A fundamental result relevant to spin chains and two-dimensional disordered systems is that the sphere sigma model with instanton coupling $\ensuremath{\theta}=\ensuremath{\pi}$ has a nontrivial low-energy fixed point and a gapless spectrum. This result is extended to two series of sigma models with $\ensuremath{\theta}=\ensuremath{\pi}:$ the $\mathrm{SU}(N)/\mathrm{SO}(N)$ sigma models flow to the $\mathrm{SU}{(N)}_{1}$ Wess-Zumino-Witten theory, while the $O(2N)/O(N)\ifmmode\times\else\texttimes\fi{}O(N)$ models flow to ${O(2N)}_{1}$ $(2N$-free Majorana fermions). These models are integrable, and the exact quasiparticle spectra and S matrices are found. One interesting feature is that charges fractionalize when $\ensuremath{\theta}=\ensuremath{\pi}.$ I compute the energy in a background field, and verify that the perturbative expansions for $\ensuremath{\theta}=0$ and \ensuremath{\pi} are the same as they must be. I discuss the flows between the two sequences of models, and also argue that the analogous sigma models with $\mathrm{Sp}(2N)$ symmetry, the $\mathrm{Sp}(2N)/U(N)$ models, flow to $\mathrm{Sp}{(2N)}_{1}.$
A fundamental result relevant to spin chains and two-dimensional disordered systems is that the sphere sigma model with instanton coupling \ensuremathθ=\ensuremathπ has a nontrivial low-energy fixed point and a gapless spectrum. This result is extended to two series of sigma models with \ensuremathθ=\ensuremathπ: the SU(N)/SO(N) sigma models flow to the SU(N)1 Wess-Zumino-Witten theory, while the O(2N)/O(N)\ifmmode×\else\texttimes\fiO(N) models flow to O(2N)1(2N-free Majorana fermions). These models are integrable, and the exact quasiparticle spectra and S matrices are found. One interesting feature is that charges fractionalize when \ensuremathθ=\ensuremathπ. I compute the energy in a background field, and verify that the perturbative expansions for \ensuremathθ=0 and \ensuremath{\pi} are the same as they must be. I discuss the flows between the two sequences of models, and also argue that the analogous sigma models with Sp(2N) symmetry, the Sp(2N)/U(N) models, flow to Sp(2N)1.
We construct and study a chiral Sachdev-Ye (SY) model consisting of N chiral SU(M)1 Wess-Zumino-Witten (WZW) models with current-current interactions among each other, which generalizes the 0+1d quantum chaotic SY spin model into 1+1d chiral system with anyon excitations. Each WZW model hosts Abelian anyons as charge excitations, and may arise as the chiral edge theory of 2+1d gapped topological phases. We solve the chiral SY model in two limits which show distinct quantum dynamics. The first limit is the case with uniform interactions at any integers N and M, which is integrable and decomposes into a chiral SU(M)N WZW model and its coset with a different speed of light. When N=M=2, the model maps to a free Majorana fermion model. The second limit is the large N and M limit with random interactions, which is solvable to the leading NM1 order, and exhibits many-body quantum chaos in the out-of-time-ordered correlation of anyons. As the interaction strength approaches the upper limit preserving the chirality, the leading velocity-dependent Lyapunov exponent of the model saturates the maximal chaos bound $2\ensuremath{\pi}/\ensuremath{\beta}$ at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$.
The low-energy limits of models with disorder are frequently described by $\ensuremath{\sigma}$ models. In two dimensions, most $\ensuremath{\sigma}$ models admit either a Wess-Zumino-Witten term or a $\ensuremath{\theta}$ term. When such a term is present the model can have a stable critical point with gapless excitations. We describe how such a critical point appears, in particular in two-dimensional superconductors with disorder. The presence of such terms is required by the underlying (anomalous) symmetries of the original electron model. This indicates that the usual symmetry classes of disordered systems in two dimensions can be further refined. Conversely, our results also indicate that models previously thought to be in different universality classes are in fact the same once the appropriate extra terms are included.
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.
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.
Motivated by recent studies of symmetry protected topological (SPT) phases, we explore the possible gapless quantum disordered phases in the (2+1)d nonlinear sigma model defined on the Grassmannian manifold U(n)×U(N−n)U(N) with a Wess-Zumino-Witten (WZW) term at level k, which is the effective low energy field theory of the boundary of certain (3+1)d SPT states. With k=0, this model has a well-controlled large-N limit, i.e. its renormalization group equations can be computed exactly with large-N. However, with the WZW term, the large-N and large-k limit alone is not sufficient for a reliable study of the nature of the quantum disordered phase. We demonstrate that through a combined large-N, large-k and ε−generalization, a stable fixed point in the quantum disordered phase can be reliably located in the large−N limit and leading order ε−expansion, which corresponds to a (2+1)d strongly interacting conformal field theory.