# Research Highlights

*Quantum resonances reveal "hidden" symmetry near quantum criticality *

Quantum phase transitions take place between distinct phases of matter at zero temperature. Experimental exploration of such phase transitions offers a unique opportunity to observe quantum states of matter with exceptional structure and properties.

One of the most theoretically studied paradigms for a quantum phase transition is the one-dimensional Ising chain in a transverse magnetic field. We have realized this system in the laboratory for the first time in experiments on the magnetic material CoNb_{2}O_{6} placed in strong magnetic fields and cooled close to absolute zero. Cobalt atoms in the material act like tiny bar magnets arranged in long chains and exemplify ferromagnetism in one dimension. When a magnetic field is applied at right angles to the aligned spin directions the spins can "quantum tunnel" between the allowed "up" and "down" orientations and at a precise value of the field those quantum fluctuations "melt" the spontaneous ferromagnetic order and a quantum critical state is reached.

Near the critical point the spin excitations were theoretically predicted nearly two decades ago to have a set of resonances with an exceptional mathematical structure due to a "hidden" symmetry described by the group E8, called by some "the most beautiful object in mathematics", which has not been seen experimentally before. Those quantum resonances in the spin chain are to be understood as the harmonic modes of vibration of the string of spins, where the string tension comes from the interactions between spins. Our experiments indeed observed such a set of quantum resonances and the ratio between the frequencies of the two lowest resonances approached the "golden ratio" near the critical point, as predicted by the E8 model. Our results emphasize that the exploration of quantum phase transitions can open up new avenues to experimentally realize otherwise inaccessible correlated quantum states of matter with complex symmetries and dynamics.

*References:*

Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry

R. Coldea, D.A. Tennant, E.M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, K. Kiefer, Science **327**, 177 (2010).

A complex symmetry arises at a spin chain's quantum critical point, Physics Today, March 2010.

'Most beautiful' math structure appears in lab for first time, New Scientist, Jan 2010.

Hidden Symmetry Revealed, This Week in Science, 8 January 2010.

*Novel charge order pattern for electrons with orbital degeneracy in triangular layers *

The presence of orbital degrees of freedom in addition to spin opens up rich and largely unexplored possibilities for complex order patterns. Particularly promising are geometrically-frustrated lattices, such as the triangular geometry, where many order patterns are degenerate (both for spins and orbitals). If the electrons are mobile even richer possibilities exist as the tendency to lift the orbital degeneracy via the symmetry-lowering local Jahn-Teller distortions competes with the tendency to remove the degenerate electrons by charge transfer between sites and therefore more complex order patterns can occur. Indeed, in recent experiments on the hexagonal delafossite AgNiO with orbitally-degenerate ) ions arranged in triangular layers, we observed an unexpected crystal superstructure where the orbital degeneracy is lifted, not by conventional Jahn-Teller orbital order, but by charge order, leading to of sites being doubly-occupied with no orbital degeneracy [red balls in Figure arranged in the centres of an electron-depleted honeycomb network (thick contour) which remains strongly metallic].

*References:*

Direct Observation of Charge Order in Triangular Metallic AgNiO2 by Single-Crystal Resonant X-Ray Scattering

G. L. Pascut, R. Coldea, P. G. Radaelli, A. Bombardi, G. Beutier, I. I. Mazin, M. D. Johannes, and M. Jansen, Phys. Rev. Lett. **106**, 157206 (2011).

Spin dynamics of the frustrated easy-axis triangular antiferromagnet 2H-AgNiO2 explored by inelastic neutron scattering

E.M. Wheeler, R. Coldea, E. Wawrzynska, T. Sörgel, M. Jansen, M.M. Koza, P. Adroguer and N. Shannon, Phys. Rev. B **79**, 104421 (2009).

Charge disproportionation and collinear magnetic order in the frustrated triangular antiferromagnet AgNiO_{2}

E. Wawrzynska, R. Coldea, E.M. Wheeler, T. Sörgel, M. Jansen, R. M. Ibberson, P. G. Radaelli, and M.M. Koza, Phys. Rev. B **77**, 094439 (2008).

Orbital Degeneracy Removed by Charge Order in Triangular Antiferromagnet AgNiO_{2}

E. Wawrzynska, R. Coldea, E.M. Wheeler, I. I. Mazin, M. D. Johannes, T. Sörgel, M. Jansen, R. M. Ibberson, and P. G. Radaelli, Phys. Rev. Lett. **99**, 157204 (2007).

*Fractional spin excitations in a frustrated two-dimensional quantum magnet *

One of the most remarkable phenomena that can occur in strongly-interacting quantum systems is the emergence of quasiparticles with fractional quantum numbers. In magnetism a key paradigm is the one-dimensional spin-1/2 Heisenberg antiferromagnetic chain where S =1 spin-waves (characteristic of 3D ordered magnets) spontaneously decay into a pair of spin-1/2 spinons that propagate as independent quasiparticles. The search for fractionalized systems in higher dimensions has been a central problem in quantum condensed matter for many decades. Our comprehensive experiments on the frustrated quantum antiferromagnet Cs_{2}CuCl_{4} have attracted a wide interest in the community as the first system with significant 2D couplings (J’/J~1/3) where experimental evidence for spinons has been observed. The figure shows the excitation spectrum observed in neutron scattering experiments, the dominant continuum of excitations with highly-dispersive boundaries is characteristic of neutron scattering by pairs of spinons.

*References:*

Extended scattering continua characteristic of spin fractionalization in the two-dimensional frustrated quantum magnet Cs_{2}CuCl_{4} observed by neutron scattering

R. Coldea, D.A. Tennant, and Z. Tylczynski, Phys. Rev. B **68**, 134424 (2003).

Direct Measurement of the Spin Hamiltonian and Observation of Condensation of Magnons in the 2D Frustrated Quantum Magnet Cs_{2}CuCl_{4}

R. Coldea, D.A. Tennant, K. Habicht, P. Smeibidl, C. Wolters, and Z. Tylczynski, Phys. Rev. Lett. **88**, 137203 (2002).

Experimental Realization of a 2D Fractional Quantum Spin Liquid

R. Coldea, D.A. Tennant, A.M. Tsvelik, and Z. Tylczynski, Phys. Rev. Lett. **86**, 1335 (2001).

*Cyclic spin exchange in the parent of high-temperature cuprate superconductors *

High-resolution mapping of the spin-wave dispersion relation in the parent of the high-temperature cuprate superconductors, La_{2}CuO_{4}, have revealed a significant modulation in the dispersion at high energies, which reveals important ring-exchange interactions among the four spins located at corners of square plaquettes in the CuO_{2} plane. The experiments provided important feedback to microscopic theories of the cuprates based on the Hubbard model where such cyclic exchanges arise. CuO_{2} planes are the second example of an important Fermi system (the nuclear magnet He^{3} is the other) where multi-spin ring exchange terms have been observed.

* References:*

Spin waves and electronic interactions in La_{2}CuO_{4}, R. Coldea, S.M. Hayden, G. Aeppli, T.G. Perring, C.D. Frost, T.E. Mason, S.W. Cheong, and Z. Fisk., Phys. Rev. Lett. **86**, 5377, (2001).