1. Background and Purpose

Frustrated quantum spin systems contain rich physical structures of nontrivial quantum states and phase transitions, and it is very important to elucidate the mechanism as a grand challenge problem in the condensed matter physics and the material design. The origin of frustration can be a geometric structure or an interaction with other degrees of freedom. Among them, the spin-lattice interaction has a large contribution for determination of effective spin-spin interaction. As a system dominated by the spin-lattice interaction, a spin-Peierls system has caught the attention for a long time. When the decrease in energy by dimerization that makes spin singlet pairs

exceeds the increase by lattice distortion, the spin-Peierls system turns into the dimer phase, which is called the spin-Peierls transition. The analysis of this system and transition will bring understanding of the role of lattice degree of freedom in condensed materials.

Two limiting cases have been considered in the spin-Peierls system: the adiabatic limit and the antiadiabatic limit. The former case is realized in some organic materials experimentally; for example, the compound TTF-CuBDT was found as the first spin-Peierls material in 1975. The low-energy physics is described by a S=1/2 one-dimensional Heisenberg chain coupled with the lattice degree of freedom in harmonic potential. Cross and Fisher analyzed this model by using the Abelian bosonization method combined with the random phase approximation in the adiabatic limit, where the quantum nature of lattice is ignored. Their theory seems effective to the organic materials because a soft phonon mode that is one of consequences of their theory was observed in experiments.

The Cross-Fisher theory is considered to be, however, not valid to the compound CuGeO$_3$ that was discovered as the first inorganic spin-Peierls material in 1993. It is because that the energy scale of phonon is the same order with that of spin, which is away from the adiabatic region, and no soft phonon was observed experimentally. This discovery triggered many theoretical approaches from the antiadiabatic limit: the perturbation expansion, the flow equation method, the unitary transformation, etc. According to these approaches, tracing out the phonon degree of freedom results in producing an effective spin frustration. Then, the quantum spins construct the dimer by this frustration effect as the famous Majumdar-Ghosh point. Interestingly, the phase diagram and the universality class of transitions are totally different between the adiabatic and antiadiabatic limits. Not only for understanding this qualitative difference but also for explaining the behavior of realistic materials away from the two limits, such as CuGeO$_3$ and MEM(TCNQ)$_2$, it is important to analyze the crossover and connection between them. A precise calculation for the intermediate region, however, has been difficult because of the complexity of the system. On these backgrounds, the purpose of this thesis are to invent a novel method that can treat the spin-Peierls system correctly and to elucidate the mechanism of the quantum phase transitions triggered by the quantum nature of lattice degree of freedom.

2. Developments of New Methods

We present three novel methods in this thesis. The first is a generic improvement of sampling efficiency in the Markov chain Monte Carlo (MCMC) method. The MCMC method is a generic tool for investigating many kinds of systems with multiple degrees of freedom. For the method to work effectively, we must consider following three key matters: the choice of ensemble, the selection of candidate states, and the optimization of transition kernel. For the first, the extended ensemble methods, such as the multicanonical method, have been applied to many hardly-sampled systems. For the second, the cluster algorithms, such as the Swendsen-Wang algorithm, can perform a global update with low or zero rejection rate. For the third, the Metropolis algorithm or the heat bath algorithm (Gibbs sampler) is widely used in practical simulations. However, these conventional methods are not optimal. In the meantime, since the invention by Metropolis and the co-workers in 1953, the MCMC method has evolved within the paradigm of the detailed balance, namely reversibility. The detailed balance is, however, not necessary condition. Instead of solving the usual algebraic problem of the detailed balance equation, we rewrite the needed task as a geometric allocation problem and always find not only a reversible solution but also an irreversible kernel with minimized rejection rate. Our approach is a new kind of optimization method that is particularly different from conventional schemes using derivatives of a cost function. As an important point, there is no additional CPU time cost in our method. This algorithm is the first versatile method that is free from the detailed balance. By applying our algorithms for the 8-state Potts model, the autocorrelation time of the single spin flip update gets more than 12 times as short as that by the Metropolis algorithm. In addition, the correlation time for the quantum Heisenberg chain with magnetic field becomes more than 100 times as short as that by the heat bath algorithm. Moreover, our algorithms can be generally extended to also continuous variables. Thus, the methods presented in this thesis should replace the conventional update procedures in all MCMC simulations. This optimization is significantly effective to also following our update method for the spin-Peierls model.

The second is the development of new quantum Monte Carlo (QMC) method for particle number nonconserving systems, such as the spin-Peierls model written by the second quantization. The QMC method that is based on the worldline representation have been applied to a wide variety of quantum spin, bosonic and fermionic systems. In recent years, the method has passed through two turning points of key improvements. One is the elimination of the decomposition error of imaginary time; Beard et al. showed the formalism where it becomes possible to simulate quantum systems directly in continuous time. The other is the invention of nonlocal worldline update methods, such as the loop algorithm and the worm algorithm. Later, Syljuasen et al. extended the worm update to a more efficient procedure, which has become the standard method now, which is called also the directed-loop algorithm. In the meanwhile, we need to treat a nonconserved particle when a good quantum number and basis set are difficult to find. In our spin-Peierls model, soft-core bosons (phonons) are not conserved by the Hamiltonian. The conventional worm (directed-loop) algorithm is, however, not applicable to nonconserved particles because of the ergodicity problem. Thus, we have extended the worm update so that it becomes possible to calculate such models. Moreover, we have successfully removed the worm bounce process, which has been a bottleneck of the method, by combining our optimized kernel as we mentioned above. By these improvements, we can treat general conserved/nonconserved quantum spins and bosons without any approximation (typically, decomposition error or occupation number cutoff of soft-core bosons). As a result, a precise Monte Carlo analysis of large-scale spin-Peierls systems has become feasible for

the first time.

The third is the invention of an estimation method for precise energy gap and of the level spectroscopy by the QMC method. Using our extended worm algorithm, we have found that the Kosterlitz-Thouless (KT) type phase transition occurs in the one dimensional spin-Peierls systems. For the KT type transition, the conventional finite-size scaling method, where the correlation length is assumed to diverge in a power-law form, fails to catch the critical point. In the meanwhile, the level spectroscopy, where a transition point and an universality class can be determined by using excitation gaps even for the KT transition, has been applied to many one-dimensional quantum systems. In this thesis, we have combined this method with the QMC technique and successfully applied to the spin-Peierls systems. For the combination, we need to calculate many excitation gaps precisely. The problem of gap estimation in the QMC method is, however, ill-posed as the inverse Laplace transformation; the conventional estimator of gap has not only statistical error but also systematic error. In order to get a precise gap, we have invented new estimator sequence that converges to a true gap value. Moreover, we have formulated new calculation procedures to measure the excitation gap of the S=0 singlet and the S=1 S$^z$=+1,-1 doublet, although only the gap of S=1 S$^z$=0 is available in the conventional approach. Thus, we can use the level spectroscopy analysis with precise gap estimations.

3. Quantum Phase Transition of Spin-Peierls Systems

We have applied above novel methods to the XXZ spin-Peierls chain and two dimensional system coupled to the quantum phonon of optical mode. In the one-dimensional XXX (Heisenberg type) spin-Peierls system, an infinitesimal spin-lattice coupling drives the ground state into the dimer state in the adiabatic limit, as the Cross-Fisher theory. On the other hand, the quantum phase transition from the Tomonaga-Luttinger liquid phase to the dimer phase occurs at a finite coupling parameter when the quantum nature of lattice is taken into account. The phase transition has been believed to be a KT type from the analogy with the frustrated J1-J2 model. The transition point has been estimated by using some analytical method, such as the flow equation method or the unitary transformation. These estimated points, however, have differed considerably between each others. We have determined the critical point much more precisely than ever by applying the developed gap-estimation method with the level spectroscopy. We have determined also the universality class, and elucidated the phase diagram of the one dimensional XXZ spin-Peierls system for the first time. For the XY-like anisotropy, the phase transition between the Tomonaga-Luttinger liquid phase and the dimmer phase occurs and the low-energy physics on the transition line is governed by the SU(2) k=1 Wess-Zumino-Witten model. For the Ising-like model, on the other hand, the transition line of the Gaussian universality class appears between the Neel phase and the dimmer phase. The whole phase diagram is qualitatively consistent to the sine-Gordon model. This correspondence is reasonable because the original spin-Peierls model can be renormalized to the frustrated J1-J2 model and further to the sine-Gordon model, according to the antiadiabatic approach. However, we have confirmed the aspect of this phase diagram still remains even in the adiabatic parameter region. This correspondence is not trivial because it implies the universality class instantaneously changes no matter how small the quantum fluctuation of lattice degree of freedom is. This finding manifests the essence of the quantum nature of lattice in the spin-Peierls systems.

We also investigated a two dimensional spin-Peierls system. For higher dimensions than one, there have been only few researches so far because the bosonization method on which most analytical approaches have relied cannot be effective. We have considered an interesting two dimensional system where the spin-Peierls chains are connected only by phonon interchain interaction without interchain spin interaction. For a strong spin-phonon coupling region of this model, a finite-temperature phase transition from the disorder phase to the dimer phase occurs. On the other hand, for a weak spin-phonon coupling region, we have found a interesting two-dimensional gapless phase that has a similar property with the one-dimensional case. This gapless excitation certainly stems from the effective spin frustration beyond the mean-field treatment. This phase is the first discovery of the two dimensional gapless quantum liquid phase by the worldline Monte Carlo method.

4. Conclusion

We have developed the three novel methods for investigating the spin-Peierls system correctly. These methods are generally applicable to other physical and statistical problems. We have applied our methods to the one and two dimensional large-scale spin-Peierls systems. As a result, we have elucidated the phase diagram of the one-dimensional XXZ spin-Peierls model and found the interesting quantum liquid phase in the two-dimensional system. These analyses are the first precise calculations of the quantum criticality not only for the spin-Peierls model but also for effectively frustrated quantum spin systems by the worldline Monte Carlo method.

物性物理学・物質科学において、フラストレート量子スピン系は新奇な量子相や相転移が生じ得る物理系としてたいへん興味深く、実験的にも理論的にも国内外で精力的に研究が行われている。そのようなスピンフラストレーションの起源の一つとして、スピン・格子相互作用が挙げられる。本論文は、新しく効率的な計算科学的手法を開発することで、量子スピン・フォノン複合系の詳細な解析を可能とし、実効的にフラストレートする量子スピン系の基底状態と臨界現象を明らかにしたものである。

第一章は緒言であり、代表的なスピン格子複合系であるスピンパイエルス系の先行研究と未解決の問題が述べられている。先行研究として、スピンと格子自由度のエネルギースケールが大きく異なる場合に対応する、断熱極限と非断熱極限の理論の間の相違点が説明されている。そして、現実物質に対応するような、両極限をつなぐクロスオーバーの解析と、理論研究の主な対象であった一次元鎖を超えた解析の必要性が述べられている。しかし、そのように複雑な系の解析はこれまで困難であった。この序章の中で、その困難と、本論文で行われた手法開発がなぜ必要だったのかが説明されている。

第二章では、本論文で用いたマルコフ連鎖モンテカルロ法において、計算効率を大幅に改善する一般的な改良法が提案されている。まず既存のアルゴリズムが紹介された後、ンテカルロ法における棄却確率を最小化することの重要性が述べられている。主要なアイデアは、これまで代数的に解かれていた確率決定問題を、幾何学的な「重みの割り当て」問題として視覚的に表現することである。この新しい表現を用いて、半世紀以上使われ続けてきた「詳細つりあい条件」を課さずとも、正しい遷移確率を構成できることが示された。またいくつかの物理的模型において、この改良により計算効率が数倍から100 倍以上改善できることが確認されている。

第三章では、まずスピンパイエルス系に対してモンテカルロ法を用いる際のこれまでの困難が説明されている。スピンと相互作用するフォノンは、ハミルトニアンにより粒子数が保存しないボゾンとして表現されるが、経路積分量子モンテカルロ法で現在最も汎用性のあるワームアルゴリズムを用いても、そのような非保存粒子を扱うことはできなかった。本章では、スピンパイエルス系のような粒子数非保存系に対しても効率的な新しい状態更新法が提案されている。またグリーン関数などの非自明な相関関数の計算法についても詳細に述べられている。

第四章では、通常の有限サイズスケーリング法が有効でないKosterlitz-Thouless 転移に対して強力な解析法であるレベルスペクトロスコピーを、モンテカルロ法と組み合わせて用いる新しい数値解析手法が提案されている。この手法は、1 次元のスピンパイエルス系などの量子相転移の解析に特に有効である。この手法の鍵はエネルギーギャップの精密測定にあるが、従来のモンテカルロ法では統計誤差のみでなく標準誤差も含まれていたため、詳細な解析は困難であった。本章では、標準誤差を系統的に小さくする新しい統計量が提案され、その有効性が示されている。さらに、ボンド交替鎖に対してエネルギーギャップの交差を測定することで、小さな系の計算から非常に効率良く転移点を求めることができることが実証されている。

第五章では、上記の手法を用いて1 次元XXZ スピンパイエルス系の基底状態相図が明らかにされた。レベルスペクトロスコピーを用いることによって、相転移の臨界指数とユニバーサリティークラスが高精度で求められている。さらに、断熱領域のパラメータであっても、非断熱極限の有効模型であるフラストレートスピン模型と定性的に相図が一致することから、非断熱極限から断熱極限まで相図は連続的に繋がっていることが結論された。またこのクロスオーバー領域に対する詳細な解析から、以前の繰り込み群の解析とは定性的にも異なる結果を導いた。

第六章では、1 次元鎖を超えた模型として、スピンパイエルス鎖を格子相互作用で繋げた系の基底状態の解析がなされている。量子モンテカルロシミュレーションと有効模型における臨界次元の解析から、鎖間相互作用にフラストレーションがない場合は無限小の鎖間相互作用により基底状態でダイマー秩序が生じると予想された。一方、鎖間の相互作用に強いフラストレーションがある場合、系は2 次元の量子的液体状態から、マクロスコピックに縮退したダイマー状態へ相転移すると予想された。これはフラストレートした2次元系が動的に1 次元的な励起を持つことに起因すると考えられ、有効相互作用の見積もりや他の模型との比較から結果の妥当性が議論された。

第七章は本論文の結言であり、新たに開発された数値手法を用いることにより、実効的にフラストレートする量子スピン系に関する詳細な解析がはじめて可能となった点が強調された。本論文は、これまで負符号問題によりモンテカルロ法が適用できなかった模型に対する一つの非常に有望な解析法を提案し、またそれを用いて、1 次元・2 次元のスピンパイエルス系のおける詳細な相図と新奇な臨界現象を明らかにしたという点で大きなインパクトがあり、高く評価される。さらに、本論文で提案された計算科学的手法は、非常に一般的なものであり、今後、物性物理分野だけでなく他の分野にも広く波及すると期待される。

よって本論文は博士(工学) の学位請求論文として合格と認められる。