Session P4: Magnetic Bose-Einstein Condensation

High field behavior of the spin-dimer compound BaCuSi$_2$O$_6$: magnon BEC and the role of crystal structure

BaCuSi$_2$O$_6$ is a quasi-2D compound comprising layers of well separated vertical Cu$^{2+}$ dimers. The material has a singlet ground state in zero magnetic field, with a gap of 3.14 meV to the lowest triplon mode. A structural distortion at 100 K leads to multiple triplon bands, separated by less than their band width. Magnetic fields in excess of $H_{c1}$ $\sim$ 23 T close the spin gap, resulting in a state characterized by long- range XY antiferromagnetic order at low temperature. Critical exponents describing the phase boundary approaching the QCP at $H_{c1}$ are consistent with BEC universal scaling, and EPR measurements confirm the absence of terms in the spin Hamiltonian that explicitly break axial symmetry down to an energy scale of 11 mK. A cross-over from 3D to 2D BEC scaling is observed below $\sim$ 0.5 K, which can be attributed to geometric frustration in the body centered crystal structure. \newline \newline *Work performed in collaboration with S. E. Sebastian, E. Samulon, N. Harrison, M. Jaime, C. D. Batista, Z. Islam, S. Hill, N. Kawashima, R. Stern, Ch. Ruegg, H. Ronnow and D. McMorrow.   

Geometric Frustration and Dimensional Reduction at a Quantum Critical Point

The universality class of a critical point depends on a few properties such as the symmetry of the underlying model, the range of the interactions, the number of components of the order parameter, and the space dimensionality. Usually these properties do not change when the system approaches its critical point. However, this is not true in general. For instance, it is natural to ask what happens when the coupling along one of the spatial dimensions vanishes right at the critical point. This is the case of a Bose Einstein Condensation quantum phase transition whenever the underlying lattice consists of layers coupled by a geometrically frustrated interaction. We will show in this talk that this results in a dimensional reduction at the quantum critical point as manifested by its critical exponents. Our theoretical predictions for the critical temperature as a function of the chemical potential correspond very well with recent measurements in BaCuSi$_{2}$O$_{6}$ [S. E. Sebastian \textit{et al}, Nature \textbf{411}, 617 (2006)].   

Quantum Magnetism and possible BEC in an organic Nickel compound

I will review recent experimental and theoretical work on the S=1 quantum magnet, NiCl$_{2}$-4SC(NH$_{2})_{2}$. [1] This compound exhibits field-induced XY antiferromagnetism for magnetic fields along the tetragonal c-axis between H$_{c1}$ = 2.1 and H$_{c2}$ = 12.6 T. The axial symmetry of the spin environment allows us to understand the quantum phase transitions at H$_{c1}$ and H$_{c2}$ in terms of Bose-Einstein condensation (BEC) of spin levels. Here the tuning parameter for BEC transition is the magnetic field and not the temperature. Specific heat, magnetocaloric effect, and magnetization data at low temperatures confirm the predicted behavior for a BEC: Hc-H$_{c1} \quad \sim $ T$^{\alpha }$ and M(H$_{c1}) \quad \sim $ T$^{\alpha }$ where $\alpha $ = 3/2. I will also present magnetostriction data [2] taken at dilution refrigerator temperatures that show significant magnetoelastic coupling and magnetic-order-induced modifications of the lattice parameters in this soft organic compound. The data are well-described by Quantum Monte Carlo calculations, allowing us to make a quantitative determination of the magnetoelastic coupling, and also extract the spin-spin correlation function from the magnetostriction data. \newline \textbf{ \newline }[1] V. S. Zapf, D. Zocco, B. R. Hansen, M. Jaime, N. Harrison, C. D. Batista, M. Kenzelmann, C. Niedermayer, A. Lacerda, and A. Paduan-Filho, Phys. Rev. Lett. 96, 077204 (2006).\newline [2] V. S. Zapf, V. Correa, C. D. Batista, T. Murphy, E. D. Palm, M. Jaime, S. Tozer, A. Lacerda, A. Paduan-Filho, ``Magnetostriction in the Bose-Einstein Condensate quantum magnet NiCl$_{2}$-4SC(NH$_{2})_{2}$,'' cond-mat/0611229.   

Magnetic phase diagram of F$_{2}$PNNNO

2-[2',6',-difluoro-4'-($N$-\textit{tert}-butyl-$N$-oxyamino)phenyl]-4,4,5,5-tetramethyl-4,5- dihydro-1$H$-imidazol-1-oxyl 3-oxide, F$_{2}$PNNNO for short, is an organic molecule containing two unpaired electrons. Residing in the $N$-\textit{tert}-butyl nitroxide and nitronyl nitroxide groups, the two $S$=1/2 spins of these electrons are ferromagnetically coupled with an exchange constant of 407 K. In a crystal, two neighboring F$_{2}$PNNNO molecules form a pair in which the nitronyl-nitroxide spins are coupled antiferromagnetically with an exchange constant of 67 K. The magnetism of F$_{2}$PNNNO is that of the spin tetramers of these molecular pairs, which in turn are antiferromagnetically coupled with an exchange constant of 7.4 K [1]. Specific-heat and magnetocaloric-effect measurements reveal a highly symmetric boundary of the ordered phase in the phase diagram, with a lower critical field of $H_{c1}$=9.46 T and an upper critical field of $H_{c2}$=15.37 T. The ordering temperature $T_{c}$ obeys a power law $T_{c }\sim (H-H_{c1})^{\alpha }$ near $H_{c1}$, with the exponent $\alpha $ approaching 2/3 in the low-temperature limit, indicative of a Bose-Einstein condensation (BEC) of $\vert S$, $S_{z}>=\vert $1,1$>$ tetramers. Near the upper critical field $H_{c2}$, where one expects a BEC of singlet $\vert $0,0$>$ tetramers in the ``vacuum'' comprising $\vert $1,1$>$ tetramers, the corresponding power-law exponent remains around 0.4. Remarkably, the temperature dependence of the specific heat indicates that the magnon dispersion is independent of magnetic field between the two critical fields. This work is in collaboration with H. Tsujii, B. Andraka, Y. Hosokoshi, and K. Inoue. [1] Y. Hosokoshi \textit{et al.}, \textit{Phys. Rev. B }\textbf{60}, 12924 (1999).   

Quasiparticle condensation and breakdown in a quantum spin liquid

Piperazinium hexachlorodicuprate (PHCC) is a frustrated bilayer antiferromagnet with a disordered quantum spin-liquid (QSL) ground state at zero field [1] and a diverse magnetic field versus temperature phase diagram which includes two field- induced quantum critical points [2]. The spin excitations in PHCC have a spectral gap of $\Delta \approx 1$~meV above which they follow a nearly 2D-isotropic dispersion with a bandwidth slightly larger than $\Delta$. Field dependent neutron scattering and thermodynamic measurements reveal a lower critical field, $H_{c1}=7.5$~T, separating the QSL phase from a three dimensional spin-ordered state and an upper critical field, $H_{c2}=37$~T, marking the onset of a saturated ferromagnetic phase. The two-dimensional antiferromagnet supports a field induced long range ordered phase well described as a Bose-Einstein condensate (BEC) embedded within a gapless quasi-two-dimensional paramagnetic regime. Inelastic neutron scattering experiments also reveal a peculiar type of hybridization of magnetic excitations in PHCC with their two- particle continuum [3], similar to the post-roton regime in superfluid helium. The excitations at this point become broadened and diffuse, no longer describable as quasiparticles. Although such effects are expected to be strongest in one- dimensional systems with gapped spectra [4], such as Haldane chains, direct observation therein is difficult due to a weak scattering structure factor in the vicinity of the quasiparticle breakdown point [5,6]. The dimer-dominated magnetism in PHCC, on the other hand, is favorable for investigating changes in quasiparticle spectra in the vicinity of their breakdown point. Our results have implications for a variety of condensed matter systems, in particular for other QSLs, where spin excitations have a bandwidth greater than the gap energy. \newline [1] M. B. Stone, {\it et al}. Phys. Rev. B {\bf 64}, 144405 (2001).\newline [2] M. B. Stone, {\it et al}. Phys. Rev. Lett. {\bf 96}, 257203 (2006).\newline [3] M. B. Stone, {\it et al}. Nature, {\bf 440}, 187 (2006).\newline [4] T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press (2005).\newline [5] S. Ma, {\it et al}. Phys. Rev. Lett. {\bf 69}, 3571 (1992).\newline [6] I. A. Zaliznyak, {\it et al}. Phys. Rev. Lett. {\bf 87}, 017202 (2001).