Nonlinear Sigma Model Research Articles

How spectrum-wide quantum criticality protects surface states of topological superconductors from Anderson localization: Quantum Hall plateau transitions (almost) all the way down

We review recent numerical studies of two-dimensional (2D) Dirac fermion theories that exhibit an unusual mechanism of topological protection against Anderson localization. These describe surface-state quasiparticles of time-reversal invariant, three-dimensional (3D) topological superconductors (TSCs), subject to the effects of quenched disorder. Numerics reveal a surprising connection between 3D TSCs in classes AIII, CI, and DIII, and 2D quantum Hall effects in classes A, C, and D. Conventional arguments derived from the non-linear σ-model picture imply that most TSC surface states should Anderson localize for arbitrarily weak disorder (CI, AIII), or exhibit weak antilocalizing behavior (DIII). The numerical studies reviewed here instead indicate spectrum-wide surface quantum criticality, characterized by robust eigenstate multifractality throughout the surface-state energy spectrum. In other words, there is an “energy stack” of critical wave functions. For class AIII, multifractal eigenstate and conductance analysis reveals identical statistics for states throughout the stack, consistent with the class A integer quantum-Hall plateau transition (QHPT). Class CI TSCs exhibit surface stacks of class C spin QHPT states. Critical stacking of a third kind, possibly associated to the class D thermal QHPT, is identified for nematic velocity disorder of a single Majorana cone in class DIII. The Dirac theories studied here can be represented as perturbed 2D Wess–Zumino–Novikov–Witten sigma models; the numerical results link these to Pruisken models with the topological angle ϑ=π. Beyond applications to TSCs, all three stacked Dirac theories (CI, AIII, DIII) naturally arise in the effective description of dirty d-wave quasiparticles, relevant to the high-Tc cuprates.

Localization of the gauged linear sigma model for KK5-branes

Abstract We study quantum aspects of the target space of the non-linear sigma model, which is a low-energy effective theory of the gauged linear sigma model (GLSM). As such, we especially compute the exact sphere partition function of the GLSM for KK$5$-branes whose background geometry is a Taub–NUT space, using the supersymmetric localization technique on the Coulomb branch. From the sphere partition function, we distill the world-sheet instanton effects. In particular, we show that, concerning the single-centered Taub–NUT space, instanton contributions exist only if the asymptotic radius of the $S^1$ fiber in the Taub–NUT space is zero.

The dressing method as non linear superposition in sigma models

We apply the dressing method on the Non Linear Sigma Model (NLSM), which describes the propagation of strings on ℝ × S2, for an arbitrary seed. We obtain a formal solution of the corresponding auxiliary system, which is expressed in terms of the solutions of the NLSM that have the same Pohlmeyer counterpart as the seed. Accordingly, we show that the dressing method can be applied without solving any differential equations. In this context a superposition principle emerges: the dressed solution is expressed as a non-linear superposition of the seed with solutions of the NLSM with the same Pohlmeyer counterpart as the seed.

Equilibrium density matrices for the 2D black hole sigma models from an integrable spin chain

This work concerns the quantum Lorentzian and Euclidean black hole non-linear sigma models. For the Euclidean black hole sigma model an equilibrium density matrix is proposed, which reproduces the modular invariant partition function from the 2001 paper of Maldacena, Ooguri and Son. For the Lorentzian black hole sigma model, using its formulation as a gauged SL(2, ℝ) WZW model, we describe the linear and Hermitian structure of its space of states and also propose an expression for the equilibrium density matrix. Our analysis is guided by the results of the study of a certain critical, integrable spin chain. In the scaling limit, the latter exhibits the key features of the Lorentzian black hole sigma model including the same global symmetries, the same algebra of extended conformal symmetry and a continuous spectrum of conformal dimensions.

Anomalous dimensions of effective theories from partial waves

On-shell amplitude methods have proven to be extremely efficient for calculating anomalous dimensions. We further elaborate on these methods to show that, by the use of an angular momentum decomposition, the one-loop anomalous dimensions can be reduced to essentially a sum of products of partial waves. We apply this to the SM EFT, and show how certain classes of anomalous dimensions have their origin in the same partial-wave coefficients. We also use our result to obtain a generic formula for the one-loop anomalous dimensions of nonlinear sigma models at any order in the energy expansion, and apply our method to gravity, where it proves to be very advantageous even in the presence of IR divergencies.

Quantum antiferromagnets near SU(N) symmetry

We study antiferromagnetic systems with enhanced “nearly-SU(N)” symmetry, which can be realized in systems of ultracold spinor atoms in optical lattices. Examples of N = 3 (for S = 1 bosons) and N = 4 (for S=32 fermions) are considered. Near the SU(N) point, the low-energy physics can be described by the CPN−1 model with an additional symmetry-breaking term lowering the symmetry down to SU(2) and favoring the Néel ordering. We show that the effective theory of such systems can be cast in the form of a nonlinear sigma model with the SO(3) matrix-valued field, which is typically obtained for frustrated magnets with non-collinear order. Further, we show that those systems possess a peculiar effect of topological binding: for a system with the underlying spin S, lowering of the symmetry from SU(2S + 1) to SU(2) leads to binding of topological unit-charge excitations of the CP2S model (skyrmions for space dimension d = 2, instantons for d = 1, and hedgehogs for d = 3) into 2S-multiplets.

Symmetry restoration and the gluon mass in the Landau gauge

We investigate the generation of a gluon screening mass in Yang-Mills theory in the Landau gauge. We propose a gauge-fixing procedure where the Gribov ambiguity is overcome by summing over all Gribov copies with some weight function. This can be formulated in terms of a local field theory involving constrained, nonlinear sigma model fields. We show that a phenomenon of radiative symmetry restoration occurs in this theory, similar to what happens in the standard nonlinear sigma model in two dimensions. This results in a nonzero gluon screening mass, as seen in lattice simulations.

Observation of second-order meron polarization textures in optical microcavities

Multicomponent Bose–Einstein condensates, quantum Hall systems, and chiral magnetic materials display twists and knots in the continuous symmetries of their order parameters known as skyrmions. Originally discovered as solutions to the nonlinear sigma model in quantum field theory, these vectorial excitations are quantified by a topological winding number dictating their interactions and global properties of the host system. Here, we report the experimental observation of a stable individual second-order meron and antimeron appearing in an electromagnetic field. We realize these complex textures by confining light into a liquid-crystal-filled cavity that, through its anisotropic refractive index, provides an adjustable artificial photonic gauge field that couples the cavity photon motion to its polarization, resulting in the formation of these fundamental vectorial vortex states of light. Our observations could help bring topologically robust room-temperature optical vector textures into the field of photonic information processing and storage.

Scaling limit of the [formula omitted] invariant inhomogeneous six-vertex model

The work contains a detailed study of the scaling limit of a certain critical, integrable inhomogeneous six-vertex model subject to twisted boundary conditions. It is based on a numerical analysis of the Bethe ansatz equations as well as the powerful analytic technique of the ODE/IQFT correspondence. The results indicate that the critical behaviour of the lattice system is described by the gauged SL(2) WZW model with certain boundary and reality conditions imposed on the fields. Our proposal revises and extends the conjectured relation between the lattice system and the Euclidean black hole non-linear sigma model that was made in the 2011 paper of Ikhlef, Jacobsen and Saleur.

Nonanalytic momentum dependence of spin susceptibility for Heisenberg magnets in the paramagnetic phase and its effect on critical exponents

We study momentum dependence of static magnetic susceptibility $\chi(q)$ in paramagnetic phase of Heisenberg magnets and its relation to critical behavior within nonlinear sigma model (NLSM) at arbitrary dimension $2<d<4$. In the first order of $1/N$ expansion, where $N$ is the number of spin components, we find $\chi(q)\propto[q^{2}+\xi^{-2}(1+f(q\xi ))]^{-1+\eta /2}$, where $\xi $ is the correlation length, $q$ is the momentum, measured from magnetic wave vector, the universal scaling function $f(x)$ describes deviation from the standard Landau-Ginzburg momentum dependence. In agreement with previous studies at large $x$ we find $f(x\gg 1)\simeq (2B_{4}/N)x^{4-d}$; the absolute value of the coefficient $B_4$ increases with $d$ at $d>5/2$. Using NLSM, we obtain the contribution of the"anomalous" term $\xi^{-2}f(q\xi )$ to the critical exponent $\nu $, comparing it to the contribution of the non-analytical dependence, originating from the critical exponent $\eta $ (the obtained critical exponents $\nu $ and $\eta $ agree with previous studies). In the range $3\leq d<4$ we find that the former contribution dominates, and fully determines $1/N$ correction to the critical exponent $\nu $ in the limit $d\rightarrow 4.$

The volume of the quiver vortex moduli space

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vacuum expectation value of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey–Kirwan residue formula) leads to the Bradlow bounds ( upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss the properties of the moduli space volume in these theories. Our formulae are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with $\mathbb{C} P^N$ target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

Ricci Flow Approach to the Cosmological Constant Problem

In order to resolve the cosmological constant problem, the notion of reference frame is re-examined at the quantum level. By using a quantum non-linear sigma model (Q-NLSM), a theory of quantum spacetime reference frame is proposed. The underlying mathematical structure is a new geometry endowed with intrinsic second central moment (variance) or even higher moments of its coordinates, which generalizes the classical Riemannian geometry based on only first moment (mean) of its coordinates. The second central moment of the coordinates directly modifies the quadratic form distance which is the foundation of the Riemannian geometry. At semi-classical level, the second central moment introduces a flow which continuously deforms the Riemannian geometry driven by its classical Ricci curvature, which is known as the Ricci flow. A generalized equivalence principle of quantum version is also proposed to interpret the new geometry endowed with at least second moment. As a consequence, the spacetime is stabilized against quantum fluctuation, and the cosmological constant problem is resolved within the framework. With an isotropic positive curvature initial condition, the long flow time solution of the Ricci flow exists, the accelerating expansion universe at cosmic scale is an observable effect of the spacetime deformation of the normalized Ricci flow. A deceleration parameter − 0.67 consistent with measurement is obtained by using the reduced volume method introduced by Perelman. Effective theory of gravity within the framework is also discussed.

Effective action of type II superstring theories at order \u03b1\u20323: NS-NS couplings

Recently, it has been shown that the minimum number of gauge invariant couplings for B-field, metric and dilaton at order α′3 is 872. These couplings, in a particular scheme, appear in 55 different structures. In this paper, up to an overall factor, we fix all parameters in type II supertirng theories by requiring the reduction of the couplings on a circle to be invariant under T-duality transformations. We find that there are 445 non-zero couplings which appear in 15 different structures. The couplings are fully consistent with the partial couplings that have been found in the literature by the four-point S-matrix element and by the non-linear Sigma model methods.

RG flows of integrable \u03c3-models and the twist function

In the study of integrable non-linear σ-models which are assemblies and/or deformations of principal chiral models and/or WZW models, a rational function called the twist function plays a central rôle. For a large class of such models, we show that they are one-loop renormalizable, and that the renormalization group flow equations can be written directly in terms of the twist function in a remarkably simple way. The resulting equation appears to have a universal character when the integrable model is characterized by a twist function.

Phases of the (2+1) Dimensional SO(5) Nonlinear Sigma Model with Topological Term.

We use the half-filled zeroth Landau level in graphene as a regularization scheme to study the physics of the SO(5) nonlinear sigma model subject to a Wess-Zumino-Witten topological term in 2+1 dimensions. As shown by Ippoliti etal. [Phys. Rev. B 98, 235108 (2019)PRBMDO2469-995010.1103/PhysRevB.98.235108], this approach allows for negative sign free auxiliary field quantum MonteCarlo simulations. The model has a single free parameter U_{0} that monitors the stiffness. Within the parameter range accessible to negative sign free simulations, we observe an ordered phase in the large U_{0} or stiff limit. Remarkably, upon reducing U_{0} the magnetization drops substantially, and the correlation length exceeds our biggest system sizes, accommodating 100 flux quanta. The implications of our results for deconfined quantum phase transitions between valence bond solids and antiferromagnets are discussed.

Static spacetimes haunted by a phantom scalar field. II. Dilatonic charged solutions

We present a method to generate static solutions in the Einstein-Maxwell system with a (phantom) dilaton field in $n(\ensuremath{\ge}4)$ dimensions, based upon the symmetry of the target space for the nonlinear sigma model. Unlike the conventional Einstein-Maxwell-dilaton system, there appears a critical value of the coupling constant for a phantom dilaton field. In the noncritical case, the target space is $\mathbb{R}\ifmmode\times\else\texttimes\fi{}\mathrm{SL}(2,\mathbb{R})/H$ with the maximal subgroup $H={\mathrm{SO}(2),\mathrm{SO}(1,1)}$, whereas in the critical case the target space becomes a symmetric pp-wave and the corresponding Killing vectors form a non-semi-simple algebra. In either case, we apply the formalism to charge up the neutral solutions and show the analytical expression for dilatonic charged versions of (i) the Fisher solution, (ii) the Gibbons solution, and (iii) the Ellis-Bronnikov solution. We discuss global structures of these solutions in detail. It turns out that some solutions contained in the Fisher and Gibbons classes possess the parallelly propagated (p.p) curvature singularities in the parameter region where all the scalar curvature invariants remain bounded. These p.p curvature singularities are not veiled by a horizon, thrusting them into physically untenable nakedly singular spacetimes. We also demonstrate that the dilatonic-charged Ellis-Bronnikov solution admits a parameter range under which the solution represents a regular wormhole spacetime in the two-sided asymptotically flat regions.

Higher derivative supersymmetric nonlinear sigma models on Hermitian symmetric spaces and BPS states therein

We formulate four-dimensional $\mathcal{N} = 1$ supersymmetric nonlinear sigma models on Hermitian symmetric spaces with higher derivative terms, free from the auxiliary field problem and the Ostrogradski's ghosts, as gauged linear sigma models. We then study Bogomol'nyi-Prasad-Sommerfield equations preserving 1/2 or 1/4 supersymmetries. We find that there are distinct branches, that we call canonical ($F=0$) and non-canonical ($F\neq 0$) branches, associated with solutions to auxiliary fields $F$ in chiral multiplets. For the ${\mathbb C}P^N$ model, we obtain a supersymmetric ${\mathbb C}P^N$ Skyrme-Faddeev model in the canonical branch while in the non-canonical branch the Lagrangian consists of solely the ${\mathbb C}P^N$ Skyrme-Faddeev term without a canonical kinetic term. These structures can be extended to the Grassmann manifold $G_{M,N} = SU(M)/[SU(M-N)\times SU(N) \times U(1)]$. For other Hermitian symmetric spaces such as the quadric surface $Q^{N-2}=SO(N)/[SO(N-2) \times U(1)])$, we impose F-term (holomorphic) constraints for embedding them into ${\mathbb C}P^{N-1}$ or Grassmann manifold. We find that these constraints are consistent in the canonical branch but yield additional constraints on the dynamical fields thus reducing the target spaces in the non-canonical branch.

Degenerate States in Nonlinear Sigma Model with U(1) Symmetry &amp;lt;sub&amp;gt;—For Study on Violation of Cluster Property&amp;lt;/sub&amp;gt;

Entanglement in quantum theory is a concept that has confused many scientists. This concept implies that the cluster property, which means no relations between sufficiently separated two events, is non-trivial. In the works for some quantum spin systems, which have been recently published by the author, extensive and quantitative examinations were made about the violation of cluster property in the correlation function of the spin operator. The previous study of these quantum antiferromagnets showed that this violation is induced by the degenerate states in the systems where the continuous symmetry spontaneously breaks. Since this breaking is found in many materials such as the high temperature superconductors and the superfluidity, it is an important question whether we can observe the violation of the cluster property in them. As a step to answer this question we study a quantum nonlinear sigma model with U(1) symmetry in this paper. It is well known that this model, which has been derived as an effective model of the quantum spin systems, can also be applied to investigations of many materials. Notifying that the existence of the degenerate states is essential for the violation, we made numerical calculations in addition to theoretical arguments to find these states in the nonlinear sigma model. Then, successfully finding the degenerate states in the model, we came to a conclusion that there is a chance to observe the violation of cluster property in many materials to which the nonlinear sigma model applies.

Classically Integrable Non-Linear Sigma Models and their Geometric Properties

General classes of non-linear sigma models originating from a specified action are developed and studied. Models can be grouped and considered within a single mathematical structure this way. The geometrical properties of these models and the theories they describe are developed in detail. The zero curvature representation of the equations of motion are found. Those representations which have a spectral parameter are of importance here. Some new models with Lax pairs which depend on a spectral parameter are found. Some particular classes of solutions are worked out and discussed.

The Higgs field and the Jordan Brans Dicke cosmology

We investigate a field theoretical approach to the Jordan–Brans–Dicke (JBD) theory extended with a particular potential term on a cosmological background by starting with the motivation that the Higgs field and the scale factor of the universe are related. Based on this relation, it is possible to come up with mathematically equivalent but two different interpretations. From one point of view while the universe is static, the masses of the elementary particles change with time. The other one, which we stick with throughout the manuscript, is that while the universe is expanding, particle masses are constant. Thus, a coupled Lagrangian density of the JBD field and the scale factor (the Higgs field), which exhibit a massive particle and a linearly expanding space in zeroth order respectively, is obtained. By performing a coordinate transformation in the field space for the reduced JBD action whose kinetic part is nonlinear sigma model, the Lagrangian of two scalar fields can be written as uncoupled for the Higgs mechanism. After this transformation, as a result of spontaneous symmetry breaking, the time dependent vacuum expectation value (vev) of the Higgs field and the Higgs bosons which are the particles corresponding to quantized oscillation modes about the vacuum, are found.