50 research outputs found
Area Law Violations and Quantum Phase Transitions in Modified Motzkin Walk Spin Chains
Area law violations for entanglement entropy in the form of a square root has
recently been studied for one-dimensional frustration-free quantum systems
based on the Motzkin walks and their variations. Here we consider a Motzkin
walk with a different Hilbert space on each step of the walk spanned by
elements of a {\it Symmetric Inverse Semigroup} with the direction of each step
governed by its algebraic structure. This change alters the number of paths
allowed in the Motzkin walk and introduces a ground state degeneracy sensitive
to boundary perturbations. We study the frustration-free spin chains based on
three symmetric inverse semigroups, \cS^3_1, \cS^3_2 and \cS^2_1. The
system based on \cS^3_1 and \cS^3_2 provide examples of quantum phase
transitions in one dimensions with the former exhibiting a transition between
the area law and a logarithmic violation of the area law and the latter
providing an example of transition from logarithmic scaling to a square root
scaling in the system size, mimicking a colored \cS^3_1 system. The system
with \cS^2_1 is much simpler and produces states that continue to obey the
area law.Comment: 40 pages, 14 figures, A condensed version of this paper has been
submitted to the Proceedings of the 2017 Granada Seminar on Computational
Physics, Contains minor revisions and is closer to the Journal version. v3
includes an addendum that modifies the final Hamiltonian but does not change
the main results of the pape
Physics on Noncommutative Spacetimes
The structure of spacetime at the Planck scale remains a mystery to this date with a lot of insightful attempts to unravel this puzzle. One such attempt is the proposition of a `pointless\u27 structure for spacetime at this scale. This is done by studying the geometry of the spacetime through a noncommutative algebra of functions defined on it. We call such spacetimes \u27noncommutative spacetimes\u27. This dissertation probes physics on several such spacetimes. These include compact noncommutative spaces called fuzzy spaces and noncompact spacetimes. The compact examples we look at are the fuzzy sphere and the fuzzy Higg\u27s manifold. The noncompact spacetimes we study are the Groenewold-Moyal plane and the Bxn plane.
A broad range of physical effects are studied on these exotic spacetimes. We study spin systems on the fuzzy sphere. The construction of Dirac and chirality operators for an arbitrary spin j is studied on both S2/F and S2 in detail. We compute the spectrums of the spin 1 and spin 3/2 Dirac operators on S2/F. These systems have novel thermodynamical properties which have no higher dimensional analogs, making them interesting models.
The fuzzy Higg\u27s manifold is found to exhibit topology change, an important property for any theory attempting to quantize gravity. We study how this change occurs in the classical setting and how quantizing this manifold smoothens the classical conical singularity. We also show the construction of the star product on this manifold using coherent states on the noncommutative algebra describing this noncommutative space.
On the Moyal plane we develop the LSZ formulation of scalar quantum field theory. We compute scattering amplitudes and remark on renormalization of this theory. We show that the LSZ formalism is equivalent to the interaction representation formalism for computing scattering amplitudes on the Moyal plane. This result is true for on-shell Green\u27s functions and fails to hold for off-shell Green\u27s functions.
With the present technology available, there is a scarcity of experiments which directly involve the Planck scale. However there are interesting low and medium energy experiments which put bounds on the validity of established principles which are thought to be violated at the Planck scale. One such principle is the Pauli principle which is expected to be violated on noncommutative spacetimes. We introduce a noncommutative spacetime called the Bxn plane to show how transitions, not obeying the Pauli principle, occur in atomic systems. On confronting with the data from experiments, we place bounds on the noncommutative parameter
Beyond fuzzy spheres
We study polynomial deformations of the fuzzy sphere, specifically given by
the cubic or the Higgs algebra. We derive the Higgs algebra by quantizing the
Poisson structure on a surface in . We find that several
surfaces, differing by constants, are described by the Higgs algebra at the
fuzzy level. Some of these surfaces have a singularity and we overcome this by
quantizing this manifold using coherent states for this nonlinear algebra. This
is seen in the measure constructed from these coherent states. We also find the
star product for this non-commutative algebra as a first step in constructing
field theories on such fuzzy spaces.Comment: 9 pages, 3 Figures, Minor changes in the abstract have been made. The
manuscript has been modified for better clarity. A reference has also been
adde
Novel quantum phases on graphs using abelian gauge theory
Graphs are topological spaces that include broader objects than discretized
manifolds, making them interesting playgrounds for the study of quantum phases
not realized by symmetry breaking. In particular they are known to support
anyons of an even richer variety than the two-dimensional space. We explore
this possibility by building a class of frustration-free and gapped
Hamiltonians based on discrete abelian gauge groups. The resulting models have
a ground state degeneracy that can be either a topological invariant, an
extensive quantity or a mixture of the two. For two basis of the degenerate
ground states which are complementary in quantum theory, the entanglement
entropy is exactly computed. The result for one basis has a constant global
term, known as the topological entanglement entropy, implying long-range
entanglement. On the other hand, the topological entanglement entropy vanishes
in the result for the other basis. Comparisons are made with similar
occurrences in the toric code. We analyze excitations and identify anyon-like
excitations that account for the topological entanglement entropy. An analogy
between the ground states of this system and the -vacuum for a
gauge theory on a circle is also drawn.Comment: 51 pages, 14 figures, Published version. Includes a section on
particle statistics on graphs and a fully worked out exampl
Generating W states with braiding operators
Braiding operators can be used to create entangled states out of product
states, thus establishing a correspondence between topological and quantum
entanglement. This is well-known for maximally entangled Bell and GHZ states
and their equivalent states under Stochastic Local Operations and Classical
Communication, but so far a similar result for W states was missing. Here we
use generators of extraspecial 2-groups to obtain the W state in a four-qubit
space and partition algebras to generate the W state in a three-qubit space. We
also present a unitary generalized Yang-Baxter operator that embeds the W
state in a -qubit space.Comment: 13 pages, Published versio