Tohru Koma

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.

We study stationary states for the nonlinear Schrodinger equation on Fibonacci lattices, which are expected to be realized by Bose-Einstein condensates loaded into an optical lattice. When the model does not have a nonlinear term, the wavefunctions and the spectrum are known to show fractal structures. Such wavefunctions are termed critical. We present a phase diagram of the energy spectrum for varying the nonlinearity. It consists of three portions: a forbidden region, the spectrum of critical states and the spectrum of stationary solitons. We show that the energy spectrum of critical states remains intact, irrespective of the nonlinearity in the large number of stationary solitons.

We study the topology of frustration in d-dimensional Ising spin glasses with d >= 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.

We study lattice SU(2) Yang-Mills theory with dimension d >= 4. The model can be expressed as a (d - 1)-dimensional O(4) nonlinear sigma model in a d-dimensional heat bath. As is well known, the nonlinear sigma model alone shows a phase transition. If the quark confinement is a consequence of the 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 nonlinear sigma 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 nonlinear sigma model by analyzing the Fokker-Planck equation. It turns out that the fluctuations indeed destroy the long-range order of the nonlinear sigma 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.

We consider many-body systems with a global U(1) symmetry on a class of lattices with the (fractal) dimension 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 orders. (c) 2007 American Institute of Physics.

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.

We re-examine the charge transport induced by a weak electric field in two-dimensional quantum Hall systems in a finite, periodic box at very low temperatures. Our model covers random vector and electrostatic potentials and electron-electron interactions. The resulting linear response coefficients consist of the time-independent term sigma(xy) corresponding to the Hall conductance and the linearly time-dependent term gamma(sy) . 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 gamma(sy) . 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 sigma(yy) corresponding to the diagonal conductance which generates dissipation of heat always vanishes. The well-known topological argument yields the integral and fractional quantization of the averaged Hall conductance sigma(xy)over gauge parameters under the assumption that there exists a spectral gap above the ground state. In addition to this fact, we show that the averaged acceleration coefficients gamma(sy) vanish under. the same assumption. In the non-interacting case, the spectral gap between the neighboring Landau levels persists if the vector and the electrostatic potentials together satisfy a certain condition, and then the Hall conductance sigma(xy) without averaging exhibits the exact integral quantization with the vanishing acceleration coefficients in the infinite volume limit. We also estimate their finite size corrections. In the interacting case, the averaged Hall conductance sigma(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. We also discuss the relation between the fractional quantum Hall effect and the Atiyah-Singer index theorem for non-Abelian gauge fields.

We study the scattering of an electron by a ferromagnetic domain wall of the quantum Heisenberg-Ising model (XXZ model) with certain boundary conditions. The spin of the electron interacts with the spins of the XXZ model by the Hund coupling. Using the exact domain wall ground states of the XXZ model, we analytically obtain the exact effective Schrödinger equation for conduction electrons. This equation coincides with a conventional phenomenological Schrödinger equation which was derived in a classical treatment of spins of a domain wall. By solving the Schrödinger equation numerically, we have calculated the transmission coefficient which is a function of the Hund coupling and of the anisotropy of the XXZ model. It turns out that the transmission coefficient is vanishing in the low energy limit for the electron.

We show that the well-known translation invariant ground states and the recently discovered kink and antikink ground states are the complete set of pure infinite-volume ground states (in the sense of local stability) of the spin-S ferromagnetic XXZ chains with Hamiltonian for all Δ &gt 1, and all S ∈ 1/2N. For the isotropic model (Δ = 1) we show that all ground states are translation invariant. For the proof of these statements we propose a strategy for demonstrating completeness of the list of the pure infinite-volume ground states of a quantum many-body system, of which the present results for the XXX and XXZ chains can be seen as an example. The result for Δ &gt 1 can also be proved by an easy extension to general S of the method used in [T. Matsui, Lett. Math. Phys. 37 (1996) 397] for the spin-1/2 ferromagnetic XXZ chain with Δ &gt 1. However, our proof is different and does not rely on the existence of a spectral gap. In particular, it also works to prove absence of non-translationally invariant ground states for the isotropic chains (Δ = 1), which have a gapless excitation spectrum. Our results show that, while any small amount of the anisotropy is enough to stabilize the domain walls against the quantum fluctuations, no boundary condition exists that would stabilize a domain wall in the isotropic model (Δ = 1). © 1997, 1998 T. Koma and B. Nachtergaele.

We study two-magnon Bethe states in the spin-1/2 XXZ chain. The string hypothesis assumes that complex rapidities of the bound states take special forms. It is known, however, that there exist ''non-string states,'' which substantially disagrees with the string hypothesis. In order to clarify their nature, we study the large-N behavior of solutions of the Bethe-Ansatz equations to obtain explicit forms of typical Bethe states, where N is the length of the chain, and apply the scaling analysis (the multifractal analysis) to the Bethe states. It turns out that the non-string states contain ''quasi-bound'' states, which in some sense continuously interpolate between extended states and localized states. The ''quasi-bound'' states can be distinguished from known three types of states, i.e., extended, localized, and critical states. Our results indicate that there might be a need to reconsider the standard classification scheme of wavefunctions.

We study statistical characterization of the many-body states in exactly solvable models with internal degrees of freedom. The models under consideration include the isotropic and anisotropic Heisenberg spin chains, the Hubbard chain, and a model in higher dimensions which exhibits the Mott metal-insulator transition. It is shown that the ground states of these systems are all described by that of a generalized ideal gas of particles (called exclusons) which have mutual-exclusion statistics, either between different rapidities or between different species. For the Bethe ansatz solvable models, the low-temperature properties are well described by the excluson description if the degeneracies due to string solutions with complex rapidities are taken into account correctly. For the Hubbard chain with strong but finite coupling, charge-spin separation is shown for thermodynamics at low temperatures. Moreover, we present an exactly solvable model in arbitrary dimensions which, in addition to giving a perspective view of spin-charge separation, constitutes an explicit example of mutual-exclusion statistics in more than two dimensions.

We study the finite-size scaling behavior of two-point correlation functions of translationally invariant many-body systems al criticality. We propose an efficient method for calculating the two-point correlation functions in the thermodynamic limit from numerical data of finite systems. Our method is most effective when applied to a two-dimensional (classical) system which possesses a conformal invariance. By using this method with numerical data obtained from exact diagonalizations and Monte Carlo simulations, we study the spin-spin correlations of the quantum spin-1/2 and -3/2 antiferromagnetic chains. In particular, the logarithmic corrections to power-law decay of the correlation of the spin-1/2 isotropic Heisenberg antiferromagnetic chain are studied thoroughly. We clarify the cause of the discrepancy in previous calculations for the logarithmic corrections. Our result strongly supports the field-theoretic predicion based on the mappings to the Wess-Zumino-Witten nonlinear sigma-model or the sine-Gordon model. We also treat logarithmic corrections and crossover phenomena in the spin-spin correlation of the spin-3/2 isotropic Heisenberg antiferromagnetic chain. Our results are consistent with the Affleck-Haldane prediction that the correlation of the spin-3/2 chain exhibits a crossover to the same asymptotic behavior as in the spin-1/2 chain.

We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite-volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Neel order and the Hubbard model with a (superconducting) off-diagonal long-range order. In the corresponding finite system, the symmetry breaking is usually ''obscured'' by ''quantum fluctuation'' and one gets a symmetric ground state with a long-range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of order N-1, where N denotes the number of sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever-increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant times N-1. A crucial feature of the particular low-lying states we consider is that they can be regarded as finite-volume counterparts of the infinite-volume ground states. By forming linear combinations of these low-lying states and the (finite-volume) ground state and by taking infinite-volume limits, we construct infinite-volume ground states with explicit symmetry breaking. We conjecture that these infinite-volume ground states are ergodic, i.e., physically natural. Our general theorems not only shed light on the nature of symmetry breaking in quantum many-body systems, but also provide indispensable information for numerical approaches to these systems. We also discuss applications of our general results to a variety of interesting examples. The present paper is intended to be accessible to readers without background in mathematical approaches to quantum many-body systems.

We extend Griffith's theorem on symmetry breaking in quantum spin systems to the situation where the order operator and the Hamiltonian do not commute with each other. The theorem establishes that the existence of a long range order in a symmetric (non-pure) infinite-volume state implies the existence of a symmetry breaking in the state obtained by applying an infinitesimal symmetry-breaking field. The theorem is most meaningful when applied to a class of quantum antiferromagnets where the existence of a long range order has been proved. by the Dyson-Lieb-Simon method. We also present a related theorem for the ground states. It is an improvement of the theorem by Kaplan, Horsch and von der Linden. Our lower bounds on the spontaneous staggered magnetization in terms of the long range order parameter take into account the symmetry of the system properly, and are likely to be saturated in general models.

We question the validity of the finite-size-correction method based on conformal field theories. It is shown that the method leads to incorrect critical exponents on some of the correlation functions for a one-dimensional free fermion model. The model exhibits the same critical behaviour as in the well-known square-lattice Ising model that is believed to be conformally invariant.

We propose a new Monte Carlo method for calculating eigenvalues of transfer matrices leading to free energies and to correlation lengths of classical and quantum many-body systems. Generally, this method can be applied to the calculation of the maximum eigenvalue of a nonnegative matrix A such that all the matrix elements of A(k) are strictly positive for an integer k. This method is based on a new representation of the maximum eigenvalue of the matrix A as the thermal average of a certain observable of a many-body system. Therefore one can easily calculate the maximum eigenvalue of a transfer matrix leading to the free energy in the standard Monte Carlo simulations, such as the Metropolis algorithm. As test cases, we calculate the free energies of the square-lattice Ising model and of the spin-1/2 XY Heisenberg chain. We also prove two useful theorems on the ergodicity in quantum Monte Carlo algorithms, or more generally, on the ergodicity of Monte Carlo algorithms using our new representation of the maximum eigenvalue of the matrix A.

We prove the bound m(s)(beta) greater-than-or-equal-to square-root 3 sigma(beta) between the spontaneous staggered magnetization m(s)(beta) and the long range order parameter sigma(beta) in the quantum Heisenberg antiferromagnets. The same bound has been known for the Heisenberg ferromagnets from the works of Griffiths, and of Dyson, Lieb, and Simon, but extensions to the antiferromagnets had been lacking for more than two decades. When combined with the results on nonvanishing sigma(beta) from the Dyson-Lieb-Simon method, our bound proves the existence of symmetry breaking in the physically natural equilibrium states and ground states obtained by applying an infinitesimal symmetry breaking field.