Yang-Lee Zeros of the Yang-Lee Model

Yang-Lee Zeros of the Yang-Lee Model

G. Mussardo SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136, Trieste, Italy International Centre for Theoretical Physics (ICTP), I-34151, Trieste, Italy    R. Bonsignori SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136, Trieste, Italy    A. Trombettoni CNR-IOM DEMOCRITOS Simulation Center, Via Bonomea 265, I-34136 Trieste, Italy SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136, Trieste, Italy
Abstract

To understand the distribution of the Yang-Lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the two-dimensional Yang-Lee model. The grand-canonical partition function of this quantum field theory, as a function of the fugacity and the inverse temperature , can be computed in terms of the Thermodynamics Bethe Ansatz based on its exact -matrix. We extract the Yang-Lee zeros in the complex plane by using a sequence of polynomials of increasing order in which converges to the grand-canonical partition function. We show that these zeros are distributed along curves which are approximate circles as it is also the case of the zeros for purely free theories. There is though an important difference between the interactive theory and the free theories, for the radius of the zeros in the interactive theory goes continuously to zero in the high-temperature limit while in the free theories it remains close to 1 even for small values of , jumping to 0 only at .

Pacs numbers: 11.10.St, 11.15.Kc, 11.30.Pb

I Introduction

Many physical quantities reveal their deeper structure by going to the complex plane. This is the case, for instance, of the analytic properties of the scattering amplitudes where the angular momentum is not longer restricted to be an integer but allowed to take any complex value giving rise in this way to the famous Regge poles [Regge, ]. Another famous example is the Yang-Lee theory of equilibrium phase transitions [YL1, , YL2, ] based on the zeros of the grand-canonical partition function in the complex plane of the fugacity: in a nutshell, the main observation of Yang and Lee was that the zeros of the grand-canonical partition functions in the thermodynamic limit usually accumulate in certain regions or curves of the complex plane, with their positive local density which changes by changing the temperature; if at a critical value , the zeros accumulate and pinch a positive value of the real axis, this is what may mark the onset of a phase transition.

As we are going to discuss extensively through the rest of the paper, the pattern of zeros of grand-canonical partition functions can be generally quite interesting and this study alone is a source of many stimulating physical and mathematical questions. If the study of the patterns of Yang-Lee zeros is then the first topic of this paper, the Yang-Lee model (and its zeros!) is our second main topic. In order to introduce such a model and present the work of this paper in its proper perspective, we need to talk about the pattern of zeros of just one particular statistical system: the Ising model. In ref. [YL2, ] Yang and Lee showed that for ferromagnetic Ising-like models, independently on the dimensionality and regularity of the lattice and also largely independently on the nature of the couplings, the zeros of the Ising model lie on the unit circle111This circle-theorem was later extended by many authors to ferromagnetic Ising model of arbitrarily high spin and with many-body spin interactions [Asano, , Suzuki, , Griffiths, , SuzukiFisher, ]. in the complex plane of the variable (where and is the external magnetic field): posing , they have the following structure (see Figure 1)

• for the zeros are placed along a ”C”, namely a symmetric arc around whose edges are at ;

• at these edges move to the real axis and pinch it;

• for the zeros densely cover the entire circle.

Kortman and Griffiths [KortmanGriffiths, ] were the first to notice that the density of the Yang-Lee zeros of the Ising model nearby the edges gives rise to a problem which has its own interest since such a density presents an anomalous behaviour with a scaling law ruled by a critical exponent

 η(θ,T)∼|θ−θ0(T)|σ,T>Tc. (1)

Such a behavior is closely analogous to the usual critical phenomena (although in this case triggered by a purely imaginary magnetic field ) and therefore Fisher [FisherEdge, ] posed the question about its effective quantum field theory and argued that, in sufficiently high dimension , this consists of a Landau-Ginzburg theory for the scalar field with euclidean action given by

 A=∫ddx[12(∂ϕ)2+i(h−h0)ϕ+igϕ3]. (2)

This action is what defines the Yang-Lee model, i.e. the quantum field theory studied in this paper, and the imaginary couplings present in such an action is what makes the Yang-Lee model a non-hermitian theory; the theory is however invariant under a CP transformations [CM, , Bender, ] and therefore its spectrum is real. At criticality, i.e. when , the corresponding fixed point of the Renormalization Group presents only one relevant operator, namely the field itself: in two dimensions Cardy showed [CardyYL, ] that all these properties are encoded into the simplest non-unitarity minimal model of Conformal Field Theories which has central charge and only one relevant operator of conformal dimension . Still in two dimensions, Cardy and Mussardo [CM, ] then argued that the action (2), regarded as deformation of this minimal conformal model by means of its relevant operator corresponds to an integrable quantum field theory. This fact has far-reaching consequences: indeed, the infinite number of conservation laws – the fingerprint of any integrable field theory [ZamZam, , Zam1, , GM, ] – implies that scattering processes of the Yang-Lee model are completely elastic and factorizable in terms of the two-body scattering matrix which was exactly computed in [CM, ]. The spectrum can be easily extracted by the poles of the -matrix and turns out to consist of only one massive particle which may be regarded as a bound state of itself. Exact Form Factors and two-point correlation functions of this massive model were later computed and discussed in [ZamFFYL, ].

Based on the -matrix scattering theory and the Thermodynamic Bethe Ansatz introduced by Al.B. Zamolodchikov [ZamTBA, ], one can then in principle determine exactly (although numerically) the free-energy of the integrable models as function of the inverse temperature and the fugacity [Fendley, , KM, ]. This is indeed what we have done in this paper for the Yang-Lee model, i.e. one of the simplest representative of integrable quantum field theories, for the purpose of studying what kind of distribution emerges for the zeros of the grand-canonical partition function of these systems. We hope these brief introductory considerations were useful to clarify the general aims of this paper and the ”recursive” use of Yang-Lee names, both to denote the zeros and the eponymous model, and the zeros of the model itself!

A comment is in order on the logic of this work. In the Yang-Lee theory of phase transitions one relates thermodynamical properties, such the free energy and the magnetization, to the density of Yang-Lee zeros . Usually the focus is on the (often approximate) determination of in order then to extract the equilibrium quantities of interest. However, the point of view we adopt to deal with integrable quantum field theories as, for instance the Yang-Lee model, is rather the opposite, since the free energy of these models are already known in terms of the Thermdoynamics Bethe Ansatz equations, and this gives the possibility to determine the properties of the zeros. We have applied such a procedure to the Yang-Lee model but it is clear that it can be applied as well to other integrable quantum field theories.

The rest of the paper is organised as follows. Section II is devoted to a brief recap of the main points of the Yang-Lee theory of the phase transition and the importance to control the distribution of the zeros of the grand-canonical partition function in order to study the thermodynamics. Section III contains a detailed discussion on the nature of the roots of polynomials which may be regarded as grand-canonical partition functions of some appropriate physical system. In Section IV we present the closed formulas of the partition functions of free bosonic and fermionic theories, both in the non-relativistic and relativistic case, and also in the presence of an harmonic trap. Sections V and VI contain a detailed discussion of the zeros of the grand-canonical partition functions of the non-relativistic and relativistic free theories respectively: apart from some peculiar features emerging from this study, the basic purpose of these Sections is to set the stage for the analysis of the interacting integrable models discussed in the later Sections. In particular, in Section VII we start recalling the basic properties of integrable quantum field theories, namely the -matrix formulation and the Thermodynamics Bethe Ansatz which allows us to recover the partition function of the integrable models. In Section VIII we study in detail the Yang-Lee zeros of the simplest quantum integrable field theories, namely the Yang-Lee model. Our conclusions are finally collected in Section IX.

Ii Yang-Lee theory of phase transitions

In order to overcome some inadequacies of the Mayer method [Mayer, ] for dealing with the condensation of a gas and also to understand better the underlying mathematical reasons behind the occurrence of phase transitions, in 1952 Yang and Lee proposed to analytically continue the grand-canonical partition function to the complex plane of its fugacity and determine the pattern of zeros of this function. Although only real values of the fugacity determine the physical value of the pressure, the magnetization or other relevant thermodynamical quantities, the overall analytic behavior of these observables can only be understood by looking at how the zeros move as a function of an external parameter such as the temperature. In this Section we are going to simply write down, mainly for future reference, the basic formulas of the Yang-Lee formalism with few extra comments.

Concerning general references on this topic, in addition to the original papers [YL1, , YL2, ] and others previously mentioned [Asano, , Suzuki, , Griffiths, , SuzukiFisher, , KortmanGriffiths, , FisherEdge, , CM, , CardyYL, ], the reader may also benefit of some standard books [Huang, , Mattis, ] or reviews such as [Bena, , Blythe, ] and references therein. As a matter of fact, the literature on the subject is immense, even ranging across several fields of physics and mathematics. For this reason we cannot definitely do justice to all authors who contributed to the development of the subject but we would like nevertheless to explicitly mention few more references which we have found particularly useful, such as the series of papers by Ikeda [Ikeda1, , IkedaF, , IkedaB, ] or by Abe [Abe1, , Abe2, ], the papers by Katsura on some analytic expressions of the density of zeros [Suzuki, ], the paper by Fonseca and Zamolodchikov [FonsecaZam, ] on the analytic properties of the free energy of the Ising field theory and some related references on this subject such as [ZamTBA, , YurovZam, , Mossa, , Wydro, , Jacobsen, , Ana, ], and finally some references on the experimental observations of the Yang-Lee zeros [Binek, ], in particular those based on the coherence of a quantum spin coupled to an Ising-type thermal bath [Wei, , Peng, ].

Yang-Lee formulation. Consider for simplicity the grand-canonical partition function of a gas made of particles with hard cores in a volume , of activity and at temperature is given by

 ΩN(z)= N∑k=01k!Zk(V,T)zk=N∏l=1(1−zzl), (3)

where is the largest number of particles that can be contained in the volume, the coefficients are the canonical partition functions of a system of particles and the is the fugacity. As a polynomial of order , has zeros in the complex plane. The thermodynamics of the system is recovered by defining, in the limit , the pressure and the density of the system as

 p(z)kT≡^f(z)=limV→∞1VlogΩN(z)=limV,N→∞1VN∑l=1log(1−zzl), (4) ρ(z)≡z^f′(z)=limV→∞1VzddzlogΩN(z)=limV,N→∞1VN∑l=1zz−zl. (5)

For extended systems the limit also enforces and therefore there will be an infinite number of zeros: these may become densely distributed in the complex plane according to their positive density function , which can be different from zero either in a region of the complex plane (such a situation we will call later area law for the zeros) or along a curve, (to which we refer to as a perimeter law). Apart from these two generic cases, it can also be that the zeros may remain isolated points in the complex plane or, rather pathologically, accumulated instead around single points.

As shown in the original papers by Yang and Lee [YL1, , YL2, ], the entire thermodynamics can be recovered in terms of the density of the zeros of the grand-canonical partition function. Notice that, for the reality of the original , this function must satisfy the property

 η(z)=η(z∗), (6)

i.e. must be symmetric with respect to the real axis. The region or the curve depend on the temperature and they change their shape by changing . Let’s initially assume that the zeros are placed on an extended area : in this case, for all points outside this region one can analytically extend the definition of the pressure as

We can split this function into its real and imaginary part

 p(z)kT≡P(z)=φ(z)+iψ(z), (8)

where its real part

 φ(z)=∫dξη(ξ)log∣∣∣1−zξ∣∣∣ (9)

involves , i.e. the Green function of the two-dimensional Laplacian operator . Therefore satisfies the Poisson equation

 Δφ(z)=2πη(z). (10)

Drawing an electromagnetic analogy, this equation implies that can be thought as the electrostatic potential generated by the the (positive) distribution of charges with density . Posing , the corresponding components of the electric field are given by

 E1=−∂φ∂x,E2=−∂φ∂y. (11)

Since in any region not occupied by the charges both and its companion are analytic functions related by the Cauchy-Riemann equations, we have

 dPdz=dφdx+idψdx=dφdx−idψdy=−¯¯¯¯E, (12)

where is the complex electric field ().

This electrostatic analogy is pretty appealing but it is important to realise that not all charge densities are appropriate for the statistical mechanics problem. For this purpose, there are indeed certain requirements to fulfil, such as: its pressure , computed according to eq.  (4), must be necessarily a continuous and positive function of , monotonically increasing, along the real axis of the variable ; at the same time, its density , computed according to eq. (5), must also be a positive but not necessarily a continuous function, increasing too along the real axis of . It may be stressed that these conditions alone may be not sufficient to define a physical systems but if they are violated the system at hand is surely unphysical. For instance, for the random distributions of zeros shown in the top of Figure 2, the corresponding pressure and density, shown on the bottom of the same figure, do not fulfil the physical conditions of positivity and monotonicity: therefore, this set of zeros shown does not correspond to any physical statistical system. We will come back again to this issue later at the end of this Section.

Ising models. The formulas (4) and (5) for the pressure and the density of a gas are also useful to express the parametric equations of state for the Ising model whose spins can assume values . This can be easily done in terms of the mapping which exists between the quantities of the Ising model with spin and the lattice gas in a volume [YL2, ]. In the following denotes the number of positive/negative spins, the magnetization of the system , the free energy of the Ising model and an external magnetic field. So, making the substitutions in eqs. (4) and (5) according to the following dictionary

 IsingmodelLatticegasN=V1−M=2ρF+H=−$p$ (13)

one can easily write down the free energy and the magnetization of the Ising spin in terms of the zeros in the variable .

Singular behavior. It may happen that in the thermodynamic limit the zeros of the grand-canonical partition function do not spread on an area but lie instead along a particular curve thus satisfying a perimeter law. Such a curve has not to be necessarily a circle but in any case the previous formulas for the pressure and the density become in this case

 p(z)kT≡^f(z)=∫Cη(s)log(1−zz(s))ds, (14) ρ(z)=z^f′(z)=z∫Cη(s)z−z(s)ds. (15)

These equations are particularly useful to characterise the nature of the phase transition which occurs when the zeros pinch the real axis at same point . Let’s indeed suppose, as it is often the case, that nearby this point the zeros lie on a smooth curve : using the electrostatic analogy, in this case we are in presence of a line distribution of the charges and therefore there will be a discontinuity in the electric field, given by the gradient of , across the line charge distribution. Denoting by and the pressure (up to factor) across the line of the zeros and using the parameter to move along this line, applying the Gauss law to the rectangle shown in Figure 3, one sees that [Bena, , Blythe, ]

 (∇φ2−∇φ1)⋅n|C=2πη(s) (16)

Using the Cauchy-Riemann equation, this relation can be also expressed as

 dds(ψ2−ψ1)|C=2πη(s). (17)

The nature of the phase transition at is then determined by the behavior of the density of the zeros nearby this point. Particularly important are two cases:

1. When we have a jump in the derivative of the pressure at , therefore we are in presence of a first order phase transition.

2. When , the density of zeros vanishes at and therefore the first derivative of the pressure is continuous at while there is a discontinuity in its second derivative. In this case we are in presence of a second order phase transition.

Zeros on a circle. Notice that the equations (14) and (15) further simplify when the curve is a circle, say of radius equal to 1: in this case the density becomes an even function of the angle , normalised as

 ∫π−πη(θ)=1, (18)

and the pressure and density are expressed as

 p(z)kT=∫π0η(θ)log(z2−2zcosθ+1)dθ, (19) ρ(z)=2z∫π0η(θ)z−cosθz2−2zcosθ+1dθ. (20)

For a circle distribution of the zeros, it is easy to find a condition on the density which ensures that both the pressure and the density are positive monotonic functions of : as shown in [Ikeda1, ], it is in fact sufficient that the density is bounded and continuous, while its derivative is a bounded, continuous and positive function. Indeed, taking the derivative of with respect to we have

 dρdz=2∫π0η(θ)2z−(1+z2)cosθ(z2−2zcosθ+1)2dθ, (21)

which, with the change of variable and an integration by part, can be expressed as

 dρdz=−4[ξη(2arctanξ)(z+1)2ξ2+(z−1)2]∞0+8∫π0ξη′(2arctanξ)(1+ξ)2((z+1)2ξ2+(z−1)2dξ. (22)

With the hypothesis that is a bounded function, the first term of this expression vanishes while the second term, as far as , is positive. Once established that the function is then an increasing monotonic function, to show that it is always positive is sufficient to calculate its value at the origin and, if not negative, the function will be indeed always positive. Since at we have , this is sufficient to show the positivity of .

Under the same hypothesis for , we can also conclude that the pressure is a positive and increasing function of : since

 ρ(z)=zdpdz=dpdlogz, (23)

the positivity of implies that is an increasing function of , i.e. of itself since the logarithmic function is a monotonic function. So, also for to prove that this is always a positive function it is sufficient to compute its value at the origin and check that it is not negative. Since , this concludes the argument.

The importance of these considerations becomes more clear once one realises that even assuming the most favourable distribution of the zeros, i.e. along a circle, the analytic expression of the densities is largely unknown. The very few cases where we have such an information include the one-dimensional Ising model with nearest-neighbor interaction [YL2, ] and various versions of the mean field solutions of the same model [Katsura, ]. As a further intriguing remark, it seems that if one is able to point out what are the conditions in order to have a proper physical density , there is a plenty of room to define new statistical model by inverting somehow the theory of Yang and Lee. We will further comment on this point in the conclusions of the paper.

Yang-Lee edge singularities. We can take advantage of the known expression of the density of zeros of the one-dimensional ferromagnetic Ising model for presenting the simplest example of Yang-Lee edge singularities. The absence of a phase transition in this one-dimensional model of course implies that must vanish in an interval around the origin. This is indeed the case and the exact expression of the distribution is given by [YL2, ]

 η(θ)=12πsinθ2√sin2θ2−sin2θ02,if|θ|>θ0 (24)

otherwise , where and is the coupling constant between two next-neighbor spins. The zeros take then the -shape of the plot on the left in Figure 1. In this example plays the role of an edge singularity and this value pinches the origin only when , namely at . Nearby the density of zeros behaves anomalously as

 g(θ)∼C√θ−θ0, (25)

(), therefore for the one-dimensional Ising model the Yang-Lee edge singularity exponent defined in eq. (1) is equal to . Consider now the expression of the magnetization in terms of the density of the zeros using the dictionary previously established

 M(z)=1−4z∫π0η(θ)z−cosθz2−2zcosθ+1dθ. (26)

Performing the integral we have

 M(z)=z−1√(z−eiθ0)(z−e−iθ0). (27)

Posing and , and expanding this formula around , we have

 M(h)∼M0(h−ihc)1/2, (28)

i.e. the value can be considered as a singular point.

Polynomials vs Series. The discussion made so far concerned with the singular behavior which emerges by increasing the order of a sequence of genuine polynomials, in particular when . But what about is the partition function is ab-initio given in terms not of a polynomial but an infinite series? Let’s say in a neighborough of is given by the infinite series

 S(z)=∞∑k=0αnzn. (29)

In this case one must be aware that analysing the Yang-Lee zeros of an expression such as in eq. (29) there may be a condensation of zeros along some positive value of the fugacity which however does not necessarily signal a phase transition of the physical system but rather the finite radius of convergence of the series itself! Imagine in fact that the series (29) could be analytically continued and that the corresponding function has the closest singularity nearby the origin at a negative value . This automatically fixes the radius of convergence of the series (29) to be and therefore if we would use such a series to define the partition function, the corresponding Yang-Lee zeros may also condensate at the positive value , even though this point is not associated to a phase transition of the actual function . A simple example of this phenomena is worked out in detail in Appendix A. We will see that similar cases also emerge in discussing the Yang-Lee zero distributions of fermionic theories whose corresponding series has alternating sign and therefore a singularity at a negative value of : their zeros however also condensate at a positive real value of .

Iii Playing with polynomials

In this Section we are going to deal extensively with the properties of the main mathematical object of this paper, namely the class of real polynomials of order in the variable

 ΩN(z)=γ0+γ1z+γ2z2+…γNzN. (30)

The coefficients of these polynomials are real and we choose hereafter . Since we are going to interpret as a generalised grand-canonical partition function of a statistical model, either classical or quantum, we pose

 ΩN(z)≡eFN(z), (31)

where we have define the so-called free-energy of the system, directly related to the pressure of the system, see eq. (4). In light of this statistical interpretation of in the following we will also express it with a different normalization of the coefficients

 ΩN(z)=N∑k=0akk!zk, (32)

with and while the higher coefficients ’s assume the familiar meaning of canonical partition functions of particles. Formulas which we derive below are sometimes more elegantly expressed in terms of the ’s although we will switch often between the two equivalent expressions (30) and (32), hoping that this will not confuse the reader.

iii.1 Sign of the coefficients

For models coming from classical statistical physics, it is easy to argue that the coefficients of the relative polynomial are generically all positive, . In this respect, consider for instance two significant examples:

• Classical Gas. A classical model of a gas in -dimension, made of particles of mass and Hamiltonian of the form

 H=N∑i=1p2i2m+∑i,ju(rij), (33)

where is the distance between the -th and -th particle. In this case the grand-canonical partition function of the system assumes the form (32), where the variable expressed by

 z=eβμ(2πmβ/h2)d2, (34)

where is the fugacity, where is the temperature and is the Planck constant here introduced to normalise the phase-space integral. Therefore in this example the coefficients are given the positive integrals

 ak=∫⋯∫dr1⋯drkexp[−β∑i,ju(rij)]>0. (35)
• Ising in a magnetic field. As a second example of classical statistical mechanics, consider the ferromagnetic Ising model in a magnetic field studied originally by Lee and Yang in [YL2, ]: in this case, with and a general two-body ferromagnetic Hamiltonian of spins in a regular lattice in arbitrary -dimensional lattice of the form

 H=−∑i,jJijσiσj−H∑iσi,Jij>0 (36)

posing

 z=e−2βH, (37)

the partition function of the model is expressed by a palindrome polynomial in this variable, i.e. a polynomial of the form (30) where the coefficients satisfy the additional condition

 γk=γN−k>0. (38)

This because is the contribution to the partition function of the Ising model in zero magnetic field coming from configurations in which the number of spins is equal to ; this contribution is evidently the same also for .

It is however important to stress that for situations which come from quantum statistics and which involve, in particular, fermionic degrees of freedom, the sign of the coefficients ’s of the polynomial may be not necessarily all positive but rather alternating. However, despite the presence of negative coefficients, in the physical domain of the variable (alias the positive real axis, ), the partition function of these systems assumes only positive values and is a monotonic function of . The request of positivity of in its physical domain will play an important role in our future analysis when we are going to truncate our expressions of the partition function.

iii.2 Patterns of the zeros

Let’s now make a preliminar discussion about the nature of the zeros of a polynomial such as before focusing our attention on a series of examples. Being a real polynomial, its N zeros are either real or grouped in pairs of complex conjugated values in the complex plane. Hence, any distribution of the zeros has to be symmetric with respect the real axis. Concerning the real zeros, when all coefficients of are positive, they cannot be obviously positive. Hence, for any finite order of the polynomial, it should exist a region which contains the whole positive real axis which is free of zeros: a positive real zero can emerge only in the limit , i.e. as an accumulation point of complex zeros and, in this case, it corresponds to a point of singularity of the grand-canonical partition function [YL1, , YL2, ].

Clearly, the overall distribution of zeros and its details depend upon the choice of the coefficients and therefore, in general, there may be a wide range of situations. We find however important to try to identify at least two important patterns of zeros, hereafter called area and perimeter distributions respectively, for which in Section III.6 we will find some criteria for their realisation. At the level of terminology, we say

• we are in presence of an area distribution when the the N zeros are distributed in an extended area of the complex plane. This is the case, for instance, of the distribution shown in the first plot in Fig. (4), obtained by taking as coefficients of just the -th prime number. We will comment more on this example later.

• we are in presence of a perimeter distribution when the N zeros are distributed along particular lines. A particular interesting case of perimeter distributions is when all the zeros are along a circle. This is the case, for instance, of the distribution of the zeros shown in the second plot in Fig. (4), obtained by considering a grand-canonical partition function with coefficients given the sequence . This and other similar cases will be discussed in more details below.

iii.3 Bounds on the absolute module of the zeros

Given a polynomial of the form (30), there are several bounds on the magnitudes of its roots. Here we simply state some of these bounds without any proof for them (it may be useful to consult the book by Prasolov [prasolov, ] as a general reference on polynomials and bounds on their roots). Some of these bounds are more stringent than others (and we list them in the order of increasing their level of refinement) but altogether they may help in getting an idea about the distribution of the roots.

• Cauchy bound. The Cauchy bound states that the roots have an absolute value less than , , where

 RC=1+1|γN|max{|γ0|,|γ1|,…|γN−1|}. (39)

This is usually the less stringent bound on the module on the roots.

• Sun-Hsieh bound. The Cauchy bound can be refined in terms of the Sun-Hsieh bound, , where

 RSH=1+12((|γN−1/γN|−1)+√(|γN−1/γN|−1)2+4a),a=max{|γk/γN|}. (40)
• Fujiwara bound. A further refinement comes from the Fujiwara bound where all roots are within the disc of radius , , where

 RK=2max⎧⎨⎩∣∣∣γN−1γN∣∣∣,∣∣∣γN−2γN∣∣∣12,⋯,∣∣∣γ1γN∣∣∣1N−1,12∣∣∣γ0γN∣∣∣1N⎫⎬⎭. (41)
• Eneström bounds. When all , we can have a lower and an upper bound of the magnitude of the roots given by

 Rd≤|zi|≤Ru, (42)

where

 Rd=min{γkγk+1},Ru=max{γkγk+1},k=0,1,…N−1. (43)

The Eneström bounds are usually the best estimate of the annulus in the complex plane where the roots are located.

• Eneström-Kakeya bound. Let us also mention the Eneström-Kakeya bound which refers to polynomials where the coefficients satisfy the condition

 γ0≤γ1≤γ2…≤γN. (44)

In this case we have that . In particular, if all coefficients are positive, the roots are within the unit disc, . Of course, when the coefficients satisfy instead the condition

 γ0≥γ1≥γ2…≥γN. (45)

when .

Let’s make a final comment on the information provided by the bounds on the modules of the zeros: these bounds refer to all roots, so if knew for instance that , it may happen that just one or few zeros has module while all the rest may have a very small module. In other words, bounds usually refer to the largest of the zeros rather than their overall distribution.

iii.4 Motion of the zeros: perturbative analysis

We can study how the distribution of the zeros of a polynomial is modified by the addition of a new root as the order of the polynomial is increased by . To this aim, consider a polynomial of the form

 ΩN(x)=ΩN−1(x)+γNxN. (46)

The addition of the new term to has two effects:

1. it creates a new zero;

2. it moves the previous ones.

Let us address the first issue by considering the equation for the zeros of the new polynomial

 ΩN−1(x)+γNxN=0. (47)

Writing it as

 x=−ΩN−1(x)γNxN−1=−γN−1γN−γN−2γN−1x+…, (48)

we see that, perturbatively in , the new root is roughly placed at

 x∗≃−γN−1γN. (49)

If is infinitesimally small with respect to the other previous coefficients, the value determined from this equation is large and therefore, self-consistently, it is justified to neglect its corrections coming from the inverse powers present in the left hand side of (48).

Let us now address the second issue, i.e. the motion of the other zeros, once again perturbatively in the quantity . Let () be one of the zeros of the polynomial . Let’s now write

 xi+δxi (50)

as the new position of the root, once the new term has been added

 ΩN(xi+δxi) = ΩN−1(xi+δxi)+γN(xi+δxi)N=0 (51) = ΩN−1(xi)+δxidΩN−1dxi+γNxNi+NγNxN−1iδxi= = δxi[dΩN−1dxi+NγNxN−1i]+γNxNi=0,

hence the displacement of the root due to the addition on the term is given by

 δxi=−γNxNi(dZN−1dxi+γNNxN−1i). (52)

iii.5 Statistical approach

In order to understand better the pattern of the zeros and, in particular, to see whether we are able to qualitatively predict if an area or a perimeter law occurs, it is worth setting up a statistical analysis of the zeros. The obvious quantities to look at are the statistical moments of the zeros defined as

 sk≡N∑l=1zkl, (53)

where can be either a positive or a negative integer. Notice that, being the polynomial real, all are real quantities.

Negative moments. Let’s see how to relate the negative moments of the zeros

 s−m≡^sm=N∑l=1(1zl)m, (54)

to the coefficients of the polynomial . In order to do so, let’s factorise the partition functions in terms of its zeros as

 ΩN=N∏l=1(1−zzl). (55)

Taking the logarithm of both sides we arrive to the familiar cluster expansion of the free-energy

 FN(z)=logΩN(z) = N∑l=1log(1−zzl)=−N∑l=1∞∑m=11m(zzl)m= ≡ ∞∑m=1bmzm.

The negative moments are then related to the cluster coefficients as [YL1, ]

 bm=−1mN∑l=1(1zl)m=−1m^sm. (56)

It is also custom to expand the free-energy in terms of the cumulants defined by

 FN(z)=lnΩN≡∞∑k=1ckk!zk. (57)

If we use for the expression (32) and the coefficients , the cumulants are given by

 c1=a1c2=a2−a21c3=a3−3a2a1+2a31⋮ (58)

As a matter of fact there exists a closed formula which relates the two sets of coefficients and . To this aim let’s introduce the determinant of a () matrix whose entries involve the first coefficients ()

 M(k)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝a110⋯0a2a11⋮a3a2(21)a10⋮⋮⋮⋮akak−1(k1)ak−2⋯(k−1k−2)a1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (59)

The final formula is then

 ck=(−1)k−1detM(k). (60)

The relevant thing to note is that the -th cluster coefficient , which is of course related to the -th moment of the inverse roots as

 ck=−(k−1)!^sk, (61)

is entirely determined only by the first coefficients of the original polynomial . This implies that, if we increase the order of the polynomial by adding to the new coefficients but keeping fixed all the previous ones, there will be an increasing number of the zeros but their overall positions are constrained by the condition that the first negative moments

 s−k≡^sk=N∑l=1(1zl)k=~N∑l=1(1zl)k,k=1,…,N (62)

before and after adding the new coefficients, remain constant.

Positive moments. Let’s now discuss the positive moments of the zeros

 sk=N∑l=1(zl)k,k>0. (63)

We can relate them to the coefficients of the polynomial as follows. Let’s first introduce the elementary symmetric polynomials defined as

 σk(x1,…,xN)=∑1≤j1≤j2≤…≤jk≤Nxj1…xjk,k=0,…,N. (64)

Notice that and . Since

 G(z)≡N∏l=1(z−zl)=N∑k=0(−1)kσk(z1,…,zN)zN−k, (65)

we can express the partition function as

 ΩN(z)=N∑m=0γmzm≡γNG(z)==γNN∑k=0(−1)kσk(z1,…,zN)zN−k. (66)

Therefore

 γm=(−1)N−mγNσN−m, (67)

and, sending , we have the final relation between the symmetric polynomials and the coefficients (or ) of the partition function

 σm=(−1)mγN−mγN=(−1)mN!(N−m)!aN−maN. (68)

Moreover, we can relate the moments to the elementary symmetric polynomials thanks to the Newton-Girard formula

 (−1)mmσm(z1,…,zN)+m∑k=1(−1)k+msk(z1,…,zN)σm−k=0, (69)

so that

 s1−σ1=0, s2−s1σ1+2σ2=0, s3−s2σ1+s1σ2−3σ3=0, (70) ⋮ sN−sN−1σ1+sN−2σ2−…+(−1)NσN=0.

A closed solution of these relations can be given in terms of a formula which employs the following determinant

 sp=∣∣ ∣ ∣ ∣ ∣ ∣∣σ110⋯02σ2σ11⋯03σ3σ2σ1⋮⋮⋮pσpσp−1σp−2⋯σ1∣∣ ∣ ∣ ∣ ∣ ∣∣. (71)

The relevant thing to notice in this case is that the positive moment is determined by the first elementary symmetric polynomials which, on the other hand, are fully determined by the last coefficients of the polynomial (see eq. (68)).

Moments: summary. So far we have seen that the negative moments of the zeros are determined by the first coefficients of the polynomial while the positive moments are determined instead by the last coefficients of

 ΩN=1+γ1z+γ2z2+…+γkzk^sk=s−k+…+γN−lzN−l+γN−l+1zN−l+1+…+γNzNsl. (72)

Obviously all the negative and positive moments higher or equal than are linearly dependent from the previous ones. To show this, let’s interpret the polynomial as the characteristic polynomial of a matrix which satisfies the same equation satisfied by the zeros themselves of

 1+γ1M+γ2M2+⋯γNMN=0, (73)

where is the identity matrix. Since , taking now the trace of the equation above, it is easy to see that the -th positive moment is linearly dependent from the previous moments

 sN=−1γN(N+γ1s1+γ2s2+⋯γN−1sN−1). (74)

Moreover, to get the linear equation which links the higher positive moment to the previous ones, it is sufficient to multiply by eq. (73) and then to take the trace of the resulting expression.

Concerning instead the linear combinations which involve the negative moments higher or equal to , it is sufficient to multiply eq. (73) by and then repeating the steps described above. For instance, the negative moment depends linearly from the previous ones as

 ^sN=−γ1^sN−1−γ2^sN−2−⋯−γNN. (75)

iii.6 Area and perimeter laws

Hereafter we are going to set a certain number of ”rules of thumb” that allow us to have a reasonable guess whether the zeros satisfy the area or the perimeter laws. The criteria make use of the bounds on the modules of the zeros, the geometrical and the arithmetic means of the zeros and also of their variance. All these quantities are easy to compute in terms of the coefficients of the partition function and therefore they provide a very economical way for trying to anticipate their distribution. In other words, we must subscribe to a reasonable compromise between the reliability of the prediction and the effort to compute the indicators on the zeros: of course, would one increase the number of computed moments of the zeros, then he/she would narrow better and better the prediction but at the cost of course to engage into the full analysis of the problem! This is precisely the origin of the compromise. One must be aware, however, of the heuristic nature of the arguments we are going to discuss below, which have not at all the status of a theorem, and therefore they must be taken with a grain of salt. Moreover, one must also be aware that not all distributions are either area or perimeter laws, since there exist cases where the roots have simultaneously both distributions or they are made by isolate points.

Geometrical Mean of the zeros. From eq. (68), taking we have

 σN≡N∏k=1zk=(−1)NN!aN=(−1)N1γN. (76)

Let’s put and take the N-th root of both terms in eq. (76): for the left hand side, we have the geometrical mean of the roots

 ⟨z⟩geom=(N∏k=1zk)1/N, (77)

while for the right-hand side we have:

 (−1)(qN)1/N. (78)

For the nature of the zeros of a real polynomial – which either pair in complex conjugate values and or are real (where can be also negative) – the product of all zeros is a real quantity which depends upon only the product of the modules of all roots. When the module of the geometrical mean is finite, say , the zeros basically must be symmetric under the mapping . Notice that the easiest way to implement such a symmetry is that the zeros are placed along a circle of radius .

Arithmetic Mean of the zeros. Given the zeros of a polynomial we can define their arithmetic mean: this is simply the first positive moment of the zeros divided by

 ⟨z⟩arith=1NN∑k=1zk=1Ns1=1Nσ1, (79)

and therefore, using eqs. (68) and (70), it is easily related to the ratio of the last two coefficients

 ⟨z⟩arith=−aN−1aN=−1NγN−1γN. (80)

For the reality of the polynomials considered in this paper it is obvious that the arithmetic mean of the zeros just depends on their real part and therefore it is a real number.

Variance of the zeros. The variance222Notice that what we call here the variance is not the familiar variance of real random variables, since our definition employs the square of the differences of complex numbers and therefore is not necessarily positive. of the zeros is defined as

 μ2=1NN∑k=1(zk−¯zarith)2. (81)

Putting and expressing the zeros in terms of their real and imaginary parts, , we have

 μ2=1NN∑k=1[(xk−a)2−y2k], (82)

while the imaginary part of vanishes for the symmetry