Spin-rotational Invariance Research Articles

Excitonic instability towards a Potts-nematic quantum paramagnet

Magnetic frustration can lead to peculiar magnetic orderings that break a discrete symmetry of the lattice in addition to the fundamental magnetic symmetries (i.e., spin rotation invariance and time-reversal symmetry). In this work, we focus on frustrated quantum magnets and study the nature of the quantum phase transition between a paramagnet and a magnetically ordered state with broken threefold (Z3) crystal rotation symmetry. We show that the transition can occur in two stages, giving rise to an intermediate nematic phase in which rotation symmetry is broken but the system remains magnetically disordered. Since the nematic transition is described by the three-state Potts model, the intermediate phase is a Z3 Potts-nematic phase. Our prediction of the existence of a Potts-nematic phase is based on an analysis of bound states formed from two-magnon excitations in the paramagnet, which become gapless while single-magnon excitations remain gapped. By considering three different lattice models, we demonstrate a generic instability towards two-magnon bound state formation in the Potts-nematic channel. We present both numerical results and a general analytical perturbative formula for the bound state binding energy similar to BCS theory. We further discuss a number of different materials that realize key features of the model considered, and thus provide promising venues for possible experimental observation. Published by the American Physical Society 2024

Strong-coupling expansion of multi-band interacting models: Mapping onto the transverse-field [formula omitted]-[formula omitted] Ising model

We investigate a class of two-dimensional two-band microscopic models in which the inter-band repulsive interactions play the dominant role. We first demonstrate three different schemes of constraining the ratios between the three types of inter-band interactions – density-density, spin exchange, and pair-hopping – that render the model free of the fermionic sign-problem for any filling and, consequently, amenable to efficient Quantum Monte Carlo simulations. We then study the behavior of these sign-problem-free models in the strong-coupling regime. In the cases where spin-rotational invariance is preserved or lowered to a planar symmetry, the strong-coupling ground state is a quantum paramagnet. However, in the case where there is only a residual Ising symmetry, the strong-coupling expansion maps onto the transverse-field J1-J2 Ising model, whose pseudospins are associated with local inter-band magnetic order. We show that by varying the band structure parameters within a reasonable range of values, a variety of ground states and quantum critical points can be accessed in the strong-coupling regime, some of which are not realized in the weak-coupling regime. We compare these results with the case of the single-band Hubbard model, where only intra-band repulsion is present, and whose strong-coupling behavior is captured by a simple Heisenberg model.

Spin functional renormalization group for quantum Heisenberg ferromagnets: Magnetization and magnon damping in two dimensions

We use the spin functional renormalization group recently developed by two of us [J. Krieg and P. Kopietz, Phys. Rev. B $\bf{99}$, 060403(R) (2019)] to calculate the magnetization $M ( H , T )$ and the damping of magnons due to classical longitudinal fluctuations of quantum Heisenberg ferromagnets. In order to guarantee that for vanishing magnetic field $H \rightarrow 0$ the magnon spectrum is gapless when the spin rotational invariance is spontaneously broken, we use a Ward identity to express the magnon self-energy in terms of the magnetization. In two dimensions our approach correctly predicts the absence of long-range magnetic order for $H=0$ at finite temperature $T$. The magnon spectrum then exhibits a gap from which we obtain the transverse correlation length. We also calculate the wave-function renormalization factor of the magnons. As a mathematical by-product, we derive a recursive form of the generalized Wick theorem for spin operators in frequency space which facilitates the calculation of arbitrary time-ordered connected correlation functions of an isolated spin in a magnetic field.

Combined use of translational and spin-rotational invariance for spin systems

Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry $C_N$ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry SU(2) can be used, where the Hamiltonian matrix is block-structured according to the total spin- and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both $C_N$ and SU(2), mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.

Heat Transport and Majorana Fermions in a Superconducting Dot-Wire System: An Exact Solution

We obtain exact expressions for the first three moments of the heat conductance of a quantum chain that crosses over from a superconducting quantum dot to a superconducting disordered quantum wire. Our analytic solution provides exact detailed descriptions of all crossovers that can be observed in the system as a function of its length, which include ballistic-metallic and metallic-insulating crossovers. The two Bogoliubov-de Gennes (BdG) symmetry classes with time-reversal symmetry are accounted for. The striking effect of total suppression of the insulating regime in systems with broken spin-rotation invariance is observed at large length scales. For a single channel system, this anomalous effect can be interpreted as a signature of the presence of the elusive Majorana fermion in a condensed matter system.

Triangular antiferromagnetism on the honeycomb lattice of twisted bilayer graphene

We present the electronic band structures of states with the same symmetry as the three-sublattice planar antiferromagnetic order of the triangular lattice. Such states can also be defined on the honeycomb lattice provided the spin density waves lie on the bonds. We identify cases which are consistent with observations on twisted bilayer graphene: a correlated insulator with an energy gap, yielding a single doubly-degenerate Fermi surface upon hole doping. We also discuss extensions to metallic states which preserve spin rotation invariance, with fluctuating spin density waves and bulk $\mathbb{Z}_2$ topological order.

Measuring space-group symmetry fractionalization in Z2 spin liquids

The interplay of symmetry and topological order leads to a variety of distinct phases of matter, the Symmetry Enriched Topological (SET) phases. Here we discuss physical observables that distinguish different SETs in the context of Z$_2$ quantum spin liquids with SU(2) spin rotation invariance. We focus on the cylinder geometry, and show that ground state quantum numbers for different topological sectors are robust invariants which can be used to identify the SET phase. More generally these invariants are related to 1D symmetry protected topological phases when viewing the cylinder geometry as a 1D spin chain. In particular we show that the Kagome spin liquid SET can be determined by measurements on one ground state, by wrapping the Kagome in a few different ways on the cylinder. In addition to guiding numerical studies, this approach provides a transparent way to connect bosonic and fermionic mean field theories of spin liquids. When fusing quasiparticles, it correctly predicts nontrivial phase factors for combining their space group quantum numbers.

Translational Symmetry and Microscopic Constraints on Symmetry-Enriched Topological Phases: A View from the Surface

The Lieb-Schultz-Mattis theorem and its higher dimensional generalizations by Oshikawa and Hastings require that translationally invariant 2D spin systems with a half-integer spin per unit cell must either have a continuum of low energy excitations, spontaneously break some symmetries, or exhibit topological order with anyonic excitations. We establish a connection between these constraints and a remarkably similar set of constraints at the surface of a 3D interacting topological insulator. This, combined with recent work on symmetry-enriched topological phases (SETs) with on-site unitary symmetries, enables us to develop a framework for understanding the structure of SETs with both translational and on-site unitary symmetries, including the effective theory of symmetry defects. This framework places stringent constraints on the possible types of symmetry fractionalization that can occur in 2D systems whose unit cell contains fractional spin, fractional charge, or a projective representation of the symmetry group. As a concrete application, we determine when a topological phase must possess a "spinon" excitation, even in cases when spin rotational invariance is broken down to a discrete subgroup by the crystal structure. We also describe the phenomena of "anyonic spin-orbit coupling", which may arise from the interplay of translational and on-site symmetries. These include the possibility of on-site symmetry defect branch lines carrying topological charge per unit length and lattice dislocations inducing on-site symmetry protected degeneracies.

Optical Emission Spectroscopy Study of Competing Phases of Electrons in the Second Landau Level.

Quantum phases of electrons in the filling factor range 2≤ν≤3 are probed by the weak optical emission from the partially populated second Landau level and spin wave measurements. Observations of optical emission include a multiplet of sharp peaks that exhibit a strong filling factor dependence. Spin wave measurements by resonant inelastic light scattering probe breaking of spin rotational invariance and are used to link this optical emission with collective phases of electrons. A remarkably rapid interplay between emission peak intensities manifests phase competition in the second Landau level.

Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic 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 one is an entanglement-based scheme, and the second one studies the system on an appropriate flat "Bieberbach" manifold to obtain the filling conditions for all 230 space groups. These approaches assume only 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.

Multifractality and electron-electron interaction at Anderson transitions

Mesoscopic fluctuations and correlations of the local density of states are studied near metal-insulator transitions in disordered interacting electronic systems. We show that the multifractal behavior of the local density of states survives in the presence of Coulomb interaction. We calculate the spectrum of multifractal exponents in $2+\ensuremath{\epsilon}$ spatial dimensions for symmetry classes characterized by broken (partially or fully) spin-rotation invariance and show that it differs from that in the absence of interaction. We also estimate the multifractal exponents at the Anderson metal-insulator transition in 2D systems with preserved spin-rotation invariance. Our results for multifractal correlations of the local density of states are in qualitative agreement with recent experimental findings.

Relativistic dynamical spin excitations of magnetic adatoms

We present a first-principles theory of dynamical spin excitations in the presence of spin-orbit coupling. The broken global spin rotational invariance leads to a new sum rule. We explore the competition between the magnetic anisotropy energy and the external magnetic field, as well as the role of electron-hole excitations, through calculations for 3$d$-metal adatoms on the Cu(111) surface. The spin excitation resonance energy and lifetime display non-trivial behavior, establishing the strong impact of relativistic effects. We legitimate the use of the Landau-Lifshitz-Gilbert equation down to the atomic limit, but with parameters that differ from a stationary theory.

Emergent quantum phases in a frustratedJ1−J2Heisenberg model on the hyperhoneycomb lattice

We investigate possible quantum ground states as well as the classical limit of a frustrated ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the three-dimensional (3D) hyperhoneycomb lattice. Our study is inspired by the recent discovery of $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$, where ${\mathrm{Ir}}^{4+}$ ions form a 3D network with each lattice site being connected to three nearest neighbors. We focus on the influence of magnetic frustration caused by the second-nearest-neighbor spin interactions. Such interactions are likely to be significant due to the large extent of $5d$ orbitals in iridates or other $5d$ transition metal oxides. In the classical limit, the ground state manifold is given by line degeneracies of the spiral magnetic-order wave vectors when ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.17$ while the collinear stripy order is included in the degenerate manifold when ${J}_{2}/{J}_{1}=0.5$. Quantum order-by-disorder effects are studied using both the semiclassical $1/S$ expansion in the spin-wave theory and the Schwinger-boson approach. In general, certain coplanar spiral orders are chosen from the classical degenerate manifold for a large fraction of the phase diagram. Nonetheless quantum fluctuations favor the collinear stripy order over the spiral orders in an extended parameter region around ${J}_{2}/{J}_{1}=0.5$, despite the spin-rotation invariance of the underlying Hamiltonian. This is in contrast to the emergence of stripy order in the Heisenberg-Kitaev model studied earlier on the same lattice, where the Kitaev-type Ising interactions are important for stabilizing the stripy order. As quantum fluctuations become stronger, $U(1)$ and ${Z}_{2}$ quantum spin liquid phases are shown to arise via quantum disordering of the N\'eel, stripy, and spiral magnetically ordered phases. The effects of magnetic anisotropy and their relevance to future experiments are also discussed.

A minimal model for “Hidden Order” in URu2Si2

We present a bare bones description of a model which may lead to a Hidden Order Phase. We use a variational approximation and find that spin-rotational invariance, periodic translational invariance and gauge invariance are broken at low temperatures. The breaking of spin-rotational invariance leads to the development of an anisotropy in the susceptibility.

Signatures of broken spin-rotational invariance in the “Hidden Ordered” compound URu2Si2?

We briefly review the status of research on the “Hidden Ordered” compound . We focus on evidence which indicates that the compound undergoes an electronic structural change from body-centred tetragonal to simple tetragonal, but which apparently does not involve any atomic displacements. The fourfold rotational invariance under rotations in the a–b plane may also be spontaneously broken by the transition. We suggest that the electronic system may be modelled by an undercompensated Anderson Lattice Model, and that the transition to the “Hidden Ordered” state is driven by nested interband correlations. We show that the model exhibits a delicate competition between the “Hidden Order” and a Neel antiferromagnetic phase, and that it might also describe the first-order transition in found by applying pressure. Furthermore, we show that the broken spin-rotational invariance of the novel phase becomes apparent when magnetic fields are applied.

Coulomb interaction parameters in bcc iron: an LDA+DMFT study

We study the influence of Coulomb interaction parameters on electronic structure and magnetic properties of paramagnetic bcc Fe by means of the local density approximation plus dynamical mean-field theory approach. We consider the local Coulomb interaction in the density-density form as well as in the form with spin rotational invariance approximated by averaging over all directions of the quantization axis. Our results indicate that the magnetic properties of bcc Fe are mainly affected by the Hund's rule coupling J rather than by the Hubbard U. By employing the constrained density functional theory approach in the basis of Wannier functions of spd character, we obtain U = 4 eV and J = 0.9 eV. In spite of the widespread belief that U = 4 eV is too large for bcc Fe, our calculations with the obtained values of U and J result in a satisfactory agreement with the experiment. The correlation effects caused by U are found to be weak even for large U = 6 eV. The agreement between the calculated and experimental Curie temperatures is further improved if J is reduced to 0.8 eV. However, with the decrease of J, the effective local magnetic moment moves further away from the experimental value.

Unconventional pairing and electronic dimerization instabilities in the doped Kitaev-Heisenberg model

We study the quantum many-body instabilities of the t−JK−JH Kitaev-Heisenberg Hamiltonian on the honeycomb lattice as a minimal model for a doped spin-orbit Mott insulator. This spin-1/2 model is believed to describe the magnetic properties of the layered transition-metal oxide Na2IrO3. We determine the ground state of the system with finite charge-carrier density from the functional renormalization group (fRG) for correlated fermionic systems. To this end, we derive fRG flow equations adapted to the lack of full spin-rotational invariance in the fermionic interactions, here represented by the highly frustrated and anisotropic Kitaev exchange term. Additionally employing a set of the Ward identities for the Kitaev-Heisenberg model, the numerical solution of the flow equations suggests a rich phase diagram emerging upon doping charge carriers into the ground-state manifold (Z2 quantum spin liquids and magnetically ordered phases). We corroborate superconducting triplet p-wave instabilities driven by ferromagnetic exchange and various singlet pairing phases. For filling δ>1/4, the p-wave pairing gives rise to a topological state with protected Majorana edge modes. For antiferromagnetic Kitaev and ferromagnetic Heisenberg exchanges, we obtain bond-order instabilities at van Hove filling supported by nesting and density-of-states enhancement, yielding dimerization patterns of the electronic degrees of freedom on the honeycomb lattice. Further, our flow equations are applicable to a wider class of model Hamiltonians.11 MoreReceived 1 May 2014Revised 30 June 2014DOI:https://doi.org/10.1103/PhysRevB.90.045135©2014 American Physical Society

Rotationally invariant parametrization of Coulomb interactions in multi-orbital Hubbard models

Multi-orbital models with strong Coulomb correlations represent a fundamental tool for the analysis of transition metal oxides, which exhibit physical properties strongly affected by the orbital degrees of freedom. In this context, specific relations between the local coupling constants must be imposed in order to preserve spin and charge rotational invariance. We present here an alternative route for the derivation of these relations, based on the use of a complete set of commuting operators which includes the pseudospin orbital operator and thus is particularly suited for the study of systems where phenomena of orbital ordering are known to play a major role. This approach is expected to be useful in investigations of multi-orbital Hubbard models analyzed within the framework of the dynamical mean field theory or in exact diagonalization studies on finite-size clusters.

The Nematic Character of a Hidden Ordered State revealed by Magnetic Fields.

ABSTRACTWe examine a novel phase of the underscreened Anderson lattice Model that might pertain to the ”Hidden Ordered” phase of URu2Si2. We show that the system breaks spin-rotational invariance below the critical temperature and spontaneously selects a preferred axis of spin quantization. As a result, the low temperature phase exhibits a magnetic anisotropy, where the electronic properties depend not only on the magnitude of the magnetic field but also on the orientation of the applied field relative to the axis of quantization. The results are discussed in the context of recent experimental findings on URu2Si2.

The underscreened anderson lattice: A model for uranium compounds

We first present a model based on the Underscreened Anderson Lattice with two 5f electrons per site to account for the Kondo-ferromagnetism co-existence observed in some actinide compounds. Then, we analyze the physics behind a novel type of phase transition that appears in the spinrotationally invariant form of the Underscreened Anderson Lattice Model. The model exhibits a continuous phase transition and below the critical temperature a pseudo-gap opens in the density of states. The transition is driven by the spin-flip part of the inter-orbital Hund’s rule interaction which produces a novel order parameter and breaks spin-rotational invariance. It is suggested that this order parameter might describe the “Hidden Order” transition found in URu2Si2.