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Register a new userPhases of unitary matrix models and lattice QCD2
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Authors
Jorge G. Russo
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We investigate the different large $N$ phases of a generalized Gross-Witten-Wadia $U(N)$ matrix model. The deformation mimics the one-loop determinant of fermion matter with a particular coupling to gauge fields. In one version of the model, the GWW phase transition is smoothed out and it becomes a crossover. In another version, the phase transition occurs along a critical line in the two-dimensional parameter space spanned by the t~Hooft coupling $lambda$ and the Veneziano parameter $tau$. We compute the expectation value of Wilson loops in both phases, showing that the transition is third-order. A calculation of the $beta $ function shows the existence of an IR stable fixed point.
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We study four-dimensional Chern-Simons theory on $D times mathbb{C}$ (where $D$ is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an analytically-continued toroidal Lie algebra. In addition, we show how certain bulk correlation functions of two and three Wilson lines can be captured by boundary correlation functions of local operators in the three-dimensional WZW model. In particular, we reproduce the leading and subleading nontrivial contributions to the rational R-matrix purely from the boundary theory.
We elucidate how integrable lattice models described by Costellos 4d Chern-Simons theory can be realized via a stack of D4-branes ending on an NS5-brane in type IIA string theory, with D0-branes on the D4-brane worldvolume sourcing a meromorphic RR 1-form, and fundamental strings forming the lattice. This provides us with a nonperturbative integration cycle for the 4d Chern-Simons theory, and by applying T- and S-duality, we show how the R-matrix, the Yang-Baxter equation and the Yangian can be categorified, that is, obtained via the Hilbert space of a 6d gauge theory.
We study a family of models for an $N_1 times N_2$ matrix worth of Ising spins $S_{aB}$. In the large $N_i$ limit we show that the spins soften, so that the partition function is described by a bosonic matrix integral with a single `spherical constraint. In this way we generalize the results of [1] to a wide class of Ising Hamiltonians with $O(N_1,mathbb{Z})times O(N_2,mathbb{Z})$ symmetry. The models can undergo topological large $N$ phase transitions in which the thermal expectation value of the distribution of singular values of the matrix $S_{aB}$ becomes disconnected. This topological transition competes with low temperature glassy and magnetically ordered phases.
We give efficient quantum algorithms to estimate the partition function of (i) the six vertex model on a two-dimensional (2D) square lattice, (ii) the Ising model with magnetic fields on a planar graph, (iii) the Potts model on a quasi 2D square lattice, and (iv) the Z_2 lattice gauge theory on a three-dimensional square lattice. Moreover, we prove that these problems are BQP-complete, that is, that estimating these partition functions is as hard as simulating arbitrary quantum computation. The results are proven for a complex parameter regime of the models. The proofs are based on a mapping relating partition functions to quantum circuits introduced in [Van den Nest et al., Phys. Rev. A 80, 052334 (2009)] and extended here.
We consider gauge theories of non-Abelian $finite$ groups, and discuss the 1+1 dimensional lattice gauge theory of the permutation group $S_N$ as an illustrative example. The partition function at finite $N$ can be written explicitly in a compact form using properties of $S_N$ conjugacy classes. A natural large-$N$ limit exists with a new t Hooft coupling, $lambda=g^2 log N$. We identify a Gross-Witten-Wadia-like phase transition at infinite $N$, at $lambda=2$. It is first order. An analogue of the string tension can be computed from the Wilson loop expectation value, and it jumps from zero to a finite value. We view this as a type of large-$N$ (de-)confinement transition. Our holographic motivations for considering such theories are briefly discussed.
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