45 research outputs found
Bulk--Boundary Correspondence for Chiral Symmetric Quantum Walks
Discrete-time quantum walks (DTQW) have topological phases that are richer
than those of time-independent lattice Hamiltonians. Even the basic symmetries,
on which the standard classification of topological insulators hinges, have not
yet been properly defined for quantum walks. We introduce the key tool of
timeframes, i.e., we describe a DTQW by the ensemble of time-shifted unitary
timestep operators belonging to the walk. This gives us a way to consistently
define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using
an "inversion symmetric" pulse sequence. For one-dimensional DTQW's with CS, we
identify the bulk ZxZ topological invariant that controls the number of
topologically protected 0 and pi energy edge states at the interfaces between
different domains, and give simple formulas for these invariants. We illustrate
this bulk--boundary correspondence for DTQW's on the example of the "4-step
quantum walk", where tuning CS and particle-hole symmetry realizes edge states
in various symmetry classes
Topological phases and delocalization of quantum walks in random environments
We investigate one-dimensional (1D) discrete time quantum walks (QWs) with
spatially or temporally random defects as a consequence of interactions with
random environments. We focus on the QWs with chiral symmetry in a topological
phase, and reveal that chiral symmetry together with bipartite nature of the
QWs brings about intriguing behaviors such as coexistence of topologically
protected edge states at zero energy and Anderson transitions in the 1D chiral
class at non-zero energy in their dynamics. Contrary to the previous studies,
therefore, the spatially disordered QWs can avoid complete localization due to
the Anderson transition. It is further confirmed that the edge states are
robust for spatial disorder but not for temporal disorder.Comment: 7 pages, 7 figure
Explicit definition of symmetry for non-unitary quantum walks with gain and loss
symmetry, that is, a combined parity and time-reversal
symmetry is a key milestone for non-Hermite systems exhibiting entirely real
eigenenergy. In the present work, motivated by a recent experiment, we study
symmetry of the time-evolution operator of non-unitary quantum
walks. We present the explicit definition of symmetry by
employing a concept of symmetry time frames. We provide a necessary and
sufficient condition so that the time-evolution operator of the non-unitary
quantum walk retains symmetry even when parameters of the model
depend on position. It is also shown that there exist extra symmetries embedded
in the time-evolution operator. Applying these results, we clarify that the
non-unitary quantum walk in the experiment does have symmetry.Comment: 14 pages, 8 figure
Relation between two-phase quantum walks and the topological invariant
We study a position-dependent discrete-time quantum walk (QW) in one
dimension, whose time-evolution operator is built up from two coin operators
which are distinguished by phase factors from and . We call
the QW the - to discern from the
two-phase QW with one defect[13,14]. Because of its localization properties,
the two-phase QWs can be considered as an ideal mathematical model of
topological insulators which are novel quantum states of matter characterized
by topological invariants. Employing the complete two-phase QW, we present the
stationary measure, and two kinds of limit theorems concerning and the , which are the
characteristic behaviors in the long-time limit of discrete-time QWs in one
dimension. As a consequence, we obtain the mathematical expression of the whole
picture of the asymptotic behavior of the walker in the long-time limit. We
also clarify relevant symmetries, which are essential for topological
insulators, of the complete two-phase QW, and then derive the topological
invariant. Having established both mathematical rigorous results and the
topological invariant of the complete two-phase QW, we provide solid arguments
to understand localization of QWs in term of topological invariant.
Furthermore, by applying a concept of , we
clarify that localization of the two-phase QW with one defect, studied in the
previous work[13], can be related to localization of the complete two-phase QW
under symmetry preserving perturbations.Comment: 50 pages, 13 figure
Bulk-Edge Correspondence for Point-Gap Topological Phases in Junction Systems
The bulk-edge correspondence is one of the most important ingredients in the
theory of topological phases of matter. While the bulk-edge correspondence is
applicable for Hermitian junction systems where two subsystems with independent
topological invariants are connected to each other, it has not been discussed
for junction systems with non-Hermitian point-gap topological phases. In this
Letter, based on analytical results obtained by the extension of non-Bloch band
theory to junction systems, we establish the bulk-edge correspondence for
point-gap topological phases in junction systems. We also confirm that almost
all the eigenstates are localized near the interface which are called the
"non-Hermitian proximity effects". One of the unique properties is that the
localization length becomes the same for both subsystems nevertheless those
model-parameters are different.Comment: 6 pages, 4 figure
Finite Size Effects and Irrelevant Corrections to Scaling near the Integer Quantum Hall Transition
We present a numerical finite size scaling study of the localization length
in long cylinders near the integer quantum Hall transition (IQHT) employing the
Chalker-Coddington network model. Corrections to scaling that decay slowly with
increasing system size make this analysis a very challenging numerical problem.
In this work we develop a novel method of stability analysis that allows for a
better estimate of error bars. Applying the new method we find consistent
results when keeping second (or higher) order terms of the leading irrelevant
scaling field. The knowledge of the associated (negative) irrelevant exponent
is crucial for a precise determination of other critical exponents,
including multifractal spectra of wave functions. We estimate ,
which is considerably larger than most recently reported values. Within this
approach we obtain the localization length exponent confirming
recent results. Our stability analysis has broad applicability to other
observables at IQHT, as well as other critical points where corrections to
scaling are present.Comment: 6 pages and 3 figures, plus supplemental material
Spin-directed network model for the surface states of weak three-dimensional topological insulators
A two-dimensional spin-directed network model is
constructed that describes the combined effects of dimerization and disorder
for the surface states of a weak three-dimensional
topological insulator. The network model consists of helical edge states of
two-dimensional layers of topological insulators which
are coupled by time-reversal symmetric interlayer tunneling. It is argued that,
without dimerization of interlayer couplings, the network model has no
insulating phase for any disorder strength. However, a sufficiently strong
dimerization induces a transition from a metallic phase to an insulating phase.
The critical exponent for the diverging localization length at
metal-insulator transition points is obtained by finite-size scaling analysis
of numerical data from simulations of this network model. It is shown that the
phase transition belongs to the two-dimensional symplectic universality class
of Anderson transition.Comment: 36 pages and 27 figures, plus Supplemental Materia
Topological Phases in a PT-Symmetric Dissipative Kitaev Chain
We study a topological phase in the dissipative Kitaev chain described by the
Markovian quantum master equation. Based on the correspondence between
Lindbladians, which generate the dissipative time-evolution, and non-Hermitian
matrices, Lindbladians are classified in terms of non-Hermitian topological
phases. We find out that the Lindbladian retains PT symmetry which is the
prominent symmetry of open systems and then all the bulk modes can have a
common lifetime. Moreover, when open boundary conditions are imposed on the
system, the edge modes which break PT symmetry emerge, and one of the edge
modes has a zero eigenvalue.Comment: 6 pages, 2 figures, accepted for publication in JPS Conference
Proceedings (LT29
Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk
Quantum walks, whose dynamics is prescribed by alternating unitary coin and
shift operators, possess topological phases akin to those of Floquet
topological insulators, driven by a time-periodic field. While there is ample
theoretical work on topological phases of quantum walks where the coin
operators are spin rotations, in experiments a different coin, the Hadamard
operator is often used instead. This was the case in a recent photonic quantum
walk experiment, where protected edge states were observed between two bulks
whose topological invariants, as calculated by the standard theory, were the
same. This hints at a hidden topological invariant in the Hadamard quantum
walk. We establish a relation between the Hadamard and the spin rotation
operator, which allows us to apply the recently developed theory of topological
phases of quantum walks to the one-dimensional Hadamard quantum walk. The
topological invariants we derive account for the edge state observed in the
experiment, we thus reveal the hidden topological invariant of the
one-dimensional Hadamard quantum walk.Comment: 11 pages, 4 figure