Kink confinement in two-dimensional field theories and one-dimensional quantum ferromagnets: Bethe-Salpeter equation approach
Final Report Abstract
The confinement phenomenon occurs when the constituents of a compound particle cannot be separated from each other and therefore cannot be observed directly. A prominent and important example in high energy physics is the confinement of quarks in hadrons, whose consistent theoretical description is still lacking. In condensed matter systems, the confinement of topological kink excitations was recently observed in one-dimensional quantum ferromagnets. The goal of the present project has been the clarification of some aspects of the confinement phenomenon within the framework of two-dimensional quantum field theory and statistical mechanics. The simplest model exhibiting confinement of this type is the Ising Field Theory (IFT) - the two-dimensional Euclidean continuum quantum field theory that corresponds to the two-dimensional lattice Ising model in the (near-critical) scaling limit close to its phase transition point. This model has Z2 symmetry, which is spontaneously broken in the ferromagnetic phase at zero magnetic field. Elementary excitations in this case are free massive particles representing kinks (also called "Quarks"), which interpolate between two energetically degenerate ferromagnetic vacua. The application of a magnetic field lifts this degeneracy and induces a long-range attraction between kinks. This leads to the confinement of kinks in pairs forming a set of bound states - the "Mesons". For small magnetic field, the masses of the mesons can be determined perturbatively. A first result of the project is the calculation of the meson masses in the IFT to third order in the small magnetic field. At third order of perturbation theory, admixtures of four-quark, six-quark, ... configurations to the quantum meson state must be taken into account. Such multi-quark quantum fluctuations lead to radiative corrections to the quark mass and the two-particle interaction between quarks, both on long and short distances. All these radiative effects contribute to the third-order corrections of the meson masses, which were obtained by means of the form factor perturbation theory. The confinement mechanism in the IFT outlined above is quite general. It is realized in many two-dimensional field theories that are invariant under a discrete symmetry group. Specifically, it is realized in the q-state Potts Field Theory (PFT) that describes the scaling limit of the two-dimensional q-state lattice Potts model. The latter generalizes the Ising model to the case of q allowed spin orientations and reduces to the latter for q=2. Confinement in the PFT has, however, some specific features which distinguish it from the IFT case. Thus, hree-quark “baryon” bound states can exist in addition to two quark meson states in the 3-state PFT. Since the three-body dynamics is known to be much more complicated than the two-body dynamics (this applies to both classical and quantum mechanics), the calculation of baryon masses in the PFT constitutes a challenging problem. A second important result of the project is the calculation of the masses of several of the lightest baryons of the 3-state PFT in the weak confinement regime to leading order in the magnetic field. The baryon masses were determined in the non-relativistic approximation from the numerical solution of the Schrödinger partial differential equation describing three quarks moving on a line that interact via a linear attractive potential. These theoretical predictions for the baryon masses in the 3-state PFT were confirmed recently by Lencsés and Takács by direct numerical calculations performed by means of the renormalization group improved truncated conformal space approach. As a third model the n-component phi-4 model on a three-dimensional slab of width L with free boundaries was studied in the limit of large n. This model is important for the theories of boundary critical phenomena, finite-size scaling and the thermodynamic Casimir effect. For the semi-infinite critical case of an infinitely thick slab at the bulk critical temperature Tc, the exact solution of this model in the infinite-n limit is known from the work of Bray & Moore. Applying a new analytical technique based on inverse scattering theory, we have extended Bray & Moore’s results for the semi-infinite geometry to all temperatures above and below Tc in the critical region. Aside from the temperature dependence of the surface free energy, which displays a logarithmic singularity at the critical point, the exact two-point order parameter correlation function on the surface has been calculated for the semi-infinite case for all temperatures in the scaling region. For the slab geometry with finite L, we have computed by numerical means the universal free-energy scaling function describing the temperature and L-dependence in the scaling region. The so-obtained free-energy scaling function exhibits all qualitative features observed in experiments on the thinning of wetting layers of the helium-4 near the lambda-point. From the viewpoint of equilibrium statistical mechanics of two-dimensional systems, there are additional aspects of the confinement phenomenon worth studying. A particular one relates to the effect of small relevant external fields, which explicitly breaks the discrete symmetry in the system. As has been shown by McCoy and Wu for the confinement transition in the ferromagnetic IFT, the application of a small magnetic field causes a breakup of the branch cut into a sequence of poles of the Fourier transformed two-point spin-spin correlation function. Another well-known characteristic of the confinement transition in the IFT is the essential singularity (the so-called droplet singularity) of the free energy as a function of the magnetic field at zero field in the ferromagnetic phase. We have found that analogs of these features, usually associated with confinement, can be found in the behavior of the surface two-point correlation function and the free energy of the n-component vector model in the large-n limit, where the inverse 1/L of the slab thickness takes the role of the magnetic field.
Publications
- Exact thermodynamic Casimir forces for an interacting three-dimensional model system in film geometry with free surfaces. EPL (Europhysics Letters), Vol. 100. 2012, Number 1, 10004.
H. W. Diehl, D. Grüneberg, M. Hasenbusch, A. Hucht, S. B. Rutkevich, and F. M. Schmidt
(See online at https://doi.org/10.1209/0295-5075/100/10004) - Large-n approach to thermodynamic Casimir effects in slabs with free surfaces. Physical Review E, Vol. 89. 2014, Issue 6, 062123.
H. W. Diehl, Daniel Grüneberg, Martin Hasenbusch, Alfred Hucht, Sergei B. Rutkevich, and Felix M. Schmidt
(See online at https://doi.org/10.1103/PhysRevE.89.062123) - On the spectrum of the discrete 1d Schrödinger operator with an arbitrary even potential.
S. B. Rutkevich
- The O(n) φ4 model with free surfaces in the large-n limit: Some exact results for boundary critical behaviour, fluctuation-induced forces and distant-wall corrections. Journal of Physics A: Mathematical and Theoretical, Vol. 47. 2014, Issue 14, article id. 145004.
H. W. Diehl, S. B. Rutkevich
(See online at https://doi.org/10.1088/1751-8113/47/14/145004) - Baryon masses in the three-state Potts field theory in a weak magnetic field. Journal of Statistical Mechanics: Theory and Experiment, Vol. 2015, P01010.
S. B. Rutkevich
(See online at https://doi.org/10.1088/1742-5468/2015/01/P01010) - Inverse scattering-theory approach to the exact n ! 1 solutions of
O(n) 4 models on films and semi-infinite systems bounded by free surfaces. Physical Review E, Vol. 91. 2015, Issue 6, 062114.
S. B. Rutkevich, H. W. Diehl.
(See online at https://doi.org/10.1103/PhysRevE.91.062114)
