In one dimension, the systems exhibits a unique property due to strong quantum uctuation. A typical example is that conduction electrons confined in one dimension become Tomonaga-Luttinger (TL) liquid instead of Fermi liquid. Models for strongly correlated electron systems such as the Hubbard and t - J model have been intensively studied in the context of high-temperature superconductors. Recent advancement of experimental techniques realizes one dimensional systems in many kinds of actual materials, e.g. artificial quantum wire manipulated by lithography technology, carbon nanotube, edge states of quantum Hall effect, atomic gas trapped in the optical lattice and quantum spin chains. Another distinctive characteristic of one-dimensional systems is the existence of powerful methods in both analytical and numerical approaches. Critical systems are classified analytically by using conformal field theory, and some models can be solved exactly through Bethe ansatz integral equation. Mapping the models to effective field theory by bosonization is also a useful technique to study low energy excitation structure and correlation functions. On the numerical side, density matrix renor-malization group (DMRG), which enables calculations for quite large size system with good accuracy, is a popular method. Other DMRG-like numerical methods including multi-scale entanglement renormalization ansatz, projected entangled-pair states and infinite time-evolving block decimation (iTEBD) have recently been invented and improved. Of course, Quantum Monte Carlo and exact diagonalization (ED) are also available. ED is especially helpful when combined with conformal field theory knowledge as finite size scaling or level spectroscopy.

In this PhD thesis, we address three topics concerning one-dimensional quantum systems. Two of them are about spin chains and the other is ultracold atomic gas. We approach these systems from analytical and numerical sides. The analytical method is field theoretical one with bosonization and conformal field theory. The employed computational techniques are ED and iTEBD. In iTEBD, the matrix product state of the system evolves in imaginary time direction through Trotter-Suzuki decomposition, leading to the ground state. A merit of this method is that an infinite-size system can be treated directly by imposing the translational symmetry on the system. The detailed principle and implementation is explained in the appendix. We briey summarize each content of three topics in the following.

(1) Determination of bosonized coefficients in dimer operators

The correspondence between spin-1/2 XXZ chain and its effective field theory has been investigated intensively. For example, while spinon velocity and TL parameter are obtained exactly by utilizing Bethe ansatz, bosonization coefficients of spin operators are non-universal and cannot be derived analytically in magnetic field. These coefficients have been determined from the compar-ison between asymptotic form of correlation function and the DMRG result. We propose a simple new method of deriving the information on uniform spin-1/2 XXZ chains from deformed chains. Namely, the coefficients of bosonized dimer operators in spin-1/2 XXZ chains are extracted by comparing the numerically calculated gap in dimerized spin chains with the gap formula of the effective sine-Gordon theory. Figure 1 shows that the numerical data are quantitatively fitted by the gap formula of sine-Gordon theory, which supports the validity of our strategy for fixing the coefficients.

We apply obtained coefficients to the determination of ground-state phase diagrams of dimerized spin chains in a magnetic field and antiferromagnetic frustrated spin ladders with a four-spin interaction. The optical response in one-dimensional Mott insulators with Peierls instability is also evaluated quantitatively.

(2) Mass ratio of excitation particle in dimerized chain with frustration

We study Heisenberg antiferromagnetic (HAF) chain with alternating exchange interaction:

in association with the content of (1). As already mentioned, it is known that this lattice model is related with effective field theory through the bosonization. According to its prediction, there are three kinds of excitation particles, a soliton, anti-soliton and breathers (bound state of soliton and anti-soliton). The lightest breather has the same mass as a soliton and antisoliton, forming a triplet, and the mass of second lightest breather is √3 times as large as that of the triplet. However, numerical calculations for S = 1/2 indicate that the mass ratio r of the second lightest breather to the triplet deviates from √3 unless J2/J~ 0.25, where the transition from TL liquid to dimerized phase happens. It indicates that r is affected by the marginal term added to sine-Gordon theory, but the quantitative explanation has not been given.

We first demonstrate that the S = 1/2 and 1 cases are understood in a unified way by using sine-Gordon theory with the marginal term. For both cases, r becomes √3 by the introduction of next-nearest neighbor coupling J2 = J2c, where the marginal term in effective field theory vanishes. The universality class of transition is TL liquid and first order for J2 < J2c and J2 > J2c, respectively. Then, the effect of the marginal term on r quantitatively is considered. We derive the formula for r as a function of δ and J2=J, combining the result of form factor perturbation theory and renormalization analysis. The result is shown in Fig. 2. The numerically calculated r is well fitted by the derived function with a single parameter.

(3) Population imbalance in two-component atomic gas in one dimension

We study two-component (or pseudo spin- 1/2 ) Bose or Fermi gases confined in one dimension, in which particles are convertible between the components. The usual weak-coupling theory based on bosonization predicts that a strong inter-component repulsion induces spontaneous population imbalance between the components, in other words, the ferro-magnetism of the pseudo-spins (see Fig. 3). However, we cannot examine any property of ferromagnetic phase and the phase transition from the weak-coupling approach since it breaks down at the transition point. To study the strong-coupling regime, we employ numerical methods (ED and iTEBD). Then it is demonstrated that the imbalanced phase contains gapped spin excitations and gapless charge excitations characterized as TL liquid.

We also uncover a crucial effect of the inter-component particle hopping on the transition to the imbalanced phase. In the absence of this hopping, the transition is of first order. With an infinitesimal intercomponent hopping, the transition becomes of Ising type. These results can be qualitatively understood from a simple perturbation theory in the strong-coupling limit. From the accurate numerical data, we determine the ground-state phase diagram in a wide parameter regime and test the reliability of the weak-coupling bosonization formalism.

Fig. 1 Numerically evaluated excitation gap of dimerized XXZ chains with exchange anisotropy Δz = 0:6 and several values of dimerization in XY-plane δxy and along Z-axis δz. Solid curves are fitting by sine-Gordon theory.

Fig. 2 (a) The case of S = 1/2 . The circle, triangle and down-pointing triangle represent numerically obtained r for δ = 0.005, 0.01 and 0.015, respectively. The solid, dashed and dashed-dotted lines are the derived formula for r as a function of δ and J2/J. (b) The case of S = 1. The circle, triangle and down-pointing triangle represent numerically obtained r for δ - δc = -0.005, 0.005 and 0.01, respectively. Here, δc is the transition point of δ from Haldane to dimerized phase. The solid line is the derived formula.

Fig. 3 Illustrations of 1D two-component gases. The two components represent two internal states of atoms [shown by different colors in (a) and (c)]. Even without internal states, a double-channel trap potential can produce a similar situation [(b) and (d)]. A strong repulsion between the components induces the population imbalance [(c) and (d)].

微視的なハミルトニアンにより格子上で定式化された量子多体系から,その巨視的,普遍的性質を抽出する事は量子統計力学の重要な課題であり,スケール極限と繰り込みのアイデアに基づく様々なアプローチ,有効場の理論が提唱されている.特に1 次元系は顕著な量子多体効果を呈し,豊富な解析的,数値的手法が適用可能であるなど,理論物理学としても豊かな分野を形成している.

本論文の主題は,1 次元量子多体系とボソン化有効場理論の対応である.ボソン化は,Tomonaga-Luttinger (TL) 模型の研究等に伴い発展してきた手法であり,これまで多くの成果を収めている.本論文は,XXZ 鎖や2 成分原子気体系などの具体的な1 次元模型を諸種の数値的手法で解析し,繰り込み群や厳密解の帰結と比較,検証する事により,ボソン化有効場の理論の有効性を高め,妥当性を補強する結果を得ている.以下,各章ごとにその内容を概説する.

第1 章では導入として,本論文の背景,動機,位置づけなどについて,数値解析の方法論にも言及しながら説明している.後半は各章ごとにその内容の要約を与え,論文全体の概観を提示している.

第2 章ではスピン1/2 XXZ 鎖のギャップレス領域での低エネルギー励起有効場理論が,自由ボソン理論(中心電荷c = 1 の共形場理論)で記述される事を解説している.理論のパラメーターとしては,XXZ 鎖では結合定数J とその異方性〓,ボソン場の理論ではスピノン速度v といわゆるTLパラメーターK であり,後者は2体スピン相関のベキ減衰の指数と簡単な関係にある.章の最後では,関連する結果としてXXZ 鎖のスピン演算子のボソン化に現れるボソン化係数を2体相関の漸近形から決定した先行研究を紹介している.

第3 章ではXXZ 模型のダイマー演算子のボソン化係数dxy, dz を決定している.これらは有効場理論による物理量の定量的評価で重要な役割を果たす定数である.方法としては,元のXXZ 模型にボンド交代的な成分を導入し,これにより生じた励起ギャップを数値対角化で求め,対応するボソン化有効場理論であるサイン・ゴルドン模型のソリトン質量公式と比較するというもので,本論文で初めて考案,適用された.主結果としてボソン化係数dxy, dz の〓 (あるいはK) 依存性を数値表としてまとめている.本章では更にこれに関する多くの問題の考察がなされている.例えば,XY 模型でのボソン化係数の解析的結果との照合や,スピン演算子のボソン化係数の決定などがある.後者については相関関数の漸近評価による先行研究があるが,本章の手法は計算がより簡便な励起ギャップからの算出を可能にしており,既知の結果との照合によりボソン化有効場理論の妥当性を補強している.本章の最後には,新たに求めたボソン化係数の応用例として,磁場中ダイマー化スピン鎖や4体相互作用のあるスピンラダーの基底状態相図,1 次元Mott 絶縁体の電気分極や光学伝導度の計算などが挙げられている.

第4 章では等方的Heisenberg 模型の結合定数にダイマー的変調(σ に比例) を導入し,更に次近接相互作用(J2 に比例) を加えた模型を考察している.スペクトルはギャップを持ち,ボソン化有効場理論のサイン・ゴルドン模型は第1, 第2 素励起の質量比が√3 である事を予言する.本章ではスピンが1/2 と1 の場合にinfinite time-evolving block decimation (iTEBD) という数値的手法を用いて上記の格子模型の励起エネルギー比r を求めている.それによるとr =√3 となるJ2 は場の理論でマージナル項が消える値に極めて近く,この点の周りで確かにサイン・ゴルドン理論による記述が妥当である事が示唆されている.また形状因子摂動論と繰り込みの議論を併せてr のδ, J2 依存性をあらわす表式を導き,数値計算の結果と良く合致する事を示している.

第5 章では1 次元2 成分原子気体に関する模型を扱っている.成分間斥力が弱い時にはボソン化有効場理論が良い記述を与える事,斥力の増大は粒子数密度が偏る密度差転移を誘起する事,特に密度差相は,2 成分をup,down とみなした擬スピン系の強磁性とTL 液体が共存する相である事等を理論的な考察から示唆し,数値的に検証している.

第6 章では論文全体の要約と展望が述べられており,今後の課題を挙げて結びとしている.

本論文は1 次元量子多体系とボソン化有効場理論との対応に多くの新しい知見を提供するものであり,学位論文として十分な内容を持っている.

なお,本論文の3 章の一部は佐藤正寛氏,5 章の一部は同氏と古川俊輔氏との共同研究に基づくものであるが,論文提出者が主体となって解析,検討を行ったもので,その寄与は十分であると判断する.

以上の事から,博士(理学)の学位を授与できると認める.