Localization in the Discrete Non-Linear Schrödinger Equation and geometric properties of the microcanonical surface
Claudio Arezzo, Federico Balducci, Riccardo Piergallini, Antonello Scardicchio, Carlo Vanoni
LLocalization in the Discrete Non-Linear Schr¨odingerEquation and the geometric properties of themicrocanonical surface
Claudio Arezzo , , Federico Balducci , , , Riccardo Piergallini ,Antonello Scardicchio , and Carlo Vanoni The Abdus Salam ICTP, Strada Costiera 11, 34151, Trieste, Italy INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy SISSA, via Bonomea 265, 34136, Trieste, Italy Universit`a di Camerino – Scuola di Scienze e Tecnologie, Via Madonna delle Carceri, 62032Camerino, Italy Universit`a di Parma – Dipartimento di Scienze Matematiche, Fisiche e Informatiche, ParcoArea delle Scienze, 53/A 43124 ParmaE-mail: [email protected]
23 February 2021
Abstract.
It is well known that, if the initial conditions have sufficiently high energy density,the dynamics of the classical Discrete Non-Linear Schr¨odinger Equation (DNLSE) on a latticeshows breaking of ergodicity, with a finite fraction of the total charge accumulating on a singlesite and residing there for times that diverge quickly in the thermodynamic limit. In this paperwe show that this kind of localization can be attributed to some geometric properties of themicrocanonical potential energy surface and that it can be associated to a phase transition inthe lowest eigenvalue of the Laplacian on said surface. We also show that the approximation ofconsidering the phase space motion on the potential energy surface only, with effective decouplingof the potential and kinetic partition functions, is justified in the large connectivity limit, orfully connected model. In this model we further observe a synchronization transition and asynchronized phase at low temperatures. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b ocalization in the DNLSE and geometric properties of the microcanonical surface
1. Introduction
In recent years the interest in non-ergodic states of matter has grown considerably, in particular, butnot only, following recent developments in the study of many-body quantum systems. The discoveryof Many-Body Localization (MBL) [1–7] has extended the phenomenon of Anderson Localization[8] to interacting systems and suggested the emergence of a dynamical phase characterized bylocal integrals of motion [9–14] in disordered quantum systems. These studies have shown apotential big impact on quantum technologies, which are now in the realm of mesoscopic quantumsystems. Further extensions of the original idea have put forward the possibility that MBL-likephysics could be observed in Josephson junctions chains [15, 16], when the initial state presentssufficiently large charge fluctuations, playing the role of quenched disorder in an otherwise cleansystem. Slow dynamics in such clean quantum systems (see also the works on quantum scars [17]), appears in a form similar to the weak ergodicity breaking characterizing spin glasses [18–22],configurational glasses [23–25], and, in particular, non-linear oscillators models as the
Fermi-Pasta-Ulam-Tsingou model [26–28] and the
Discrete Non-Linear Schr¨odinger Equation (DNLSE) [29].The latter, furthermore, has proven to be a rich theoretical playground, in which one finds thatthe equivalence of the microcanonical and canonical Gibbs ensembles breaks down at high energy[30–32]; one observes breather-like excitations [33–40], self-localization [41–43] and, last but notleast, weak ergodicity breaking [44, 45].The similarities between the quantum MBL phenomenon and the classical localizationphenomena in the DNLSE have spurred our interest and so we have set to investigate the origins ofthe latter, in particular to highlight similarities and differences. We stress again that we will focuson clean systems with (cid:126) ≡
0: the interplay between disorder, quantum-mechanical localization andnonlinear effects in DNLS-like models has been the subject of vast research (see e.g. [46, 47] andreferences therein), but the physics behind it is appreciably different.Basing on previous works, and in particular on [30–32], as the main result of this study, wefind that the origin of the localization phenomenon at high energy density can be traced to thebehaviour of the gap of the Laplace operator on the ( N − Z = Z momentum Z position , as it happens for examplewhen looking at gases or liquids, where typically the phase-space variables ( p, x ) appear each inits own term: H ( p, x ) = K ( p ) + V ( x ). We therefore first show that expansion around infinitetemperature of the free energy (and, consequently, of all relevant observables) gets contributionsfrom the kinetic energy only at O (1 /κ ), where κ is the connectivity of the graph on which theDNLSE is considered (see Eqs. (1)–(3) to fix the notation). Therefore, if one considers a fullyconnected model, the assumption of neglecting the kinetic term is completely justified.We solve the fully connected model finding that the “infinite temperature phase”, in whichthe free energy becomes essentially given by the potential term alone, extends all the way downto a finite temperature T s = 2 g ( g is the strength of the kinetic term and v that of the potential).At temperatures lower than T s , or equivalently energy densities ε < ε s = 1 . . . . (with theparameters g = 1, v = 2 used throughout this paper), the model enters a synchronized phase in which the phases φ i (see Eqs. (2) and (3)) stop rotating independently from each other andeventually move together at T = 0 (energy density ε GS = v/ − g = − T s the motion of the phases φ i is incoherent and the charges moverandomly on the microcanonical potential energy surface.Secondly, after having highlighted the importance of the microcanonical potential energysurface, we study the topology of such manifold. We prove, using both stratified Morse theoryand a more direct geometrical approach, that the manifold undergoes a series of critical points(critical in the language of Morse theory, not of statistical physics) but it remains connected untilenergy densities ε = vN/ N/
2, which means super-extensive energy. The infinite-temperaturelocalization phase transition (previously studied in [30, 33–36, 38, 39, 42, 45, 48, 49]), taking placeat energy density ε c = v = 2, therefore, is not due to a breakdown of connectivity in such manifold. ocalization in the DNLSE and geometric properties of the microcanonical surface LocalizedSynchronized Paramagnetic
Only potential energyPotential and kinetic energy ε ε c = 2 ε s =1.481… ε GS = − 1 Figure 1.
Phase diagram of the fully connected DNLS model for v = 2 and g = 1. The region ε > ε c corresponds to non-positive temperatures and localized dynamics; the region ε s < ε < ε c to ergodic incoherent dynamics for the phases φ i (Eq. (2)); the region ε < ε s to coherentdynamics for the same phases. Rather, we attribute it to the change in the scaling with N of the smallest, non-zero eigenvalue λ of (minus) the Laplace operator on the microcanonical surface (the smallest eigenvalue λ = 0corresponds to the uniform distribution on the manifold). Namely, for ε < ε c = 2 (the numericsagrees with the thermodynamic calculation within errors) we have λ = O (1), while for ε > ε c wehave λ ∼ e − γN . The function γ ( ε ) ≥ ε → ε + c with critical exponent close to 2.We conjecture that this transition is related to an entropic effect for the motion of a particleon the microcanonical energy surface. In other words, the volume of the regions of phase spaceclose to an imbalanced configuration (i.e. when one of the charges much larger than the others)becomes much larger, at energies ε > ε c , than the volume of balanced configurations. We suggestthat this mechanism, and the link with the behaviour of the smallest eigenvalue of the Laplacianon the microcanonical surface, is generic for DNLSE with different choices of graphs and potentials(as also indicated by the results of [48, 49]).The paper is organized as follows. In Sec. 2 we introduce the DNLS model and briefly reviewsome known results. In Sec. 3 we set up an high-temperature expansion and show that, forany dimensionality, the infinite temperature point corresponds always to ε c = v = 2. In Sec.4 we perform instead an expansion in the kinetic term of the Hamiltonian, and prove that, forlarge connectivity, hopping can be completely neglected in a finite neighbourhood of ε = ε c . InSec. 5 we inspect more closely the reasons why hopping is sub-leading, finding out that there is asynchronization phase transition in the fully connected model at ε s = 1 . . . . . We discuss in detailthe implications of such phase transition. In Sec. 6 we describe (almost) rigorously the topologyof the potential energy surface, leaving to Appendix A the flawless, yet less insightful proof. Also,we provide some intuitive explanation of the connection of the geometry with the behavior of thegap of the Laplacian. In Sec. 7 we switch to the numerical study of the dynamics of the model: weprovide strong evidence that at ε = 2 also a dynamical phase transition takes place, thus implyingthat the gap of the Laplacian on the potential energy surface closes as described above. Finally,in Sec. 8 we discuss the implications of our findings and speculate on future directions.
2. The model
In this work, we want to show that the mechanism of ergodicity breaking at high energy density forthe DNLSE is very general, and depends only on the particular form of the potential energy andcharge conservation laws. For this reason, we consider the DNLS model on an arbitrary, regulargraph G : H = − gκ N (cid:88) i,j =1 A ij (cid:0) ψ ∗ i ψ j + ψ ∗ j ψ i (cid:1) + v N (cid:88) i =1 | ψ i | . (1)Here, the ψ i , ψ ∗ i are complex fields that live on the vertices i of G , and are canonically conjugated:their Poisson brackets read { ψ i , ψ ∗ j } = δ ij . Then, g , v are non-negative parameters, which we willeventually set to g = 1 and v = 2; for the time being, however, it is convenient to allow them tovary. Finally, κ is the connectivity of G and A its adjacency matrix, so each entry A ij is either 0( ij disconnected) or 1 ( ij connected). Notice that, thanks to the 1 /κ normalization of the kinetic ocalization in the DNLSE and geometric properties of the microcanonical surface H = O ( N ) for any G , even in the limit of fully connected graph κ ∼ N → ∞ , without havingto rescale g .Apart from the energy, there is another natural conservation law to take into account: definingthe charge Q := (cid:80) i | ψ i | , it holds { Q, H } = 0. Without loss of generality, we choose to work with Q ≡ N fixed from now on (or, more generally, with the average charge fixed).We define the energy density ε := H/N . Previous works (e.g. [30, 31]) have shown that ε = ε c = v corresponds to the T = ∞ limit of the model on a chain when coupled to a thermalreservoir at temperature T , and that at ε > ε c the microcanonical distribution is not equivalent tothe Gibbs distribution. Moreover, other works have shown that the dynamics of a chain ceases tobe ergodic in this non-Gibbs phase, with the charges localizing on single sites in solitons rather thanmoving around. This has been seen both by using a simplified stochastic evolution algorithm, forall ε ≥ ε c [38, 39], and with Hamiltonian dynamics [44] (although in this latter work the thresholdis put at ε (cid:39) . v ).It is important to note, however, that in the proof of the canonical/microcanonicalinequivalence [31] the kinetic term plays no role so, in particular, no role is played by the geometry ofthe lattice (or graph) on which the DNLSE is set. This simplification is justified for asymptoticallylarge ε , since the kinetic term is proportional to | ψ | , while the potential term is proportional to | ψ | and is dominating. One wonders then whether the same happens at any finite temperature,that is to say in a left neighbourhood of ε c . We will prove that it does, at least in the approximationof large connectivity.
3. Infinite temperature limit
We now start exploring the limit T → ∞ . We start with computing the canonical partition functionof the Hamiltonian (1) which, assuming ergodicity, should tell us about the behavior of the systemfor ε ≤
2. For later convenience, we employ a dimensionless chemical potential µ : Z ( N, β, µ ) = (cid:90) [ dψ dψ ∗ ] e − βH + µQ , where [ dψ dψ ∗ ] := (cid:81) i π dψ i dψ ∗ i and β := 1 /T (fixing k B ≡ (cid:40) ψ i = √ q i e iφ i ψ ∗ i = √ q i e − iφ i (2)with q i ≥ φ i ∈ [0 , π ]. The measure becomes [ dψ dψ ∗ ] = (cid:81) Ni =1 12 π dq i dφ i =: [ dq dφ ], thecharge Q = (cid:80) Ni =1 q i and the Hamiltonian H = − gκ N (cid:88) i,j =1 A ij √ q i q j cos( φ i − φ j ) + v N (cid:88) i =1 q i . (3)Let us denote the thermal average of an observable A as (cid:104) A (cid:105) β,µ := 1 Z (cid:90) [ dq dφ ] e − βH + µQ A. Then one can inspect the infinite temperature limit, β →
0, by considering the expansion (cid:104) A (cid:105) β,µ = (cid:104) A (cid:105) ,µ − β (cid:104) (cid:104) AH (cid:105) ,µ − (cid:104) A (cid:105) ,µ (cid:104) H (cid:105) ,µ (cid:105) + O ( β ) . The first thing to do is to adjust the chemical potential to have a fixed average charge (recall ourchoice in Sec. 2): N ≡ (cid:104) Q (cid:105) β,µ (cid:39) (cid:104) Q (cid:105) ,µ − β (cid:104) (cid:104) QH (cid:105) ,µ − (cid:104) Q (cid:105) ,µ (cid:104) H (cid:105) ,µ (cid:105) (cid:39) N (cid:104) q (cid:105) ,µ − β (cid:104) v (cid:0) N (cid:104) q (cid:105) ,µ + N ( N − (cid:104) q (cid:105) ,µ (cid:104) q (cid:105) ,µ (cid:1) − N (cid:104) q (cid:105) ,µ v N (cid:104) q (cid:105) ,µ (cid:105) (cid:39) − Nµ + β N vµ ocalization in the DNLSE and geometric properties of the microcanonical surface µ (cid:39) − βv . In the computation we have used the fact that the averages involving thekinetic energy vanish by symmetry, and (cid:104) q k (cid:105) ,µ = k ! / ( − µ ) k .Now we focus on the internal energy. We already have found (cid:104) H (cid:105) ,µ = N v/µ ; thus we justneed (cid:104) H (cid:105) ,µ . The only non-zero angular integrals that figure in (cid:104) H (cid:105) ,µ are (cid:90) [ dφ ] cos( φ i − φ j ) cos( φ k − φ l ) = 12 ( δ ik δ jl + δ il δ jk );therefore we find (cid:104) H (cid:105) ,µ = 4 g κ N κ (cid:104) q (cid:105) ,µ + v (cid:2) N (cid:104) q (cid:105) ,µ + N ( N − (cid:104) q (cid:105) ,µ (cid:3) where we recall κ is the connectivity of the graph. The final result is ε ( β ) = v − β (cid:18) v + 4 g κ (cid:19) + O ( β ) (4)From this expression we see that the result of the non-interacting case ε ( β = 0) = v is not modifiedby the presence of the hopping on any graph geometry. We notice also that the kinetic energyterm (measured by g ) contributes only with O ( g /κ ) and therefore vanishes to this order in themean-field, fully connected limit κ → ∞ . We explore such limit in the next section.
4. Large connectivity limit
As noted at the end of the last section, in Eq. (4) the O ( β ) correction to the internal energy densitybecomes independent of g in the limit of large connectivity κ → ∞ . Indeed, one can verify that allthe terms in the expansion involving the hopping are subleading in κ . The situation is reminiscent ofthe Thouless-Anderson-Palmer (TAP) high temperature expansion of the Sherrington-Kirkpatrickmodel [50]. Alongside with TAP, we can expand the free energy density f := − log( Z ) /βN inpowers of 1 /κ f = f + 1 κ f + O (cid:18) κ (cid:19) , (5)where f is the free energy density at κ → ∞ , or g = 0, while we can express f (and successiveorders too) as a sum of diagrams.To see it, start by expanding in powers of g the full free energy density: − βf N = log (cid:90) [ dq dφ ] e − βH + µQ ∞ (cid:88) k =0 k ! (cid:20) gβκ (cid:88) ij A ij √ q i q j cos( φ i − φ j ) (cid:21) k =: − βf N + ∞ (cid:88) (cid:96) =1 (cid:18) gβκ (cid:19) (cid:96) (cid:88) Λ ∈D (cid:96) Λ (6)where consistently we denote by a “0” subscript quantities that are evaluated at g = 0. Equation(6) is our definition of the diagrams Λ ∈ D (cid:96) , that we also show graphically in Fig. 2. More precisely,at each order (cid:96) of the effective coupling constant g/κ we have averages (cid:90) [ dφ ] cos( φ i − φ i ) cos( φ i − φ i ) · · · cos( φ i (cid:96) − − φ i (cid:96) ) N (cid:89) j =1 (cid:104) q n j / (cid:105) , where n j is the multiplicity with which index j appears in the string i i · · · i (cid:96) . We notice that(i) since we are expanding a logarithm, by the linked-cluster theorem each diagram Λ ∈ D (cid:96) mustconsist of one connected piece only;(ii) for the angular integration not to yield 0, each φ i must appear an even number of times; inparticular this means that all the diagrams in D (cid:96) must be closed and each vertex must havean even number of legs (see Fig. 2); ocalization in the DNLSE and geometric properties of the microcanonical surface simple loops
Allowed diagrams Λ ∈ D (cid:96) in the expansion (6) up to order (cid:96) = 4. Including countingfactors, they evaluate (from left to right) to
Nκ/ Nκ ( κ − / Nκ ( κ − κ − / Nκ ( κ − / Nκ/
8. Circled in red are the one-loop, connected diagrams of which there are one per eachorder (cid:96) : these contribute to lowest order in 1 /κ . Notice also that there is no watermelon diagramat O (( g/κ ) ) because of point (ii) in the text. (iii) the permutation symmetry of the couples i p i p +1 yields a factor (cid:96) ! /S Λ , where S Λ is thesymmetry factor of the diagram Λ. Therefore, according to the usual arguments this cancelsthe 1 /(cid:96) ! in the expansion of the exponential, leaving the symmetry factor in the denominator;(iv) the permutation symmetry within each couple i p i p +1 of the two indices yields a factor 2 foreach pair, and so a factor 2 (cid:96) in total;(v) the angular integration for simple loops evaluates to 2 − (cid:96) , while multiple loops give a resultdepending on the geometry (e.g. in Fig. 2 the first three diagrams are simple loops and receiverespectively a factor 1 /
2, 1 / /
8, while the fourth receives a factor 1 / / (cid:96) the simple loops (e.g. the diagrams circled in red in Fig. 2) are the least suppressed by κ . Indeed, since they are composed by the maximum number of distinct points, the factor κ (cid:96) inthe denominator of Eq. (6) is compensated by the ∼ N κ (cid:96) − possible choices of the points. Havingnoted this feature, we can explicitly compute f : the angular integration yields a factor 2 − (cid:96) (asnoted in point (v) before), the symmetry factors are S Λ = 2 (cid:96) , and only the averages (cid:104) q (cid:105) = 1appear. Therefore one has − βf = ∞ (cid:88) (cid:96) =2 (cid:18) gβκ (cid:19) (cid:96) N κ (cid:96) − (cid:96) (cid:96) (cid:96) − = − gβ − log(1 − gβ ) , (7)with the sum starting from (cid:96) = 2 because there is no diagram at order 1.At this point, we can give a physical interpretation to eqs. (5)–(7). In the large connectivitylimit κ → ∞ , the extensive contribution to the free energy is always regular and independent ofthe hopping between different sites. Moreover, as long as β < β s := (2 g ) − , the sub-extensivecontribution can be forgotten. However, at β = β s the sub-extensive part f diverges and there isa phase transition: interactions must be taken into account and to go beyond one needs to addressthe problem non-perturbatively.
5. Solution of the fully-connected model
To go beyond perturbation theory, we can compute the partition function of the fully-connectedmodel κ = N − N , we have Z MF ( N, β, µ ) = (cid:90) [ dq dφ ] exp (cid:110) − βv (cid:88) i q i + µ (cid:88) i q i + 2 βgN (cid:16) (cid:88) i √ q i e iφ i (cid:17)(cid:16) (cid:88) i √ q i e − iφ i (cid:17)(cid:111) . We expand (cid:16) (cid:88) i √ q i e iφ i (cid:17)(cid:16) (cid:88) i √ q i e − iφ i (cid:17) = (cid:16) (cid:88) i √ q i cos φ i (cid:17) + (cid:16) (cid:88) i √ q i sin φ i (cid:17) , ocalization in the DNLSE and geometric properties of the microcanonical surface Z MF ( N, β, µ ) = N π (cid:90) [ dq dφ ] (cid:90) dy dy exp (cid:110) − βv (cid:88) i q i − N y + y ) + µ (cid:88) i q i + 2 (cid:112) βg y (cid:88) i √ q i cos φ i + 2 (cid:112) βg y (cid:88) i √ q i sin φ i (cid:111) . Now all the q , φ integrals are factorized, and the basics constituents are of the form12 π (cid:90) dq dφ exp (cid:104) − βvq / µq + (cid:112) βgq ( y − e iφ + y + e − iφ ) (cid:105) with y ± = y ± iy . We can perform first the angular part:12 π (cid:90) φ dφ e z ( y − e iφ + y + e − iφ ) = 12 π (cid:90) φ dφ (cid:88) k ≥ z k y k − k ! e ikφ (cid:88) (cid:96) ≥ z (cid:96) y (cid:96) + (cid:96) ! e − i(cid:96)φ = (cid:88) k ≥ ( z y + y − ) k ( k !) = I (2 z √ y + y − ) ,I being the modified Bessel function of the first kind. Thus, defining J ( β, µ, Y ) := (cid:90) ∞ dq e − βvq / µq I (2 (cid:112) βgqY )with Y := y + y − = y + y , we arrive at Z MF ( N, β, µ ) = N (cid:90) ∞ dY e − NY/ N log J ( β,µ,Y ) . (8)When performing this integral in the N → ∞ limit, if the saddle point is within the domain ofintegration Y ≥
0, one can use the saddle point method, otherwise one needs to integrate by partsaround the lower limit of integration Y = 0. In any case, the free energy density is f ( β, µ, Y ) = β − ( Y / − log J ) , where Y solves the saddle-point equation (with the above proviso)12 = 1 J ∂J∂Y . (9)It also is convenient to trade µ for the (average) total charge (cid:104) Q (cid:105) = N :1 = 1 J ∂J∂µ . (10)Equations (9)–(10) can be easily solved numerically by iteration for any desired β .It is worth noting that eqs. (9)–(10) can be obtained also by extremization of the grand-canonical free energy density a ( β, µ, Y ) = f ( β, µ, Y ) + µ/β. (11)It turns out that the extreme point of a is a saddle (see Fig. 3) which moves in the ( µ, Y ) plane as β is changed.Another way of rewriting eqs. (9)–(10) is by interpreting J ( β, µ, Y ) as a partition function forthe variable q , which thus acquires the probability density p ( q ) = 1 J e − βvq / µq I (2 (cid:112) βgqY ) . (12) ocalization in the DNLSE and geometric properties of the microcanonical surface - - μ - - μ Figure 3.
Contour plot of the grand-canonical free energy density a (Eq. (11)), with g = 1and v = 2, for β = 1 . β = 0 . Y as the temperature is increased. Then, the two equations (9)–(10) take (respectively) the form (cid:115) Y βg = (cid:28) √ q I (2 √ βgqY ) I (2 √ βgqY ) (cid:29) p (13)1 = (cid:104) q (cid:105) p . (14)These last expressions are convenient to control the limits β → ∞ and β →
0. Indeed, as β → ∞ the problem simplifies and the probability concentrates around the saddle point q = 1 (Eq.(14)). One can also expand the Bessel functions (as long as, self-consistently, Y (cid:29) /β ) for largearguments, and substituting q = 1 in Eq. (13) gives (cid:115) Y βg = (cid:104)√ q (1 + · · · ) (cid:105) p = ⇒ Y = 4 gβ + O ( β ) . (15)Also, imposing Eq. (14) explicitly on Eq. (12), one gets µ = β ( v − g ) + O ( β ) . (16)For small β , instead, one can expand the Bessel functions for small argument (as long as thisreturns self-consistently βY (cid:28) (cid:115) Y βg = (cid:28) √ q (cid:20) ( βgqY ) / −
12 ( βgqY ) / + · · · (cid:21)(cid:29) p = ( βgY ) / (cid:104) q (cid:105) p −
12 ( βgY ) / (cid:104) q (cid:105) p + · · · . There are two solutions: Y = 0 , Y = 2 βg − βg ) (cid:104) q (cid:105) p . (17)The second solution is negative for β < β s = (2 g ) − , so in this region one must stick with Y (sincethe Y integral in Eq. (8) is on the positive domain). As β (cid:38) β s , instead, Y becomes the correctsolution, until the condition βY (cid:28) β and large- β approximations.To connect with the diagrammatic expansion done in Sec. 4, we notice that the critical value β s = (2 g ) − is the same given by the radius of convergence of perturbation theory for f , thesub-extensive contribution to the free energy. We are now in position to give an interpretation tothe phase transition taking place at T s = 1 /β s : it is the temperature below which the angles φ i nomore average to zero, but start acquiring a common orientation. Indeed, on one hand − g ∂∂g ( βf ) = β gN (cid:88) i (cid:54) = j (cid:10) √ q i q j cos( φ i − φ j ) (cid:11) ; ocalization in the DNLSE and geometric properties of the microcanonical surface Y g ( v g ) numericlarge small Figure 4.
Saddle-point value of Y found upon solving eqs. (9)–(10) by iteration, with g = 1 and v = 2 (black solid line). For β < (2 g ) − = 0 . Y , while at larger values of β it becomes Y (see Eq. (17)). For comparison, we show the approximate solutions at β → ∞ and βY → β → ∞ approximation, to the order obtained in eqs.(15)–(16), still needs a O (1) term to be fixed. Inset:
Corresponding values found for µ . on the other hand, by differentiating the saddle-point free energy, − g ∂∂g ( βf ) = g J ∂J∂g = Y J ∂J∂Y = Y . The comparison of the last two equations implies Y = 4 βgN (cid:88) i (cid:54) = j (cid:10) √ q i q j cos( φ i − φ j ) (cid:11) . We conclude that, as β → ∞ , the angles must all point in the same direction (albeit thelatter can move in time randomly). Indeed, recalling that q concentrates around 1, in orderto find the asymptotic, low temperature behaviour Y (cid:39) βg one needs that all the phases align: (cid:104) cos( φ i − φ j ) (cid:105) →
1, so φ i ≡ φ for all i = 1 , , . . . , N . This is the statistical mechanics signature ofa synchronized phase [51, 52], in which all fields have a common phase and the fluctuations of theamplitudes are negligible. The synchronization phase transition is second-order, with the orderparameter Y growing linearly close to β s = (2 g ) − .We can also express the above observations in terms of the energy density ε . At T = 0 thesystem is in the ground state, with energy density ε GS = v/ − g : this readily follows from our β → ∞ expansion of the free energy. At the synchronization transition T = T s = 2 g , instead, theenergy density can be found numerically by imposing Y = 0 and fixing µ from Eq. (10): for g = 1and v = 2 we find ε s = 1 . . . . (see also Fig. 1).
6. Topological structure of the potential energy surface
We now start focusing on the region ε ≥ ε c = v . Having completely lost the spatial structure givenby the hopping for any graph geometry , the model has become effectively non-interacting . For thisreason, we can also fix v ≡ H = N ε ) and charge ( Q = N ). We will show in this Section thatsaid microcanonical surface remains connected while passing a series of critical points accordingto stratified Morse theory, disconnecting only at the last point which occurs at extensive energydensity (so super-extensive energy). This proof excludes definitely that the dynamical transitionobserved at ε c = 2 is due to the manifold becoming disconnected.We again change variables from the ψ i ’s to the local charges q i = | ψ i | (see also Eq. (2)), thatare the only combinations of the ψ i ’s entering in the conservation laws. Thus, we are left with the ocalization in the DNLSE and geometric properties of the microcanonical surface Figure 5.
Two different views of the same stereographic projection of the manifold M ε (Eq.(18)), for N = 5 and ε = 2. While this projection respects the topology of the manifold, clearlyits metric structure is altered. Colors depend on the distance from the vertices of ∆ N − , varyingfrom blue for the five 0-handles around those vertices to green for the median sections of the ten1-handles. equations N (cid:80) Ni =1 q i = 1 N (cid:80) Ni =1 q i = εq i ≥ ∀ i = 1 , , . . . , N. (18)These equations define a ( N − M ε (see Fig.5 for a visual impression of the case N = 5), naturally embedded in R N , whose central role wasrecognized already in [53]. The topology of this manifold undergoes a series of changes as ε varies,which can be outlined by stratified Morse theory in the following way (see Appendix A for a moredetailed description).The first and the last equation in (18) represent the affine simplex ∆ N − ⊂ R N spanned bythe vectors N e , . . . , N e N , where e , . . . , e N is the canonical base of R N . Hence, M ε is non-emptyfor 1 ≤ ε ≤ N . Moreover, M ε is a small ( N − N when ε approaches 1, while it is the disjoint union of N small ( N − N − ,when ε approaches N .In order to see what happens for the intermediate values of ε , think of the boundary ∂ ∆ N − as astratified space, whose strata are its open sub-simplices of ∆ N − , and observe that ϕ : ∂ ∆ N − → R given by ϕ ( q ) = (cid:107) q (cid:107) /N is a stratified Morse function, meaning that it restricts to a Morse functionon every stratum. Then, for every 1 < ε < N , the radial projection from the barycenter of ∆ N − ,that is the vector e + · · · + e N , induces a stratified diffeomorphism between M ε and the suplevelset M ε ( ϕ ) = { q ∈ ∂ ∆ N − | ϕ ( q ) ≥ ε } ⊂ ∂ ∆ N − , according to the second equation in (18).Morse theory tells us that the topology of M ε ∼ = M ε ( ϕ ) changes only at the critical valuesof the restrictions of ϕ to the strata of ∂ ∆ N − . Such critical values have the form N/k with1 < k < N . Indeed, for each k we have (cid:0) Nk (cid:1) corresponding non-degenerate critical points ofindex N − k − k − N − . Thisimplies that for δ > M N/k − δ ( ϕ ) can be obtained by attaching (cid:0) Nk (cid:1) narrow( k − M N/k + δ ( ϕ ). Each of these ( k − N − C N − whichis the product of a ( k − C k − = Cl(Σ − M N/k + δ ( ϕ )) (where Cl stands for the closureoperator) for a ( k − N − and a small ( N − k − C N − k − such that C N − ∩ M N/k + δ ( ϕ ) = ∂ C k − × C N − k − .As a consequence, for every k = 1 , . . . , N − N/ ( k + 1) < ε < N/k the ( N − M ε ( ϕ ) is a regular neighborhood, meaning an ( N − k − N − in ∂ ∆ N − . In particular, recalling the homeomorphism M ε ∼ = M ε ( ϕ ), we can concludethat M ε has N connected components for N/ < ε ≤ N , while it is connected for 1 ≤ ε ≤ N/ N >
4, at ε = 2 the manifold M ε has gone througha series of gluing handles procedures described above, yet remaining connected. This raises theintriguing problem of characterizing M also from a purely geometrical point of view. Of coursethe simplest geometrical invariant of M ε is its volume. Since vol( M ) = vol( M N ) = 0, we know ocalization in the DNLSE and geometric properties of the microcanonical surface ε = N/ − δ ε = N/ ε = N/ δp Figure 6.
The figure concerns the case N = 4. Everything is depicted in the ( N − A N − given by the first equation in (18). In yellow M ε ⊂ S N − ε and in red thesuplevel set M ε ( ϕ ) ⊂ Bd ∆ N − . by continuity that there exists ε ( N ) ∈ (1 , N ) which maximizes vol( M ε ). Recall that Boltzmann’slaw entails S ( ε ) = log(vol( M ε )), and also that it holds1 T = 1 N dSdε .
This in turn implies that a stationary point for the volume arises at infinite temperature, i.e. for ε = 2 (value that is correct only in the limit N → ∞ ), as already argued. This classical observationhas been significantly strengthened in [31] for the model under consideration, where it is observedthat ε = 2 is indeed lim N →∞ ε ( N ) and moreover ε ( N ) = 2 + O ( N − / ). We believe that a directgeometric analysis of the behaviour of vol( M ε ) would be very interesting by itself since it couldshed light on various other aspects of the problem studied.While we leave this task for future investigation, we now observe that thanks to the Morse-theoretic description above, we can quantify the volume contribution of each handle attachmentthrough any critical value of ε = N/k , 1 < k < N . Indeed, given ε = N/k − δ and p a singularpoint in M N/k , we can look at the projection Π from the barycenter B of the symplex of aneighborhood of p in the sphere S N − inside the ( N − A N − givenby the first equation in (18) onto the tangent space to this sphere (see Figs. 6 and 7).As argued above, p is a non-degenerate critical point of index N − k − k − k − of ∆ N − and hence we can choose coordinates( x , . . . , x k − , y , . . . , y N − k − ) on T p ( S N − ) in such a way that x = ( x , . . . , x k − ) parametrizeΠ(∆ k − ), and y = ( y , . . . , y N − k − ) span its orthogonal complement. By intersecting Π( M ε ) witha ( N − C r centered in the origin of T p ( S N − ) with faces parallel to the coordinate axis,we are led to estimate the rate change of the local effect on the volume of the handle-attachmentprocedure (vol( Y δ,r ) as shown in Fig. 7). This can be done observing that such region is boundedby a function W δ ( | x | ), which is at first order quadratic in | x | , being the image via Π of the profileof the sphere, and s.t. W δ (0) = δ / + O ( δ ). It is now a straightforward computation to see thatvol( Y δ,r ) = C ( N, k ) r k − δ ( N − k − / + h . o . (19)for some constant C ( N, k ). The above computation holds for any k = 2 . . . N −
1, and singles outyet another peculiarity of the value ε = 2, corresponding to k = N/ N ). Infact, this is the only situation in which the contributions coming from the two factors of the handle C k − × C N − k − are of the same order.We can also focus on another natural geometric invariant of M ε , namely its first non-zeroeigenvalue of the Laplacian for the curved metric induced on M ε (with zero Neumann boundaryconditions). We will provide strong evidence for the following intriguing (and hard) geometric Problem.
Having set γ ( ε ) := − lim N →∞ N log λ , we have (cid:40) γ = 0 ε ≤ γ > ε > . ocalization in the DNLSE and geometric properties of the microcanonical surface Π(Δ k −1 ) = ℝ k −1 ℝ N − k −1 p δ ε = Nk ε = Nk − δ ℂ r Y δ , r r Γ W δ Figure 7.
Image of the projection via Π on T p ( S N − ) of a neighborhood of a singular point p in M N/k . Γ W δ is the profile of Π( M N/k − δ ) in T p ( S N − ). In blue the handle attachment Y δ,r . Providing fine estimates for the first eigenvalue of the Laplacian is well known to be a subtle(and important) problem in geometric analysis. The present situation seems particularly interestingand original also from a purely mathematical point of view for the concurrence of the value ε = 2 as special value both for the volume and λ , a coincidence that certainly deserves furtherunderstanding on the mathematical side.We believe, however, that the simple observation in Eq. (19) could be a first step towards theunderstanding of the coincidence stated above. Indeed, a very much conjectural, and simplifiedpicture of why the charges become localized could be based on the counting of “useful” and “useless”directions when crossing the handles connecting two different localized configurations. To be morespecific, say that one starts in a configuration with q = O ( N ) and all the others q j = O (1), j (cid:54) = 1, and wants to reach one in which q = O ( N ) and all the others q k = O (1), k (cid:54) = 2. Then,the transition happens when the number of “useless” directions ( R N − k − in Fig. 7) overcomes thenumber of “useful” directions ( R k − in the same figure). One can make as well a connection withthe question of entropic barriers in spin-glass dynamics (on this topic see [54] and the recent work[55]). We leave this connection for future investigations.
7. A Brownian dynamics on the potential energy surface and the gap of theLaplacian
In order to extract the first non-zero eigenvalue of the Laplacian, we resorted to studying thecorrelation functions of a Brownian motion on the surface M ε . Indeed, being the diffusionequation described by the Laplacian, it is known that the late decay of the correlation functionsof coordinates (e.g. the charges q i ) gives its first non-zero eigenvalue. Hence, we pick as a startingpoint a random vector (cid:126)q that satisfies all the conditions in (18) (this can be easily done by repeatedlyprojecting on the three distinct manifolds defined by each constraint, until they are all obeyed), andlet it evolve by free diffusion on M ε up to a final time T f . Specifically, at each Monte Carlo stepwe update the position as (cid:126)q ( t + dt ) = (cid:126)q ( t ) + d (cid:126)W , where dW i are i.i.d. Gaussian random variabless.t. (cid:104) dW i (cid:105) = 0 and (cid:104) dW i (cid:105) = dt , dt being small ‡ ; and then we enforce again the constraints untilthey are all satisfied § .We believe it is important to emphasize that our dynamics is fundamentally different from thatof [31, 38, 39]. In these works, the basic Monte Carlo step was the redistribution of charge withina triplet of consecutive sites (employing periodic boundary conditions, PBC). Specifically, a triplet ‡ Notice that with this normalization (cid:107) (cid:126)q ( t + dt ) − (cid:126)q ( t ) (cid:107) = O ( √ N ) √ dt and the relative Fokker-Plank equation is Eq.(21), which does not contain any explicit factor of N . Different scalings of dq can be easily obtained by rescalingtime. § We have explicitly checked that this last step does not introduce jumps, i.e. the projection of (cid:126)q ( t ) + d (cid:126)W onto M ε produces a point close to (cid:126)q ( t ). ocalization in the DNLSE and geometric properties of the microcanonical surface t G ( t ) (a) N =2560 N =1280 N =640 N =320 N =160 N =80 t G ( t ) e t / (b) N =10240 N =5120 N =2560 N =640 N =160 Figure 8. (a) Correlation function, Eq. (20), as a function of physical time t with dt = 0 . ε = 1 . ε = 2 .
05 (near-critical region) anddashed-dotted lines to ε = 2 . ε = 2. The decay is slower than an exponential(and becomes slower as N is increased), as shows the comparison with the black dashed line.For each N we performed a fit log G ( t ) = − t/τ − log(1 + ( t/t ) z ), finding values of τ , t and z that we report in Fig. 10. Each curve is obtained by averaging over at least 50000 different runs. ( q i , q i +1 , q i +2 ) was updated to a randomly chosen new triplet ( q (cid:48) i , q (cid:48) i +1 , q (cid:48) i +2 ), with the constraintthat the transformation ( q i , q i +1 , q i +2 ) (cid:55)−→ ( q (cid:48) i , q (cid:48) i +1 , q (cid:48) i +2 ) could be performed continuously in thesubsystem defined by the three charges only, and without violating the (local) charge and energyconstraints. This Monte Carlo algorithm, while providing a good description for the dynamics ofa chain (which was the main concern of those papers), does not connect with the dynamics in anyother dimensionality, where the localization (or Gibbs/non-Gibbs) transition takes place as well.Since it would be difficult to connect this “chain” dynamics to the Brownian motion (which insteadis related to the eigenvalues of the Laplacian), we decided to simulate directly the latter.In Fig. 8 we show the time evolution of the (connected) correlation function G ( t ) := 1 N N (cid:88) i =1 (cid:10) q i ( t + t (cid:48) ) q i ( t (cid:48) ) (cid:11) t (cid:48) − N N (cid:88) i =1 (cid:10) q i ( t + t (cid:48) ) (cid:11) t (cid:48) (cid:10) q i ( t (cid:48) ) (cid:11) t (cid:48) (20)where the angular brackets denote averaging wrt. the time variable in the subscript: (cid:10) A ( t (cid:48) ) (cid:11) t (cid:48) := lim T f →∞ T f (cid:90) T f A ( t (cid:48) ) dt (cid:48) . At t = 0, G (0) = ( ε −
1) because of the second constraint in Eq. (18). Taking t → ∞ , instead, if thedynamics on the manifold is ergodic it holds G ( ∞ ) = 0. Conversely, if ergodicity is broken eitherbecause M ε is disconnected in pieces or because the dynamics is effectively confined in a smallerregion, it holds G ( t ) → const >
0. According to the discussion before, M ε becomes disconnectedfor ε > N/
2. At this point there is a geometric obstruction to ergodicity: a trajectory startingin a neighborhood of, say, q = O ( N ) cannot reach the neighborhood of any other q i = O ( N )with i (cid:54) = 1. Therefore, for these (large) values of ε the correlation function does not get to 0 as t → ∞ . Before then (viz. for any finite N , and any ε < N/
2) there is always a finite time scale τ , after which the function G ( t ) does get close to 0. This correlation time is a good proxy for anequilibrium time (since the charges q i are the only observables of the systems).We also note that G ( t ) must decay exponentially in t (after, of course, a possible initialtransient). This is due to the fact that a diffusion equation is associated to the Brownian motion: ∂P ( q, t ) ∂t = ∆ P ( q, t ) , (21) ocalization in the DNLSE and geometric properties of the microcanonical surface N (a) =2.70=2.50=2.30=2.10=1.90=1.70 Figure 9. (a) Correlation times τ extracted from the exponential decay of the correlationfunction, Eq. (20): we can see that τ diverges in the thermodynamic limit as ε becomes greaterthan 2 (not all datasets are shown here to improve readability). We have also performed fits(dashed lines): for ε >
2, we employed log τ = γN + c , from which we extracted the γ ’s presentedin the right panel. For ε <
2, instead, since τ is almost constant with N we accounted for finite-size effects by fitting τ = w/ log( N ) + τ : the τ ’s obtained are displayed in the inset of the rightpanel.(b) Exponent γ as a function of ε (black dots). It can be clearly seen that γ = 0 within errors for ε <
2, while γ > ε >
2. The orange, dashed line is a fit of the form log γ = η log( ε −
2) + h ,yielding η = 1 . ± . h = − . ± . Inset : The value of τ (cid:39) τ diverges logarithmicallyin the limit ε → − (see, for a comparison at the critical point, Fig. 10a). The dashed line is afit τ = − ζ log(2 − ε ) + u , from which we find ζ = 1 . ± .
02 and u = 0 . ± . where the Laplacian has the usual definition in curvilinear coordinates ∆ := g − / ∂ a ( g / g ab ∂ b ).The smallest eigenvalue of − ∆ is λ = 0, and the corresponding (properly normalized) eigenvectoris nothing but the uniform (microcanonical) distribution P ( q, t → ∞ ) = φ ( q ). Since theLaplacian on a compact Riemannian manifold has a pure point spectrum, the first eigenvalue λ > λ =: 1 /τ, controls the asymptotic decay of P ( q, t ) (cid:39) φ ( q ) + c φ ( q ) e − t/τ + ... . In particular, this means that G ( t ) ∼ e − t/τ for large t ,as claimed before.The exponential decay can be seen clearly in Fig. 8a. Within the exponential form, one candistinguish two cases: for ε < N → ∞ ; for ε ≥ longer timescale and there is no obvious limit N → ∞ . This corresponds to the following statement onthe spectrum of the Laplacian: for ε < N → ∞ , while for ε ≥ N . We find numerically that the gap closes exponentially with N : τ (cid:39) e γN forsome rate γ >
0, see Fig. 9. This fact implies that ε = 2 is a dynamical critical point, at whichthe dynamics becomes scale-invariant [56].More precisely, for ε < τ is constant with N and grows with ε , ultimatelydiverging logarithmically as ε → − (see the inset of Fig. 9b). For ε >
2, it is γ ( ε ) which growswith ε : γ ∼ ( ε − η with η (cid:39) . ± . ε = 2 the dynamics is ergodic and G ( t ) → N and also for the limit N → ∞ ; to the right, instead, the relaxation becomes progressivelyslower as N increases and in the limit N → ∞ it holds G ( t ) → const >
0. The limiting functionalform at ε = 2 must be a function decaying to 0, but slower than an exponential. We checkedthat, in a small right neighborhood of ε = 2, the fitting function G ( t ) = ( ε − e − t/τ / (1 + ( t/t ) z )works pretty accurately with z ∈ [0 . , .
0] depending on the values of N (see Figs. 8b and 10 for thedetails). However, for this estimate to be useful one must have t (cid:28) τ → ∞ , to ensure a sufficientlylarge fitting window. Since τ grows with N (albeit only logarithmically) while t decreases (seeFig. 10a), it will eventually hold t (cid:28) τ . Unfortunately, this crossover takes place roughly atthe largest system sizes we were able to simulate, so the values of z we can extract cannot beconsidered precise. By using only the points in the “asymptotic region” (shaded region in Fig. 10), z (cid:39) .
0, while smaller values of N (non-shaded region) we have a considerably smaller z (cid:39) . ocalization in the DNLSE and geometric properties of the microcanonical surface N t i m e s c a l e s (a) t N z (b)160 N N Figure 10. (a) Typical timescales of the exponential ( τ ) and power-law ( t ) decay of thecorrelation function at the critical point ε = 2, found from the fits of the correlation function G ( t ) (Fig. 8b). We see that the crossover to power-law decay takes place at N (cid:38) z .(b) Corresponding values of the dynamical critical exponent z . The errorbars represent the fiterrors, that surely underestimate the strong fluctuations at finite N (see also [31]). Therefore,we can only present two possible fits, one excluding the smallest N points and one the largest,and give a value of z respectively z (cid:39) . z (cid:39) .
75. Since the smallest N points have t > τ (as seen in panel (a)), which is a clear pre-asymptotic behavior, we would tend to discard themin favour of the 3 largest N points in the dataset. The precise values of the critical exponents clearly requires further numerical investigations. Weend by noticing that the form of G ( t ) at criticality can be related to the distribution ρ ( λ ) of theeigenvalues of the Laplacian at ε = 2, which must be of the form ρ ( λ ) ∼ λ z − near λ = 0.
8. Conclusions and Outlook
We have studied the mechanism for weak ergodicity breaking at high energy densities in a fullyconnected DNLS model. We have shown that, whatever the interactions between sites are, they canbe neglected for ε ≥ ε s = 1 . ... that corresponds to a finite temperature T s = 2 g = 2. We are lefttherefore with a purely potential model, whose physical properties reflect the geometrical propertiesof the potential energy surface and therefore are subject to a localization transition at infinitetemperature (corresponding to ε c = v = 2). After proving that the microcanonical, potentialenergy surface is connected for all extensive energies (therefore energy densities of O (1)), we showthat the localization transition is due to a phase transition in the order parameter γ = − (log λ ) /N ,where λ is the smallest non-zero eigenvalue of − ∆, the laplacian on the (curved) equipotentialsurface. For ε < γ = 0 and for ε > γ ∼ ( ε − η with η around 2. Thisputs on firmer ground the connection between the works on thermodynamics (like [31]) and thoseon the dynamics (like [38, 45]). The approximation in which one can neglect the kinetic energyis exact on the fully connected model, and one can imagine that it is a good approximation for afinite-dimensional lattice. Whether this is compelling for a d = 1 chain geometry (where numericalintegrations of the Hamilton equations seem to show that the transition is at a higher value of theenergy density [44]) we cannot say.The transition taking place at ε c = 2 makes the equilibration time τ change from O (1) toexponentially large in N , τ ∼ e γN : the phenomenology is very similar to that of the MBL-likephase of Josephson junction arrays [15, 16], and of quantum glasses as well [57–60]. However, thenature of the present transition seems to be of entropic origin: the volume of the region of phasespace around any given localized configuration is exponentially larger than the volume connectingtwo localized configurations, therefore making the passage from one localized configuration toanother exponentially unlikely. Quantum mechanical localization, in contrast, is a consequence ofinterference and it vanishes when (cid:126) →
0. It is also tempting to notice that the lowest eigenvalueof − ∆ becoming exponentially small in a large parameter is precisely what happens in localized EFERENCES (cid:126) . Previous works have shownthat, for (cid:126) (cid:54) = 0 and at least in a 1d geometry, transport is strongly suppressed as T → ∞ [61] andnon-Gibbs state exist for ε ≥
9. Acknowledgements
We would like to thank Sergio Caracciolo, Rosario Fazio, Giacomo Gradenigo and Sergej Flach fordiscussion during the early stages of the work. C.V. thanks also ICTP for hospitality during theinitial part of this work. This work is supported by the Trieste Institute of Quantum Technolgies(TQT).
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EFERENCES Appendix A. The topology of the equipotential manifolds
Here, we want to provide a detailed description of the topology of the manifolds M ε determinedby equation (18), as outlined in Sec. 6 in terms of Morse theory. Actually, in order to avoid thetechnicalities of stratified Morse theory, we prefer to adopt a different, more direct approach, usingonly basic notions and results of piecewise topology, for which we refer to [62].According to (18), we can think of M ε as the intersection between the sphere S N − ε ⊂ R N centered at the origin of radius √ εN and the simplex ∆ N − ⊂ R N affinely spanned by the vectors N e , . . . , N e N , where e , . . . , e N is the canonical base of R N .Given any k -dimensional sub-simplex Σ = (cid:104) N e i , . . . , N e i k +1 (cid:105) ⊂ ∆ N − with 0 ≤ k ≤ N − b (Σ) = N ( e i + · · · + e i k +1 ) / ( k + 1) be the barycenter of Σ. We indicate by Γ the barycentricsubdivision of ∆ N − , whose k -simplices are given by (cid:104) b (Σ i ) , . . . , b (Σ i k +1 ) (cid:105) for any ascending chainΣ i ⊂ · · · ⊂ Σ i k +1 of sub-simplices of ∆ N − . Moreover, for every 0 ≤ k ≤ N −
1, let Γ k and Γ k denote the sub-complexes of Γ consisting of all simplices (cid:104) b (Σ i ) , . . . , b (Σ i (cid:96) +1 ) (cid:105) such thatdim Σ i j ≤ k and dim Σ i j ≥ k + 1, respectively.We observe that Γ k coincides with the barycentric subdivision of the k -skeleton ∆ N − k of∆ N − , hence dim Γ k = k , while Γ k is the sub-complex of Γ consisting of all the simplices that aredisjoint from ∆ N − k , and dim Γ k = N − k −
2. Furthermore, Γ can be expressed as the affine joinΓ = Γ k ∗ Γ k in R N , that is for every simplex Σ ∈ Γ we have Σ = (Σ ∩ Γ k ) ∗ (Σ ∩ Γ k ). Then, thereis a well-defined pseudo-radial projection π k : Γ − Γ k → Γ k , which collapses Σ − Γ k to Σ ∩ Γ k forevery simplex Σ of Γ.Now, we can start our description of M ε . First of all, we note that M ε is empty for ε < ε > N , while it consists of the single point b (∆ N − ) for ε = 1 and of the N vertices of ∆ N − for ε = N .When ε ranges in the interval [1 , N ] the sphere S N − ε meets transversally each subsimplexof ∆ N − , except for ε = N/k with k = 1 , . . . , N , in which case S N − ε is tangent to all the (cid:0) Nk (cid:1) subsimplices of ∆ N − of dimension k − M ε can beendowed with a structure of stratified space, whose strata are the components of the intersectionsof S N − ε with the open simplices of ∆ N − . Moreover, such structure is the same up to smoothisomorphism for all ε in each open interval ( N/ ( k + 1) , N/k ) with k = 1 , . . . , N − ε ∈ ( N/ ( k + 1) , N/k ) with k = 1 , . . . , N −
1. In order to describe M ε , we considerthe affine subspace A N − ⊂ R N spanned by N e , . . . , N e N , the ( N − S N − ε = S N − ε ∩ A N − , and the ( N − B N − ε = B Nε ∩ A N − bounded by S N − ε in A N − , where B Nε ⊂ R N is the N -cell centered at the origin of radius √ εN .The inequality ε > N/ ( k + 1) implies that Γ k − ⊂ Int B N − ε , since all the vertices of Γ k − belong to Int B Nε . On the other hand, the inequality ε < N/k implies that Γ k − ⊂ A N − − B N − ε ,being d (0 , Σ) = (cid:107) b (Σ) (cid:107) = N/k for any ( k − N − . Therefore, S N − ε transversally meets in a single point each segment (cid:104) p, q (cid:105) ⊂ Σ with p ∈ Γ k − , q ∈ Γ k − and Σa simplex of Γ. Hence, M ε is pseudo-radially equivalent, to the boundary Bd N (Γ k − , Γ) of aregular neighborhood N (Γ k − , Γ) of Γ k − in Γ. Finally, due to the inclusion Γ k − ⊂ Bd Γ, we canconclude that M ε is topologically equivalent to a regular neighborhood N (Γ k − , Bd Γ) of Γ k − inBd Γ. Notice that N (Γ k − , Bd Γ) coincides with the suplevel set M ε ( ϕ ) considered in Sec. 6.Of course, up to radial projection in A N − centered at b (∆ N − ), we can identify Bd ∆ N − with S N − ε and Γ k − with a sub-complex Γ k − ,ε ⊂ S N − ε , in such a way that M ε turns out tobe topologically equivalent to a regular neighborhood N (Γ k − ,ε , S N − ε ) of Γ k − ,ε in S N − ε . Then,there is a collapse M ε (cid:38) Γ k − ,ε ∼ = Γ k − .The argument above also applies to the case of ε = N/k , with the only difference that in thiscase the regular neighborhoods N (Γ k − , Bd Γ) and N (Γ k − ,ε , S N − ε ) are relative to the 0-dimen-sional subcomplex { b (Σ) | Σ is a ( k − N − } , but still we have M ε (cid:38) Γ k − ,ε ∼ = Γ k − .Summarizing, M consists of a single point and M ε is homotopically equivalent to ∆ N − k − forevery N/ ( k + 1) < ε ≤ N/k and k = 1 , . . . , N −
1. In particular, M ε is connected non-empty if1 ≤ ε ≤ N/