We discuss the lattice formulation of the 't Hooft surface, that is, the two-dimensional surface operator of a dual variable. The 't Hooft surface describes the world sheets of topological vortices. We derive the formulas to calculate the expectation value of the 't Hooft surface in the multiple-charge lattice Abelian Higgs model and in the lattice non-Abelian Higgs model. As the first demonstration of the formula, we compute the intervortex potential in the charge-2 lattice Abelian Higgs model.
As a color confinement mechanism, a dual Meissner effect due to Abelian monopoles involved in QCD has been discussed so far in various ways. But still there is an important problem unsolved. It is gauge invariance of the schemes or, in other words, the reason why non-Abelian color confinement is explained by means of the Abelian dual Meissner effect. Here it is shown that a random Abelian method based on the violation of non-Abelian Bianchi identity (VNABI) could prove SU(3) invariance and explain why color-singlets alone can survive in the confinement phase of QCD in the framework of the Abel…
In this paper we use lattice simulation to study four dimensional $N=4$ super Yang-Mills (SYM) theory. We have focused on the three color theory on lattices of size $12^4$ and for 't Hooft couplings up to $λ=40.0$. Our lattice action is based on a discretization of the Marcus or GL twist of $N=4$ SYM and retains one exact supersymmetry for non-zero lattice spacing. We show that lattice theory exists in a single non-Abelian Coulomb phase for all 't Hooft couplings. Furthermore the static potential we obtain from correlators of Polyakov lines is in good agreement with that obtained from holograp…
We apply to a lattice version of the 't~Hooft model, QCD in two space-time dimensions for large number of colours, a method recently proposed to obtain an effective mesonic action starting from the fundamental, fermionic one. The idea is to pass from a canonical, operatorial representation, where the low-energy states have a direct physical interpretation in terms of a Bogoliubov vacuum and its corresponding quasiparticle excitations, to a functional, path integral representation, via the formalism of the transfer matrix. In this way we obtain a lattice effective theory for mesons in a self-co…
The quantum phase transition between $\mathbb{Z}_{2}$ plaquette valence bound solid (PVBS) and superfluid (SF) phases on the planar pyrochlore lattice (square ice) is under debate, because the conventional deconfined theory does not support continuous one, but the numerical evidence is still not solid. Here, we propose the system can be effectively described by a 2+1 dimensional Abelian-Higgs model which may host the mixed 't Hooft anomaly, so that the deconfinement can exist on the domain walls at the transition point. To verify it, we study the spin excitation spectra by combining stochastic…
We construct a non-trivial $U(1)/\mathbb{Z}_q$ principal bundle on~$T^4$ from the compact $U(1)$ lattice gauge field by generalizing Lüscher's constriction so that the cocycle condition contains $\mathbb{Z}_q$ elements (the 't~Hooft flux). The construction requires an admissibility condition on lattice gauge field configurations. From the transition function so constructed, we have the fractional topological charge that is $\mathbb{Z}_q$ one-form gauge invariant and odd under the lattice time reversal transformation. Assuming a rescaling of the vacuum angle $θ\to qθ$ suggested from the Witten …
The symmetries and dynamics of simple chiral $SU(N)$ gauge theories, with matter Weyl fermions in a two-index symmetric tensor and $N+4$ anti-fundamental representations, are examined, by taking advantage of the recent developments involving the ideas of generalized symmetries, gauging of discrete center 1-form symmetries and mixed 't Hooft anomalies. This class of models are particularly interesting because the conventional 't Hooft anomaly matching constraints allow a chirally symmetric confining vacuum, with no condensates breaking the $U(1) \times SU(N+4)$ flavor symmetry, and with certain…
We study the phase diagram of the $(1+1)$-dimensional Gross-Neveu model with both $g_x^2(\barψψ)^2$ and $g_y^2(\barψiγ_5ψ)^2$ interaction terms on a spatial lattice. The continuous chiral symmetry, which is present in the continuum model when $g_x^2=g_y^2$, has a mixed 't~Hooft anomaly with the charge conservation symmetry, which guarantees the existence of a massless mode. However, the same 't~Hooft anomaly also implies that the continuous chiral symmetry is broken explicitly in our lattice model. Nevertheless, from numerical matrix product state simulations we find that, for certain paramete…
We study a four-dimensional $U(1)$ gauge theory with the $θ$ angle, which was originally proposed by Cardy and Rabinovici. It is known that the model has the rich phase diagram thanks to the presence of both electrically and magnetically charged particles. We discuss the topological nature of the oblique confinement phase of the model at $θ=π$, and show how its appearance can be consistent with the anomaly constraint. We also construct the $SL(2,\mathbb{Z})$ self-dual theory out of the Cardy-Rabinovici model by gauging a part of its one-form symmetry. This self-duality has a mixed 't Hooft ano…
A topological $θ$-term in gauge theories, including quantum chromodynamics in 3+1 dimensions, gives rise to a sign problem that makes classical Monte Carlo simulations impractical. Quantum simulations are not subject to such sign problems and are a promising approach to studying these theories in the future. In the near term, it is interesting to study simpler models that retain some of the physical phenomena of interest and their implementation on quantum hardware. For example, dimensionally-reducing gauge theories on small spatial tori produces quantum mechanical models which, despite being …
Numerical Hints for Dyon Condensation at $θ=2π$ via Wilson-'t Hooft Loops in $SU(2)$ Yang-Mills Theory
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Department of Physics, and Research and Education Center for Natural Sciences, Keio University（庆应义塾大学物理系及自然科学研究教育中心）；RIKEN Center for Computational Science（理化学研究所计算科学中心）；Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所跨学科理论与数学科学中心（iTHEMS））；Information Technology Center, The University of Tokyo（东京大学信息中心）；Department of Advanced Materials Science, The University of Tokyo（东京大学先进材料科学系）；Department of Information and Mathematical Sciences, Tokyo Woman’s Christian University（东京女子基督教大学信息数学科学系）；Department of Physics, Kyoto University（京都大学物理学系）
The charge-$q$ Schwinger model is the $(1+1)$-dimensional quantum electrodynamics (QED) with a charge-$q$ Dirac fermion. It has the $\mathbb{Z}_q$ $1$-form symmetry and also enjoys the $\mathbb{Z}_q$ chiral symmetry in the chiral limit, and there is a mixed 't Hooft anomaly between those symmetries. We numerically study the charge-$q$ Schwinger model in the lattice Hamiltonian formulation using the density-matrix renormalization group (DMRG). When applying DMRG, we map the Schwinger model to a spin chain with nonlocal interaction via Jordan-Wigner transformation, and we take the open boundary …
In this paper we present results from numerical simulations of N=4 super Yang-Mills for two color gauge theory over a wide range of 't Hooft coupling $0<λ\le 30$ using a supersymmetric lattice action \cite{Catterall:2009it}. Numerical study of this lattice theory has been stymied until recently by both sign problems and the occurrence of lattice artifact phases at strong coupling. We have recently developed a new action that appears capable of solving both problems. The resulting action possesses just $SU(2)$ rather than $U(2)$ gauge symmetry. By explicit computations of the fermion Pfaffian w…
