22,222 research outputs found
Phase transitions and configuration space topology
Equilibrium phase transitions may be defined as nonanalytic points of
thermodynamic functions, e.g., of the canonical free energy. Given a certain
physical system, it is of interest to understand which properties of the system
account for the presence of a phase transition, and an understanding of these
properties may lead to a deeper understanding of the physical phenomenon. One
possible approach of this issue, reviewed and discussed in the present paper,
is the study of topology changes in configuration space which, remarkably, are
found to be related to equilibrium phase transitions in classical statistical
mechanical systems. For the study of configuration space topology, one
considers the subsets M_v, consisting of all points from configuration space
with a potential energy per particle equal to or less than a given v. For
finite systems, topology changes of M_v are intimately related to nonanalytic
points of the microcanonical entropy (which, as a surprise to many, do exist).
In the thermodynamic limit, a more complex relation between nonanalytic points
of thermodynamic functions (i.e., phase transitions) and topology changes is
observed. For some class of short-range systems, a topology change of the M_v
at v=v_t was proved to be necessary for a phase transition to take place at a
potential energy v_t. In contrast, phase transitions in systems with long-range
interactions or in systems with non-confining potentials need not be
accompanied by such a topology change. Instead, for such systems the
nonanalytic point in a thermodynamic function is found to have some
maximization procedure at its origin. These results may foster insight into the
mechanisms which lead to the occurrence of a phase transition, and thus may
help to explore the origin of this physical phenomenon.Comment: 22 pages, 6 figure
Topological string entanglement
We investigate how topological entanglement of Chern-Simons theory is
captured in a string theoretic realization. Our explorations are motivated by a
desire to understand how quantum entanglement of low energy open string degrees
of freedom is encoded in string theory (beyond the oft discussed classical
gravity limit). Concretely, we realize the Chern-Simons theory as the
worldvolume dynamics of topological D-branes in the topological A-model string
theory on a Calabi-Yau target. Via the open/closed topological string duality
one can map this theory onto a pure closed topological A-model string on a
different target space, one which is related to the original Calabi-Yau
geometry by a geometric/conifold transition. We demonstrate how to uplift the
replica construction of Chern-Simons theory directly onto the closed string and
show that it provides a meaningful definition of reduced density matrices in
topological string theory. Furthermore, we argue that the replica construction
commutes with the geometric transition, thereby providing an explicit closed
string dual for computing reduced states, and Renyi and von Neumann entropies
thereof. While most of our analysis is carried out for Chern-Simons on S^3, the
emergent picture is rather general. Specifically, we argue that quantum
entanglement on the open string side is mapped onto quantum entanglement on the
closed string side and briefly comment on the implications of our result for
physical holographic theories where entanglement has been argued to be crucial
ingredient for the emergence of classical geometry.Comment: 48 pages + appendices, many tikz fgures. v2: added clarification
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Chern-Simons Theory and Topological Strings
We review the relation between Chern-Simons gauge theory and topological
string theory on noncompact Calabi-Yau spaces. This relation has made possible
to give an exact solution of topological string theory on these spaces to all
orders in the string coupling constant. We focus on the construction of this
solution, which is encoded in the topological vertex, and we emphasize the
implications of the physics of string/gauge theory duality for knot theory and
for the geometry of Calabi-Yau manifolds.Comment: 46 pages, RMP style, 25 figures, minor corrections, references adde
Spectral dimension of quantum geometries
The spectral dimension is an indicator of geometry and topology of spacetime
and a tool to compare the description of quantum geometry in various approaches
to quantum gravity. This is possible because it can be defined not only on
smooth geometries but also on discrete (e.g., simplicial) ones. In this paper,
we consider the spectral dimension of quantum states of spatial geometry
defined on combinatorial complexes endowed with additional algebraic data: the
kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the
effects of topology and discreteness of classical discrete geometries are
studied in a systematic manner. We look for states reproducing the spectral
dimension of a classical space in the appropriate regime. We also test the
hypothesis that in LQG, as in other approaches, there is a scale dependence of
the spectral dimension, which runs from the topological dimension at large
scales to a smaller one at short distances. While our results do not give any
strong support to this hypothesis, we can however pinpoint when the topological
dimension is reproduced by LQG quantum states. Overall, by exploring the
interplay of combinatorial, topological and geometrical effects, and by
considering various kinds of quantum states such as coherent states and their
superpositions, we find that the spectral dimension of discrete quantum
geometries is more sensitive to the underlying combinatorial structures than to
the details of the additional data associated with them.Comment: 39 pages, 18 multiple figures. v2: discussion improved, minor typos
correcte
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