arXiv subjects

Mahito Kohmoto

Publications and source records attributed to Mahito Kohmoto.

At least 19 recordsLinked to original sources

arXiv2013

Periodic Landau gauge and Quantum Hall effect in twisted bilayer graphene

Energy versus magnetic field (Hofstadter butterfly diagram) in twisted bilayer graphene is studied theoretically. If we take the usual Landau gauge, we cannot take a finite periodicity even when the magnetic flux through a supercell is a rational number. We show that the \textit{periodic} Landau gauge, which has the periodicity in one direction, makes it possible to obtain the Hofstadter butterfly diagram. Since a supercell can be large, magnetic flux through a supercell normalized by the flux quantum can be a fractional number with a small denominator, even when a magnetic field is not extremely strong. As a result, quantized Hall conductance can be a solution of nontrivial Diophantine equation.

cond-mat.mes-hall↗

arXiv2012

The Spectral Shift Function and The Friedel Sum Rule

We study the relationship between the spectral shift function and the excess charge in potential scattering theory. Although these quantities are closely related to each other, they have been often formulated in different settings so far. Here we first give an alternative construction of the spectral shift function, and then we prove that the spectral shift function thus constructed yields the Friedel sum rule.

math-ph↗

arXiv2011

Electric-field induced penetration of edge states at the interface between monolayer and bilayer graphene

The edge states in the hybrid system of single-layer and double-layer graphene are studied in the tight-binding model theoretically. The edge states in one side of the interface between single-layer and double-layer graphene are shown to penetrate into the single-layer region when the perpendicular electric field is applied, while they are localized in the double-layer region without electric field. The edge states in another side of the interface are localized in the double-layer region independent of the electric field. This field-induced penetration of the edge states can be applied to switching devices. We also find a new type of the edge states at the boundary between single-layer and the double-layer graphene.

cond-mat.mes-hall↗

arXiv2011

Edge states and topological phases in non-Hermitian systems

Topological stability of the edge states is investigated for non-Hermitian systems. We examine two classes of non-Hermitian Hamiltonians supporting real bulk eigenenergies in weak non-Hermiticity: SU(1,1) and SO(3,2) Hamiltonians. As an SU(1,1) Hamiltonian, the tight-binding model on the honeycomb lattice with imaginary on-site potentials is examined. Edge states with ReE=0 and their topological stability are discussed by the winding number and the index theorem, based on the pseudo-anti-Hermiticity of the system. As a higher symmetric generalization of SU(1,1) Hamiltonians, we also consider SO(3,2) models. We investigate non-Hermitian generalization of the Luttinger Hamiltonian on the square lattice, and that of the Kane-Mele model on the honeycomb lattice, respectively. Using the generalized Kramers theorem for the time-reversal operator Theta with Theta^2=+1 [M. Sato et al., arXiv:1106.1806], we introduce a time-reversal invariant Chern number from which topological stability of gapless edge modes is argued.

cond-mat.mes-hall↗

arXiv2011

Time-Reversal Symmetry in Non-Hermitian Systems

For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys Θ^2=-1, but no such a degeneracy exists when Θ^2=+1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when Θ^2=+1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split quaternion in a class of non-hermitian Hamiltonians. As concrete examples, we examine in detail NxN Hamiltonians with N=2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.

cond-mat.stat-mech↗

arXiv2009

Zero modes, energy gap, and edge states of anisotropic honeycomb lattice in a magnetic field

We present systematic study of zero modes and gaps by introducing effects of anisotropy of hopping integrals for a tight-binding model on the honeycomb lattice in a magnetic field. The condition for the existence of zero modes is analytically derived. From the condition, it is found that a tiny anisotropy for graphene is sufficient to open a gap around zero energy in a magnetic field. This gap behaves as a non-perturbative and exponential form as a function of the magnetic field. The non-analytic behavior with respect to the magnetic field can be understood as tunneling effects between energy levels around two Dirac zero modes appearing in the honeycomb lattice, and an explicit form of the gap around zero energy is obtained by the WKB method near the merging point of these Dirac zero modes. Effects of the anisotropy for the honeycomb lattices with boundaries are also studied. The condition for the existence of zero energy edge states in a magnetic field is analytically derived. On the basis of the condition, it is recognized that anisotropy of the hopping integrals induces abrupt changes of the number of zero energy edge states, which depend on the shapes of the edges sensitively.

cond-mat.mes-hall↗

arXiv2009

Wave propagation through Cantor-set media: Chaos, scaling, and fractal structures

Propagation of waves through Cantor-set media is investigated by renormalization-group analysis. For specific values of wave numbers, transmission coefficients are shown to be governed by the logistic map, and in the chaotic region, they show sensitive dependence on small changes of parameters of the system such as the index of refraction. For other values of wave numbers, our numerical results suggest that light transmits completely or reflects completely by the Cantor-set media ${\rm C}_{\infty}$. It is also shown that transmission coefficients exhibit a local scaling behavior near complete transmission if the complete transmission is achieved at a wave number $κ=κ^*$ with a rational $κ^*/π$. The scaling function is obtained analytically by using the Euler's totient function, and the local scaling behavior is confirmed numerically.

cond-mat.mes-hall↗

arXiv2008

The Hall conductance, topological quantum phase transition and the Diophantine equation on honeycomb lattice

We consider a tight-binding model with the nearest neighbour hopping integrals on the honeycomb lattice in a magnetic field. Assuming one of the three hopping integrals, which we denote t_a, can take a different value from the two others, we study quantum phase structures controlled by the anisotropy of the honeycomb lattice.For weak and strong t_a regions, respectively, the Hall conductances are calculated algebraically by using the Diophantine equation. Except for a few specific gaps, we completely determine the Hall conductances in these two regions including those for subband gaps. In a weak magnetic field, it is found that the weak t_a region shows the unconventional quantization of the Hall conductance, σ_{xy}=-(e^2/h)(2n+1), (n=0,\pm 1,\pm 2,...), near the half-filling, while the strong t_a region shows only the conventional one, σ_{xy}=-(e^2/h)n,(n=0,1,2,...). From topological nature of the Hall conductance, the existence of gap closing points and quantum phase transitions in the intermediate t_a region are concluded. We also study numerically the quantum phase structure in detail, and find that even when t_a=1, namely in graphene case, the system is in the weak t_a phase except when the Fermi energy is located near the van Hove singularity or the lower and upper edges of the spectrum.

cond-mat.mes-hall↗

arXiv2008

Quantized spin Hall effect in Helium three-A and other p-wave paired Fermi systems

In this paper we propose the quantized spin Hall effect (SHE) in the vortex state of a rotating p-wave paired Fermi system in an inhomogeneous magnetic field and in a weak periodic potential. It is the three dimensional extension of the spin Hall effect for a 3He-A superfluid film studied in Ref. [1]. It may also be considered as a generalization of the 3D quantized charge Hall effect of Bloch electrons in Ref. [2] to the spin transport. The A-phase of 3He or, more generally, the p-wave paired phase of a cold Fermi atomic gas, under suitable conditions should be a good candidate to observe the SHE, because the system has a conserved spin current (with no spin-orbit couplings).

cond-mat.supr-con↗

arXiv2008

Localization problem of the quasiperiodic system with the spin orbit interaction

We study one dimensional quasiperiodic system obtained from the tight-binding model on the square lattice in a uniform magnetic field with the spin orbit interaction. The phase diagram with respect to the Harper coupling and the Rashba coupling are proposed from a number of numerical studies including a multifractal analysis. There are four phases, I, II, III, and IV in this order from weak to strong Harper coupling. In the weak coupling phase I all the wave functions are extended, in the intermediate coupling phases II and III mobility edges exist, and accordingly both localized and extended wave functions exist, and in the strong Harper coupling phase IV all the wave functions are localized. Phase I and Phase IV are related by the duality, and phases II and III are related by the duality, as well. A localized wave function is related to an extended wave function by the duality, and vice versa. The boundary between phases II and III is the self-dual line on which all the wave functions are critical. In the present model the duality does not lead to pure spectra in contrast to the case of Harper equation.

cond-mat.stat-mech↗

arXiv2008

Gauge fields, quantized fluxes and monopole confinement of the honeycomb lattice

Electron hopping models on the honeycomb lattice are studied. The lattice consists of two triangular sublattices, and it is non-Bravais. The dual space has non-trivial topology. The gauge fields of Bloch electrons have the U(1) symmetry and thus represent superconducting states in the dual space. Two quantized Abrikosov fluxes exist at the Dirac points and have fluxes $2pi$ and $-2pi$, respectively. We define the non-Abelian SO(3) gauge theory in the extended 3$d$ dual space and it is shown that a monopole and anti-monoplole solution is stable. The SO(3) gauge group is broken down to U(1) at the 2$d$ boundary.The Abrikosov fluxes are related to quantized Hall conductance by the topological expression. Based on this, monopole confinement and deconfinement are discussed in relation to time reversal symmetry and QHE. The Jahn-Teller effect is briefly discussed.

cond-mat.mes-hall↗

arXiv2007

Adiabatic Process and Chern Numbers

We study quantum processes with two or more adiabatic parameters. When the parameters are compactified, a derivative of the Hamiltonian gives a first Chern number, hence an integer. This topological formulation is applied to the AC Josephson effect and the spin Hall effect in semiconductors.

cond-mat.mes-hall↗

arXiv2007

Zero modes and the edge states of the honeycomb lattice

The honeycomb lattice in the cylinder geometry with zigzag edges, bearded edges, zigzag and bearded edges (zigzag-bearded), and armchair edges are studied. The tight-binding model with nearest-neighbor hoppings is used. Edge states are obtained analytically for these edges except the armchair edges. It is shown, however, that edge states for the armchair edges exist when the the system is anisotropic. These states have not been known previously. We also find strictly localized states, uniformly extended states and states with macroscopic degeneracy.

cond-mat.mes-hall↗

arXiv2007

Fluctuations in Chemical Gelation

We study a chemical gelation model in two dimensions which includes both monomer aggregations and bond fluctuations. Our numerical simulation shows that a sol-gel transition occurs when an initial monomer concentration is above a critical concentration. Fractal aggregates grow until the sol-gel transition occurs. After the gelation, however, bond fluctuations break the fractal structure and a novel inhomogeneous gel fibre network appears instead. A pore size distribution of the inhomogeneous structure shows the existence of hierarchical structures in the gel phase. It is also found that slow dynamics appear near the critical concentration.

cond-mat.soft↗

arXiv2007

Distribution of spectral-flow gaps in the Rashba model with disorder: a new universality

We report a study of disordered electron systems with spin-orbit coupling on a cylinder using methods of random matrix ensembles. With a threading flux turned on, the single particle levels will generally avoid, rather than cross, each other. Our numerical study of the level-avoiding gaps in the disordered Rashba model demonstrates that the normalized gap distribution is of a universal form, independent of the random strength and the system size. For small gaps it exhibits a linear behavior, while for large gaps it decays exponentially. A framework based on matrix mechanical models is suggested, and is verified to reproduce the universal linear behavior at small gaps. Thus we propose to use the distribution of the spectral-flow gaps associated with flux insertion as a new way to characterize 2d random systems with spin-orbit coupling. The relevance and qualitative implications for spin (Hall) transport are also addressed.

cond-mat.mes-hall↗

arXiv2007

Braid Group, Gauge Invariance and Topological Order

Topological order in two-dimensional systems is studied by combining the braid group formalism with a gauge invariance analysis. We show that flux insertions (or large gauge transformations) pertinent to the toroidal topology induce automorphisms of the braid group, giving rise to a unified algebraic structure that characterizes the ground-state subspace and fractionally charged, anyonic quasiparticles. Minimal ground state degeneracy is derived without assuming any relation between quasiparticle charge and statistics. We also point out that noncommutativity between large gauge transformations is essential for the topological order in the fractional quantum Hall effect.

cond-mat.str-el↗

arXiv2006

Friedel Sum Rule as a Trace Formula

We examine the Friedel sum rule which states that the "excess charge" due to a single impurity potential in a metal is equal to a sum of phase shifts for scatterings of electrons by the impurity. For finite volume, the ``excess charge" is given by the difference between total numbers of levels in the Fermi sea with and without the impurity potential. However, a sequence of the "excess charge" for finite volume is not necessarily bounded in the infinite volume limit, as was pointed out by Kirsch. In order to circumvent this difficulty, we define "excess charge" directly for the infinite volume. The Friedel sum rule is proven to hold for the "excess charge" thus defined.

math-ph↗

arXiv2006

Quantum Hall effect and the topological number in graphene

Recently unusual integer quantum Hall effect was observed in graphene in which the Hall conductivity is quantized as $σ_{xy}=(\pm 2, \pm 6, \pm 10, >...) \times \frac{e^2}{h}$, where $e$ is the electron charge and $h$ is the Planck constant. %\cite{Novoselov2005,Zheng2005}, %although it can be explained in the argument of massless Dirac fermions, To explain this we consider the energy structure as a function of magnetic field (the Hofstadter butterfly diagram) on the honeycomb lattice and the Streda formula for Hall conductivity. The quantized Hall conductivity is obtained to be odd integer, $\pm1, \pm3, \pm5, ...$ times two (spin degrees of freedom) when a uniform magnetic field is as high as 30T for example. When the system is anisotropic and described by the generalized honeycomb lattice, Hall conductivity can be quantized to be any integer number. We also compare the results with those for the square lattice under extremely strong magnetic field.

cond-mat.dis-nn↗