Session C5: Frustrated Magnetism: Theory

Functional renormalization group - a new approach to frustrated quantum magnetism

The experimental and theoretical investigation of quantum spin systems has become one of the central disciplines of contemporary condensed matter physics. From an experimental viewpoint, the field has been significantly fueled by the recent synthesis of novel strongly correlated materials with exotic magnetic or quantum paramagnetic ground states. From a theoretical perspective, however, the numerical treatment of realistic models for quantum magnetism in two and three spatial dimensions still constitutes a serious challenge. This particularly applies to frustrated systems, which complicate the employment of established methods. This talk intends to propagate the pseudofermion functional renormalization group (PFFRG) as a novel approach to determine large size ground state correlations of a wide class of spin Hamiltonians. Using a diagrammatic pseudofermion representation for quantum spin models, the PFFRG performs systematic summations in all two-particle fermionic interaction channels, capturing the correct balance between classical magnetic ordering and quantum fluctuations. Numerical results for various frustrated spin models on different 2D and 3D lattices are reviewed, and benchmarked against other methods if available.   

Filling constraints for spin-orbit coupled insulators in symmorphic and non-symmorphic crystals

We determine conditions on the filling of electrons in a crystalline lattice to obtain the equivalent of a band insulator - a gapped insulator with neither symmetry breaking nor fractionalized excitations. We allow for strong interactions, which precludes a free particle description. Previous approaches that extend the Lieb-Schultz-Mattis argument invoked spin conservation in an essential way, and cannot be applied to the physically interesting case of spin-orbit coupled systems. Here we introduce two approaches, the first an entanglement based scheme, while the second studies the system on an appropriate flat ‘Bieberbach’ manifold to obtain the filling conditions for all 230 space groups. These approaches only assume time reversal rather than spin rotation invariance. The results depend crucially on whether the crystal symmetry is symmorphic. Our results clarify when one may infer the existence of an exotic ground state based on the absence of order, and we point out applications to experimentally realized materials. Extensions to new situations involving purely spin models are also mentioned.   

Quantum Dimer Model: Phase Diagrams

We present new theoretical analysis of the Quantum Dimer Model. We study dimer models on square, cubic and triangular lattices and we reproduce their phase diagrams (which were previously known only numerically). We show that there are several types of dimer liquids and solids. We present preliminary analysis of several other models including doped dimers and planar spin ice, and some results on the Kagome and hexagonal lattices.   

Characterizing excitation statistics in fractionalized phases through spectral functions

Characterizing topologically ordered phases of matter involves identifying the statistics of their emergent anyonic excitations. We show that the exchange statistics of excitations show characteristic signatures in experimentally relevant spectral functions. Drawing motivation from models of gapped quantum spin liquids and fractional Chern insulators which possess fractionalized anyonic excitations, we consider a model with gapped two particle and three particle abelian anyonic excitations. We show that the low energy part of spectral functions can show a robust behaviour from which the statistics of the excitations can be obtained.   

From M\"{o}bius aromaticity to gapped spin liquids

Motivated by the concept of M\”{o}bius aromatics in organic chemistry, the Hubbard model on ring-shaped molecules has been shown previously to support a fragile Mott insulator (FMI) ground state, which is distinct from a conventional insulator through its nontrivial transformation properties under point-group symmetry operations. In this talk, we discuss two-dimensional lattices of weakly-coupled FMI molecules belonging to multi-dimensional irreducible representations of the molecular point group. The low-energy effective Hamiltonians map onto quantum compass models with broken spin SU(2) symmetry. On the triangular lattice, the ground state develops long-range magnetism, which corresponds to a charge-ordered state of the molecules. On the honeycomb lattice, interestingly, we find a non-degenerate gapped spin-liquid ground state that preserves all spatial symmetries but transforms nontrivially under point-group operations. Our microscopic model therefore realizes an intrinsically interacting fermionic symmetry protected topological (SPT) phase.   

Tetrahedral Spin Crystal to a Chiral Spin Liquid: Frustration-induced quantum melting

Motivated by the recent interest in interacting topological phases, we study the Haldane-Hubbard model which is shown to host a Mott insulating state with chiral tetrahedral magnetic texture. Frustration-induced melting of this spin crystal leads to a chiral spin liquid. We discuss the properties of these phases, and the Chern-Simons-Higgs theory of the intervening exotic quantum critical point.   

Symmetric tensor networks and practical simulation algorithms to sharply identify classes of quantum phases distinguishable by short-range physics

Phases of matter are sharply defined in the thermodynamic limit. One major challenge of accurately simulating quantum phase diagrams of interacting quantum systems is due to the fact that numerical simulations usually deal with the energy density, a local property of quantum wavefunctions, while identifying different quantum phases generally rely on long-range physics. In this paper we construct generic fully symmetric quantum wavefunctions under certain assumptions using a type of tensor networks: projected entangled pair states, and provide practical simulation algorithms based on them. We find that quantum phases can be organized into crude classes distinguished by short-range physics, which is related to the fractionalization of both on-site symmetries and space-group symmetries. Consequently, our simulation algorithms, which are useful to study long-range physics as well, are expected to be able to sharply determine crude classes in interacting quantum systems efficiently. Examples of these crude classes are demonstrated in half-integer quantum spin systems on the kagome lattice. Limitations and generalizations of our results are discussed.   

Numerical studies of AKLT valence bond solids in one, two and three dimensions

The fixed-point valence bond solids of Affleck, Kennedy, Lieb and Tasaki (the so-called AKLT states) have become archetypes of symmetry protected topological order. These states are constructed by first placing $M$ valence bonds on each pair of neighboring lattice points, and then symmetrizing the $Mz$ resulting spin-1/2 degrees of freedom at each lattice site into a combined spin-$S$ degree of freedom with $2S=Mz$ (where $M$ is the multiplicity of the AKLT state and $z$ is the lattice coordination number). Using Monte Carlo sampling of the AKLT wavefunctions in the loop gas framework, we directly calculate correlation functions and energy gap estimators for these states in one, two and three dimensions. We also study the behavior of the so-called strange correlator, which has been proposed as a measure of symmetry protected topological order.   

Theory of supersymmetry ``protected'' topological phases of isostatic lattices and highly frustrated magnets

I generalize the theory of phonon topological band structures of isostatic lattices to highly frustrated antiferromagnets. I achieve this with a discovery of a many-body supersymmetry (SUSY) in the phonon problem of balls and springs which also applies to geometrically frustrated magnets. The Witten index of the SUSY model, when restricted to the single body problem (meaningful for linearized phonons), is then shown to be the Calladine-Kane-Lubensky index of mechanical structures that forms the cornerstone of the phonon topological band structure theory. ``Spontaneous supersymmetry breaking'' is then identified as the need to gap all modes in the bulk to create the topological state. The many-body SUSY formulation shows that the topology is not restricted to a band structure problem but extends to systems of coupled bosons and fermions that are in principle also realizable in solid state systems. The analogus supersymmetry of the magnon problem turns out to be particularly useful for highly frustrated magnets with the kagome family of antiferromagnets an analog of topological isostatic lattices. Thus, a solid state realization of the theory of phonon topological band structure may be found in highly frustrated magnets. However, our results show that this topology is protected not   

Non-Kondo Mechanism of Resistivity Minimum in Frustrated Itinerant Mangets

Frustration can induce novel phenomena in the transport properties of itinerant magnets. The "amount of frustration" is typically quantified by the $|\Theta_{CW}|/T_{C}$ ratio. A large value of this ratio corresponds to a broad temperature regime $T_C < T < \Theta_{CW}$, where the spins are in spin liquid state, i.e., the magnetic structure factor is not flat, as in the gas ($T>\Theta_{CW}$) state and it does not contain Bragg peaks, as in the ordered or "solid" state at $T \lt T_C $. We demonstrate that when interaction between magnetic moments is mediated by the conduction electrons, the electronic resistivity increases upon lowering temperature, due to enhanced scattering rate for $k \leq 2k_F$. To illustrate this phenomenon we consider a triangular Kondo lattice model with classical local moments. By using both analytical and numerical methods, we unambiguously demonstrate that the electronic resistivity grows upon lowering temperature inside the spin liquid regime. This growth necessarily leads to a resistivity minimum when electron-electron and electron-phonon scattering are included. We note that the origin of this resistivity minimum is radically different from the well-known minimum induced by the Kondo effect.   

Vortex Crystals with Chiral Stripes in Itinerant Magnets

Noncoplanar spin textures in itinerant magnets are generating increasing interest because of the associated spin Berry phase, which induces a tremendous effective magnetic field on the itinerant electrons. Such noncoplanar spin textures appear frequently in itinerant magnets, even with vanishingly small spin-orbit coupling. We explore a generic condition for noncoplanar spin ordering, with a focus on ``frustration" in itinerant magnets, that is characterized by multiple global maxima in the magnetic susceptibility. In a simple square Kondo lattice model, we find that a noncoplanar vortex-antivortex crystal with a one-dimensional modulation of spin scalar chirality becomes stable in a wide range of electron filling fraction~[1]. The unexpected result is obtained by careful analyses of higher-order terms in the perturbative expansion in terms of the Kondo exchange coupling and the degree of noncoplanarity, as well as numerical simulation based on the Langevin and stochastic Landau-Lifshitz-Gilbert dynamics with the kernel polynomial method. [1] R. Ozawa, S. Hayami, K. Barros, G.-W. Chern, Y. Motome, and C. D. Batista, preprint (arXiv:1510.06830).   

Lifting mean field degeneracies in anisotropic spin systems

We propose a method for calculating the fluctuation contribution to the free energy of anisotropic spin systems with generic bilinear superexchange magnetic Hamiltonian based on the Hubbard-Stratonovich transformation. We show that this contribution splits the set of mean field degenerate states with rotational symmetry, and chooses states with the order parameter directed along lattice symmetric directions as the true ground states. We consider the simple example of Heisenberg-compass model on cubic lattice to show that depending on the relative strength of the compass and Heisenberg interactions the spontaneous magnetization is pinned to either one of the cubic directions or one of the cubic body diagonals with a intermediate phase in between where the minima and maxima of the free energy interchange.