Landau-Ginzburg Potential Research Articles

Order of the SU(Nf)×SU(Nf) chiral transition via the functional renormalization group

Renormalization group flows of the SU(Nf)×SU(Nf) symmetric Ginzburg-Landau potential are calculated for a general number of flavors, Nf. Our approach does not rely on the ε expansion, but uses the functional renormalization group, formulated directly in d=3 spatial dimensions, with the inclusion of all possible (perturbatively) relevant and marginal operators, whose number is considerably larger than those in d=4. We find new, potentially infrared stable fixed points spanned throughout the entire Nf range. By conjecturing that the thermal chiral transition is governed by these “flavor continuous” fixed points, stability analyses show that for Nf≥5 the chiral transition is of second order, while for Nf=2, 3, 4, it is of first order. We argue that the UA(1) anomaly controls the strength of the first-order chiral transition for Nf=2, 3, 4, and makes it almost indistinguishable from a second-order one, if it is sufficiently weak at the critical point. This could open up a new strategy to investigate the strength of the UA(1) symmetry breaking around the critical temperature. Published by the American Physical Society 2024

The Proper Landau–Ginzburg Potential, Intrinsic Mirror Symmetry and the Relative Mirror Map

Given a smooth log Calabi–Yau pair (X, D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau–Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (X, D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of Fan et al. (Sel Math (NS) 25(4): Art. 54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z) to compute two-point relative invariants. When D is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of m Gräfnitz et al. (2022) that the proper Landau–Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.

Existence of heteroclinic and saddle-type solutions for a class of quasilinear problems in whole ℝ2

In this work, we use variational methods to prove the existence of heteroclinic and saddle-type solutions for a class quasilinear elliptic equations of the form [Formula: see text] where [Formula: see text] is a N-function, [Formula: see text] is a periodic positive function and [Formula: see text] is modeled on the Ginzburg–Landau potential. In particular, our main result includes the case of the potential [Formula: see text], which reduces to the classical double well Ginzburg–Landau potential when [Formula: see text], that is, when we are working with the Laplacian operator.

Probing the Ginzburg–Landau Potential for Lasers Using Higher-order Photon Correlations

Lasing transition is known to be analogous to the second-order phase transition. Furthermore, for some cases, it is possible to define the Ginzburg-Landau (GL) potential, and the GL theory predicts the photon statistical properties of lasers. However, the GL potential for lasers is surprising, because lasers are operating far from equilibrium. In this paper, we theoretically examine the validity of the GL theory for lasers in terms of various parameters, particularly, the ratio between photon and carrier lifetimes. For this purpose, we use stochastic rate equations and higher-order photon correlation functions. With higher-order photon correlation measurements, we can check whether or not laser dynamics are described by the GL theory. We demonstrate that, for low-$\beta$ lasers, the GL theory is applicable even when the photon lifetime is comparable to the carrier lifetime and that photon-carrier relaxation oscillation is the fundamental origin of the breakdown of the GL theory, which can be understood in the framework of center manifold reduction.

Second-order chiral phase transition in three-flavor quantum chromodynamics?

We calculate the renormalization group flows of all perturbatively renormalizable interactions in the three-dimensional Ginzburg-Landau potential for the chiral phase transition of three-flavor quantum chromodynamics. On the contrary to the common belief we find a fixed point in the system that is able to describe a second-order phase transition in the infrared. This shows that long-standing assumptions on the transition order might be false. If the transition is indeed of second order, our results may hint that the axial $U(1)$ symmetry restores at the transition temperature.

An Abelian Higgs model of pulsed field magnetization in superconductors

Pulsed field magnetization leads to trapped magnetic field persistent for long times. We present a one-dimensional model of the interaction between an electromagnetic wave and a superconducting slab based on the Maxwell-Ginzburg-Landau (Abelian Higgs) theory. We first derive the model starting from a Lagrangian coupling the electromagnetic field with the Ginzburg-Landau potential for the superconductor. Then we explore numerically its capabilities by applying a Gaussian vector potential pulse and monitoring usual quantities such as the modulus and the phase of the order parameter. We also introduce defects in the computational domain. We show that the presence of defects enhances the remanent vector potential and diminishes the modulus of the order parameter, in agreement with existing experiments.

Homological mirror symmetry of CPn and their products via Morse homotopy

We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many examples, the derived category Db(coh(X)) of coherent sheaves on a toric manifold X is compared with the Fukaya–Seidel category of the Milnor fiber of the corresponding Landau–Ginzburg potential. We instead consider the dual torus fibration π: M → B of the complement of the toric divisors in X, where B̄ is the dual polytope of the toric manifold X. A natural formulation of homological mirror symmetry in this setup is to define Fuk(M̄) a variant of the Fukaya category and show the equivalence Db(coh(X))≃Db(Fuk(M̄)). As an intermediate step, we construct the category Mo(P) of weighted Morse homotopy on P≔B̄ as a natural generalization of the weighted Fukaya–Oh category proposed by Kontsevich and Soibelman [Symplectic geometry and mirror symmetry (Seoul, 2000) (World Science Publishing, 2001), p. 203]. We then show that a full subcategory MoE(P) of Mo(P) generates Db(coh(X)) for the cases X is a complex projective space and their products.

Hodge-theoretic mirror symmetry for toric stacks

Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.

Reconstructing GKZ via Topological Recursion

In this article, a novel description of the hypergeometric differential equation found from Gel’fand–Kapranov–Zelevinsky’s system (referred to as GKZ equation) for Givental’s J-function in the Gromov–Witten theory will be proposed. The GKZ equation involves a parameter $$\hbar $$, and we will reconstruct it as a quantum curve from the classical limit $$\hbar \rightarrow 0$$ via the topological recursion. In this analysis, the spectral curve (referred to as GKZ curve) plays a central role, and it can be described by the critical point set of the mirror Landau–Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes phenomenon for the equivariant $${\mathbb {C}}\mathbf{P }^{1}$$ model, and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis, we will study Dubrovin’s conjecture for this equivariant model.

Existence of heteroclinic solutions for a class of problems involving the fractional Laplacian

In this paper, we study the existence of heteroclinic solution for a class of nonlocal problems of the type [Formula: see text] where [Formula: see text], [Formula: see text] are continuous functions verifying some technical conditions. For example [Formula: see text] can be asymptotically periodic and potential [Formula: see text] can be the Ginzburg–Landau potential, that is, [Formula: see text].

Duality and an exact Landau-Ginzburg potential for quasi-bosonic Chern-Simons-Matter theories

It has been conjectured that Chern-Simons (CS) gauged ‘regular’ bosons in the fundamental representation are ‘level-rank’ dual to CS gauged critical fermions also in the fundamental representation. Generic relevant deformations of these conformal field theories lead to one of two distinct massive phases. In previous work, the large N thermal free energy for the bosonic theory in the unHiggsed phase has been demonstrated to match the corresponding fermionic results under duality. In this note we evaluate the large N thermal free energy of the bosonic theory in the Higgsed phase and demonstrate that our results, again, perfectly match the predictions of duality. Our computation is performed in a unitary gauge by integrating out the physical excitations of the theory — i.e. W bosons — at all orders in the ’t Hooft coupling. Our results allow us to construct an exact quantum effective potential for overline{phi}phi , the lightest gauge invariant scalar operator in the theory. In the zero temperature limit this exact Landau-Ginzburg potential is non-analytic at overline{phi}phi = 0. The extrema of this effective potential at positive overline{phi}phi solve the gap equations in the Higgsed phase while the extrema at negative overline{phi}phi solve the gap equations in the unHiggsed phase. Our effective potential is bounded from below only for a certain range of x6 (the parameter that governs sextic interactions of ϕ). This observation suggests that the regular boson theory has a stable vacuum only when x6 lies in this range.

Asymptotic behavior of critical points of an energy involving a loop-well potential

ABSTRACTWe describe the asymptotic behavior of critical points of when 𝜀→0. Here, W is a Ginzburg–Landau-type potential, vanishing on a simple closed curve Γ. Unlike the case of the standard Ginzburg–Landau potential , studied by Bethuel, Brezis and Hélein, we do not assume any symmetry on W or Γ. To overcome the difficulties due to the lack of symmetry, we develop new tools which might be of independent interest.

Critical magnetic fields in a superconductor coupled to a superfluid

We study a superconductor that is coupled to a superfluid via density and derivative couplings. Starting from a Lagrangian for two complex scalar fields, we derive a temperature-dependent Ginzburg-Landau potential, which is then used to compute the phase diagram at nonzero temperature and external magnetic field. This includes the calculation of the critical magnetic fields for the transition to an array of magnetic flux tubes, based on an approximation for the interaction between the flux tubes. We find that the transition region between type-I and type-II superconductivity changes qualitatively due to the presence of the superfluid: the phase transitions at the upper and lower critical fields in the type-II regime become first order, opening the possibility of clustered flux tube phases. These flux tube clusters may be realized in the core of neutron stars, where superconducting protons are expected to be coupled to superfluid neutrons.

Mixing of charged and neutral Bose condensates at nonzero temperature and magnetic field

It is expected that in the interior of compact stars a proton superconductor coexists with and couples to a neutron superfluid. Starting from a field-theoretical model for two complex scalar fields - one of which is electrically charged - we derive a Ginzburg-Landau potential which includes entrainment between the two fluids and temperature effects from thermal excitations of the two scalar fields and the gauge field. The Ginzburg-Landau description is then used for an analysis of the phase structure in the presence of an external magnetic field. In particular, we study the effect of the superfluid on the flux tube phase by computing the various critical magnetic fields and deriving an approximation for the flux tube interaction. As a result, we point out differences to the naive expectations from an isolated superconductor, for instance the existence of a first-order flux tube onset, resulting in a more complicated phase structure in the region between type-I and type-II superconductivity.

Towards mirror symmetry for varieties of general type

The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau–Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau–Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main result shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf.

Effect of Aspect Ratio and Boundary Conditions in Modeling Shape Memory Alloy Nanostructures with 3D Coupled Dynamic Phase-Field Theories

The behavior of shape memory alloy (SMA) nanostructures is influenced by strain rate and temperature evolution during dynamic loading. The coupling between temperature, strain, and strain rate is essential to capture inherent thermomechanical behavior in SMAs. In this paper, we propose a new 3D phase-field model that accounts for two-way coupling between mechanical and thermal physics. We use the strain-based Ginzburg-Landau potential for cubic-to-tetragonal phase transformations. The variational formulation of the developed model is implemented in the isogeometric analysis framework to overcome numerical challenges. We have observed a complete disappearance of the out-of-plane martensitic variant in a very high aspect ratio SMA domain as well as the presence of three variants in equal portions in a low aspect ratio SMA domain. The dependence of different boundary conditions on the microstructure morphology has been examined energetically. The tensile tests on rectangular prism nanowires, using the displacement based loading, demonstrate the shape memory effect and pseudoelastic behavior. We have also observed that higher strain rates, as well as the lower aspect ratio domains, resulting in high yield stress and phase transformations occur at higher stress during dynamic axial loading.

On Landau–Ginzburg systems, quivers and monodromy

Let X be a toric Fano manifold and denote by C r i t ( f X ) ⊂ ( C ∗ ) n the solution scheme of the corresponding Landau–Ginzburg system of equations. For toric Del-Pezzo surfaces and various toric Fano threefolds we define a map L : C r i t ( f X ) → P i c ( X ) such that E L ( X ) : = L ( C r i t ( f X ) ) ⊂ P i c ( X ) is a full strongly exceptional collection of line bundles. We observe the existence of a natural monodromy map M : π 1 ( L ( X ) ∖ R X , f X ) → A u t ( C r i t ( f X ) ) where L ( X ) is the space of all Laurent polynomials whose Newton polytope is equal to the Newton polytope of f X , the Landau–Ginzburg potential of X , and R X ⊂ L ( X ) is the space of all elements whose corresponding solution scheme is reduced. We show that monodromies of C r i t ( f X ) admit non-trivial relations to quiver representations of the exceptional collection E L ( X ) . We refer to this property as the M -aligned property of the maps L : C r i t ( f X ) → P i c ( X ) . We discuss possible applications of the existence of such M -aligned exceptional maps to various aspects of mirror symmetry of toric Fano manifolds.

Multiphase phase field theory for temperature- and stress-induced phase transformations

Thermodynamic Ginzburg-Landau potential for temperature- and stress-induced phase transformations (PTs) between $n$ phases is developed. It describes each of the PTs with a single order parameter without an explicit constraint equation, which allows one to use an analytical solution to calibrate each interface energy, width, and mobility; reproduces the desired PT criteria via instability conditions; introduces interface stresses; and allows for a controlling presence of the third phase at the interface between the two other phases. A finite-element approach is developed and utilized to solve the problem of nanostructure formation for multivariant martensitic PTs. Results are in a quantitative agreement with the experiment. The developed approach is applicable to various PTs between multiple solid and liquid phases and grain evolution and can be extended for diffusive, electric, and magnetic PTs.

3D coupled thermo-mechanical phase-field modeling of shape memory alloy dynamics via isogeometric analysis

The paper focuses on numerical simulation of the 3D phase-field (PF) equations for modeling cubic-to-tetragonal martensitic transformations in shape memory alloys (SMAs), their complex microstructures and thermo-mechanical behavior. The straightforward solution to the fourth-order diffuse interface 3D PF equations, based on the Landau–Ginzburg potential, is numerically solved using an isogeometric analysis. We present microstructure evolution in different geometries of SMA nanostructures under temperature-induced phase transformations to illustrate the geometrical flexibility, accuracy and robustness of our approach. The simulations successfully capture the dynamic thermo-mechanical behavior of SMAs observed experimentally.

Formal pseudodifferential operators and Witten’s r-spin numbers

Abstract We derive an effective recursion for Witten’s r-spin intersection numbers, using Witten’s conjecture relating r-spin numbers to the Gel’fand–Dikii hierarchy. Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions). We use these closed-form descriptions to prove Harer–Zagier’s formula for the Euler characteristic of ℳ g , 1 {\mathcal{M}_{g,1}} . Finally, we extend Witten’s series expansion formula for the Landau–Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brézin and Hikami.