Renormalization Group, Hidden Symmetries and Approximate Ward Identities in the Xyz Model

Abstract

Using renormalization group methods, we study the Heisenberg-Ising XY Z chain in an external magnetic field directed as the z axis, in the case of small coupling J 3 in the z direction. We study the asymptotic behaviour of the spin space-time correlation function in the direction of the magnetic field and the singularities of its Fourier transform.The work is organized in two parts. In the present paper an expansion for the ground state energy and the effective potential is derived, which is convergent if the running coupling constants are small enough. In the subsequent paper, by using hidden symmetries of the model, we show that this condition is indeed verified, if J 3 is small enough, and we derive an expansion for the spin correlation function. We also prove, by means of an approximate Ward identity, that a critical index, related with the asymptotic behaviour of the correlation function, is exactly vanishing.1.4 In this paper we develop a rigorous renormalization group analysis for the XY Z Hamiltonian in its fermionic form (some "not optimal" bounds for the correlation function Ω 3 (x) were already found in [M2]). As we said before, Ω 3 (x) can be obtained from the exact solution only in the case J 3 = 0, when the fermionic theory is a non interacting one. In particular, if x = (x, 0) and |ux| << 1, (1.8) and a more detailed analysis of the "small" terms in the r.h.s. (in order to prove that their derivatives of order n decay as |x| −n ), show that Ω 3 (x, 0) is a sum of "oscillating" functions with frequencies (np F )/π mod 1, n = 0, ±1, where p F = arccos(−h); this means that its Fourier transform has to be a smooth function, even for u = 0, in the neighborhood of any momentum k = 0, ±2p F . These frequencies are proportional to p F , so they depend only on the external magnetic field h.If J 3 = 0, a similar property is satisfied for the leading terms in the asymptotic behaviour, as we shall prove, but the value of p F depends in general also on u and J 3 . For example, if u = 0, the Hamiltonian (1.5) is equal, up to a constant, to the Hamiltonian of a free fermion gas with Fermi momentum p F = arccos(J 3 − h) plus an interaction term proportional to J 3 .As it is well known, the interaction modifies the Fermi momentum of the system by terms of order J 3 and it is convenient (see [BG], for example), in order to study the interacting model, to fix the Fermi momentum to an interaction independent value, by adding a counterterm to the hamiltonian. We proceed here in a similar way, that is we fix p F and h 0 so thatand we look for a value of ν, depending on u, J 3 , h 0 , such that, as in the J 3 = 0 case, the leading terms in the asymptotic behaviour of Ω 3 L,β (x) can be represented as a sum of oscillating functions with frequencies (np F )/π mod 1, n = 0, ±1.

Universality Relations in Non-solvable Quantum Spin Chains

“…The parameter ν is chosen as a function of λ and p F , so that the singularity of the Fourier transform of the two-point function corresponding to H is fixed at k = (±p F , 0); the first equation in (23) gives the value of h corresponding, in the model (3), to the chosen value of p F . On the contrary, the parameter δ is an unknown function of λ and p F , whose value is determined by requiring that, in the renormalization group analysis, the corresponding marginal term flows to 0; this implies that v s is the sound velocity even for the full Hamiltonian H. It is well known that the correlations of the quantum spin chain can be derived by the following Grassmann integral, see [12]:…”

“…are defined in a way analogous to the definition of Ω(p) in (12). Moreover, we define the Fourier transforms of G 2,1 α , α = ρ, j, so that…”

“…The Fourier transforms G 2 th;ω (k) and G 0,2 th,ρ,ρ (p) of G 2 th;ω (y, z) and G 2,0 th,ρ,ρ (x, y) are defined in a way analogous to the definition of Ω(p) in (12). Moreover, we define the Fourier transforms of G 2,1 th,α;ω , α = ρ, j, as in (28).…”

“…(2.90) of [12]) has the same support and scaling properties as f j (k). Hence, if make the field rescaling ψ → [ Z j−1 / Z j ]ψ and we call V (j) ( Z j−1 ψ ≤j ) the new effective potential, we can write (71) in the form…”

“…The proof of the lemma is based on the RG analysis of the Grassmann integrals (24) and (29), described in [12,13] and [17,19], respectively.…”