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arXiv:1310.4821

Ising Spectroscopy II: Particles and poles at T>Tc

Alexander B. Zamolodchikov

Published Oct 17, 2013• 16 citations
arXivHTMLPDF

Abstract

I discuss particle content of the Ising field theory (the scaling limit of the Ising model in a magnetic field), in particular the evolution of its mass spectrum under the change of the scaling parameter. I consider both real and pure imaginary magnetic field. Here I address the high-temperature regime, where the spectrum of stable particles is relatively simple (there are from one to three particles, depending on the parameter). My goal is to understand analytic continuations of the masses to the domain of the parameter where they no longer exist as the stable particles. I use the natural tool -- the $2\to 2$ elastic scattering amplitude, with its poles associated with the stable particles, virtual and resonance states in a standard manner. Concentrating attention on the "real" poles (those corresponding to stable and virtual states) I propose a scenario on how the pattern of the poles evolves from the integrable point $T=T_c,\ H\neq 0$ to the free particle point $T>T_c,\ H=0$, and then, along the pure imaginary $H$, to the Yang-Lee critical point. Waypoints along this evolution path are located using TFFSA data. I also speculate about likely behavior of some of the resonance poles.

DOI
10.48550/arxiv.1310.4821

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I discuss particle content of the Ising field theory (the scaling limit of the Ising model in a magnetic field), in particular the evolution of its mass spectrum under the change of the scaling parameter. I consider both real and pure imaginary magnetic field. Here I address the high-temperature regime, where the spectrum of stable particles is relatively simple (there are from one to three particles, depending on the parameter). My goal is to understand analytic continuations of the masses to the domain of the parameter where they no longer exist as the stable particles. I use the natural tool -- the 2→22\to 22→2 elastic scattering amplitude, with its poles associated with the stable particles, virtual and resonance states in a standard manner. Concentrating attention on the "real" poles (those corresponding to stable and virtual states) I propose a scenario on how the pattern of the poles evolves from the integrable point T=Tc, H≠0T=T_c,\ H\neq 0T=Tc​, H=0 to the free particle point T>Tc, H=0T>T_c,\ H=0T>Tc​, H=0, and then, along the pure imaginary HHH, to the Yang-Lee critical point. Waypoints along this evolution path are located using TFFSA data. I also speculate about likely behavior of some of the resonance poles.
Abstract We study the “three particle coupling” Γ111(ξ) {\Gamma}_{11}^1\left(\xi \right) Γ111​(ξ) Γ111ξ , in 2dIsing Field Theory in a magnetic field, as the function of the scaling parameterξ:=h/(−m)15/8, wherem∼Tc−Tandh∼Hare scaled deviation from the critical temperature and scaled external field, respectively. The “φ3coupling” Γ111 {\Gamma}_{11}^1 Γ111​ Γ111 is defined in terms of the residue of the 2 → 2 elastic scattering amplitude at its pole associated with the lightest particle itself. We limit attention to the High-Temperature domain, so thatmis negative. We suggest “standard analyticity”: (Γ111)2 {\left({\Gamma}_{11}^1\right)}^2 (Γ111​)2 Γ1112 , as the function ofu:=ξ2, is analytic in the whole complexu-plane except for the branch cut from – ∞ to –u0≈ – 0.03585, the latter branching point –u0being associated with the Yang-Lee edge singularity. Under this assumption, the values of Γ111 {\Gamma}_{11}^1 Γ111​ Γ111 at any complexuare expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of Γ111 {\Gamma}_{11}^1 Γ111​ Γ111 near the Yang-Lee branching point, as well as its known asymptotic atu →+∞. The resulting dispersion relation agrees well with known exact data, and with numerics obtained via Truncated Free Fermion Space Approach. This work is part of extended project of studying the S-matrix of the Ising Field Theory in a magnetic field.
The study of the equation of state of the two-dimensional Ising model is initiated by use of the recently calculated nnn-point spin-correlation functions. In the scaling region where $T\ensuremath{\rightarrow}{T}_{c}$ and $h=B{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{15}{8}}(\frac{H}{k{T}_{c}})$ is of order 1 (BBB is a lattice-dependent constant) the free energy below Tc{T}_{c}Tc​ is of the form $f(T, h)\ensuremath{-}f(T, 0)=\mathrm{const}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{2}$ $\ensuremath{\Sigma}{n=1}^{\ensuremath{\infty}}(\frac{1}{n!}){C}_{n}^{\ensuremath{-}}{h}^{n}$ where the lattice-dependent constant is chosen so that ${C}_{1}^{\ensuremath{-}}=1$. The constant ${C}_{2}^{\ensuremath{-}}$ has been studied previously. In this paper we study ${C}_{3}^{\ensuremath{-}}$ and ${C}_{4}^{\ensuremath{-}}$ and find as a first approximation $\frac{{C}_{3}^{\ensuremath{-}}}{3!}\ensuremath{\sim}\ensuremath{-}\frac{11}{72{\ensuremath{\pi}}^{2}}$ and $\frac{{C}_{4}^{\ensuremath{-}}}{4!}\ensuremath{\sim}\frac{5}{189\ensuremath{\pi}}+\frac{59}{162{\ensuremath{\pi}}^{3}}$. These are in close agreement with the low-temperature-series values of -0.01538 and 0.0195.
We reexamine the problem of particle decay within the Hamiltonian formalism. By deforming contours of integration, the survival amplitude is expressed as a sum of purely exponential contributions arising from the simple poles of the resolvent on the second sheet plus a background integral along a complex contour $\ensuremath{\Gamma}$ running below the location of the poles. We observe that the time dependence of the survival amplitude in the small-time region is strongly correlated to the asymptotic behavior of the energy spectrum of the system; we compute the small-time behavior of the survival amplitude for a wide variety of asymptotic behaviors. In the special case of the Lee model, using a formal procedure of analytic continuation, we show that a complete set of complex energy eigenvectors of the Hamiltonian can be associated with the poles of the resolvent and the background contour $\ensuremath{\Gamma}$. These poles and points along $\ensuremath{\Gamma}$ correspond to the discrete and the continuum states, respectively. In this context, each unstable particle is associated with a well-defined object, which is a discrete generalized eigenstate of the Hamiltonian having a complex eigenvalue, with its real and negative imaginary parts being the mass and half-width of the particle, respectively. Finally, we briefly discuss the analytic continuation of the scattering amplitude within this generalized scheme, and note the appearance of "redundant poles" which do not correspond to discrete solutions of the modified eigenvalue problem.
We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 to 2 S-matrix element to the complex s-plane. Focusing primarily on the "high temperature" regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.