Haut PDF Free energy topological expansion for the 2-matrix model.

Free energy topological expansion for the 2-matrix model.

Free energy topological expansion for the 2-matrix model.

In parallel, the hermitian two-matrix model (2MM) solutions have been obtained, at early stages, in the planar limit of the 1/N-expansion [34,21]. It was however almost immediately observed [5, 18, 21] that the 2MM solution in the planar limit enjoys the same geometrical properties as the 1MM solution; only the spectral curve becomes an arbitrary algebraic curve, not just an hyperellitic curve arising in the 1MM case. The subsequent progress was however hindered by that the corresponding master loop equation in the 2MM case cannot immediately be expressed in terms of correlation functions alone. Nevertheless, using the geometrical properties of this equation, the solution in the first subleading order has been constructed [19, 20] on the base of knowledge of Bergmann tau function on Hurwitz spaces. However, the general belief arose that the 2MM case should not be very much different from the 1MM case, that is, we expect to find all ingredients of the 1MM solution in the 2MM case. Next step towards constructing the topological expansion of the 2MM was performed in [24], where the first variant of the diagrammatic technique for the correlation functions in the 2MM case was constructed. In the present paper we improve the technique of [24] (actually effectively simplifying it) to accommodate the action of the loop insertion operator. In fact, we demonstrate that the diagrammatic technique for the 2MM case is even closer to the one in the 1MM case that was before: in particular, we need only three-valent vertices, and the additional operator H we need to obtain the free energy turns out to be of the same origin as the one in the 1MM.
En savoir plus

32 En savoir plus

Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants.

Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants.

of the matrices, and could not be computed by standard methods. The first explicit computations were obtained in [4] and [17]. Here in this paper, we show how to compute the topological expansion of a family of mixed correlation functions of the 2-matrix model. In a coming work [23], we shall show how to compute all mixed correlations, and introduce a link with group theory and Bethe ansatz (this is a generalization of [22]).

38 En savoir plus

Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

Recently, it has progressively become clear that large N expansion of random matrix models has a strong link with algebraic geometry [27]. The free energy and correlation functions have been computed in terms of properties of an algebraic curve. The large N limit of the 1-point correlation function (called the resolvent) is solution of an algebraic equation, which thus defines an algebraic curve. There have been many works which computed free energy and correlation functions in terms of that algebraic curve. The leading order resolvent and free energy were computed in the 1-cut case (algebraic curve of genus zero) in the pioneering work of [7], then some recursive method for computing correlation functions and free energy to all orders in 1/N were invented by [3, 4]. Those methods were first limited to 1-matrix case and 1-cut.
En savoir plus

50 En savoir plus

Invariants of algebraic curves and topological expansion

Invariants of algebraic curves and topological expansion

What is more interesting is to see what is Z N (E) for curves not coming from the large N limit of the loop equations of a matrix model. We study in details a few examples. - The double scaling limit of a matrix model. It has been well known since [51, 26], that if we fine tune the parameters of a matrix model so that the algebraic curve E develops a singularity, the free energies become singular and the most singular part of the free energies form the KP-hierarchy τ -function (KdV hierarchy for the 1-matrix model). We show, by looking at the double scaling limit of matrix models, that the KP τ -function (resp. KdV τ -function), coincides with our definition for the classical limit of the (p, q) systems (resp. (p, 2)).
En savoir plus

94 En savoir plus

Loop equations for the semiclassical 2-matrix model with hard edges

Loop equations for the semiclassical 2-matrix model with hard edges

Let us also remark that the function eq.2-27 E(x, y) is unchanged under the exchange x ↔ y, V1 ↔ V2, s ↔ ˜ s, which is the generalization of Matytsin’s duality property [17]: X(Y (x)) = x (4-1) The consequences of that algebraic equation, can then be studied. This is done for instance in [3]. One can also expect to generalize the works of [9], or [7], or [8], i.e. the computation of all correlation functions and their 1/N 2 expansion, and further, compute the expansion of the free energy [10, 11].
En savoir plus

21 En savoir plus

A matrix model for the topological string II: The spectral curve and mirror geometry

A matrix model for the topological string II: The spectral curve and mirror geometry

matrix model M X and to compare it to the mirror curve S X of X. Recall that M X is a chain of matrices matrix model with non-polyno- mial potential. The problem of determining the spectral curve of simple matrix models, such as the 1-matrix model, was solved long ago [ 4 , 27 ]. The problem becomes more difficult as the complexity of the matrix model increases. A pro- cedure for determining the spectral curve for a chain of matrices with polyno- mial potentials exists in the literature [ 2 , 3 ], and is rather easy to apply to short chains of matrice, and when the potentials have low degree. It can be extended in a straightforward manner to potentials whose derivative is rational. Here, our matrix model M X is a chain of matrices of arbitrary length, and the poten-
En savoir plus

41 En savoir plus

Topological expansion of the chain of matrices

Topological expansion of the chain of matrices

fields also have some combinatorial interpretations, and have been studied for various applications. The most famous is the Kontsevich integral, which is the generating function for intersection numbers [33, 24, 19]. Here, we solve this more general model. Multimatrix model also play an important role in quantum gravity and string theory, where they play the role of a regularized discrete space-time. The 1-matrix model, counts discrete surfaces without color, and is a model for quantum gravity without matter, whereas the chain of matrices counts discrete surfaces with n colors, and is interpreted as a model of quantum gravity with some matter field [27, 10, 1, 29, 30, 13], namely a matter which can have n possible states. More recently, matrix models have played a role in topological string theory [12].
En savoir plus

25 En savoir plus

Evaluating the flow stress of aerospace alloys for tube hydroforming process by free expansion testing

Evaluating the flow stress of aerospace alloys for tube hydroforming process by free expansion testing

modeled as a rigid body and the tube as a deformable material. In the model, a surface-to-surface contact algorithm was ap- plied to the interface between the tube and the die with Coulomb’s friction set to 0.05 in accordance with different reported values in the literature for an unlubricated condition [ 4 , 22 ]. To mimic the experimental loading conditions in the FE model, the internal pressure was applied on each element of the meshed tube and increased linearly, while fixing the end nodes of the tube to emulate the no end feeding condition used during the free expansion process. The simulation results for the bulge height versus internal pressure were computed up to the maximum bulge height obtained experimentally. It is noteworthy that in the present work, the effect of anisotropy and, in the case of SS 304L only, the welded seam on the mechanical response was not considered in the FE model of the free expansion process. To this end, the effective true stress–true stain data employed in the FE model were based on the formulations given in Table 2 (spline profile) for the different alloys studied in this work. To understand the influ- ence of different material models on the FE simulation results, the uniaxial true stress–true strain curves as formulated in Table 3 were also implemented in the free expansion model. In addition the influence of the bulge profile on the FE simulation results was considered using the formulations in Table 4 (elliptical profile).
En savoir plus

13 En savoir plus

Rapid prediction of solvation free energy. 2. The first-shell hydration (FiSH) continuum model

Rapid prediction of solvation free energy. 2. The first-shell hydration (FiSH) continuum model

These results highlight the benefits of ISCD-dependent Born radii to account for charge asymmetry effects, as well as the improvements afforded by training on real molecules for the nonlinear model. The linear correction function appears to provide robust and competitive results when compared with the nonlinear correction functions. However, as presented in the Theory and Implementation section, our computational experiments on spheres clearly show that correction should be nonlinear with respect to the induced surface charge density. The linear function most likely gives good results since the ISCD range explored by the neutral molecules in our data sets is rather narrow (between -0.01 and +0.01 e/Å 2 , see Figure S2, Supporting Information), for which the linear approximation is still applicable (see Figure 2). With charged molecules, the range of ISCD will be expanded and the linear correction will most likely fail. For example, the nitrogen of a terminal alkyl ammonium would have an ISCD of around -0.025 e/Å 2 , which falls outside of the linear region seen in Figure 2 and justifies the use of a nonlinear function. A more complete study (outside the scope of this work) is needed for charged molecules, but for now, the nonlinear model seems most appropriate because of its greater generality. Given the comparable performances obtained with the two nonlinear correction functions, the rational function (eq 5) is preferred over the arctan + Gaussian function (eq 4) due to a lower number of fitted parameters and will be featured for the rest of the paper. The correlation between the FiSH continuum electrostatic component and the explicit-solvent LIE electrostatic term for the training, testing, and SAMPL1 data sets is shown in Figure 4.
En savoir plus

17 En savoir plus

The sinelaw gap probability, Painlevé 5, and asymptotic expansion by the topological recursion

The sinelaw gap probability, Painlevé 5, and asymptotic expansion by the topological recursion

Math´ ematiques, Montr´ eal. 2 ? Institut de physique th´ eorique, CEA Saclay, France. 3 Abstract: The goal of this article is to rederive the connection between the Painlev´ e 5 integrable system and the universal eigenvalues correlation functions of double-scaled hermi- tian matrix models, through the topological recursion method. More specifically we prove, to all orders, that the WKB asymptotic expansions of the τ -function as well as of determinan- tal formulas arising from the Painlev´ e 5 Lax pair are identical to the large N double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bulk. In other words, we rederive the “sine-law” universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to O(N −5 ).
En savoir plus

39 En savoir plus

Topological expansion and boundary conditions

Topological expansion and boundary conditions

The result seems to have a nice combinatorial interpretation, as all the possibilities of drawing interfaces (between the + and - spins of the Ising model) in all possible ways. However, a combinatorial derivation is missing. Also, our result can have interpretations in conformal field theories when one goes to the so called double-scaling-limit [6, 5], and should be compared with recent results from Liouville theory [19, 16, 2]. In particular, in [2], our formula for planar disc amplitudes is interpreted in terms of the interactions of long folded strings and it would be interesting to check the non-planar cases as well.
En savoir plus

26 En savoir plus

Topological superconductivity in the one-dimensional interacting Creutz model

Topological superconductivity in the one-dimensional interacting Creutz model

4.1. Phase diagram Figure 1 shows the topological phase diagram for different values of interaction U , combining results from the mean-field analysis and DMRG simulations. The agreement between both approaches is overall very good. The main standout feature of the phase diagram is that the topological phase and its associated Majorana bound states at zero energy are robust to interactions. In fact, the parameter range for which a topological Majorana phase is stabilized globally expands, upon increasing the Hubbard coupling. However, there is also a region in the phase diagram (|∆/w|, |g/w| < 1), where the Hubbard interaction is detrimental to the Majoranas. The mean-field phase diagram captures the topological transitions of the model rather accurately for small and moderate interactions. At strong interactions, the mean field keeps a very good estimate of the topological transition at small g/w, while at large g/w it tends to overestimate the extension of the Majorana phase. The “spin-orbit” coupling g tends to delocalize the electrons and leads to quantum fluctuations in the particle number. This explains, at a qualitative level, the divergence of the mean field results from the “exact” DMRG results at large “spin-orbit” coupling. A priori, this overall very good agreement may seem surprising in a quantum 1D system, where fluctuations are expected to be pronounced. However, let us recall that U(1) charge and the SU(2) spin-rotation symmetries are broken in this model, and hence International Conference on Strongly Correlated Electron Systems 2014 (SCES2014) IOP Publishing Journal of Physics: Conference Series 592 (2015) 012133 doi:10.1088/1742-6596/592/1/012133
En savoir plus

8 En savoir plus

Free-energy-dissipative schemes for the Oldroyd-B model

Free-energy-dissipative schemes for the Oldroyd-B model

1.3 Outline of the paper and results We will show that it is possible to build numerical schemes discretizing the Oldroyd-B system (1)– (2) such that solutions to those discretizations satisfy a free energy estimate similar to that estab- lished in [24, 20] for smooth solutions to the continuous equations. Our approach bears similarity with [36], where the authors also derive a discretization that preserves an energy estimate satisfied at the continuous level, and with [32], where another discretization is proposed for the same energy estimate as in [36]. Yet, unlike the estimates in [36, 32], our estimate, the so-called free energy estimate derived in [24, 20], ensures the long-time stability of solutions. As mentioned above, long-time computations are indeed often used to obtain a stationary state.
En savoir plus

57 En savoir plus

Topological expansion of the β-ensemble model and quantum algebraic geometry in the sectorwise approach

Topological expansion of the β-ensemble model and quantum algebraic geometry in the sectorwise approach

CNRS, URA 2306, F-91191 Gif-sur-Yvette, France. Abstract We solve the loop equations of the β-ensemble model analogously to the solution found for the Hermitian matrices β = 1. For β = 1, the solution was expressed using the algebraic spectral curve of equation y 2 = U (x). For arbitrary β, the spectral curve converts into a Schr¨ odinger equation ((~∂) 2 −U(x))ψ(x) = 0 with ~ ∝ ( √ β −1/ √ β)/N . This paper is similar to the sister paper I, in particular, all the main ingredients specific for the algebraic solution of the problem remain the same, but here we present the second approach to finding a solution of loop equations using sectorwise definition of resolvents. Being technically more involved, it allows to define consistently the B-cycle structure of the obtained quantum algebraic curve (a D-module of the form y 2 −U(x), where [y, x] = ~) and to construct explicitly the correlation functions and the corresponding symplectic invariants F h , or the terms of the free energy, in 1/N 2 -expansion at arbitrary ~.
En savoir plus

59 En savoir plus

Master loop equations, free energy and correlations for the chain of matrices.

Master loop equations, free energy and correlations for the chain of matrices.

Each Mk lies in the potential well Vk and is linearly coupled to its neigh- boors Mk−1 and Mk+1. M0 and MN have only one neighboor, and thus the chain is open. So far, the problem of the closed chain has remained unsolved. The multimatrix model is a generalization of the 2-matrix model. It was often considered in the context of 2-dimensional quantum gravity and string theory. Its critical points are known to represent the minimal conformal field theories, characterized by a pair of integers (p, q). It is known that one can get a (p, q) critical point with a multimatrix model where N = q − 2. The necessity of studying multimatrix models can be understood from the fact that the one matrix models contains only the critical models with q ≤ 2.
En savoir plus

56 En savoir plus

Large N expansion of the 2-matrix model, multicut case.

Large N expansion of the 2-matrix model, multicut case.

A physical picture is that the support of the large N average density of eigenvalues of the matrix M 1 is made of g + 1 intervals [a i , b i ], i = 1, . . . , g + 1, and for each i, A i is a contour which encloses [a i , b i ] in the trigonometric direction, and which does not enclose the other a j or b j with j 6= i. eq. (3.3) means that the interval [a i , b i ] contains a

34 En savoir plus

Topological expansion in isomorphisms with random walks for matrix valued fields

Topological expansion in isomorphisms with random walks for matrix valued fields

xTrpΦpxqq TrpΦpyqqy U β,n “ nGpx, yq.  4. Proofs 4.1. Proof of Theorem 3.2. We give the proof of Theorem 3.2, which contains Theorem 3.1 as a special case. A possible approach would be to develop the product of traces on the left-hand side of the isomorphism identity, apply Theorem 2.1, and then recombine the terms to get the right-hand side of the isomorphism identity. This works well for β P t1, 2u (see [Zvo97] for one matrix integrals), but for β “ 4 this approach is less tractable because of the non-commutativity of quaternions. Instead, we will rely on a recurrence over the numbers of edges |ν|{2, similar to that in [BP09].
En savoir plus

24 En savoir plus

Phase structure of topological insulators by lattice strong-coupling expansion

Phase structure of topological insulators by lattice strong-coupling expansion

In this paper, we focus on such an effect of electron correlation on the phase structure of topological insulators, described by the boundary between the topological insulator and the normal insulator phases. We formulate the models of topological insulators in terms of massive lattice femions: in 2-dimensions (2D), we take the Kane–Melé model for quantum spin Hall (QSH) in- sulator, one of the classes of topological insulators in 2D, which is a straightforward extension of the conventional tight-binding model of graphene. In 3D, we make use of the Wilson fermion on the hypothetical square lattice. As in the previous studies on graphene, we incorporate the electron-electron interaction mediated by the electromagnetic field in terms of lattice gauge theory, and analyze the emergent orders by the strong coupling expansion methods. As a result, we find a new phase with an in-plane antiferromagnetism in 2D, which appears by the similar mechanism as that of the pion condensate phase (so-called “Aoki phase”) in lattice QCD with Wilson fermion [9]. On the other hand, such a phase does not appear in 3D, and the electron correlation results in the shifting of the topological phase boundary [10].
En savoir plus

8 En savoir plus

A Matrix Model for the Topological String I: Deriving the Matrix Model

A Matrix Model for the Topological String I: Deriving the Matrix Model

conjecture is that it provides a systematic method for computing the topological string partition function, genus by genus, away from the large radius limit, and without having to solve differential equations. This conjecture was motivated by the fact that symplectic invariants have many intriguing properties reminiscent of the topological string free energies. They are invariant under transformations S → ˜ S which conserve the symplectic form dx ∧ dy = d˜ x ∧ d˜ y, whence their name [19]. They satisfy holomorphic anomaly equations [41], they have an integrable structure similar to Givental’s formulae [42, 43, 44, 45, 46], they satisfy some special geometry relations, WDVV relations [47], and they give the Witten-Kontsevich theory as a special case [19, 48].
En savoir plus

36 En savoir plus

Topological model for machining of parts with complex shapes

Topological model for machining of parts with complex shapes

2 recovery is associated to transition shapes which generally have low curvature and small areas. It is performed by cutting tools with small diameters [7]. Implementation of the HSM process therefore requires the development of a complex process planning that is very different from that presented in most of the work on prismatic parts machining [8] [9]. Indeed a process planning for prismatic parts defines the sequence of simple machining operations, each including the geometrical feature and machining parameters related to machine tool, cutting tool and cutting conditions [10]. Sequence of operations is based on features accessibility constructed from the adjacency graph describing parts topology [11]. In the case of dies, it is difficult to identify geometrical features which are the base of the process planning. Several works have been made in the extraction of geometric characteristics of complex shapes. They are generally oriented design support of parts [12] [13], determination of cutting tool orientation to improve cutting conditions 14] and tools paths generation without planning or scheduling [15] [16]. In the case this paper is focused, complex shapes processed correspond to those found on the forging dies. Works results in this field allow identifying machined surfaces, cutting tools used, but with difficulty machining limits. At this stage, basic machining operations can be generated but it is difficult to ensure that tools paths are gouge-free and to optimize trajectories, in particular those out of material. Adjacency graph of surfaces and shapes associated with an evaluation of their sizes and positions is expected to solve the difficulties. But the concavity test used to build adjacency graph for prismatic parts is not applicable to complex shapes (large variations in curvature and normal direction). It is therefore appropriate to see how topological information required to generate process planning of complex shapes can be extracted or defined from CAD model.
En savoir plus

23 En savoir plus

Show all 10000 documents...