Graphene is a monoatomic layer material composed of carbon atoms, arranged in the honeycomb lattice structure. It has been studied intensely as a two-dimensional electronic system easy to create and observe, since its first experimental isolation by Geim and Novoselov in 2004. Charge carriers on monolayer graphene can be described as massless Dirac fermions at long wavelength limit in the vicinity of two "Dirac points" in the momentum space, known as the "Dirac cone (valley)" structure. This nature makes graphene attractive from the viewpoint of quantum field theories as well as of materials physics.

However, the Dirac cone picture may break down in the presence of a strong electron-electron interaction, where the excitations are more likely to be scattered to the momentum region far from the Dirac cones. Such a strong coupling is supposed to be achieved in a vacuum-suspended graphene, where the Coulomb interaction is effectively enhanced from the ordinary quantum electrodynamics (QED), by the discrepancy between the propagation speed of the charge carriers (Fermi velocity, vF=c/300) and that of photons (speed of light, c). Under such a strong electron-electron interaction, some symmetry of the system may get spontaneously broken, driving the system into an excitonic insulator. This mechanism is analogous to the spontaneous breaking of chiral symmetry and dynamical fermion mass generation in strongly coupled field theories, such as quantum electrodynamics (QCD). In graphene, however, its microscopic lattice structure may host various gap-opening patterns (see Fig.1):

(a) Charge density wave (CDW): One triangular sublattice of the honeycomb lattice is more occupied by electrons than the other sublattice. It breaks the discrete chiral symmetry defined by two sublattices. It can be induced explicitly by some external substrates, like silicon carbide or boron nitride.

(b) Spin density wave (SDW): One sublattice is occupied by electrons with a certain spin direction, while the other sublattice by the opposite spin direction. The system shows antiferromagnetism. It breaks both the sublattice symmetry and the spin SU(2) symmetry.

(c) Kekule distortion: Nearest-neighbor hopping amplitude becomes non-uniform with a certain pattern larger than the unit cell, breaking the translational invariance partially. It can be induced explicitly by some adatoms on the layer.

(d) Haldane/Kane-Mele flux: Next-to nearest neighbor hopping with a complex amplitude is induced by an effective "magnetic flux", breaking the time-reversal symmetry. Haldane flux preserves the SU(2) spin symmetry, inducing an anomalous quantum Hall conductivity. Kane-Mele flux, which breaks the spin symmetry, corresponds to spin-orbit interaction, and induces a so-called "quantum spin Hall" effect.

The interplay among these ordering patterns, induced either spontaneously or externally, is still little known under the sufficiently strong electron-electron interaction. Motivated by those problems in monolayer graphene, we study the phase structure of graphenelike systems with a sufficiently strong electron-electron interaction in this thesis. We assume that the interaction is mediated by the electromagnetic field like quantum electrodynamics (QED), and investigated the interplay effects among various ordering patterns (either spontaneous or explicit) listed above.

In Section 2, we construct a lattice gauge theory model defined on the graphenelike honeycomb lattice. We convert the conventional tight-binding Hamiltonian into path integral formalism, and define the lattice effective action, with the effect of electron-electron interaction included in terms of U(1) gauge field. This is the lattice gauge theory description of the so-called "reduced QED", where the fermions are confined in the (2+1)-dimensional plane, while the photons (gauge field) propagate in the (3+1)-dimensional space. This effective model can host the ordering patterns shown above, which are characteristic to the honeycomb lattice, except for those preserving the full SU(2) spin symmetry.

In the first half of Section 3, we observe the phase structure of the graphenelike gauge theory constructed above by using the techniques of strong coupling expansion of lattice gauge theory, inspired by the analyses of QCD phase structure. Strong coupling expansion decomposes the long-range Coulomb interaction into the sum of 4-fermi local interaction terms. The leading order in the expansion gives the on-site interaction, which may lead to the spontaneous sublattice symmetry breaking (SLSB), while the next-to leading order yields the nearest-neighbor repulsion, which can contribute to the renormalization of the Fermi velocity and the spontaneous Kekule distortion. We map the phase diagram of the system with these spontaneous orders at the mean-field level, by varying the ratio between the amplitudes of terms in the expansion (see Fig.2). It should be noted that the Kekule distortion is classified into two phases (KD1/KD2), characterized by the sign of its amplitude. The difference of two Kekule distortion phase originates from the parabolic band far from the Dirac points, which is neglected in the Dirac cone approximation.

In the second half of Section 3, we focus on the competition between the spontaneous sublattice symmetry breaking seen above and the externally introduced orders, within the strong coupling limit of lattice gauge theory. Here we incorporate Kekule distortion and Kane-Mele flux (spin-orbit interaction) as the explicit orders. In the presence of a sufficiently large Kekule distortion, the spontaneous order is suppressed and the sublattice symmetry gets restored, which agrees with the analysis within the Dirac cone approximation. On the other hand, when the distortion amplitude is small enough, the sublattice symmetry breaking order grows quadratically as a function of the distortion amplitude, which cannot be seen within the Dirac cone approximation (see Fig.3). Therefore, microscopic effect from the lattice structure is crucial for the interplay of these orders, as long as the Kekule distortion amplitude is small compared to the energy scale characterized by the lattice structure.

The spin-orbit interaction, which breaks the sublattice symmetry, SU(2) spin symmetry, and the valley (pseudospin) inversion symmetry, also competes with the spontaneous antiferromagnetic order, which breaks the sublattice and spin symmetry but preserves the valley inversion symmetry. The spin-orbit coupling tilts the antiferromagnetic order towards the XY-plane, away from the direction originally pointed by the spin-orbit term. Such an interplay between the normal antiferromagnetic order and the spin-orbit coupling occasionally shifts the topological phase structure of the system, which can contribute to quantum spin Hall effect (see Fig.4). This phase structure is also related with the parity-breaking phase of lattice QCD with Wilson fermions, which is called "Aoki phase". We can give a conjecture about the phase structure of quantum spin Hall system at finite coupling, from the analogy with the phase structure of lattice QCD.

In Section 4, we attempt to give a clue about the breaking/restoration of the exact SU(2) spin symmetry in the honeycomb lattice system, which cannot be observed in the lattice gauge theory due to the artifact of lattice discretization. Here we employ the extended Hubbard model, which includes the 4-point interaction terms similar to those derived from the strong coupling expansion of lattice gauge theory. We observe the phase structure of this model by solving the variational gap equations including CDW and SDW orders. Variational gap equation, which is optimized on the basis of Jensen-Peierls inequality, is advantageous over the ordinary mean-field gap equation derived by Hubbard-Stratonovich transformation in that competition among various order parameters can be taken into account. In the absence of the external staggered potential, the phases are classified into CDW, SDW, and semi-metallic phases. The on-site (Hubbard) repulsion favors SDW, while the nearest-neighbor (NN) repulsion favors CDW. The external staggered potential enhances the CDW phase, suppressing the SDW phase (see Fig.5). The spin SU(2) symmetry is broken in the SDW phase, and the band degeneracy of two spin states is correspondingly shifted in the presence of the external staggered potential: band gap amplitudes and Fermi velocities of the electrons with up/down spin become different (see Fig.6). This phenomenon may have an effect on the spin transport, such as the filtering of spin components.

We expect that all the analytical results above can be tested experimentally by the direct observation of band structure like angle-resolved photoemission spectroscopy (ARPES), or the measurement of electronic transport properties like charge/spin Hall effects. Our results can be applied to the correlated fermion systems on the honeycomb lattice, such as graphene, cold atoms on optical lattice, and the topological insulators, with a small modification. The application of our methods to the bilayer graphene systems, where the spontaneously gapped phase has already been observed experimentally, is a future problem.

Fig.1: Schematic picture of the gap-opening orders characteristic to the honeycomb lattice.

Fig.2: Phase diagram displaying the competition among spontaneous orders: SLSB, KD1 and KD2. Vertical axis represents the strength of leading order (on-site interaction), while the horizontal axis the next-to leading order (nearest neighbor repulsion).

Fig.3: Behavior of spontaneous SLSB order σ(Δ) in the presence of the external Kekule distortion Δ. "Full band" is calculated with the full band structure, while "Dirac cone" is obtained by Dirac cone approximation.

Fig.4: Phase diagram in the presence of spin-orbit coupling (t') and the normal antiferromagnetic (AF) order (σ1). There appears a new "tilted AF" phase by the effect of electron correlation.

Fig.5: Phase diagram of the extended Hubbard model, with the on-site (Hubbard) repulsion U and the NN repulsion V. External staggered potential m eventually suppresses the SDW phase.

Fig.6: Behavior of the gap amplitude Δσ (left) and the Fermi velocity renormalization factor Zσ for each spin state σ=↑,↓, as functions of the external staggered potential m, where the on-site interaction U=6.0t and the NN interaction V=0.5t. Band degeneracy is shifted in the SDW phase.

本学位論文は6 章からなり、1章はグラフェンおよび、この系における多体効果についての序論および本論文の概要、2章は蜂の巣格子に関する有効理論の解説、3章は主たる部分であり、格子ゲージ理論における強結合展開と、蜂の巣格子への適用、4章はスピンのSU(2) 対称性を保った平均場近似、5 章は本論文で得られた結論および今後の展望を述べている。

グラフェン(炭素原子一層が蜂の巣格子をなす物質)は、そのバンド構造がディラック電子と見なせる部分を含むために、20 世紀中頃から理論的に興味をもたれてきた。2000 年代初頭からグラフェンが実際の試料として得られるようになり、その特異な物性が注目された。2010 年にグラフェンのテーマがノーベル物理学賞を受賞した前後からは、物性物理学の大きな分野が創成されている。ディラック電子というのは、元来は場の理論において電子を相対論的に扱う際に生じるものであるが、一般には質量をもった相対論的粒子である。グラフェンで実現するのは、質量がゼロのディラック電子であり、それに伴う物性の特異性が興味の焦点となった。このディラック電子とのアナロジーからは、この問題を、素粒子・原子核物理学的な観点、特にハドロン物理学の観点から扱うと、どのような物理が構築できるか、という興味深い点が生じる。グラフェンは、この観点からも多くの素粒子・原子核理論物理学者の興味を惹き、精力的な研究が行われている。

本学位論文の主眼は、グラフェンの蜂の巣格子に関する物理に啓発され、ハドロン物理学でスタンダードとなっている格子ゲージ理論を蜂の巣格子上で行うとどうなるかを調べた点である。興味の焦点は二点あり、(a) グラフェンそのものは、炭素という軽元素系であるために、電子間相互作用(電子相関の強さ)は小さいが、理論的な問題として、相互作用が強いとしたときに、蜂の巣格子上の荷電フェルミオン系がどのような多体状態をとるか、という問題がある。これは、物性物理学においては、Hubbard 模型のような格子フェルミオン模型を用いて精力的に調べられている。この問題を、蜂の巣格子を用いた格子ゲージ理論で扱ったのが本論文の第一の眼目である。現実のグラフェンにおいては、炭素のpπ 軌道が蜂の巣格子状に並んだ強束縛模型上で多体効果を考えるのが良い模型となるが、蜂の巣格子ゲージ理論では、電子が原子軌道に束縛されていることや、pπ 軌道以外の軌道により電子間クーロン相互作用が遮蔽される効果を無視して、1=r相互作用する荷電フェルミオン系に対する作用において((3+1) 次元の)電磁場のベクトル・ポテンシャルに関する部分を消去することにより、電子の有効ラグランジアンを求めることになる。

(b) 蜂の巣格子のバンド構造においては、Brillouin 帯のK 点、K' 点と呼ばれる二点の近傍においてディラック電子的になるために、グラフェンを場の理論の観点から扱う際にはディラック場を導入して調べることが多いが、本論文ではそうではなく、格子ゲージ理論における格子を蜂の巣格子にとることを考える。このために、電子間相互作用の行列要素は、K 点、K' 点を結ぶものが考慮されるために、その効果に興味がもたれる。

学位申請者は、上記の二点を眼目として、蜂の巣格子ゲージ理論を調べ、以下の結果を得た。(1)多体状態として現れ得ると思われる、電荷秩序相(CDW)、反強磁性状態(SDW)、2原子を結ぶbond が、ケクレ・パターン状に強弱をとる構造(KD1, KD2)、一種のflux phase としてHaldaneがゼロ磁場中での量子ホール効果のトポロジカル相として考えた構造(flux) を考えた。上記の蜂の巣格子ゲージ理論において、荷電フェルミオンと電磁場との結合が強いとする強結合展開を行い、この初項と次項まで取り入れた模型で考察した。この結果、この2項の強さに対する相図として、SDW, KD1, KD2 が現れるものが得られた。また、flux を外場としてかけた場合の相図も得られ、この中には、反強磁性相とトポロジカル相の間に、傾いた反強磁性相(量子色力学(QCD)の青木相に対応)も存在することが示された。

以上のように、本学位論文では、通常は単純な正方格子、立方格子などで行われる格子ゲージ理論を、蜂の巣格子という特徴的な格子に対して実行した初めての例であり、興味深い結果が得られ、荷電フェルミオン多体系の物理の発展に資することが期待される。現実のグラフェンの物性との対応を見るには、グラフェンを記述する強束縛多体模型に現れる物理量と、本研究での格子ゲージ理論に現れるパラメータとの関連付けをする必要があるが、これは将来の課題であろう。なお、本論文の一部は初田哲男教授、木村太郎氏、Gordon W. Semenoff 氏との共同研究であるが、論文提出者が主体となって研究したものであり、論文提出者の寄与が十分であると判断される。したがって、審査員全員により、博士(理学)を授与できると認める。