arXiv subjects

Tohru Koma

Publications and source records attributed to Tohru Koma.

At least 37 records · Page 2Linked to original sources

arXiv2019

Quantization of Conductance in Quasi-Periodic Quantum Wires

We study charge transport in the Peierls-Harper model with a quasi-periodic cosine potential. We compute the Landauer-type conductance of the wire. Our numerical results show the following: (i) When the Fermi energy lies in the absolutely continuous spectrum that is realized in the regime of the weak coupling, the conductance is quantized to the universal conductance. (ii) For the regime of localization that is realized for the strong coupling, the conductance is always vanishing irrespective of the value of the Fermi energy. Unfortunately, we cannot make a definite conclusion about the case with the critical coupling. We also compute the conductance of the Thue-Morse model. Although the potential of the model is not quasi-periodic, the energy spectrum is known to be a Cantor set with zero Lebesgue measure. Our numerical results for the Thue-Morse model also show the quantization of the conductance at many locations of the Fermi energy, except for the trivial localization regime. Besides, for the rest of the values of the Fermi energy, the conductance shows a similar behavior to that of the Peierls-Harper model with the critical coupling.

cond-mat.mes-hall↗

arXiv2018

The Noncommutative Index Theorem and the Periodic Table for Disordered Topological Insulators and Superconductors

We study a wide class of topological free-fermion systems on a hypercubic lattice in spatial dimensions $d\ge 1$. When the Fermi level lies in a spectral gap or a mobility gap, the topological properties, e.g., the integral quantization of the topological invariant, are protected by certain symmetries of the Hamiltonian against disorder. This generic feature is characterized by a generalized index theorem which is a noncommutative analogue of the Atiyah-Singer index theorem. The noncommutative index defined in terms of a pair of projections gives a precise formula for the topological invariant in each symmetry class in any dimension ($d \ge 1$). Under the assumption on the nonvanishing spectral or mobility gap, we prove that the index formula reproduces Bott periodicity and all of the possible values of topological invariants in the classification table of topological insulators and superconductors. We also prove that the indices are robust against perturbations that do not break the symmetry of the unperturbed Hamiltonian.

math-ph↗

arXiv2018

Decay of Superconducting Correlations for Gauged Electrons in Dimensions $D\le 4$

We study lattice superconductors coupled to gauge fields, such as an attractive Hubbard model in electromagnetic fields, with a standard gauge fixing. We prove upper bounds for a two-point Cooper pair correlation at finite temperatures in spatial dimensions $D\le 4$. The upper bounds decay exponentially in three dimensions, and by power law in four dimensions. These imply absence of the superconducting long-range order for the Cooper pair amplitude as a consequence of fluctuations of the gauge fields. Since our results hold for the gauge fixing Hamiltonian, they cannot be obtained as a corollary of Elitzur's theorem.

math-ph↗

arXiv2017

A New Numerical Method for $\mathbb{Z}_2$ Topological Insulators with Strong Disorder

We propose a new method to numerically compute the $\mathbb{Z}_2$ indices for disordered topological insulators in Kitaev's periodic table. All of the $\mathbb{Z}_2$ indices are known to be derived from the index formulae which are expressed in terms of a pair of projections introduced by Avron, Seiler, and Simon. For a given pair of projections, the corresponding index is determined by the spectrum of the difference between the two projections. This difference exhibits remarkable and useful properties, as it is compact and has a supersymmetric structure in the spectrum. These properties make it possible to numerically determine the indices of disordered topological insulators highly efficiently. The method is demonstrated for the Bernevig-Hughes-Zhang and Wilson-Dirac models whose topological phases are characterized by a $\mathbb{Z}_2$ index in two and three dimensions, respectively.

cond-mat.mes-hall↗

arXiv2016

Two No-Go Theorems on Superconductivity

We study lattice superconductors such as attractive Hubbard models. As is well known, Bloch's theorem asserts absence of persistent current in ground states and equilibrium states for general fermion systems. While the statement of the theorem is true, we can show that the theorem cannot exclude possibility of a surface persistent current. Such a current can be stabilized by boundary magnetic fields which do not penetrate into the bulk region of a superconductor, provided emergence of massive photons, i.e., Meissner effect. Therefore, we can expect that a surface persistent current is realized for a ground/equilibrium state in the sense of stability against local perturbations. We also apply Elitzur's theorem to superconductors at finite temperatures. As a result, we prove absence of symmetry breaking of the global $U(1)$ phase of electrons for almost all gauge fields. These observations suggest that the nature of superconductivity is the emergence of massive photons rather than the symmetry breaking of the $U(1)$ phase of electrons.

math-ph↗

arXiv2016

The $Z_2$ Index of Disordered Topological Insulators with Time Reversal Symmetry

We study disordered topological insulators with time reversal symmetry. Relying on the noncommutative index theorem which relates the Chern number to the projection onto the Fermi sea and the magnetic flux operator, we give a precise definition of the $Z_2$ index which is a noncommutative analogue of the Atiyah-Singer $Z_2$ index. We prove that the noncommutative $Z_2$ index is robust against any time-reversal symmetric perturbation including disorder potentials as long as the spectral gap at the Fermi level does not close.

math-ph↗

arXiv2015

Topological Current in Fractional Chern Insulators

We consider interacting fermions in a magnetic field on a two-dimensional lattice with the periodic boundary conditions. In order to measure the Hall current, we apply an electric potential with a compact support. Then, due to the Lorentz force, the Hall current appears along the equipotential line. Introducing a local current operator at the edge of the potential, we derive the Hall conductance as a linear response coefficient. For a wide class of the models, we prove that if there exists a spectral gap above the degenerate ground state, then the Hall conductance of the ground state is fractionally quantized without averaging over the fluxes. This is an extension of the topological argument for the integrally quantized Hall conductance in noninteracting fermion systems on lattices.

math-ph↗

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↗

arXiv2010

Fluctuations Destroying Long-Range Order in SU(2) Yang-Mills Theory

We study lattice SU(2) Yang-Mills theory with dimension $d\ge 4$. The model can be expressed as a $(d-1)$-dimensional O(4) non-linear $σ$-model in a $d$-dimensional heat bath. As is well known, the non-linear $σ$-model alone shows a phase transition. If the quark confinement is a consequence of absence of a phase transition for the Yang-Mills theory, then the fluctuations of the heat bath must destroy the long-range order of the non-linear $σ$-model. In order to clarify whether this is true, we replace the fluctuations of the heat bath with Gaussian random variables, and obtain a Langevin equation which yields the effective action of the non-linear $σ$-model through analyzing the Fokker-Planck equation. It turns out that the fluctuations indeed destroy the long-range order of the non-linear $σ$-model within a mean field approximation estimating a critical point, whereas for the corresponding U(1) gauge theory, the phase transition to the massless phase remains against the fluctuations.

hep-lat↗

arXiv2010

Algebraic Topology of Spin Glasses

We study topology of frustration in $d$-dimensional Ising spin glasses with $d\ge 2$ with nearest-neighbor interactions. We prove the following: For any given spin configuration, the domain walls on the unfrustration network are all transverse to a frustrated loop on the unfrustration network, where a domain wall is defined to be a connected element of the collection of all the $(d-1)$-cells which are dual to the bonds having an unfavorable energy, and the unfrustration network is the collection of all the unfrustrated plaquettes. These domain walls are topologically nontrivial because they are all related to the global frustration of a loop on the unfrustration network. Taking account of the thermal stability for the domain walls, we can explain the numerical results that three or higher dimensional systems exhibit a spin glass phase, whereas two-dimensional ones do not. Namely, in two dimensions, the thermal fluctuations of the topologically nontrivial domain walls destroy the order of the frozen spins on the unfrustration network, whereas they do not in three or higher dimensions. This may be interpreted as a global, topological effect of the frustrations.

math-ph↗

arXiv2008

Widths of the Hall Conductance Plateaus

We study the charge transport of the noninteracting electron gas in a two-dimensional quantum Hall system with Anderson-type impurities at zero temperature. We prove that there exist localized states of the bulk order in the disordered-broadened Landau bands whose energies are smaller than a certain value determined by the strength of the uniform magnetic field. We also prove that, when the Fermi level lies in the localization regime, the Hall conductance is quantized to the desired integer and shows the plateau of the bulk order for varying the filling factor of the electrons rather than the Fermi level.

math-ph↗

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↗

arXiv2005

Spectral Gap and Exponential Decay of Correlations

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D<2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.

math-ph↗

arXiv2005

Spectral Gap and Decay of Correlations in U(1)-Symmetric Lattice Systems in Dimensions D<2

We consider many-body systems with a global U(1) symmetry on a class of lattices with the (fractal) dimensions D<2 and their zero temperature correlations whose observables behave as a vector under the U(1) rotation. For a wide class of the models, we prove that if there exists a spectral gap above the ground state, then the correlation functions have a stretched exponentially decaying upper bound. This is an extension of the McBryan-Spencer method at finite temperatures to zero temperature. The class includes quantum spin and electron models on the lattices, and our method also allows finite or infinite (quasi)degeneracy of the ground state. The resulting bounds rule out the possibility of the corresponding magnetic and electric long-range order.

math-ph↗

arXiv2004

Revisiting the Charge Transport in Quantum Hall Systems

We reexamine the charge transport induced by a weak electric field in two-dimensional quantum Hall systems in a finite, periodic box at very low temperatures. The resulting linear response coefficients consist of the time-independent term $σ_{xy}$ corresponding to the Hall conductance and the linearly time-dependent term $γ_{sy}\cdot t$ in the transverse and longitudinal directions $s=x,y$ in a slow switching limit for adiabatically applying the initial electric field. The latter terms $γ_{sy}\cdot t$ are due to the acceleration of the electrons by the uniform electric field in the finite and isolated system, and so the time-independent term $σ_{yy}$ corresponding to the diagonal conductance always vanishes. The well known topological argument yields the integral and fractional quantization of the averaged Hall conductance $\bar{σ_{xy}}$ over gauge parameters under the assumption on the existence of a spectral gap above the ground state. In addition to this fact, we show that the averaged acceleration coefficients $\bar{γ_{sy}}$ are vanishing under the same assumption. In the non-interacting case, the spectral gap between the neighbouring Landau levels persists if the vector and the electrostatic potentials together satisfy a certain condition, and then the Hall conductance $σ_{xy}$ without averaging exhibits the exact integral quantization in the infinite volume limit with the vanishing acceleration coefficients. We also estimate their finite size corrections. In the interacting case, the averaged Hall conductance $\bar{σ_{xy}}$ for a non-integer filling of the electrons is quantized to a fraction not equal to an integer under the assumption that the potentials satisfy certain conditions in addition to the gap assumption.

math-ph↗

arXiv2002

The spectral gap for the ferromagnetic spin-J XXZ chain

We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain. Our main theorem is that there is a non-vanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about the magnitude of this gap, about its behavior for large Δ, about its overall behavior as a function of Δand its dependence on the ground state, about the scaling of the gap and the structure of the low-lying spectrum for large J, and about the existence of isolated eigenvalues in the excitation spectrum. By combining information obtained by perturbation theory, numerical, and asymptotic analysis we arrive at a number of interesting conjectures. The proof of the main theorem, as well as some of the numerical results, rely on a comparison result with a Solid-on-Solid (SOS) approximation. This SOS model itself raises interesting questions in combinatorics, and we believe it will prove useful in the study of interfaces in the XXZ model in higher dimensions.

math-ph↗

arXiv2000

Detecting Gapless Excitations above Ferromagnetic Domain Walls

In a two or three dimensional ferromagnetic XXZ model, a low energy excitation mode above a magnetic domain wall is gapless, whereas all of the usual spin wave excitations moving around the whole crystal are gapful. Although this surprising fact was already proved in a mathematically rigorous manner, the gapless excitations have not yet been detected experimentally. For this issue, we show theoretically that the gapless excitations appear as the dynamical fluctuations of the experimental observable, magnetoresistance, in a ferromagnetic wire. We also discuss other methods (e.g., ferromagnetic resonance and neutron scattering) to detect the gapless excitations experimentally.

cond-mat.mes-hall↗

arXiv2000

Resistance Jumps and Hysteresis in Ferromagnetic Wires

A phenomenological approach based on the principle of minimum heat generation is proposed for understanding experiments for metallic wires with a ferromagnetic domain wall controlled by a magnetic field. This approach elucidates the origin of negative jump and of hysteresis in the magnetoresistance experiments. To justify our approach, we computed the transmission probability of a single electron through a wire. Under certain conditions, the presence of a domain wall enhances the transmission through an impurity potential. Combining this result with the principle of minimum heat generation explains the counterintuitive negative resistance jump.

cond-mat.mes-hall↗