**Topological states in magnetic insulators:**
**Hall effect of triplons in a dimerized quantum magnet**

Edwin Hall discovered the effect named after him in 1879 when he placed a gold foil in a magnetic field and passed a current through it. He found that the magnetic field induces a voltage perpendicular to the applied current. Indeed, the Hall effect is easily explained by taking into account the deflection of charge carriers by a magnetic field. In the 1980's, Klaus von Klitzing revisited the Hall effect at low temperatures and observed that the Hall voltage increases in steps as the magnetic field is increased. He showed that the Hall conductance is 'quantized' — it takes values that are integer multiples of a fundamental constant. This path-breaking discovery earned him the Nobel prize in 1985.
The quantization of Hall conductance is intimately tied to topology, the field of mathematics which studies smooth deformations of geometric objects. A cube and a sphere are 'topologically equivalent', i.e., a cube can be smoothly deformed into a sphere. Whereas, a cube cannot be deformed into a torus without rupturing the surface. A torus, thus, belongs to a different 'topological class'. The Hall conductance turns out to be a quantum number that characterizes the topology of electronic wavefunctions in a magnetic field. The observed jumps in the Hall conductance are akin to rupturing the surface to go from one topological class to another.

Topology also appears in 'quantum magnets' —
magnetic systems wherein the quantization of spin plays a central role. We investigated Sr_{2}Cu(BO_{3})_{2}, an insulator in which the constituent spins are pairwise entangled, i.e., pairs of electrons interact in such a way that their spins cancel each other. An excitation of this system corresponds to breaking a pair —
we showed that these excitations are topological in nature.

At low temperatures, the excitations show features very similar to electrons in the quantum Hall effect. The bulk remains insulating, while the edges carry Hall currents. As these excitations carry no electrical charge, there is no Hall voltage. However, as they still carry energy, the Hall current will transport heat from one edge of the sample to another. The topological nature of the excitations can be tuned with a small applied magnetic field. At a critical value of the field, the excitations form a new kind of 'Dirac cone', forming a magnetic analogue of graphene. This Dirac cone corresponds to rupturing the surface to change the topology. Indeed, if the magnetic field is increased beyond the critical value, the excitations lose their topological character.
Beyond theoretical interest, the topologically protected edge states are potentially useful in the field of spintronics. They are ideally suited for devices in which the electrons themselves are static, but the spin can be transported as in a wire. Such currents will not be dissipated by heating and the device itself can work at higher speeds. This discovery suggests new systems for spintronics and new ways to use spin currents to process and store information.

J. Romhányi, **K. Penc**, R. Ganesh,
Nat. Comm. **6**, 6805 (2015).
[arXiv:1406.1163]

**Berry phase induced dimerization in one-dimensional quadrupolar systems
**

We investigate the effect of the Berry phase on quadrupoles that occur, for example, in the low-energy description of spin models. Specifically, we study here the one-dimensional bilinear-biquadratic spin-one model. An open question for many years about this model is whether it has a nondimerized fluctuating nematic phase. The dimerization has recently been proposed to be related to Berry phases of the quantum fluctuations. We use an effective low-energy description to calculate the scaling of the dimerization according to this theory and then verify the predictions using large scale density-matrix renormalization group simulations, giving good evidence that the state is dimerized all the way up to its transition into the ferromagnetic phase. We furthermore discuss the multiplet structure found in the entanglement spectrum of the ground state wave functions.

S. Hu, A. M. Turner, **K. Penc**, F. Pollmann,
Phys. Rev. Lett. **113**, 027202 (2014).
[arXiv:1401.3246]

**Spin-orbital quantum liquid on the honeycomb lattice**

In addition to low-energy spin fluctuations, which distinguish them from band
insulators, Mott insulators often possess orbital degrees of freedom. This can
occur when the degeneracy of atomic levels is not completely lifted by the
crystalline environment, as for the 3d levels of copper in an octahedral
environment. In most cases, a phase transition takes place at low temperature to
a phase in which the crystalline environment of the atoms is less symmetric so
as to completely lift the degeneracy, a mechanism known as Jahn-Teller
distortion that leads to orbital order and to purely magnetic low energy
fluctuations. However, this needs not be the case, and the search for
spin-orbital models that could realize a spin-orbital quantum liquid in which
spins and orbitals fluctuate down to the lowest temperatures has been an active
area of research for the past fifteen years. In this paper, we provide strong
evidence that this is the case for the symmetric Kugel-Khomskii spin-orbital
model, a minimal model of spin-1/2 Mott insulators with two-fold orbital
degeneracy, when it lives on the honeycomb lattice. This conclusion is based on
convergent indications from state-of-the-art numerical and variational
approaches, and it provides an interesting starting point to understand the
recently discovered spin-orbital liquid behavior of Ba_{3}CuSb_{2}O_{9}. The present
results are also relevant to cold atoms: They suggest to choose optical lattices
with honeycomb geometry in the search for quantum liquids in ultra-cold
four-color fermionic atoms.

P. Corboz, **M. Lajkó**, A. M. Läuchli, **K. Penc**, F. Mila,
Phys. Rev. X **2**, 041013 (2012).
[arXiv:1207.6029]

**Spin-stretching modes in non-centrosymmetric magnets: spin-wave excitations in the multiferroic Ba _{2}CoGe_{2}O_{7}**

In this paper we describe some unusual collective spin excitations in crystals of low symmetry. Such excitations coexist with usual spin waves but instead of precession, spins change their length periodically and hence we call them `spin-stretching modes'. It turns out that materials with spin-stretching modes acquire new optical properties. Spin-stretching modes are excited by oscillating magnetic field and in the absence of inversion symmetry even by the electric field of light. We studied spin stretching-modes in detail in the multiferroic Ba

_{2}CoGe

_{2}O

_{7}, in which there is a magnetoelectric coupling between electric polarization and magnetization. Due to this coupling, the spin-stretching modes are also excited by the electric field of irradiation, and we observe a strong absorption of THz light by electron spin resonance and THz spectroscopy in magnetic fields. These observations are in very good agreement with our theory, that also suggests that spin-stretching modes are present in a broad class of ordered spin systems with strong single-ion anisotropy. This effect could be utilized to manipulate THz light with dc electric and magnetic fields.

Selected as Research Highlight of the European Magnetic Field Laboratory:

*Directional Dichroism and Spin-Wave Excitations in the Multiferroic Material BCGO .*

**K. Penc**,

**J. Romhányi**, T. Rõõm, U. Nagel, Á. Antal, T. Fehér, A. Jánossy, H. Engelkamp, H. Murakawa, Y. Tokura, D. Szaller, S. Bordács, I. Kézsmárki: Phys. Rev. Lett

**108**, 257203 (2012). [arXiv:1202.3996]

**Quantum Ice**

Water ice comprises a loosely-packed lattice of water molecules, held together by hydrogen bonds. This structure hides a puzzle — chemical bonding alone does not select a unique orientation of the water molecules. As a result each water molecule has a finite ground state entropy _{0} ≈ k_{B} ln 3/2, in violation of the third law of thermodynamics. The same degeneracy, and the same contradiction, arises in problems of frustrated charge order on the pyrochlore lattice, and in the family of rare-earth magnets collectively known as spin ice. Of particular interest at the moment are "quantum spin ice" materials, where large quantum fluctuations may permit tunnelling between different ice states. Here we show how such tunnelling can lift the degeneracy of a spin or charge ice, stabilising a unique "quantum ice" ground state. This quantum ice has excitations described by the Maxwell action of 3+1 – dimensional quantum electrodynamics. We further show how such a quantum ice state might be distinguished in neutron scattering experiments on a spin ice material.

Nic Shannon, Olga Sikora, Frank Pollmann, **Karlo Penc**, Peter Fulde,
Phys. Rev. Lett **108**, 067204 (2012).
[arXiv:1105.4196]

**Exact ground states with deconfined gapless excitations for the 3 leg spin-1/2 tube**

We consider a 3-leg spin-1/2 ladder with periodic boundary conditions (a spin tube) with a Hamiltonian given by two projection operators, one on the triangles, and the other on the square plaquettes on the side of the tube, that can be written in terms of Heisenberg and four spin ring exchange interactions. Depending on the relative strength of these two operators, we identify 3 phases: (i) for strongly antiferromagnetic exchange on the triangles, an exact dimerized ground state wave function with a gapped spectrum can be given as an alternation of spin and chirality valence bonds between nearest triangles; (ii) for ferromagnetic exchanges on the triangle we recover the phase of the spin-3/2 Heisenberg chain; (iii) between these two phases a gapless incommensurate phase exists. Furthermore, we explicitly construct an exact ground state wave function with two deconfined domain walls and gapless excitation spectrum at the quantum phase transition point between the incommensurate and dimerized phase.

**Miklós Lajkó**, Philippe Sindzingre, **Karlo Penc**,
Phys. Rev. Lett **108**, 017205 (2012).
[arXiv:1107.5501]

**Simultaneous dimerization and SU(4) symmetry breaking of 4-color fermions on the square lattice**

Using infinite project entangled pair states (iPEPS), exact diagonalization, and flavor-wave theory, we show that the SU(4) Heisenberg model undergoes a spontaneous dimerization on the square lattice, in contrast to its SU(2) and SU(3) counterparts, which develop Neel and three-sublattice stripe-like long-range order. Since the ground state of a dimer is not a singlet for SU(4) but a 6-dimensional irrep, this leaves the door open for further symmetry breaking. We provide evidence that, unlike in SU(4) ladders, where dimers pair up to form singlet plaquettes, here the SU(4) symmetry is additionally broken, leading to a gapless spectrum in spite of the broken translational symmetry.

Philippe Corboz, Andreas M. Läuchli, **Karlo Penc**, Matthias Troyer, Frédéric Mila,
Phys. Rev. Lett **107**, 215301 (2011).
[arXiv:1108.2857]

**Effect of Dzyaloshinskii-Moriya interactions on the phase diagram and magnetic
excitations of SrCu _{2}(BO_{3})_{2}**

The orthogonal dimer structure of the SrCu

_{2}(BO

_{3})

_{2}spin-1/2 magnet provides a realization of the Shastry-Sutherland model. Using a dimer-product variational wave function, we map out the phase diagram of the Shastry-Sutherland model including anisotropies. Based on the variational solution, we construct a bond-wave approach to obtain the excitation spectra as a function of the magnetic field. The characteristic features of the experimentally measured neutron and ESR spectra are reproduced, like the anisotropy-induced zero-field splittings and the persistent gap at higher fields.

**Judit Romhányi**, Keisuke Totsuka,

**Karlo Penc**, Phys. Rev. B

**83**, 024413 (2011).

**Three-Sublattice Ordering of the SU(3) Heisenberg Model of Three-Flavor Fermions on the Square and Cubic Lattices**

Combining a semiclassical analysis with exact diagonalizations, we show that the ground state of the SU(3) Heisenberg model on the square lattice develops three-sublattice long-range order. This surprising pattern for a bipartite lattice with only nearest-neighbor interactions is shown to be the consequence of a subtle quantum order-by-disorder mechanism. By contrast, thermal fluctuations favor two-sublattice configurations via entropic selection. These results are shown to extend to the cubic lattice, and experimental implications for the Mott-insulating states of three-flavor fermionic atoms in optical lattices are discussed.

**Tamás A. Tóth**, Andreas M. Läuchli, Frédéric Mila, **Karlo Penc**,
Phys. Rev. Lett **105**, 265301 (2010).

** Nematic, vector-multipole, and plateau-liquid states in the classical O(3) pyrochlore antiferromagnet with biquadratic interactions in applied magnetic field**

The classical bilinear-biquadratic nearest-neighbor Heisenberg antiferromagnet on the pyrochlore lattice does not exhibit conventional Neel-type magnetic order at any temperature or magnetic field. Instead spin correlations decay algebraically over length scales
r~T^{-1/2},
behavior characteristic of a Coulomb phase arising from a strong local constraint. Despite this, its thermodynamic properties remain largely unchanged if Neel order is restored by the addition of a degeneracy-lifting perturbation, e.g., further neighbor interactions.
Here we show how these apparent contradictions can be resolved by a proper understanding of way in which long-range Neel order emerges out of well-formed local correlations and identify nematic and vector-multipole orders hidden in the different Coulomb phases of the model. So far as experiment is concerned, our results suggest that where long-range interactions are unimportant, the magnetic properties of Cr spinels which exhibit half-magnetization plateaux may be largely independent of the type of magnetic order present.

Nic Shannon, **Karlo Penc**, Yukitoshi Motome,
Phys. Rev. B **81**, 184409 (2010).

**A quantum liquid with deconfined fractional excitations in three dimensions**

Excitations which carry "fractional" quantum numbers
are known to exist in one dimension in polyacetylene, and in two dimensions,
in the fractional quantum Hall effect. Fractional excitations have also
been invoked to explain the breakdown of the conventional theory of
metals in a wide range of three-dimensional materials. However the
existence of fractional excitations in three dimensions remains highly controversial. In this Letter we report direct numerical evidence for the existence of a quantum liquid phase supporting fractional excitations in a concrete, three-dimensional microscopic model - the quantum dimer model on a diamond lattice. We demonstrate explicitly that the energy cost of separating
fractional monomer excitations vanishes in this liquid phase, and that its energy spectrum matches that of the Coulomb phase in (3+1) dimensional quantum electrodynamics.

Olga Sikora, Frank Pollman, Nic Shannon, **Karlo Penc**, Peter Fulde,
Phys. Rev. Lett. **103**, 247001 (2009).

** Doped singlet-pair crystal in the Hubbard model on the checkerboard lattice
**

In the limit of large nearest-neighbor and on-site Coulomb repulsions, the Hubbard model on the planar pyrochlore lattice maps, near quarter filling, onto a doped, quantum, fully packed loop model. The phase diagram exhibits at quarter filling a quantum state of matter, the resonating singlet-pair crystal, an insulating phase breaking lattice symmetry. Properties of a few doped holes or electrons are investigated. In contrast to the doped quantum antiferromagnet, phase separation is restricted to very small hopping, leaving an extended
"window" for superconducting pairing. However, the latter is more fragile for large hopping than in the case of the antiferromagnet. The existence of a fermionic supersolid is discussed.

Didier Poilblanc, ** Karlo Penc**, Nic Shannon,
Phys. Rev. B **75**, 220503(R) (2007).

**Ising phases of Heisenberg ladders in a magnetic field**

We show that Dzyaloshinskii-Moriya (DM) interactions can substantially modify the phase diagram of spin-1/2 Heisenberg ladders in a magnetic field provided they compete with exchange. For nonfrustrated ladders, they induce a local magnetization along the DM vector that turns the gapless intermediate phase into an Ising phase with broken translational symmetry, while for frustrated ladders, they extend the Ising order of the half-integer plateau to the surrounding gapless phases of the purely Heisenberg case. Implications for experimental ladder and dimer systems are discussed.

**Karlo Penc**, Jean-Baptiste Fouet, Shin Miyahara, Oleg Tchernyshyov, Frédéric Mila,
Phys. Rev. Lett. **99**, 117201/1-4 (2007).

**Quadrupolar Phases of the S=1 Bilinear-Biquadratic Heisenberg Model on the Triangular Lattice**

Using mean-field theory, exact diagonalizations, and SU(3) flavor theory, we have precisely mapped out the phase diagram of the S=1 bilinear-biquadratic Heisenberg model on the triangular lattice in a magnetic field, with emphasis on the quadrupolar phases and their excitations. In particular, we show that ferroquadrupolar order can coexist with short-range helical magnetic order, and that the antiferroquadrupolar phase is characterized by a remarkable 2/3 magnetization plateau, in which one site per triangle retains quadrupolar order while the other two are polarized along the field. Implications for actual S=1 magnets are discussed.

Andreas M. Läuchli, Frédéric Mila, **Karlo Penc**,
Phys. Rev. Lett. **97**, 087205 (2006).

**Half-Magnetization Plateau Stabilized by Structural Distortion in the Antiferromagnetic Heisenberg Model on a Pyrochlore Lattice**

Magnetization plateaux, visible as anomalies in magnetic susceptibility at low temperatures, are one of the hallmarks of frustrated magnetism. We show how an extremely robust half-magnetization plateau can arise from coupling between spin and lattice degrees of freedom in a pyrochlore antiferromagnet, and develop
a detailed symmetry of analysis of the simplest possible scenario for such a plateau state. The application of this theory to the spinel oxides CdCr_{2}O_{4} and HgCr_{2}O_{4}, where a robust half-magnetization plateau has been observed, is discussed.

** Karlo Penc **, Nic Shannon, Hirouki Shiba,
Phys. Rev. Lett. **93**, 197203 (2004).

**Cyclic exchange, isolated states and spinon deconfinement in an XXZ Heisenberg model
on the checkerboard lattice**

The antiferromagnetic Ising model on a checkerboard lattice has an ice-like ground state manifold with extensive degeneracy. and, to leading order in J_{xy}, deconfined spinon excitations. We explore the role of cyclic exchange arising at order J_{xy}^{2}/J_{z} on the ice states and their associated spinon excitations. By mapping the original problem onto an equivalent quantum six-vertex model, we identify three different phases as a function of the chemical potential for flippable plaquettes — a phase with long range Neel order and confined spinon excitations, a non-magnetic state of resonating square plaquettes, and a quasi-collinear phase with gapped but deconfined spinon excitations. The relevance of the results to the square-lattice quantum dimer model is also discussed.

Nic Shannon, Gregoire Misguich, **Karlo Penc **
Phys. Rev. B **69**, 220403(R)/1-4 (2004).

**Fractional charges in pyrochlore lattices
**

A pyrochlore lattice is considered where the average electron number of lectrons per site is half-integer, concentrating on the case of exactly half an electron per site. Strong on-site repulsions are assumed, so that all sites are either empty or singly occupied. Where there are in addition strong nearest-neighbour repulsions, a tetrahedron rule comes into effect, as previously suggested for magnetite. We show that in this case, there exist excitations with fractional charge (+/-) e/2. These are intimately connected with the high degeneracy of the ground state in the absence of kinetic energy terms. When an additional electron is inserted into the system, it decays into
two point like excitations with charge -e/2, connected by a Heisenberg spin chain which carries the electron's spin.

Peter Fulde, ** Karlo Penc **, Nic Shannon,
Annalen der Physik **11**, 892 (2002).