There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

This article reviews the physics of high-temperature superconductors from the point of view of the doping of a Mott insulator. The basic electronic structure of cuprates is reviewed, emphasizing the physics of strong correlation and establishing the model of a doped Mott insulator as a starting point. A variety of experiments are discussed, focusing on the region of the phase diagram close to the Mott insulator (the underdoped region) where the behavior is most anomalous. The normal state in this region exhibits pseudogap phenomenon. In contrast, the quasiparticles in the superconducting state are well defined and behave according to theory. This review introduces Anderson's idea of the resonating valence bond and argues that it gives a qualitative account of the data. The importance of phase fluctuations is discussed, leading to a theory of the transition temperature, which is driven by phase fluctuations and the thermal excitation of quasiparticles. However, an argument is made that phase fluctuations can only explain pseudogap phenomenology over a limited temperature range, and some additional physics is needed to explain the onset of singlet formation at very high temperatures. A description of the numerical method of the projected wave function is presented, which turns out to be a very useful technique for implementing the strong correlation constraint and leads to a number of predictions which are in agreement with experiments. The remainder of the paper deals with an analytic treatment of the $t\text{\ensuremath{-}}J$ model, with the goal of putting the resonating valence bond idea on a more formal footing. The slave boson is introduced to enforce the constraint againt double occupation and it is shown that the implementation of this local constraint leads naturally to gauge theories. This review follows the historical order by first examining the U(1) formulation of the gauge theory. Some inadequacies of this formulation for underdoping are discussed, leading to the SU(2) formulation. Here follows a rather thorough discussion of the role of gauge theory in describing the spin-liquid phase of the undoped Mott insulator. The difference between the high-energy gauge group in the formulation of the problem versus the low-energy gauge group, which is an emergent phenomenon, is emphasized. Several possible routes to deconfinement based on different emergent gauge groups are discussed, which leads to the physics of fractionalization and spin-charge separation. Next the extension of the SU(2) formulation to nonzero doping is described with a focus on a part of the mean-field phase diagram called the staggered flux liquid phase. It will be shown that inclusion of the gauge fluctuation provides a reasonable description of the pseudogap phase. It is emphasized that $d$-wave superconductivity can be considered as evolving from a stable U(1) spin liquid. These ideas are applied to the high-${T}_{c}$ cuprates, and their implications for the vortex structure and the phase diagram are discussed. A possible test of the topological structure of the pseudogap phase is described.

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Doping a Mott insulator: Physics of high-temperature superconductivity

This article reviews the effort to understand the physics of high temperature superconductors from the point of view of doping a Mott insulator. The basic electronic structure of the cuprates is reviewed, emphasizing the physics of strong correlation and establishing the model of a doped Mott insulator as a starting point. A variety of experiments are discussed, focusing on the region of the phase diagram close to the Mott insulator (the underdoped region) where the behavior is most anomalous. We introduce Anderson's idea of the resonating valence bond (RVB) and argue that it gives a qualitative account of the data. The importance of phase fluctuation is discussed, leading to a theory of the transition temperature which is driven by phase fluctuation and thermal excitation of quasiparticles. We then describe the numerical method of projected wavefunction which turns out to be a very useful technique to implement the strong correlation constraint, and leads to a number of predictions which are in agreement with experiments. The remainder of the paper deals with an analytic treatment of the t-J model, with the goal of putting the RVB idea on a more formal footing. The slave-boson is introduced to enforce the constraint of no double occupation. The implementation of the local constraint leads naturally to gauge theories. We give a rather thorough discussion of the role of gauge theory in describing the spin liquid phase of the undoped Mott insulator. We next describe the extension of the SU(2) formulation to nonzero doping. We show that inclusion of gauge fluctuation provides a reasonable description of the pseudogap phase.

Doping a Mott Insulator: Physics of High Temperature Superconductivity

This review presents a summary and evaluation of the experimental properties of unconventional superconductivity in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ as they were known in the spring of 2002. At the same time, the paper is intended to be useful as an introduction to the physics of spin-triplet superconductivity. First, the authors show how the normal-state properties of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ are quantitatively described in terms of a quasi-two-dimensional Fermi liquid. Then they summarize its phenomenological superconducting parameters in the framework of the Ginzburg-Landau model, and discuss the existing evidence for spin-triplet pairing. After a brief introduction to the vector order parameter, they examine the most likely symmetry of the triplet state. The structure of the superconducting energy gap is discussed, as is the effect of symmetry-breaking magnetic fields on the phase diagram. The article concludes with a discussion of some outstanding issues and desirable future work. Appendixes on additional details of the normal state, difficulty in observing the bulk Fermi surface by angle-resolved photoemission, and the enhancement of superconducting transition temperature in a two-phase ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}\ensuremath{-}\mathrm{Ru}$ system are included.

The superconductivity of Sr 2 RuO 4 and the physics of spin-triplet pairing

A theoretical review is given on high temperature superconductivity in copper oxides (cuprates) by focusing on the hole doping cases based on the view that it is realized in carrier doped Mott insulators, as noted by Anderson in the initial stage. From the detailed knowledge of electronic states deduced from experiments that showed the undoped parent case is Mott insulators (charge transfer type insulators, to be precise) and that the hole doping is mainly on oxygen sites, the t?J model, as derived by Zhang and Rice, is shown to be a canonical model for hole doped cuprates and values of various parameters of the model have been assessed. Results of many different numerical methods so far obtained for this t?J model, especially the variational Monte Carlo method, have clearly indicated the stability of the -wave superconductivity at absolute zero for the parameter region of actual experimental interest and the particular doping dependences of the condensation energy of superconductivity reflecting particular features of doped Mott insulators. For finite temperatures, on the other hand, the field theoretical slave-boson approximation based on the spin (spinons) and charge (holons) separations and the gauge fields as a glue combining them predicts qualitatively particular features of the existence of characteristic crossover temperatures of the spin singlet of the resonating valence bond (RVB) state, TRVB and the onset of Bose condensation of holons, TB, triggering coherent motion of electrons as convoluted particles of spinons and holons. The considerations based on the gauge field indicate that the onset temperature of superconductivity, Tc, is the lower one of these two, i.e. either TB (overdoped cases) or TRVB (underdoped cases), respectively. These characteristic features of the 'phase diagram' at finite temperatures are in overall agreement with various experimental observations, especially with the existence of spin-gap or pseudo-gap phases. In more detailed examinations of the underdoped region, the antiferromagnetic long-range order and superconductivity show a very intricate relationship at low temperatures depending on the system; they coexist as clarified in the inner layer of Hg-1245 but spin glass states intervene between them in La2?xSrxCuO4 (LSCO). It is argued that these differences can be attributed to the different degrees of disorder. Actually, theories based on the t?J model have also predicted the coexistence of antiferromagnetism and superconductivity in the ground state of clean systems. On the other hand, interesting experimental findings of large Nernst effect and 'Fermi arc' in LSCO and impurity effects in YBCO have prompted the necessity of theoretical investigations of electronic states of lightly doped Mott insulators in the presence of strong disorder.

https://iopscience.iop.org/article/10.1088/0034-4885/71/3/036501/pdf

The t–J model for the oxide high-Tc superconductors

We study the interplay between stripes and d-wave superconductivity in the two-dimensional t-t'-J model using a variational Monte Carlo method. The next-nearest-neighbor hopping t' < 0 stabilizes the stripe states around 1/8 hole doping rate. We find that stripes and spatially oscillating superconductivity coexist depending on parameters. The superconducting orders are enhanced at the hole stripe regions. Although the energy differences are relatively small, the stripe state in which the phases between adjacent superconducting stripes are the opposite (antiphase) is also stabilized. We consider the possibility that the antiphase coexistence may explain the weakness of the c-axis Josephson couplings in the La 1 . 6 - x Nd 0 . 4 Sr x CuO 4 .

Stripe states with spatially oscillating d-wave superconductivity in the two-dimensional t-t'-J model.

The phase diagram of the one-dimensional t-J model is investigated by analyzing the results of exact diagonalization and the exact solutions at J/t=0 and 2. Phase separation takes place above a critical value of J around ${\mathit{J}}_{\mathit{c}}$/t=2.5--3.5 depending on the electron density. In the small-J region, Tomonaga-Luttinger liquid theory holds and its correlation exponents are calculated as a function of J/t and the electron density. Superconducting correlations become dominant in a region between the solvable case (J/t=2) and phase separation. A spin-gap region is also found at low density.

Phase diagram of the one-dimensional t-J model.

This is the peer reviewed version of the following article: Journal of the Physical Society of Japan 75 (2006): 051009, which has been published in final form at http://dx.doi.org/10.1143/JPSJ.75.051009

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Theoretical Aspects of Charge Ordering in Molecular Conductors

Theoretical studies on charge ordering phenomena in quarter-filled molecular (organic) conductors are reviewed. Extended Hubbard models including not only the on-site but also the inter-site Coulomb repulsion are constructed in a straightforward way from the crystal structures, which serve for individual study on each material as well as for their systematic understandings. In general the inter-site Coulomb interaction stabilizes Wigner crystal-type charge ordered states, where the charge localizes in an arranged manner avoiding each other, and can drive the system insulating. The variety in the lattice structures, represented by anisotropic networks in not only the electron hopping but also in the inter-site Coulomb repulsion, brings about diverse problems in low-dimensional strongly correlated systems. Competitions and/or co-existences between the charge ordered state and other states are discussed, such as metal, superconductor, and the dimer-type Mott insulating state which is another typical insulating state in molecular conductors. Interplay with magnetism, e.g., antiferromagnetic state and spin gapped state for example due to the spin-Peierls transition, is considered as well. Distinct situations are pointed out: influences of the coupling to the lattice degree of freedom and effects of geometrical frustration which exists in many molecular crystals. Some related topics, such as charge order in transition metal oxides and its role in new molecular conductors, are briefly remarked.

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Masao Ogata

Papers

The t–J model for the oxide high-Tc superconductors

Stripe states with spatially oscillating d-wave superconductivity in the two-dimensional t-t'-J model.

Phase diagram of the one-dimensional t-J model.

Theoretical Aspects of Charge Ordering in Molecular Conductors

Theoretical Aspects of Charge Ordering in Molecular Conductors