Low energy configurations of topological singularities in two dimensions: A Γ-convergence analysis of dipoles

2019 ◽  
Vol 22 (03) ◽  
pp. 1950019
Author(s):  
Lucia De Luca ◽  
Marcello Ponsiglione
Keyword(s):  
Convergence Analysis ◽  
Two Dimensions ◽  
Low Energy ◽  
Ginzburg Landau ◽  
Energy Plus ◽  
Γ Convergence ◽  
Flat Norm ◽  
Topological Singularities ◽  
Energy Configurations ◽  
Small Zero

This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the core radius approach and the Ginzburg–Landau energy. Denoting by [Formula: see text] the length scale parameter in such models, we focus on the [Formula: see text] energy regime. It is well known that, for configurations whose energy is bounded by [Formula: see text], the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying [Formula: see text] energy, plus a measure supported on small zero-average sets. Loosely speaking, on such sets the vorticity measure is close, with respect to the flat norm, to zero-average clusters of positive and negative masses. Here, we perform a compactness and [Formula: see text]-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale [Formula: see text], for [Formula: see text]), which vanish in the flat convergence and whose energy contribution has, so far, been neglected. Our results refine and contain as a particular case the classical [Formula: see text]-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order [Formula: see text] with [Formula: see text].

Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach

2014 ◽  
Vol 214 (1) ◽  
pp. 269-330 ◽  
Author(s):  
Roberto Alicandro ◽  
Lucia De Luca ◽  
Adriana Garroni ◽  
Marcello Ponsiglione
Keyword(s):  
Two Dimensions ◽  
Γ Convergence ◽  
Topological Singularities

scholarly journals Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture

2013 ◽  
Vol 13 (5&6) ◽  
pp. 393-429
Author(s):  
Matthew Hastings
Keyword(s):  
Ground State ◽  
Coding Theory ◽  
Energy State ◽  
Ground States ◽  
Coarse Graining ◽  
Quantum Circuit ◽  
Technical Condition ◽  
Two Dimensions ◽  
Low Energy ◽  
Coarse Grained

We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have ``trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states\cite{bv}. Using a coarse-graining procedure, this implies that any such Hamiltonian with bounded range interactions in one dimension has a trivial ground state. In this paper, we further explore the question of which Hamiltonians have trivial ground states. We define an ``interaction complex" for a Hamiltonian, which generalizes the notion of interaction graph and we show that if the interaction complex can be continuously mapped to a $1$-complex using a map with bounded diameter of pre-images then the Hamiltonian has a trivial ground state assuming one technical condition on the Hamiltonians holds (this condition holds for all stabilizer Hamiltonians, and we additionally prove the result for all Hamiltonians under one assumption on the $1$-complex). While this includes the cases considered by Ref.~\onlinecite{bv}, we show that it also includes a larger class of Hamiltonians whose interaction complexes cannot be coarse-grained into the case of Ref.~\onlinecite{bv} but still can be mapped continuously to a $1$-complex. One motivation for this study is an approach to the quantum PCP conjecture. We note that many commonly studied interaction complexes can be mapped to a $1$-complex after removing a small fraction of sites. For commuting projector Hamiltonians on such complexes, in order to find low energy trivial states for the original Hamiltonian, it would suffice to find trivial ground states for the Hamiltonian with those sites removed. Such trivial states can act as a classical witness to the existence of a low energy state. While this result applies for commuting Hamiltonians and does not necessarily apply to other Hamiltonians, it suggests that to prove a quantum PCP conjecture for commuting Hamiltonians, it is worth investigating interaction complexes which cannot be mapped to $1$-complexes after removing a small fraction of points. We define this more precisely below; in some sense this generalizes the notion of an expander graph. Surprisingly, such complexes do exist as will be shown elsewhere\cite{fh}, and have useful properties in quantum coding theory.

scholarly journals A phase-field approximation of the Steiner problem in dimension two

2019 ◽  
Vol 12 (2) ◽  
pp. 157-179 ◽  
Author(s):  
Antonin Chambolle ◽  
Luca Alberto Davide Ferrari ◽  
Benoit Merlet
Keyword(s):  
Phase Field ◽  
Transportation Problem ◽  
Numerical Approximation ◽  
Unit Length ◽  
Field Approximation ◽  
Two Dimensions ◽  
Steiner Problem ◽  
Limit Case ◽  
Fixed Parameter ◽  
Γ Convergence

AbstractIn this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form {1+\beta\,\theta}, where θ denotes the amount of transported mass and {\beta>0} is a fixed parameter (notice that the limit case {\beta=0} corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ({\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}}) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of {\{\mathcal{F}_{\varepsilon}\}} as {\varepsilon\downarrow 0}. Our functionals are modeled on the Ambrosio–Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.

On Geometric Models for Interphase Boundaries

1990 ◽  
Vol 187 ◽  
Author(s):  
Hans J. Fecht
Keyword(s):  
Atomic Structure ◽  
Chemical Interaction ◽  
Low Energy ◽  
Geometric Models ◽  
Lattice Matching ◽  
Metal Metal ◽  
Interphase Boundaries ◽  
Static Distortion ◽  
Lock In ◽  
Energy Configurations

AbstractThe energy interphase boundaries can be described as a function of the lattice matching at the interface between two adjacent crystals, the chemical interaction and the interfacial entropy of the boundaries. Geometric models relating the energy of interphase boundaries (metal/metal and metal/non-metal) to their atomic structure can be based on the static distortion wave concept. This approach constitutes the physical basis for the lock-in model and the planar CSL-model proposed previously to describe the low energy configurations of such interfaces.

STUDIES OF FINDING LOW ENERGY CONFIGURATIONS IN OFF-LATTICE PROTEIN MODELS

2006 ◽  
Vol 05 (03) ◽  
pp. 587-594 ◽  
Author(s):  
JINGFA LIU ◽  
WENQI HUANG
Keyword(s):  
Lower Energy ◽  
Gradient Descent ◽  
Energy Landscape ◽  
Three Dimensional ◽  
Ground States ◽  
Low Energy ◽  
Protein Models ◽  
Energy Configurations ◽  
Energy Landscape Paving ◽  
Lattice Protein Models

We studied two three-dimensional off-lattice protein models with two species of monomers, hydrophobic and hydrophilic. Low energy configurations in both models were optimized using the energy landscape paving (ELP) method and subsequent gradient descent. The numerical results show that the proposed methods are very promising for finding the ground states of proteins. For all sequences with lengths 13 ≤ n ≤ 55, the algorithm finds states with lower energy than previously proposed putative ground states.

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