Yang-Lee edge singularities in the large N limit

We discuss the next-to-leading (but dominant for dimension less than six) corrections to the large-X behavior of the magnetization at the Yang-Lee edge singularity. The X dependence of the corresponding amplitude is valid for all dimensions below six.


I. INTRODUCTION
The role of zeros of the partition function was first pointed out by Yang and Lee. ' The location of these zeros in the complex chemical-potential plane, for the case of a fluid, or in the complex magnetic-field plane, in the case of a spin system, plays a crucial role in the onset of phase transitions. For an Ising system, Yang and Lee showed that all of the zeros lie on the imaginary magnetic field H axis and as long as T & T, it is generally believed that they do not come down to h =0. For a general O(N) spin system, one only knows that for T & T, the partition function is free of zeros in a strip~ImH~& ho. We do, however, believe that the location of these zeros for the general case is, as in the Ising model, along the imaginary axis. The point H =iho(T) is a branch point of the partition function, referred to as the Yang-Lee edge singularity. For h =ImH &Hp, the density at the zeros is expected to vary as (IrrMho) resulting in a magnetization ' presented in Sec. IV.) The nonuniformity of the large Nlimit has been remarked on previously and it is the purpose of this paper to clarify this point. We will show that below six dimensions, and for large, but finite N, the dominant singular part of the magnetization is expected to behave as (1. 4) with f ( x) behaving as a constant for x~co, and as x ' for x~0, with a crossover exponent ttp=4/(6 D), which is divergent when D reaches 6.
We obtain an expression for the exponent c valid for all dimensions less than six, c =(2 -4tr)/(6 -D) . This can be reconciled with the N = Oo result if one assumes for D &6, hoh, and N ' small, a scaling form where' Mo(ih) consists of a regular function of h and functions less singular than M". "g. This separation into regular and singular pieces is valid in the neighborhood of ihp.
There is an accumulation of evidence ' that the critical exponent tT is independent of temperature (as long as T )T,), the number of components of the O(N) spin system, and of course the details of the lattice. These results obtain in mean-field theory and large-X calculations. It is the latter that lead to a seemtnyly paradoxical result. The large-N calculation yields o.= -, for all dimensions. However, for finite N, the e expansion for D =6e, ' series expansions for D=2 and D=3, and exact results for D =0 and D =1 yield an exponent 0 decreasing from the value of -, ' at D=6 down to o= --, ' for D=l and cJ= -1 at D =0. (Details of the case D =0 will be II. LARGE-W RESULTS +VNH fd xs (2.1) where s and H have been scaled as appropriate to this limit. We use continuum notation with a lattice spacing or momentum cutoff implied. The Dirac 5 functions are replaced by their integral representations For large N the spin system will be studied by a saddle-point method. The partition function is (2.9) Thus, in the limit, first N~&a, then h~ho, the critical exponent cr is -,. However, the above discussion makes the saddle-point method suspect just in the region h -+hp. We should note in passing that as T approaches T, both mp and hp approach zero. Thus, at the critical magnetic field we are faced with a massless iy field theory, where the cubic couplings are nonlocal. The range of nonlocality is 1/m i. As long as T & T"m&+0 and we may view 1/mi as a new lattice constant. In the critical region we are interested in large distances, larger than 1/mi. In this regime the field theory can be considered as local. Above six dimensions the masslessness of the theory is innocuous and we recover the value o.= -, ; below six dimensions, the infrared behavior is crucial and a will deviate from this value.
The critical Hamiltonian will be independent of T and N, reflecting itself in the fact that o. is also independent of these parameters. Details will be presented in the next section.

III. CONTRIBUTIONS OF FLUCTUATIONS TO THE MAGNETIZATION
The effective-field theory for the long-wavelength fluctuations around the saddle point is obtained from Eqs.
The arguments leading to Eq. (3.7) form a special case of the general ones developed by Parisi to obtain the bare coupling-constant dependence. At first glance, we expect the above results to be valid for 4 & D & 6. Below four dimensions, other operators may become relevant and spoil the simple scaling ideas used to obtain Eq. (3.7). We shall, however, in Sec. IV confirm (3.g) in low dimension, so that we expect it to be valid for all dimensions less than six.
S =N f d x -, ' ( Vp)'+ -, ' p'p'+ -'r(p'+ . The dependence of this action on the magnetic field is contained in the effective mass p. Thus, the contribution of the fluctuation to the magnetization is 1 2 , (q ). p Using the equations of motion we find that  and its contribution yields a magnetization with a critical field ho --1 and o.= -, '. The N dependence of the relative coefficients of the above contribution to the magnetization and that due to the simple zeros of the Bessel function is in accord with our previous discussion.
The saddle point gives the correct asymptotic behavior in N of the Bessel function as long as h&1. For h -1, we again find that this saddle point disappears and higher terms in the expansion must be retained. Keeping up to terms cubic in xxo we find (for h -1) where -x& is the first negative zero of the Airy function.
The quantity in the second set of parentheses has the expected behavior, going tox for small x, and to 1 for x~oo. The exponent P = -, ' agrees with (1.5).

B. One dimension
A one-dimensional field theory corresponding to Eq. (3.3) may be considered as a problem in quantum mechanics with the Hamiltonian 2 2 ' 6 (4.6) The free energy is the "ground-state energy" of this system. In order to evaluate the relevant part of the magneti- and treat A