17 research outputs found
Holographic Symmetries and Generalized Order Parameters for Topological Matter
We introduce a universally applicable method, based on the bond-algebraic
theory of dualities, to search for generalized order parameters in disparate
systems including non-Landau systems with topological order. A key notion that
we advance is that of {\em holographic symmetry}. It reflects situations
wherein global symmetries become, under a duality mapping, symmetries that act
solely on the system's boundary. Holographic symmetries are naturally related
to edge modes and localization. The utility of our approach is illustrated by
systematically deriving generalized order parameters for pure and
matter-coupled Abelian gauge theories, and for some models of topological
matter.Comment: v2, 10 pages, 3 figures. Accepted for publication in Physical Review
B Rapid Communication
Unified approach to Quantum and Classical Dualities
We show how classical and quantum dualities, as well as duality relations
that appear only in a sector of certain theories ("emergent dualities"), can be
unveiled, and systematically established. Our method relies on the use of
morphisms of the "bond algebra" of a quantum Hamiltonian. Dualities are
characterized as unitary mappings implementing such morphisms, whose even
powers become symmetries of the quantum problem. Dual variables -which were
guessed in the past- can be derived in our formalism. We obtain new
self-dualities for four-dimensional Abelian gauge field theories.Comment: 4+3 pages, 3 figure
Symmetry and Topological Order
We prove sufficient conditions for Topological Quantum Order at both zero and
finite temperatures. The crux of the proof hinges on the existence of
low-dimensional Gauge-Like Symmetries (that notably extend and differ from
standard local gauge symmetries) and their associated defects, thus providing a
unifying framework based on a symmetry principle. These symmetries may be
actual invariances of the system, or may emerge in the low-energy sector.
Prominent examples of Topological Quantum Order display Gauge-Like Symmetries.
New systems exhibiting such symmetries include Hamiltonians depicting
orbital-dependent spin exchange and Jahn-Teller effects in transition metal
orbital compounds, short-range frustrated Klein spin models, and p+ip
superconducting arrays. We analyze the physical consequences of Gauge-Like
Symmetries (including topological terms and charges), discuss associated
braiding, and show the insufficiency of the energy spectrum, topological
entanglement entropy, maximal string correlators, and fractionalization in
establishing Topological Quantum Order. General symmetry considerations
illustrate that not withstanding spectral gaps, thermal fluctuations may impose
restrictions on certain suggested quantum computing schemes and lead to
"thermal fragility". Our results allow us to go beyond standard topological
field theories and engineer systems with Topological Quantum Order.Comment: 10 pages, 2 figures. Minimal changes relative to published version-
most notably the above shortened title (which was too late to change upon
request in the galley proofs). An elaborate description of all of the results
in this article appeared in subsequent works, principally in
arXiv:cond-mat/0702377 which was published in the Annals of Physics 324, 977-
1057 (2009
Repulsive interactions in quantum Hall systems as a pairing problem
23 págs.; 4 figs.; 5 tabs. ; PACS number(s): 73.43.Cd, 02.30.Ik, 74.20.RpA subtle relation between quantum Hall physics and the phenomenon of pairing is unveiled. By use of second quantization, we establish a connection between (i) a broad class of rotationally symmetric two-body interactions within the lowest Landau level and (ii) integrable hyperbolic Richardson-Gaudin-type Hamiltonians that arise in (px+ipy) superconductivity. Specifically, we show that general Haldane pseudopotentials (and their sums) can be expressed as a sum of repulsive noncommuting (px+ip y)-type pairing Hamiltonians. The determination of the spectrum and individual null spaces of each of these noncommuting Richardson-Gaudin-type Hamiltonians is nontrivial yet is Bethe ansatz solvable. For the Laughlin sequence, it is observed that this problem is frustration free and zero-energy ground states lie in the common null space of all of these noncommuting Hamiltonians. This property allows for the use of a new truncated basis of pairing configurations in which to express Laughlin states at general filling factors. We prove separability of arbitrary Haldane pseudopotentials, providing explicit expressions for their second quantized forms, and further show by explicit construction how to exploit the topological equivalence between different geometries (disk, cylinder, and sphere) sharing the same topological genus number, in the second quantized formalism, through similarity transformations. As an application of the second quantized approach, we establish a >squeezing principle> that applies to the zero modes of a general class of Hamiltonians, which includes but is not limited to Haldane pseudopotentials. We also show how one may establish (bounds on) >incompressible filling factors> for those Hamiltonians. By invoking properties of symmetric polynomials, we provide explicit second quantized quasihole generators; the generators that we find directly relate to bosonic chiral edge modes and further make aspects of dimensional reduction in the quantum Hall systems precise. © 2013 American Physical Society.This work has been partially supported by the National
Science Foundation under NSF Grants No. DMR-1206781
(A.S.) and No. DMR-1106293 (Z.N.), and by the Spanish
MICINN Grant No. FIS2012-34479. G.O. would like to
thank the Max-Planck-Institute in GarchingPeer Reviewe
Exact results on the Kitaev model on a hexagonal lattice: spin states, string and brane correlators, and anyonic excitations
In this work, we illustrate how a Jordan-Wigner transformation combined with
symmetry considerations enables a direct solution of Kitaev's model on the
honeycomb lattice. We (i) express the p-wave type fermionic ground states of
this system in terms of the original spins, (ii) adduce that symmetry alone
dictates the existence of string and planar brane type correlators and their
composites, (iii) compute the value of such non-local correlators by employing
the Jordan-Wigner transformation, (iv) affirm that the spectrum is
inconsequential to the existence of topological quantum order and that such
information is encoded in the states themselves, and (v) express the anyonic
character of the excitations in this system and the local symmetries that it
harbors in terms of fermions.Comment: 14 pages, 7 figure
Partons as unique ground states of quantum Hall parent Hamiltonians: The case of Fibonacci anyons
We present microscopic, multiple Landau level, (frustration-free and positive
semi-definite) parent Hamiltonians whose ground states, realizing different
quantum Hall fluids, are parton-like and whose excitations display either
Abelian or non-Abelian braiding statistics. We prove ground state energy
monotonicity theorems for systems with different particle numbers, demonstrate
S-duality in the case of toroidal geometry and establish an exact zero-energy
mode counting. The emergent Entangled Pauli Principle, introduced in Phys. Rev.
B 98, 161118(R) (2018) and which defines the "DNA" of the quantum Hall fluid,
is behind the exact determination of the topological characteristics of the
fluid, including charge and braiding statistics of excitations, and effective
edge theory descriptions. When the closed-shell condition is satisfied, the
densest (i.e., the highest density and lowest total angular momentum)
zero-energy mode is a unique parton state. As a corollary, it follows that the
Moore-Read Pfaffian and Read-Rezayi states (both of which may be expressed as
linear combinations of parton-like states) cannot be densest ground states of
two-body parent Hamiltonians. We conjecture, based on the algebra of
polynomials in holomormorphic and anti-holomorphic complex variables, that
parton-like states span the subspace of many-body wave functions with the
two-body -clustering property, that is, wave functions with th-order
coincidence plane zeroes. We illustrate our framework by presenting a parent
Hamiltonian whose excitations are rigorously proven to be Fibonacci anyons and
show how to extract the DNA of the fluid whose entanglement pattern manifests
in the form of a matrix product state.Comment: 49 pages, 17 figure
Dualities and the phase diagram of the -clock model
A new "bond-algebraic" approach to duality transformations provides a very
powerful technique to analyze elementary excitations in the classical
two-dimensional XY and -clock models. By combining duality and Peierls
arguments, we establish the existence of non-Abelian symmetries, the phase
structure, and transitions of these models, unveil the nature of their
topological excitations, and explicitly show that a continuous U(1) symmetry
emerges when . This latter symmetry is associated with the appearance
of discrete vortices and Berezinskii-Kosterlitz-Thouless-type transitions. We
derive a correlation inequality to prove that the intermediate phase, appearing
for , is critical (massless) with decaying power-law correlations.Comment: 48 pages, 5 figures. Submitted to Nuclear Physics
On thermalization in Kitaev's 2D model
The thermalization process of the 2D Kitaev model is studied within the
Markovian weak coupling approximation. It is shown that its largest relaxation
time is bounded from above by a constant independent of the system size and
proportional to where is an energy gap over the
4-fold degenerate ground state. This means that the 2D Kitaev model is not an
example of a memory, neither quantum nor classical.Comment: 26 page
Breakdown of a perturbed Z_N topological phase
We study the robustness of a generalized Kitaev's toric code with Z_N degrees
of freedom in the presence of local perturbations. For N=2, this model reduces
to the conventional toric code in a uniform magnetic field. A quantitative
analysis is performed for the perturbed Z_3 toric code by applying a
combination of high-order series expansions and variational techniques. We
provide strong evidences for first- and second-order phase transitions between
topologically-ordered and polarized phases. Most interestingly, our results
also indicate the existence of topological multi-critical points in the phase
diagram.Comment: 27 pages, 10 figure
Generalized Toric Codes Coupled to Thermal Baths
We have studied the dynamics of a generalized toric code based on qudits at
finite temperature by finding the master equation coupling the code's degrees
of freedom to a thermal bath. As a consequence, we find that for qutrits new
types of anyons and thermal processes appear that are forbidden for qubits.
These include creation, annihilation and diffusion throughout the system code.
It is possible to solve the master equation in a short-time regime and find
expressions for the decay rates as a function of the dimension of the
qudits. Although we provide an explicit proof that the system relax to the
Gibbs state for arbitrary qudits, we also prove that above a certain crossing
temperature, qutrits initial decay rate is smaller than the original case for
qubits. Surprisingly this behavior only happens with qutrits and not with other
qudits with .Comment: Revtex4 file, color figures. New Journal of Physics' versio