Certain geometric properties of submanifolds of configuration space are
numerically investigated for classical lattice phi^4 models in one and two
dimensions. Peculiar behaviors of the computed geometric quantities are found
only in the two-dimensional case, when a phase transition is present. The
observed phenomenology strongly supports, though in an indirect way, a recently
proposed topological conjecture about a topology change of the configuration
space submanifolds as counterpart of a phase transition.Comment: REVTEX file, 4 pages, 5 figure

C.N. Yang

D.H.E. Gross

J. Milnor

J.A. Thorpe

K. Binder

L. Caiani

L. Casetti

Lapo Casetti

Lionel Spinelli

M. Cerruti-Sola

M. P. do Carmo

M. W. Hirsch

M.C. Firpo

Marco Pettini

Roberto Franzosi

English

arXiv

Franzosi, Roberto

Casetti, Lapo

Spinelli, Lionel

Pettini, Marco

HAL AMU

Topological aspects of geometrical signatures of phase transitions

Crossref

Hal-Diderot

Certain geometric properties of submanifolds of configuration space are numerically investigated for classical lattice [Formula Presented] models in one and two dimensions. Peculiar behaviors of the computed geometric quantities are found only in the two-dimensional case, when a phase transition is present. The observed phenomenology strongly supports, though in an indirect way, a recently proposed topological conjecture about a topology change of the configuration space submanifolds as counterpart to a phase transition. © 1999 The American Physical Society

Franzosi, R.

Casetti, L.

Spinelli, L.

Pettini, M.

Archivio della Ricerca - Università degli Studi di Siena

21 May 2023Franzosi, R., Casetti, L., Spinelli, L., Pettini, M. (1999). Topological aspects of geometrical signatures ofphase transitions. PHYSICAL REVIEW E, 60(5), R5009-R5012 [10.1103/PhysRevE.60.R5009].Topological aspects of geometrical signatures of phase transitionsPublished:DOI:10.1103/PhysRevE.60.R5009Terms of use:Open Access(Article begins on next page)The terms and conditions for the reuse of this version of the manuscript are specified in the publishingpolicy. Works made available under a Creative Commons license can be used according to the terms andconditions of said license.For all terms of use and more information see the publisher's website.Availability:This version is availablehttp://hdl.handle.net/11365/1231278 since 2023-04-28T09:45:37ZOriginal:This is a pre print version of the following article:arXiv:cond-mat/9810180v1  [cond-mat.stat-mech]  15 Oct 1998Topological aspects of geometrical signatures of phase transitionsRoberto Franzosi1,5,a, Lapo Casetti2,b , Lionel Spinelli3,5,c, and Marco Pettini4,5,d1Dipartimento di Fisica, Università di Firenze, Largo Enrico Fermi 2, 50125 Firenze, Italy2 Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino, Italy3Centre de Physique Théorique du C.N.R.S., Luminy Case 907, 13288 Marseille Cedex 9, France4Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy5 Istituto Nazionale per la Fisica della Materia, Unità di Ricerca di Firenze, Firenze, Italy(July 24, 2021)Certain geometric properties of submanifolds of configu-ration space are numerically investigated for classical latticeϕ4 models in one and two dimensions. Peculiar behaviors ofthe computed geometric quantities are found only in the two-dimensional case, when a phase transition is present. Theobserved phenomenology strongly supports, though in an in-direct way, a recently proposed topological conjecture abouta topology change of the configuration space submanifolds ascounterpart of a phase transition.PACS numbers: 05.70.Fh; 64.60.-i; 02.40.-k ; 05.20.-yIn Statistical Mechanics phase transitions are asso-ciated with the appearence of the so-called Yang-Lee(real)zeros [1] of the grand-partition function entailingsingular temperature-dependence of thermodynamic ob-servables. However in the Yang-Lee theory the necessaryconditions for the existence of real zeros remain unspec-ified. It has been recently suggested [2,3] that the ther-modynamic singularities might have their deep origin insome major topological change in configuration space, i.e.in a non-trivial structure of the support of the equilibriumstatistical measure.This topological conjecture [4] has been put forwardheuristically within the framework of a numerical in-vestigation of the Hamiltonian dynamical counterpart ofphase transitions. An interesting outcome of such investi-gations was the clear evidence of a peculiar temperature-behavior of the largest Lyapunov exponent at the phasetransition point. This was observed in lattice scalar andvector ϕ4 models [3,5], in two and three dimensional XYmodels [2], in the Θ-transition of homopolymeric chains[6], and - analytically - in the mean-field XY model [7].Moreover, in the light of a Riemannian geometrizationof Hamiltonian dynamics – where Lyapunov exponentsare related with average curvature properties of subman-ifolds of configuration space [8–10] – also the tempera-ture dependence of abstract geometric observables hasbeen investigated. Among them the averages of curva-ture fluctuations (that enter the analytic formula for thelargest Lyapunov exponent [10]) exhibit a cusp-like pat-tern. The peak coincides with the phase transition point.Qualitatively similar peaks of curvature fluctuations havebeen reproduced in abstract geometric models (familiesof surfaces of R3 where the variation of a parameter leadsto a change of the topological genus), whence the heuris-tic argument about the topological meaning of the peaksof configuration-space-curvature fluctuations at a phasetransition. These fluctuations have been obtained as timeaverages, computed along the dynamical trajectories ofthe Hamiltonian systems under investigation. Now, timeaverages of geometric observables are usually found inexcellent agreement with ensemble averages [2,3,10,11]therefore one could argue that the mentioned singular-like patterns of the averages of geometric observables aresimply the precursors of truly singular patterns due tothe tendency of the measures of all the statistical ensem-bles to become singular in the limitN → ∞ when a phasetransition is present. In other words, geometric observ-ables, like any other “honest” observable, already at fi-nite N would feel the eventually singular character of thestatistical measures, and if this was the true explanationwe could not attribute the cusp-like patterns of curvaturefluctuations to special geometric features of configurationspace. Hence the motivation of the present paper. Weaim at elucidating this important point by working outpurely geometric informations about the submanifolds (tobe specified below) of configuration space, independentlyof the statistical measures. In the present paper a newstep forward is done in the direction of supporting thetopological conjecture mentioned above.Our geometrical framework is the configuration spaceM of systems whose degrees of freedom are uncostrainedand are real numbers, thus M = RN .Our statistical framework is the canonical ensemblewhose volume in M is given by the configurational par-tition function ZC =∫∏Ni=1 dqi exp[−V (q)], where q =(q1, . . . , qN ) ∈ RN . Finally, our topological framework iselementary Morse theory [12,13]. Let us here recall itsbasic idea with the help of Fig.1. The “U–shaped” cylin-der of Fig.1 is the ambient manifold M where a functionV : M 7→ R, smooth and bounded below, is defined to bethe height of any point of M with respect to the floor-plane. For any given value u of the function V two kindsof submanifolds are determined: the level sets Σu of all1the points x ∈ M for which V (x) = u, and Mu, the partof M below the level u, i.e. the set of all points x ∈ Msuch that V (x) ≤ u. The remarkable fact of Morse the-ory is that from the knowledge of all the critical pointsof V , i.e. those points where ∇V = 0, and of their Morseindexes, i.e. the number of negative eigenvalues of theHessian of V , one can infer the topological structure ofthe manifolds Mu, provided that the critical points arenondegenerate, i.e. with nonvanishing eigenvalues of theHessian of V . Two such points are marked in Fig.1, atthe bottom of M and at some intermediate level uc forwhich Σuc is an eight-shaped curve. It is evident fromFig.1 that the manifolds Mu<uc are not diffeomorphicto the manifolds Mu>uc : the formers are homeomorphicto a disk and the latters are homeomorphic to a cylin-der. The same happens (in general) to the boundaries Σuthat here are circles for u < uc and become the topolog-ical union of two circles for u > uc. This simple exampledisplays the general fact that passing a critical level of aMorse function is in one-to-one relation with a topologychange. A critical level is a surface Σu that contains oneor more critical points.Let us now consider the configuration space M = RNof a physical system and its potential V as the Morsefunction. The interesting things are supposed to oc-cur below some large value u so that the corresponding(large) subset M ⊂ M is compact. Then M and all itssubmanifolds Mu are given a Riemannian metric g. Onall these manifolds (Mu, g) there is a standard invariantvolume element: dη =√det(g)dq1 . . . dqN .In order to study the topology of the family {(Mu, g)}we should find, analytically or numerically, all the crit-ical points of V , but at large N this is a formidabletask, therefore we have approached this problem as fol-lows. Generalizing a simple geometric example reportedin [2,3], we have computed the total degree of secondvariation σ2K of the Gaussian curvature, i.e. σ2K =〈K2G〉Σu − 〈KG〉2Σuwhere 〈·〉 stands for integration overthe surface Σu, as a function of u in the neighborhoodof a critical point. This is possible in general, indepen-dently of the potential V , because any Morse functioncan be parametrized in the neighborhood of a criticalpoint, located at x0 ∈ M , by means of the so-calledMorse chart, i.e. a system of local coordinates {xi} suchthat V (x) = V (x0) −∑ki=1 x2i +∑Ni=k+1 x2i (k is theMorse index). Then standard formulae for Gauss cur-vature of hypersurfaces of RN [14] can be used to ex-plicitly work out KG and σ2K . The intersection of anhypersphere of unit radius - centered around u = 0(the critical point) - with each Σu is used to boundthe domain of integration. The numerically tabulatedresults are reported in Fig.2 and show that σ2K devel-ops a sharp, singular peak as the critical surface is ap-proached. It seems therefore reasonable to apply thisgeometric probing of the presence of critical points, andhence of topology changes, to the study of possible topol-ogy changes of the manifolds (Mu, g). In fact the mani-folds Mu inheritate - by somewhat smoothing them - thepeculiar geometric properties of the {Σu}u≤uc – due tothe presence of critical points – because of the relation∫Mufdη =∫ u0dv∫Σvf |Σvdω/‖∇V ‖, where dω is the in-duced measure on Σv and f a generic function [15]. Thesurface Σuc defined by V (x) = uc is a degenerate quadric,therefore close to it some of the principal curvatures [14]of the surfaces Σu≃uc tend to diverge. Such a divergenceis generically detected by any function of the principalcurvatures and thus, for practical computational reasons,instead of Gauss curvature (which is the product of allthe principal curvatures) we shall consider the total sec-ond variation of the scalar curvature R (i.e. the sum ofall the possible products of two principal curvatures) ofthe manifolds (Mu, g), according to the definitionσ2R(u) = [Vol(Mu)]−1∫Mudη (R− ̺u)2(1)̺u = [Vol(Mu)]−1∫MudηR (2)with R = gkjRlklj , where Rlkij is the Riemann curvaturetensor [16], and Vol(Mu) =∫Mudη.Let us now consider the family of submanifolds(Mu, g) associated with the so-called ϕ4 model, on a d-dimensional lattice Zd, described by the potential func-tionV =∑α∈Zd(−µ22q2α +λ4!q4α)+∑〈αβ〉∈Zd12J(qα − qβ)2 (3)where 〈αβ〉 stands for nearest-neighboring sites. We con-sider d = 1, 2; this system has a discrete Z2-symmetryand short-range interactions, therefore in d = 1 there isno phase transition whereas in d = 2 there is a symmetry-breaking transition (this system has the same universal-ity class of the 2d Ising model).The potential in Eq.(3) defines the subsets Mu of con-figuration space, these subsets are given the structure ofRiemannian manifolds (Mu, g) by endowing all of themwith the same metric tensor g. However, as far as wewant to learn something about the topology of thesemanifolds, the choice of the metric g is arbitrary. Wehave therefore chosen three different types of metrics, oneconformally flat and the others non-conformal, accordingto a compromise between simplicity and non-triviality.They are: i) g(1)µν = [A − V (q)]δµν , i.e. a conformal de-formation of the euclidean flat metric δµν , A > 0 is anarbitrary constant larger than u; ii) g(2)µν and g(3)µν arenon-conformal metrics defined by(g(k)µν ) =f (k) 0 10 I 01 0 1 , k = 2, 3 (4)2where I is the N − 2 dimensional identity matrix, g(2) isobtained by setting f (2) = 1N∑α∈Zd q4α +A, and g(3) bysetting f (3) = 1N∑α∈Zd q6α+A, with A > 0, and α labelsthe N lattice sites of a linear chain (d = 1) or of a squarelattice (d = 2, N = n×n). Simple algebra [16] yields thescalar curvature function for each metric:R(1) = (N − 1)[△V(A− V )2−‖∇V ‖2(A− V )3(N4−32)](5)R(k) =1(f (k) − 1)[‖∇̃f (k)‖22(f (k) − 1)− △̃f (k)], k = 2, 3 (6)where ∇ and △ are euclidean gradient and laplacian re-spectively; ∇̃ and △̃ do not contain the derivatives ∂/∂qαwith α = 1 (d = 1) or α = (1, 1) (d = 2).We constructed an ad-hoc MonteCarlo algorithm tosample the geometric measure dη by means of the stan-dard “importance sampling” method [17], then we ap-plied it to the computation of σ2R(u), given by Eq.(2),for the one- and two- dimensional lattice ϕ4 model de-fined in (3) with the following choice of the parameters:λ = 0.6, µ2 = 2, J = 1. The values of R are com-puted according to Eqs.(5) and (6). In order to locate thephase transition that occurs in the two-dimensional case,we have computed 〈V 〉 vs T by means of both Monte-Carlo averaging with the canonical configurational mea-sure, and Hamiltonian dynamics (by adding to V a stan-dard kinetic energy term). In the latter case the temper-ature T is given by the average kinetic energy per degreeof freedom, and 〈V 〉 is obtained as time average. Fig.3shows a perfect agreement between time and ensembleaverages, thus we worked out Fig.3 by computing 200time averages because they converge much faster thanensemble ones. The phase transition point is well visibleat uc = 〈V 〉 ≃ 3.75.In Figs.4 and 5 we synoptically report the patterns ofσ2R(u) for the one and two dimensional cases obtainedat different lattice size with g(1) (Fig.4), and obtained atgiven lattice size with g(2,3) (Fig.5). Peaks of σ2R(u) ap-pear at uc - the value of 〈V 〉 at the phase transition point- in the two-dimensional case, whereas only monotonicpatterns are found in the one-dimensional case, where nophase transition is present.“Singular”, cuspy patterns of σ2R(u) (with the meaningthat such attributes can have for numerical results) arefound independently of any possible statistical mechani-cal effect, and independently of the geometric structuregiven to the family {Mu} by the metric tensors chosen.Though within the well evident limits of numerical simu-lations and of a limited choice of different metrics, our re-sults suggest that the “singular” patterns are most likelyto have their origin at a deeper level than the geomet-ric one, i.e. at the topological level. Hence the observedphenomenology strongly hints at the occurrence of somemajor change in the topology of the configuration-space-submanifolds {Mu} in correspondence with a second-order phase transition. Finally, the expression ”majortopology change” is to suggest that a change of the co-homological type of the Mu - or Σu - might well be anecessary but not sufficient condition for a phase tran-sition, and that in any case some “big” change has tooccur.It is a pleasure to thank E.G.D. Cohen, M. Rasetti andG. Vezzosi for their continuous interest in our work andfor useful comments and suggestions.a Also at INFN, Sezione di Firenze, Italy. E-mail: fran-zosi@fi.infn.itb E-mail: lapo@polito.itc Present address: Osservatorio Astrofisico di Arcetri,Largo E. Fermi 5, 50125 Firenze, Italy. E-mail:spinelli@arcetri.astro.itd Also at INFN, Sezione di Firenze, Italy. E-mail: pet-tini@arcetri.astro.it[1] C.N. Yang and T.D. Lee, Phys. Rev. 87, 404 (1952).[2] L. Caiani, L. Casetti, C. Clementi and M. Pettini, Phys.Rev. Lett. 79, 4361 (1997).[3] L. Caiani, L. Casetti, C. Clementi, G. Pettini, M. Pettiniand R. Gatto, Phys. Rev. E57, 3886 (1998).[4] Here topology is meant in the sense of the de Rham’scohomology.[5] L. Caiani, L. Casetti, and M. Pettini, J. Phys. A: Math.Gen. 31, 3357 (1998).[6] C. Clementi, A. Maritan, and M. Pettini, preprint.[7] M.C. Firpo, Phys. Rev. E57, 6599 (1998).[8] M. Cerruti-Sola, R. Franzosi, and M. Pettini, Phys. Rev.E56, 4872 (1997).[9] L. Casetti, R. Livi, and M. Pettini, Phys. Rev. Lett. 74,375 (1995).[10] L. Casetti, C. Clementi, and M. Pettini, Phys. Rev. E54,5969 (1996), and references cited therein.[11] L. Casetti and M. Pettini, Phys. Rev. E48, 4320 (1993).[12] J. Milnor, Morse Theory, (Annals of Mathematics Stud-ies, Princeton University Press, Princeton, 1969).[13] M. W. Hirsch, Differential Topology, (Springer-Verlag,New York, 1976).[14] J.A. Thorpe, Elementary Topics in Differential Geome-try, (Springer-Verlag, New York, 1979).[15] This classic co-area formula can be found in : H. Federer,Geometric Measure Theory, (Springer, Berlin, 1969).[16] M. P. do Carmo, Riemannian Geometry (Birkhäuser,Boston-Basel, 1992).[17] K. Binder, MonteCarlo Methods in Statistical Physics,(Springer-Verlag, Berlin-New York, 1979).FIG. 1. Illustration of the relationship between topologyand critical points. The U-shaped manifold M is born at P0.The level surfaces Σu and the parts of M below them - Mu- change topology when u exceeds the height of the criticalpoint P1.3FIG. 2. Variance of Gauss curvature vs u close to a criticalpoint. σ2/NK is reported because it is dimensionally homoge-neous to the scalar curvature. Here N = dim(Σu) = 100,and Morse indexes are: k = 1, 15, 33, 48, represented by solid,dotted, dashed, long-dashed lines respectively.FIG. 3. Average potential energy vs temperature for the2d lattice ϕ4 model. Lattice size N = 20 × 20. The solidline, made out of 200 points, refers to time averages. Fullcircles represent MonteCarlo estimates of canonical ensembleaverages. The dotted lines locate the phase transition.FIG. 4. Variance of the scalar curvature of Mu vs u com-puted with the metric g(1). Full circles correspond to the1d-ϕ4 model with N = 400. Open circles refer to the 2d-ϕ4model with N = 20 × 20 lattice sites, and full triangles referto 40× 40 lattice sites (whose values are rescaled for graphicreasons).FIG. 5. Variance of the scalar curvature of Mu vs u com-puted for the ϕ4 model with: metric g(2) in 1d, N = 400(open triangles); metric g(2) in 2d, N = 20 × 20 (full trian-gles); metric g(3) in 1d, N = 400 (open circles); metric g(3) in2d, N = 20× 20 (full circles).4P1P0Fig. 1  Franzosi et al."Topological aspects of ..."u<uMM u>uu<uΣu>uΣuΣcccccu

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