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Register a new userLarge-N CP(N-1) sigma model on a finite interval: physical boundary effects
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We analyze the two-dimensional CP(N-1) sigma model defined on a finite space interval L, with various boundary conditions, in the large N limit. With the Dirichlet boundary condition at the both ends, we show that the system has a unique phase, which smoothly approaches in the large L limit the standard 2D CP(N-1) sigma model in confinement phase, with a constant mass generated for the n(i) fields. We study the full functional saddle-point equations for finite L, and solve them numerically. The latter reduces to the well-known gap equation in the large L limit. It is found that the solution satisfies actually both the Dirichlet and Neumann conditions.
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We compute the free energy in the presence of a chemical potential coupled to a conserved charge in the effective SU(N)xSU(N) scalar field theory to third order for asymmetric volumes in general d-dimensions, using dimensional regularization. We also compute the mass gap in a finite box with periodic boundary conditions.
Noncompact SO(1,N) sigma-models are studied in terms of their large N expansion in a lattice formulation in dimensions d geq 2. Explicit results for the spin and current two-point functions as well as for the Binder cumulant are presented to next to leading order on a finite lattice. The dynamically generated gap is negative and serves as a coupling-dependent infrared regulator which vanishes in the limit of infinite lattice size. The cancellation of infrared divergences in invariant correlation functions in this limit is nontrivial and is in d=2 demonstrated by explicit computation for the above quantities. For the Binder cumulant the thermodynamic limit is finite and is given by 2/(N+1) in the order considered. Monte Carlo simulations suggest that the remainder is small or zero. The potential implications for ``criticality and ``triviality of the theories in the SO(1,N) invariant sector are discussed.
We investigate the lattice ${mathbb C} P^{N-1}$ sigma model on $S_{s}^{1}$(large) $times$ $S_{tau}^{1}$(small) with the ${mathbb Z}_{N}$ symmetric twisted boundary condition, where a sufficiently large ratio of the circumferences ($L_{s}gg L_{tau}$) is taken to approximate ${mathbb R} times S^1$. We find that the expectation value of the Polyakov loop, which is an order parameter of the ${mathbb Z}_N$ symmetry, remains consistent with zero ($|langle Prangle|sim 0$) from small to relatively large inverse coupling $beta$ (from large to small $L_{tau}$). As $beta$ increases, the distribution of the Polyakov loop on the complex plane, which concentrates around the origin for small $beta$, isotropically spreads and forms a regular $N$-sided-polygon shape (e.g. pentagon for $N=5$), leading to $|langle Prangle| sim 0$. By investigating the dependence of the Polyakov loop on $S_{s}^{1}$ direction, we also verify the existence of fractional instantons and bions, which cause tunneling transition between the classical $N$ vacua and stabilize the ${mathbb Z}_{N}$ symmetry. Even for quite high $beta$, we find that a regular-polygon shape of the Polyakov-loop distribution, even if it is broken, tends to be restored and $|langle Prangle|$ gets smaller as the number of samples increases. To discuss the adiabatic continuity of the vacuum structure from another viewpoint, we calculate the $beta$ dependence of ``pseudo-entropy density $proptolangle T_{xx}-T_{tautau}rangle$. The result is consistent with the absence of a phase transition between large and small $beta$ regions.
In this paper we study the dynamical generation of mass in the Lorentz-violating $CP^{(N-1)}$ model defined in two and three-dimensional aether-superspace. We show that even though the model presents a phase structure similar to the usual, Lorentz invariant case, the dynamically generated mass by quantum corrections has a dependence on the Lorentz violating background properties, except for spacelike LV vector parameter. This is to be contrasted with the behavior of the quantum electrodynamics in the two-dimensional aether-superspace, where the dynamical generation of mass was shown to exhibit an explicit dependence on the aether parameters in every possible case.
We consider properties of the inhomogeneous solution found recently for mbox{$mathbb{CP}^{,N-1}$} model. The solution was interpreted as a soliton. We reevaluate its energy in three different ways and find that it is negative contrary to the previous claims. Hence, instead of the solitonic interpretation it calls for reconsideration of the issue of the true ground state. While complete resolution is still absent we show that the energy density of the periodic elliptic solution is lower than the energy density of the homogeneous ground state. We also discuss similar solutions for the ${mathbb{O}}(N)$ model and for SUSY extensions.
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