ThirdOrder Phase Transition: Random Matrices and Screened Coulomb Gas with Hard Walls
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Abstract
Consider the free energy of a ddimensional gas in canonical equilibrium under pairwise repulsive interaction and global confinement, in presence of a volume constraint. When the volume of the gas is forced away from its typical value, the system undergoes a phase transition of the third order separating two phases (pulled and pushed). We prove this result (i) for the eigenvalues of onecut, offcritical random matrices (loggas in dimension \(d=1\)) with hard walls; (ii) in arbitrary dimension \(d\ge 1\) for a gas with Yukawa interaction (aka screened Coulomb gas) in a generic confining potential. The latter class includes systems with Coulomb (long range) and delta (zero range) repulsion as limiting cases. In both cases, we obtain an exact formula for the free energy of the constrained gas which explicitly exhibits a jump in the third derivative, and we identify the ‘electrostatic pressure’ as the order parameter of the transition. Part of these results were announced in Cunden et al. (J Phys A 51:35LT01, 2018).
Keywords
Random matrices Coulomb and Yukaw gases loggases Phase transitions Extreme value statistics Large deviations Potential theory1 Introduction and Statement of Results
Phase transitions—points in the parameter space which are singularities in the free energy—generically occur in the study of ensembles of random matrices, as the parameters in the joint probability distribution of the eigenvalues are varied [9]. The aim of this paper is to characterise the pulledtopushed phase transition (defined later) in random matrices with hard walls and, more generally, in systems with repulsive interaction in arbitrary dimensions.
Remark 1
Having clarified the physical meaning of the meanfield functional (1.4), we now turn to the problem addressed in this paper.
Notational remark
Throughout the paper \(\mathscr {P}(B)\) denotes the set of probability measures whose support lies in \(B\subset \mathbb {R}^d\). The Euclidean ball of radius R centred at 0 is denoted by \(B_R=\{x\in \mathbb {R}^d:x=(x_1^2+\cdots +x_d^2)^{1/2}\le R\}\). If F is a formula, then \(\mathbb {1}_{F}\) is the indicator of the set defined by the formula F. We also use the notation \(a\wedge b=\min \{a,b\}\).
1.1 Formulation of the Problem
 (i)
In the unconstrained problem (\(B_R=\mathbb {R}^d\)) the global minimiser \(\rho _{R_{\star }}\) is supported on the ball \(B_{R_{\star }}\);
 (ii)
If \(R>R_{\star }\), the constraint in (1.13) is immaterial (\(B_R\) contains \(B_{R_\star }\)), and hence the equilibrium measure is \(\rho _R=\rho _{R_{\star }}\) and \(F(R)=0\). This is the socalled pulled phase, borrowing a terminology suggested in [48];
 (iii)
If \(R<R_{\star }\) the system is in a pushed phase, the constraint is effective, and the equilibrium energy of the system increases \(\mathscr {E}[\rho _R]\ge \mathscr {E}[\rho _{R_\star }]\).
 (iv)
At \(R=R_{\star }\) the gas undergoes a phase transition and the free energy F(R) displays a nonanalytic behaviour. Typically at microscopic scales one expects a crossover function separating the pushed and pulled phases.
 (i)
For the loggas \(\varPhi (x)=\log x\) in dimension \(d=1\) (eigenvalues of onecut, offcritical random matrices with hard walls);
 (ii)For the Yukawa gas \(\varPhi (x)=\varPhi _d(x)\) within arbitrary dimension \(d\ge 1\), including its limiting cases \(m\rightarrow 0\) (Coulomb gas) and \(a\rightarrow 0\) (Thomas–Fermi gas). In (1.16), \(K_\nu \) denotes the modified Bessel function of the second kind.$$\begin{aligned} \varPhi _d(x)=\frac{1}{a^2 2^{\frac{d}{2}1}}\frac{1}{\varGamma \left( \frac{d}{2}\right) }\left( \frac{m}{ax}\right) ^{\frac{d}{2}1}K_{\frac{d}{2}1}\left( \frac{mx}{a}\right) \end{aligned}$$(1.16)
1.2 Extreme Eigenvalues of Random Matrices and LogGases with Hard Walls
1.2.1 Hermitian Random Matrices
Similar phase transitions of the pulledtopushed type have been observed in several physics models related to random matrices [13, 47], including largeN gauge theories [3, 36, 57, 64], longest increasing subsequences of random permutations [39], quantum transport fluctuations in mesoscopic conductors [12, 33, 34, 62, 63], nonintersecting Brownian motions [28, 58], entanglement measures in a bipartite system [17, 26, 27, 49], random tilings [10, 11], random landscapes [29], and the tail analysis in the KPZ problem [41]. (See also the recent popular science articles [5, 65]).
In this paper we derive a general explicit formula for the free energyF(R) of a loggas in dimension\(d=1\)in presence of hard walls, and we prove the universality of the thirdorder phase transition for onecut, offcritical matrix models. This proves the prediction arising from the extreme value statistics criterion formulated in [47].
While here we examine problems with radial symmetry in both the potential and the hard walls as they constitute a paradigmatic framework and allow for a systematic treatment, the electrostatic interpretation we present in Sect. 1.4.2 below enjoys a wider range of application. In Remark 5, indeed, we will show how the formulae and conclusions concerning the symmetric case carry over without extra efforts to nonsymmetric potentials or random matrices with a single hardwall as well.
We consider potentials V(x) satisfying the following assumptions.
Assumption 1
V(x) is \(C^{3}(\mathbb {R})\), symmetric \(V(x)=V(x)\), strictly convex and satisfies \(\liminf _{x\rightarrow \infty }\frac{V(x)}{\log x}>1\).
We remark that strictly convex and superlogarithmic V(x)’s are in the class of onecut, offcritical potentials.
We are going to present and discuss two theorems for the loggas that will be proven in Sect. 3.
Theorem 1
 (i)there exists a unique probability measure that is solution of the constrained minimisation problem (1.11) for the energy functional (1.4), and it takes the formwhere \(P_R(x)\) and \(Q(x)= \lim _{R\uparrow R_{\star }}P_{R}(x)/(R^2x^2)\) are nonnegative on the support \([R,R]\) and \([R_{\star },R_{\star }]\), respectively.$$\begin{aligned} \mathrm {d}\rho _R(x)= {\left\{ \begin{array}{ll} \displaystyle \frac{1}{\pi }\frac{P_R(x)}{\sqrt{R^2x^2}}\mathbb {1}_{x< R}\,\mathrm {d}x &{}\quad \text {if }\, R< R_{\star } \mathrm{(}pushed\,phase\mathrm{)}\\ \displaystyle \frac{1}{\pi }Q(x)\sqrt{R_{\star }^2x^2}\mathbb {1}_{x\le R_{\star }}\,\mathrm {d}x &{}\quad \text {if }\, R\ge R_{\star } \mathrm{(}pulled\,phase\mathrm{)}, \end{array}\right. } \end{aligned}$$(1.29)
 (ii)An explicit expression of \(P_R(x)\) (and Q(x)) is as follows. Denote by \(c_n(R)\) the Chebyshev coefficients of V(Rx). i.e.,Then, the equilibrium measure (1.29) is uniquely determined as$$\begin{aligned} c_n(R)=\frac{1}{h_n}\int _{1}^1 \frac{V(R x)T_n(x)}{\sqrt{1x^2}}\mathrm {d}x. \end{aligned}$$(1.30)The critical radius \(R_{\star }\) is the smallest positive solution of the equation$$\begin{aligned} P_R(x)=1\sum _{n\ge 1}nc_n(R)T_{n}(x/R). \end{aligned}$$(1.31)$$\begin{aligned} \sum _{n\ge 1}nc_{n}(R_{\star })=1. \end{aligned}$$(1.32)
On the other hand in the pushed phase, the density is strictly positive in its support and has an integrable singularity at the hard walls \(\pm R\) (cf. with the GUE case (1.23)). See Fig. 1.
A corollary of formulae (1.29)–(1.31) is the first main result of this paper.
Theorem 2
Remark 2
In the statement of the results, the family of strictly convex potentials is not the largest class of potentials where the above result holds true. What is really required is that in the pulled phase the associated equilibrium measure is supported on a single interval, and vanishes as a square root at the endpoints. (An explicit characterisation of these conditions is quite complicated.) More precisely, Theorem 1 is true when V(x) is a onecut potential. ‘Onecut’ means that the equilibrium measure is supported on a single bounded interval. Theorem 2 is valid under the hypotheses that V(x) is onecut and offcritical potential. ‘Offcritical’ means that \(P_{R_{\star }}(R_{\star })=0\) but \(P_{R_{\star }}'(R_{\star })\ne 0\) so that \(\rho _{R_{\star }}(x)\sim \sqrt{R_{\star }^2x^2}\) at the edges. This is certainly true for strictly convex potentials. For ‘critical’ potentials the pulledtopushed transition is weaker than thirdorder (see Eq. (3.18) in the proof). These potentials are however ‘exceptional’ in the ‘onecut’ class. The problem for ‘multicut’ matrix models remains open.
Example 1
Theorem 2 and the previous discussion might lead to conclude that the universality of the thirdorder phase transition is inextricably related to the presence of a Tracy–Widom distribution separating the pushed and pulled phases. A hint that this is not the case comes from the study of extreme statistics of nonHermitian matrices whose eigenvalues have density in the complex plane.
1.2.2 NonHermitian Random Matrices
1.3 Beyond Random Matrices: Yukawa Interaction in Arbitrary Dimension
1.3.1 Coulomb and Thomas–Fermi Gas
Theorem 3
In the pulled phase, the equilibrium density of the Coulomb gas is supported on the ball of radius \(R_{\star }\) and there is no accumulation of charge on the surface (\(c(R)=0\) for \(R\ge R_{\star }\)). In the pushed phase, the equilibrium density in the bulk does not change, while an excess charge (\(c(R)>0\) for \(R<R_{\star }\)) accumulates on the surface.
For the Thomas–Fermi gas, \(R_\star \)—the edge in the pulled phase—is determined by the condition that the gas density vanishes on the surface, i.e. \(\mu (R_{\star })=v(R_{\star })\) for \(R\ge R_{\star }\). In the pushed phase \(R<R_{\star }\), the chemical potential increases to keep the normalisation of \(\rho _R\), but there is no accumulation of charge on the surface, i.e. singular components in the equilibrium measure (otherwise the energy would diverge).
1.3.2 Yukawa Gas in Generic Dimension
The ubiquity of this transition calls for a comprehensive theoretical framework, which should be valid irrespective of spatial dimension d and the details of the confining potential V, and for the widest class of repulsive interactions \(\varPhi \).
Assumption 2
V(x) is \(C^{3}(\mathbb {R}^d)\), radially symmetric \(V(x)=v(x)\), with v increasing and strictly convex.
The solution of the constrained equilibrium problem for the Yukawa interaction (announced in [16]) is stated below.^{3}
Theorem 4
In particular: (i) in the pulled phase the equilibrium measure is absolutely continuous with respect to the Lebesgue measure on \(\mathbb {R}^d\); (ii) when the gas is ‘pushed’, the density in the bulk increases by a constant and a singular component builds up on the surface of the ball \(B_R\).
Theorem 4 implies the second main result in this paper : the universality of the jump in the third derivative of excess free energy of a Yukawa gas with constrained volume. This universality extends to the limit cases \(m\rightarrow 0\) (Coulomb gas) and \(a\rightarrow 0\) (Thomas–Fermi gas) solved in [15, 16], respectively. See Remark 3 below.
Remark 3
Theorem 5
Remark 4
Example 2
1.4 Order Parameter of the Transition
The universal formulae (1.34) and (1.65) for the free energy F(R) are remarkably simple. It feels natural to ask whether it is possible to derive them from a simpler physical argument. Moreover, it would be desirable to express the free energy in terms of a quantity that captures the nonanalytic behaviour in the vicinity of the transition. This quantity, traditionally called order parameter of the transition, must be zero in one phase and nonzero in the other phase.
In the following, we outline a heuristic argument that reproduces formulae (1.34) and (1.65), and identifies the ‘electrostatic pressure’ as the order parameter of the transition.
1.4.1 Electrostatic Pressure: Screened Coulomb Interaction
How to compute the pressure exerted by the surface’s field on itself? The argument that follows is similar to the one used to evaluate the ‘electrostatic pressure’ on a layer of charges, e.g. the surface of a charged conductor. This is a problem that some textbooks in electrostatics occasionally mention (see the classical books of Jackson [38, Section 3.13] and Purcell [51, Section 1.14]), but which is rarely discussed due to the difficulty of making the argument rigorous for conductors of generic shapes. To stress the analogy and for the lack of a better terminology, in the following we will keep using the expressions ‘electrostatic’ pressure, force, field, Gauss’s law, etc. even though the interaction we are considering is Yukawa (screened Coulomb).
1.4.2 Electrostatic Pressure: Random Matrices
When the system is confined in a ball \(B_r\) at density \(\rho _r\), the pressure is given by the normal force per unit length. The force \(F_n(x)\) at point x is equal to the charge \(\rho _r(x)\mathrm {d}x\) in the infinitesimal segment \(\mathrm {d}x\) around x times the electric field. To proceed in the computation it is convenient to use complex coordinates (recall that \(\log x\) is the Coulomb interaction in dimension \(d=2\)).
At the edge \(x=r\) (similar considerations for \(x=r\)) the situation is more delicate. By symmetry, the total field \(E_{\text {edge}}\) generated by \(\rho _r(x)\) at \(x=r\) must be directed along the xaxis, but must be zero for \(x<r\epsilon \). Therefore, repeating the argument of the previous section, the field experienced by the ‘hole’ at the edge \(x=r\) is half the field generated by \(\rho _r(x)\) at \(r+\epsilon \), i.e. \(E_{\text {hole}}=(1/2)E_{\text {edge}}\).
Remark 5
Remark 6
The calculation of the work done in a compression of a loggas in dimension \(d=1\) bears a strong resemblance to a way to calculate the work (energy) per unit fracture length for a crack propagating in a continuous medium. In linear elastic fracture mechanics, Cherepanov [8] and Rice [54] have independently developed a line integral called the Jintegral that is contour independent. The usefulness of this integral comes about when the contour encloses the cracktip region, as this is where the most intense (actually divergent) fields are found (c.f. the edge of the cut in the loggas). Evaluating the Jintegral then gives the variation of elastic energy. An analogous integral in electrostatics has been discovered later [31].
It is likely that the computations outlined above can be recast in the language of linear elastic mechanics/electrostatics. The link between ‘eigenvalues of random matrices’ and the ‘theory of fractures’ suggested here will be explored in future works.
1.5 Outline of the Paper
The rest of the article is organised as follows. In Sect. 2 we recall some general variational arguments for the solution of the constrained equilibrium problem. Section 3 contains the proof of Theorems 1 and 2 for the loggas. In Sect. 4 we present the proof of Theorem 4 (equilibrium problem for Yukawa interaction) and Theorem 5 (universality of the jump in the third derivative of the free energy). Finally, the Appendices A and B contain the proof of some technical lemmas.
2 Variational Approach to the Constrained Equilibrium Problem
We resort to a variational argument to derive necessary and sufficient conditions for \(\rho _R\) to be the minimiser of the energy functional \(\mathscr {E}\) over \(\mathscr {P}(B_R)\). (These arguments are not new at all. They appear in many different forms and specialisations in the literature, see, e.g., [4, 21, 46].)
3 Proof of Theorems 1 and 2
 if the walls are not active (pulled phase) the density is supported on \({\text {supp}}\rho _{R_{\star }}=[R_{\star },R_{\star }]\) with \(R_{\star }\) solution ofand the density is given by Tricomi’s formula [60]$$\begin{aligned} \frac{1}{\pi }\int _{R_{\star }}^{+R_{\star }}\frac{V'(x)}{\sqrt{R_{\star }^2x^2}}\mathrm {d}x=1, \end{aligned}$$(3.1)(3.2)

if the walls are active (pushed phase) the density is supported on \({\text {supp}}\rho _{R}=[R,R]\) and is given by (3.2) with the replacement \(R_{\star }\mapsto R\).
Lemma 1
The Chebyshev polynomials satisfy the following electrostatic formula (for a proof see Appendix B).
Lemma 2
4 Proof of Theorems 4 and 5
A systematic analysis of the equilibrium problem for the screened Coulomb interaction does not seem to have appeared in the existing literature. Here we solve the problem (Theorem 4). In this Section we put forward a sensible ansatz for \(\rho _R\) depending on two parameters (chemical potential \(\mu \) and surface charge c), that we then prove to be the minimiser by imposing the EL conditions that fix \(\mu =\mu (R)\) and \(c=c(R)\). The only thing that needs to be checked is the positivity of the candidate solution (Remark 7). In solving the problem, we will make the most of its spherical symmetry. A key technical ingredient in the derivation will be a shell integration lemma (Lemma 3), the analogue of Newton’s formula (4.20) for a Yukawa interaction in generic dimension \(d\ge 1\).
4.1 General Form of the Constrained Minimisers
The usual strategy in these minimisation problems is to look for a candidate solution of the EL conditions (2.7). The condition \(\mathscr {E}_2>0\) guarantees that the saddlepoint is the minimiser in \(\mathscr {P}(B_R)\).
One can prove the absence, at equilibrium, of condensation of particles in the bulk (absence of \(\delta \)components). To see this, one compares the energy of a density containing a \(\delta \)function in the bulk to one where the \(\delta \)function has been replaced by a narrow, symmetric mollification \(\delta _{\epsilon }\) (see [4, Section 3.2.1] for details). This argument fails on the boundary where it is not possible to consider symmetric mollifications \(\delta _{\epsilon }\) contained in the support.
Remark 7
Before embarking in the calculations leading to the explicit formulae for \(\mu (R)\) and c(R), we derive the universal formula of the free energy (Theorem 5) assuming Theorem 4.
Proof of Theorem 5
4.2 Chemical Potential and Excess Charge
The chemical potential \(\mu (R)\) and the excess charge c(R) are fixed by the normalisation of \(\rho _R\) and the EL conditions.
Lemma 3
Proof
See Appendix B. \(\square \)
Remark 8
Lemma 4
Proof
Footnotes
Notes
Acknowledgements
The research of FDC is supported by ERC Advanced Grant 669306. PV acknowledges the stimulating research environment provided by the EPSRC Centre for Doctoral Training in CrossDisciplinary Approaches to NonEquilibrium Systems (CANES, EP/L015854/1). ML was supported by Cohesion and Development Fund 2007–2013  APQ Research Puglia Region “Regional program supporting smart specialization and social and environmental sustainability  FutureInResearch.” PF was partially supported by by Istituto Nazionale di Fisica Nucleare (INFN) through the project “QUANTUM.” FDC, PF and ML were partially supported by the Italian National Group of Mathematical Physics (GNFMINdAM). The authors would like to thank S. N. Majumdar and Y. V. Fyodorov for useful discussions, and G. Schehr and Y. Levin for remarks on the first version of the paper. We also thank E. Katzav for drawing our attention to the similarity between some of our calculations and the Jintegral of linear elastic fracture mechanics theory.
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