704 research outputs found
An Isomonodromy Cluster of Two Regular Singularities
We consider a linear matrix ODE with two coalescing regular
singularities. This coalescence is restricted with an isomonodromy condition
with respect to the distance between the merging singularities in a way
consistent with the ODE. In particular, a zero-distance limit for the ODE
exists. The monodromy group of the limiting ODE is calculated in terms of the
original one. This coalescing process generates a limit for the corresponding
nonlinear systems of isomonodromy deformations. In our main example the latter
limit reads as , where is the -th Painlev\'e equation. We
also discuss some general problems which arise while studying the
above-mentioned limits for the Painlev\'e equations.Comment: 44 pages, 8 figure
Quantum simulators, continuous-time automata, and translationally invariant systems
The general problem of finding the ground state energy of lattice
Hamiltonians is known to be very hard, even for a quantum computer. We show
here that this is the case even for translationally invariant systems. We also
show that a quantum computer can be built in a 1D chain with a fixed,
translationally invariant Hamitonian consisting of nearest--neighbor
interactions only. The result of the computation is obtained after a prescribed
time with high probability.Comment: partily rewritten and important references include
Minimum construction of two-qubit quantum operations
Optimal construction of quantum operations is a fundamental problem in the
realization of quantum computation. We here introduce a newly discovered
quantum gate, B, that can implement any arbitrary two-qubit quantum operation
with minimal number of both two- and single-qubit gates. We show this by giving
an analytic circuit that implements a generic nonlocal two-qubit operation from
just two applications of the B gate. We also demonstrate that for the highly
scalable Josephson junction charge qubits, the B gate is also more easily and
quickly generated than the CNOT gate for physically feasible parameters.Comment: 4 page
Preparing ground states of quantum many-body systems on a quantum computer
Preparing the ground state of a system of interacting classical particles is
an NP-hard problem. Thus, there is in general no better algorithm to solve this
problem than exhaustively going through all N configurations of the system to
determine the one with lowest energy, requiring a running time proportional to
N. A quantum computer, if it could be built, could solve this problem in time
sqrt(N). Here, we present a powerful extension of this result to the case of
interacting quantum particles, demonstrating that a quantum computer can
prepare the ground state of a quantum system as efficiently as it does for
classical systems.Comment: 7 pages, 1 figur
Ettingshausen effect due to Majorana modes
The presence of Majorana zero-energy modes at vortex cores in a topological
superconductor implies that each vortex carries an extra entropy , given
by , that is independent of temperature. By utilizing this
special property of Majorana modes, the edges of a topological superconductor
can be cooled (or heated) by the motion of the vortices across the edges. As
vortices flow in the transverse direction with respect to an external imposed
supercurrent, due to the Lorentz force, a thermoelectric effect analogous to
the Ettingshausen effect is expected to occur between opposing edges. We
propose an experiment to observe this thermoelectric effect, which could
directly probe the intrinsic entropy of Majorana zero-energy modes.Comment: 16 pages, 3 figure
Necessary Condition for the Quantum Adiabatic Approximation
A gapped quantum system that is adiabatically perturbed remains approximately
in its eigenstate after the evolution. We prove that, for constant gap, general
quantum processes that approximately prepare the final eigenstate require a
minimum time proportional to the ratio of the length of the eigenstate path to
the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We
also give a necessary condition for the adiabatic approximation that depends on
local properties of the path, which is appropriate when the gap varies.Comment: 5 pages, 1 figur
Non-Abelian statistics as a Berry phase in exactly solvable models
We demonstrate how to directly study non-Abelian statistics for a wide class
of exactly solvable many-body quantum systems. By employing exact eigenstates
to simulate the adiabatic transport of a model's quasiparticles, the resulting
Berry phase provides a direct demonstration of their non-Abelian statistics. We
apply this technique to Kitaev's honeycomb lattice model and explicitly
demonstrate the existence of non-Abelian Ising anyons confirming the previous
conjectures. Finally, we present the manipulations needed to transport and
detect the statistics of these quasiparticles in the laboratory. Various
physically realistic system sizes are considered and exact predictions for such
experiments are provided.Comment: 10 pages, 3 figures. To appear in New Journal of Physic
Anatomy of fermionic entanglement and criticality in Kitaev spin liquids
We analyze in detail the effect of nontrivial band topology on the area-law behavior of the entanglement entropy in Kitaev's honeycomb model. By mapping the translationally invariant 2D spin model onto 1D fermionic subsystems, we identify those subsystems responsible for universal entanglement contributions in the gapped phases and those responsible for critical entanglement scaling in the gapless phases. For the gapped phases, we analytically show how the topological edge states contribute to the entanglement entropy and provide a universal lower bound for it. For the gapless semimetallic phases and topological phase transitions, the identification of the critical subsystems shows that they fall always into the Ising or the XY universality classes. As our study concerns the fermionic degrees of freedom in the honeycomb model, qualitatively similar results are expected to apply also to generic topological insulators and superconductors
Parafermionic edge zero modes in Z_n-invariant spin chains
A sign of topological order in a gapped one-dimensional quantum chain is the
existence of edge zero modes. These occur in the Z_2-invariant Ising/Majorana
chain, where they can be understood using free-fermion techniques. Here I
discuss their presence in spin chains with Z_n symmetry, and prove that for
appropriate coupling they are exact, even in this strongly interacting system.
These modes are naturally expressed in terms of parafermions, generalizations
of fermions to the Z_n case. I show that parafermionic edge zero modes do not
occur in the usual ferromagnetic and antiferromagnetic cases, but rather only
when the interactions are chiral, so that spatial-parity and time-reversal
symmetries are broken.Comment: 22 pages. v2: small changes, added reference
Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors
One of the defining properties of the conventional three-dimensional
("-", or "spin-orbit"-) topological insulator is its
characteristic magnetoelectric effect, as described by axion electrodynamics.
In this paper, we discuss an analogue of such a magnetoelectric effect in the
thermal (or gravitational) and the magnetic dipole responses in all symmetry
classes which admit topologically non-trivial insulators or superconductors to
exist in three dimensions. In particular, for topological superconductors (or
superfluids) with time-reversal symmetry which lack SU(2) spin rotation
symmetry (e.g. due to spin-orbit interactions), such as the B phase of He,
the thermal response is the only probe which can detect the non-trivial
topological character through transport. We show that, for such topological
superconductors, applying a temperature gradient produces a thermal- (or mass-)
surface current perpendicular to the thermal gradient. Such charge, thermal, or
magnetic dipole responses provide a definition of topological insulators and
superconductors beyond the single-particle picture. Moreover we find, for a
significant part of the 'ten-fold' list of topological insulators found in
previous work in the absence of interactions, that in general dimensions the
effective field theory describing the space-time responses is governed by a
field theory anomaly. Since anomalies are known to be insensitive to whether
the underlying fermions are interacting or not, this shows that the
classification of these topological insulators is robust to adiabatic
deformations by interparticle interactions in general dimensionality. In
particular, this applies to symmetry classes DIII, CI, and AIII in three
spatial dimensions, and to symmetry classes D and C in two spatial dimensions.Comment: 16 pages, 2 figure
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