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**Floquet topological phases in correlated electron systems**

**Norio Kawakami (Kyoto Univ.)**

We here discuss two topics on Floquet topological phases induced by laser irradiation in correlated electron systems. We first propose a possible way to realize topological superconductivity with application of laser light to superconducting cuprate thin films. Applying Floquet theory to a model of d-wave superconductors with Rashba spin-orbit coupling, we derive an effective model and discuss its topological nature. Interplay of the Rashba spin-orbit coupling and the laser light effect induces the synthetic magnetic fields, thus making the system gapped. Then the system acquires the topologically non-trivial nature which is characterized by Chern numbers. The effective magnetic fields do not create the vortices in superconductors, and thus the proposed scheme provides a promising way to dynamically realize a topological superconductor in cuprates. We further study the nature of laser-irradiated Kondo insulators. Applying Floquet theory to a periodic Anderson model, we find two generic effects induced by laser light. One is the dynamical localization, which suppresses hopping and hybridization and the other is the laser-induced hopping and hybridization, which can be interpreted as a synthetic spin-orbit coupling or magnetic field. In topological Kondo insulators, linearly polarized laser light realizes phase transitions between trivial, weak topological, and strong topological Kondo insulators, whereas circularly polarized laser light breaks time-reversal symmetry and induces Weyl semimetallic phases.

**Spin separation in the fractional topological insulator**

**Kwon Park (KIAS)**

Fractional topological insulators (FTIs) have been proposed as being composed of two independent copies of the fractional Chern insulator (FCI) with opposite Chern numbers for different spins, preserving the time-reversal symmetry as a whole. An important question is if the correlation between electrons with different spins can be really ignored. To address this question, we investigate the effects of correlation in the presence of spin-dependent holomorphicity, i.e., electrons of one spin species reside in the holomorphic lowest Landau level, while those of the other in the antiholomorphic counterpart. By constructing and performing exact diagonalization of an appropriate model Hamiltonian, here, we show that generic, strongly correlated, fractionally filled states with spin-dependent holomorphicity cannot be described as two independent copies of the FQHS, suggesting that FTIs in the lattice cannot be described as those of the FCI either. ly filled states in this system are generally compressible except at half filling, where an insulating state called the half-filled spin-holomorphic FTI occurs. It is predicted that the half-filled spin-holomorphic FTI is susceptible to an inherent spontaneous symmetry breaking, leading to the spatial separation of spins.

**Generalized Compatibility Relation: Atiyah-Hirzebruch Spectral Sequence in Band Topology**

**Ken Shiozaki (RIKEN)**

We study the Atiyah-Hirzebruch spectral sequence (AHSS) for equivariant K-theory in the context of band theory. The AHSS is known to be a mathematical tool to compute a cohomology theory. We found that the AHSS is a natural framework to study the band topology: it gives us a systematic description of bulk gapless phases and approximates the K-group. The AHSS can be considered as the suitable generalization of the compatibility relation. Using the AHSS, we got the complete classification of topological invariants for 230 space groups in class A and AIII. Some new topological invariants we discovered will be presented.

**EMUS-QMC: Elective Momentum Ultra-Size Quantum Monte Carlo Method**

**Yang Qi (Fudan Univ.)**

One bottleneck of quantum Monte Carlo (QMC) simulation of strongly correlated electron systems lies at the scaling relation of computational complexity with respect to the system sizes. For generic lattice model of interacting fermions, the best methodology at hand still scales with βN^3 where β is the inverse temperature and N is the system size. Such scaling behavior has greatly hampered the accessibility of the universal infrared (IR) physics of many interesting correlated electron models at (2+1)D, let alone (3+1)D. To reduce the computational complexity, we develop a new QMC method with inhomogeneous momentum-space mesh, dubbed elective momentum ultra-size quantum Monte Carlo (EQMC). In this method, the fermion determinant is written in momentum-space, where more attention is paid towards the k-points associated with the IR physics (in this case, the so-called hot-spots), such that the computational complexity is reduced to Nf^33 where Nf is the volume of momentum patches around hot-spots. Speedup, to the level of 10^3, can be easily achieved in EQMC as it is easy to have N/Nf ∼ 10. We demonstrate the power of this method with a model of antiferromagnetic itinerant quantum critical point, realized in frustrated transverse-field triangle lattice Ising model coupled to Fermi surface. The system size of 48 × 48 × 48 (L × L × β, more than 5 times larger than the previous investigations) are comfortably accessed with EQMC. Spin fluctuations introduce effective interactions among fermions and the fermions in return render the bare bosonic critical point into a different universality with Hertz-Mills type exponents. With much larger system sizes, the antiferromagnetic itinerant quantum critical scaling is unveiled with unprecedingly high accuracy.

**Cooper pair spin current in SrRuO3 / Sr2RuO4 heterostructure**

**Suk Bum Chung (Univ. of Seoul)**

The spin-triplet superconductor by definition should involve spin ordering that gives rise to the spin collective phenomena such as the spin collective modes and the spin supercurrent. However, in the superconducting phase in the best-known candidate material, Sr2RuO4, only the latter has been observed just for the mesoscopic sample [1]. I will show how the recently fabricated heterostructure of bulk Sr2RuO4 and the ferromagnetic SrRuO3 provides a particularly natural probe for detecting both types of spin collective phenomena. The spin supercurrent that can be injected naturally into Sr2RuO4 from SrRuO3 can be used to obtain an exceptionally high-quality spin valve, while applying an AC bias on SrRuO3 can drive the Sr2RuO4 spin collective modes [2].

[1] J Jang, D J Ferguson, V Vakaryuk, R Budakian, **SBC**, P M Goldbart, and Y Maeno, Science **331** (2011), 186

[2] **SBC**, S K Kim, K H Lee, and Y Tserkovnyak, arXiv: 1802.01599

**Symmetry based indicator of band topology**

**Haruki Watanabe (Univ. Of Tokyo)**

Symmetry does not only protect topological phases but also helps us diagnosing the topological properties of the system. In this talk, we will discuss how the symmetry representations of band insulators are related to their topology and surface states.

**Multipolar order and superconductivity in Pr(TM)2(Al,Zn){20} Kondo Materials**

**SungBin Lee (KAIST)**

In heavy fermions, many exotic phenomena coexist such as hidden order, unconventional metal and superconductivity. In particular, Pr based Kondo materials Pr(TM)_2(Al,Zn)_{20} exhibit unique multipolar order and superconductivity.[1-4] Motivated by recent experimental results, we will focus on the multipolar ordering of Pr^{3+} and discuss possible phase transitions, in addition to the field effect. Then we will also discuss the unique feature of superconductivity driven by quadrupolar order fluctuations.

**Non-equilibrium control of the effective free energy landscape in a frustrated magnet **

**Yuan Wan (IOP)**

Geometrically frustrated magnets often possess accidentally degenerate ground states at zero temperature. At low temperature, thermal fluctuations lift the accidental degeneracy and tend to stabilize ground states with maximal entropy. This phenomenon, known as “order by disorder”, underlines the fluctuation contribution to the free energy landscape in frustrated magnets.

In this talk, I show that one can control such free energy landscape in a non-equilibrium setting. In a frustrated magnet with precessional dynamics, the system’s slow drift motion within the degenerate ground state manifold is governed by the fast modes out of the manifold. Exciting these fast modes generates a tuneable effective free energy landscape with minima located at thermodynamically unstable portions of the ground state manifold. I demonstrate this phenomenon on pyrochlore XY antiferromagnet, where a magnetic field pulse is sufficient for controlling the effective free energy landscape at non-equilibrium.

**Magnetic-field induced topological semimetals near a quantum critical point **

**of pyrochlore iridates**

**Bohm Jung Yang (Seoul Nat'l Unv.)**

Motivated by the recent experimental observation of anomalous magneto-transport properties near the Mott quantum critical point (QCP) of pyrochlore iridates, we study the generic topological band structure near the QCP in the presence of magnetic field. We have found that the competition between different energy scales can generate various topological semimetal phases near the QCP. Here the central role is played by the presence of a quadratic band crossing (QBC) with four-fold degeneracy in the paramagnetic band structure. Due to the large band degeneracy and strong spin-orbit coupling, the degenerate states at the QBC can show an anisotropic Zeeman effect as well as the conventional isotropic Zeeman effect. Through the competition between three different magnetic energy scales including the exchange energy betwen Ir ions and two Zeeman energies, various topological semimetals can be generated near the QCP. Moreover, we have shown that these three magnetic energy scales can be controlled by modulating the magnetic multipole moment (MMM) of the cluster of spins in a unitcell, which can couple to the intrinsic MMM of the degenerate states at the QBC.We propose the general topological band structure under magnetic field achievable near the QCP, which would facilitate the experimental discovery of novel topological semimetal states in pyrochlore iridates.

**Spin and Thermal Excitations in Kitaev-type Frustrated Magnets**

**Youhei Yamaji (Univ. Of Tokyo)**

Topological states of matters in strongly correlated electron systems are characterized by emergence of fractionalized quasiparticles. The Majorana fermions are examples of the fractionalized excitations, which have recently attracted much attention as ingredients of topological quantum computations. Intensive studies on a possible realization of the Majorana fermions in crystalline solids have been triggered by the pioneering finding of the spin liquid ground state in the Kitaev model and theoretical proposal on the realization of the Kitaev model in iridium oxides. Recently, a ruthenium chloride, α-RuCl3, has been an important Majorana hunting field. As the simplest effective Hamiltonian of α-RuCl3, the Kitaev-Γ model has been studied and shown that the ground state is non magnetic and adiabatically connected to the ground state of the Kitaev model. We further found the plateau in temperature dependence of entropy and spin excitation continuum at finite temperatures, which show a significant crossover from the Kitaev limit to the Γ limit.

**Lieb-Schultz-Mattis Theorem and Topological Phase**

**Gil Young Cho (POSTECH)**

The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy states from a lattice model cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if the lattice translation symmetry and particle number conservation are strictly imposed. In this talk, we will compare the one-dimensional gapless states enforced by the LSM theorem and the boundaries of one-higher dimensional strong symmetry-protected topological (SPT) phases from the perspective of quantum anomalies. We first note that, they can be both described by the same low-energy effective field theory with the same effective symmetry realizations on low-energy modes, wherein non-on-site lattice translation symmetry is encoded as if it is a local symmetry. In spite of the identical form of the low-energy effective field theories, we show that the quantum anomalies of the theories play different roles in the two systems. In particular, we find that the chiral anomaly is equivalent to the LSM theorem, whereas there is another anomaly, which is not related to the LSM theorem but is intrinsic to the SPT states. As an application, we extend the conventional LSM theorem to multiple-charge multiple-species problems and construct several exotic symmetric insulators. We also find that the (3+1)d chiral anomaly provides only the perturbative stability of the gapless-ness local in the parameter space. We will conclude the talk with a few future research topics stemming from the discussed topics.

**Local magnetic resonance of Kitaev’s spin liquid**

**Masafumi Udagawa (Gakushuin Univ.)**

Recently, Kitaev’s model is drawing considerable attention as a platform to study quantum spin liquid, and several compounds are proposed as candidates of this system. Among many non-trivial properties, the Kitaev’s spin liquid phases host unusual excitations: spins are fractionalized into itinerant Majorana fermions and Visons, and the latter behave as abelian/non-abelian anyons. The understanding of these elementary excitations is essential to diagnosing the formation of Kitaev’s spin liquid in actual materials. In this contribution, we present an analytical solution of dynamical correlation function for Kitaev’s spin liquid. With this solution, we will address how local disturbance of the system, such as impurity, manifests itself in the thermodynamic and dynamical properties. In particular, we will focus on the zero-energy state appearing around the diluted site, and clarify its nature in relation to the fractionalized excitations.

**Interplay between Topology, Symmetry, and Coulomb interaction**

**Eun-Gook Moon (KAIST)**

The discovery of topological phases shed new lights on our understanding in physics. Almost complete understanding has been achieved in non-interacting systems, but interacting systems with non-trivial topology, on the other hand, have intrinsic complexity from the Coulomb interaction. One important question is how the long-range Coulomb interaction affects topological phases and phase transitions. In this talk, we partially answer the question by using the renormalization group technique. We explicitly show that conventional wisdoms in relativistic systems such as emergence of the Lorentz invariance do not hold at low energy physics. In other words, a low energy theory may not be described by a relativistic quantum field theory. Our results demonstrate that correlation effects should be considered from the beginning in a class of topological phases and phase transitions