Using new as well as known results on dimerized quantum spin chains with frustration, we are able to infer some properties on the low-energy spectrum of the O (3) Nonlinear Sigma Model with a topological θ-term. In particular, for sufficiently strong coupling, we find a range of values of θ where a singlet bound state is stable under the triplet continuum. On the basis of these results, we propose a new renormalization group flow diagram for the Nonlinear Sigma Model with a θ-term.
Using new as well as known results on dimerized quantum spin chains with frustration, we are able to infer some properties on the low-energy spectrum of the O (3) Nonlinear Sigma Model with a topological θ-term. In particular, for sufficiently strong coupling, we find a range of values of θ where a singlet bound state is stable under the triplet continuum. On the basis of these results, we propose a new renormalization group flow diagram for the Nonlinear Sigma Model with a θ-term.
We study the strong-coupling expansion of a Euclidean lattice formulation of the two-dimensional O(3) \ensuremath{\sigma}-model at nonzero topological angle \ensuremath{\theta}, mapping it onto a solid-on-solid model with well-known critical behavior. The results are consistent with those predicted by the mapping of the Hamiltonian formulation of the \ensuremath{\sigma} model onto quantum spin chains. The model has a gapless phase at \ensuremath{\theta}=\ensuremath{\pi} and weak coupling in which a condensate of Coleman's ``half-asymptotic'' particles exists.
It is known that the U ( 2 ) Wess-Zumino-Witten model is dual to the free fermion theory in two dimensions via non-Abelian bosonization. While it is decomposed into the S U ( 2 ) Wess-Zumino-Witten model and a free compact boson, the former is believed to be equivalent to the O ( 3 ) nonlinear sigma model with the theta term at θ = π . In this work, we reexamine this duality through the lens of nonperturbative renormalization group (RG) flow. We analyze the RG flow structure of the O ( 3 ) nonlinear sigma model with the theta term in two dimensions using the functional renormalization group. Our results reveal a nontrivial fixed point with a nonzero value of the topological coupling. The scaling dimensions (critical exponents) at this fixed point suggest the realization of a duality between the O ( 3 ) nonlinear sigma model with the theta term and the free fermion theory, indicating that these models belong to the same universality class.
We prove a theorem on the O(3) nonlinear \ensuremath{\sigma} model with the topological \ensuremath{\theta} term which states that the grround-state energy at \ensuremath{\theta}=\ensuremath{\pi} is always higher than the ground-state energy at \ensuremath{\theta}=0, for the same value of the coupling constant g. Provided that the nonlinear \ensuremath{\sigma} model gives the correct description for the Heisenberg spin chains in the large-s limit, this theorem makes a definite prediction relating the ground-state energies of the half-integer- and the integer-spin chains. The ground-state energies obtained from the exact Bethe Ansatz solution for the spin-(1/2 chain and the numerical diagonalizaton on the spin-1,- (3/2, and -2 chains support this prediction.
Microscopic models of quantum antiferromagnets are investigated on the basis of a mapping onto effective low-energy Hamiltonians. Lattice effects are carefully taken into account and their role is discussed. We show that the presence of an external staggered magnetic field modifies in a nontrivial way the usual mapping onto the nonlinear sigma model, leading to the appearance of terms neglected in previous works. Our analysis is compared with Lanczos diagonalizations of S=1 Heisenberg chains in a staggered field, confirming the validity of the single-mode approximation for the evaluation of the dynamical structure factor. The results are relevant for the interpretation of experiments in quasi-one-dimensional compounds. Microscopic realizations of SU(4) spin chains are also discussed in the framework of spin-orbital lattice systems. The low-energy physics is shown to be described by sigma models with topological angle $\ensuremath{\theta}$ in one dimension. This mapping strongly suggests that the one-dimensional CP3 model (with $\ensuremath{\theta}=\ensuremath{\pi})$ undergoes a second-order phase transition as a function of the coupling.
We consider two sets of related models: initially, these are SU(2) antiferromagnetic spin chains with N sites of spin S, and the O(3) nonlinear sigma model in two dimensions with topological coefficient Θ a multiple of π (and later, the extensions of these with any semisimple Lie group symmetry). It is known that, in a continuum description, the low-energy behavior of the spin chain is given by the sigma model with Θ=2πS. We study these models with N odd and with antiperiodic (A) boundary condition (b.c.), respectively, which correspond. The A b.c. in the sigma model involves the Z2 inversion symmetry n→−n, and amounts to a flux of a Z2 gauge field through a spacetime torus; summing over the two b.c.s for each direction would amount to gauging the Z2 inversion symmetry. We show directly that, if and only if (−1)Θ/π=−1, the gauging cannot be carried out; there is an 't Hooft anomaly. The partition function for the A b.c. exists, but is not gauge invariant; consequently, the sum over b.c.s cannot be made modular invariant. The gauged model would be a sigma model with target space RP2≅S2/Z2, and hence this model does not exist for Θ=π (mod 2π). A related result is that, using semiclassical quantization, in the spin chain we obtain the known values of the ground-state crystal momentum, which at leading order depend only on N modulo 4 and 2S modulo 2. For a large class of spin chains and associated sigma models we find similar results, but now (−1)Θ/π is replaced by the value ±1 of the square of the time-reversal operator acting on a single spin, which is still determined by the coefficients of the topological terms, in a way that depends on the symmetry group.
On the infrared limit of the O(3) nonlinear σ-model at θ=π
2D nonlinear sigma models with Hermitian symmetric target admit a theta-term, which couples the field theory to the topological charge of its instanton gas. At the special coupling theta = pi, by what is nowadays attributed to a coupling-constant anomaly of Lieb-Schultz-Mattis type, such models have a degenerate ground state. Yet, the details of their non-trivial infrared limit have remained open in general. Here we suggest that non-perturbative renormalization group flow into the strong-coupling regime induces strong fluctuations of the theta-parameter, with the consequence that the instanton density is suppressed, the target-space topology effectively altered, and the target-space metric driven off reality and into geometrostasis. Assuming this heuristic scenario and combining it with a Cauchy process of target-space deformation, we present a detailed argument that the O(3) nonlinear sigma model at theta = pi, known to be the effective field theory for critical antiferromagnetic quantum spin chains with large half-integer spin, renormalizes to the conformal field theory of a U(1) boson compactified at a special radius with SU(2) symmetry. A closely related scenario applies to Pruisken's nonlinear sigma model for the integer quantum Hall transition.