability of Quantum Error Correction (QEC) for a quantum code. For certain QEC codes, such as the well-known Kitaev's toric/surface code, one can find a mapping of the QEC decoding problem onto a statistical mechanics model for a given noise model. The investigation of the threshold probability then corresponds to that of the phase diagram of the mapped statistical mechanics model. This can be studied by Monte Carlo simulation of the statistical mechanics model. In~\cite{Rispler}, we investigate the effects of realistic noise models on the toric/surface code in two dimensions together with syndrome measurement noise and introduce the random coupled-plaquette gauge model, 3-dimensional

Z(2) \times Z(2)

lattice gauge theory. This new Z(2) gauge theory model captures main aspects of toric/surface code under depolarizing and syndrome noise. In these proceedings, we mainly focus on the aspects of Mont Carlo simulation and discuss preliminary results from Monte Carlo simulations of mapped classes of Z(2) lattice theories.