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arXiv:1212.3763 · Physical Review B

Spatially resolved magnetization in the Bose-Einstein condensed state of BaCuSi 2 O 6 : Evidence for imperfect frustration

S. Krämer, Nicolas Laflorencie, Raivo Stern, M. Horvatić, C. Berthier, Hiroyuki Nakamura, T. Kimura, Frédéric Mila

Published May 23, 2013• 17 citations
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Abstract

In order to understand the nature of the two-dimensional Bose-Einstein condensed (BEC) phase in BaCuSi${}_{2}$O${}_{6}$, we performed detailed ${}^{63}$Cu and ${}^{29}$Si NMR above the critical magnetic field, ${H}_{c1}=23.4$ T. The two different alternating layers present in the system have very different local magnetizations close to ${H}_{c1}$; one is very weak, and its size and field dependence are highly sensitive to the nature of inter-layer coupling. Its precise value could only be determined by ``on-site'' ${}^{63}$Cu NMR, and the data are fully reproduced by a model of interacting hard-core bosons in which the perfect frustration associated to tetragonal symmetry is slightly lifted, leading to the conclusion that the population of the less populated layers is not fully incoherent but must be partially condensed.

DOI
10.1103/physrevb.87.180405

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In order to understand the nature of the two-dimensional Bose-Einstein condensed (BEC) phase in BaCuSi2{}_{2}2​O6{}_{6}6​, we performed detailed 63{}^{63}63Cu and 29{}^{29}29Si NMR above the critical magnetic field, Hc1=23.4{H}_{c1}=23.4Hc1​=23.4 T. The two different alternating layers present in the system have very different local magnetizations close to Hc1{H}_{c1}Hc1​; one is very weak, and its size and field dependence are highly sensitive to the nature of inter-layer coupling. Its precise value could only be determined by ``on-site'' 63{}^{63}63Cu NMR, and the data are fully reproduced by a model of interacting hard-core bosons in which the perfect frustration associated to tetragonal symmetry is slightly lifted, leading to the conclusion that the population of the less populated layers is not fully incoherent but must be partially condensed.
We present a 63,65Cu^{63,65}\mathrm{Cu}63,65Cu and 29Si^{29}\mathrm{Si}29Si NMR study of the quasi-2D coupled spin 1/2 dimer compound BaCuSi2O6{\mathrm{BaCuSi}}_{2}{\mathrm{O}}_{6}BaCuSi2​O6​ in the magnetic field range 13−−26T13--26\phantom{\rule{0.3em}{0ex}}\mathrm{T}13−−26T and at temperatures as low as 50mK50\phantom{\rule{0.3em}{0ex}}\mathrm{mK}50mK. NMR data in the gapped phase reveal that below 90K90\phantom{\rule{0.3em}{0ex}}\mathrm{K}90K different intradimer exchange couplings and different gaps $({\ensuremath{\Delta}}_{\mathrm{B}}∕{\ensuremath{\Delta}}_{\mathrm{A}}=1.16)$ exist in every second plane along the ccc axis, in addition to a planar incommensurate (IC) modulation. 29Si^{29}\mathrm{Si}29Si spectra in the field induced magnetic ordered phase reveal that close to the quantum critical point at Hc1=23.35T{H}_{\mathrm{c}1}=23.35\phantom{\rule{0.3em}{0ex}}\mathrm{T}Hc1​=23.35T the average boson density n‾\overline{n}n of the Bose-Einstein condensate is strongly modulated along the ccc axis with a density ratio for every second plane ${\overline{n}}_{\mathrm{A}}∕{\overline{n}}_{\mathrm{B}}\ensuremath{\simeq}5$. An IC modulation of the local density is also present in each plane.
Building on recent neutron and NMR experiments, we investigate the field-induced exotic criticality observed in the frustrated compound BaCuSi2O6 using a frustrated model with two types of bilayers. A semiclassical treatment of the effective hard-core boson model shows that perfect interlayer frustration leads to a 2D-like critical exponent varphi=1 without logarithmic corrections and to a 3D low temperature phase with different but nonvanishing triplet populations in both types of bilayers. These further suggest a simple phenomenology in terms of a field-dependent transverse coupling in the context of which we reproduce the entire field-temperature phase diagram with quantum Monte Carlo simulations.
We synthesized single crystals of composition Ba2CuSi2O6Cl2{\mathrm{Ba}}_{2}{\mathrm{CuSi}}_{2}{\mathrm{O}}_{6}{\mathrm{Cl}}_{2}Ba2​CuSi2​O6​Cl2​ and investigated their quantum magnetic properties. The crystal structure is closely related to that of the quasi-two-dimensional (2D) dimerized magnet BaCuSi2O6{\mathrm{BaCuSi}}_{2}{\mathrm{O}}_{6}BaCuSi2​O6​ also known as Han purple. Ba2CuSi2O6Cl2{\mathrm{Ba}}_{2}{\mathrm{CuSi}}_{2}{\mathrm{O}}_{6}{\mathrm{Cl}}_{2}Ba2​CuSi2​O6​Cl2​ has a singlet ground state with an excitation gap of $\mathrm{\ensuremath{\Delta}}/{k}_{\mathrm{B}}=20.8$ K. The magnetization curves for two different field directions almost perfectly coincide when normalized by the ggg factor except for a small jump anomaly for a magnetic field perpendicular to the ccc axis. The magnetization curve with a nonlinear slope above the critical field is in excellent agreement with exact-diagonalization calculations based on a 2D coupled spin-dimer model. Individual exchange constants are also evaluated using density functional theory (DFT). The DFT results demonstrate a 2D exchange network and weak frustration between interdimer exchange interactions, supported by weak spin-lattice coupling implied from our magnetostriction data. The magnetic-field-induced spin ordering in Ba2CuSi2O6Cl2{\mathrm{Ba}}_{2}{\mathrm{CuSi}}_{2}{\mathrm{O}}_{6}{\mathrm{Cl}}_{2}Ba2​CuSi2​O6​Cl2​ is described as the quasi-2D Bose-Einstein condensation of triplets.
Quantum phase transition at the saturation field is studied for a class of frustrated quantum antiferromagnets. The considered models include (i) the ${J}_{1}\mathrm{\text{\ensuremath{-}}}{J}_{2}$ frustrated square-lattice antiferromagnet with J2=12J1{J}_{2}=\frac{1}{2}{J}_{1}J2​=21​J1​ and (ii) the nearest-neighbor Heisenberg antiferromagnet on a face centered cubic lattice. In the fully saturated phase the magnon spectra for the two models have lines of degenerate minima. Transition into a partially magnetized state is treated via a mapping to a dilute gas of hard-core bosons and by complementary spin-wave calculations. Momentum dependence of the exact four-point boson vertex removes the degeneracy of the single-particle excitation spectra and selects the ordering wave vectors at $(\ensuremath{\pi},\ensuremath{\pi})$ and $(\ensuremath{\pi},0,0)$ for the two models. We predict a unique form for the magnetization curve $\ensuremath{\Delta}M=S\ensuremath{-}M\ensuremath{\simeq}{\ensuremath{\mu}}^{(d\ensuremath{-}1)/2}(\mathrm{log}\ensuremath{\mu}{)}^{(d\ensuremath{-}1)}$, where $\ensuremath{\mu}$ is a distance from the quantum critical point.
Bose Einstein Condensation of Magnons in BaCuSi2_{2}2​O6_{6}6​: An experimental perspective
Han Purple, a pigment first obtained in ancient China, is one of the earliest known synthetic pigments. Also naturally occurring, as a mineral, it is known as colinowensite (Cwn). Its chemical formula is BaCuSi2_{2}2​O6_{6}6​, and its structure is a layered cyclosilicate in which magnetic Cu2+^{2+}2+ ions (S = 1/2) form dimers arranged on a square lattice, making it also the first known synthetic metal dimer compound. Most interesting magnetic properties arise from a strong intradimer spin coupling, accompanied by weaker interdimer interactions within and between Cu-dimer layers. In zero or small magnetic fields, BaCuSi2_{2}2​O6_{6}6​ remains a quantum paramagnet. However, under high magnetic fields between 23 and 49 Tesla -- about a million times stronger than Earth's magnetic field -- it undergoes magnetic ordering at subliquid Helium temperatures into an almost ideal easy-plane (XY) antiferromagnetic state regarded as a realization of a Bose-Einstein condensate of magnons. Within this experimentally accessible field range, BaCuSi2_{2}2​O6_{6}6​ serves as an extraordinary playground for testing predictions of quantum many-body physics.
We study Bose-Einstein condensation (BEC) in three-dimensional two-component bosonic gases, characterizing the universal behaviors of the critical modes arising at the BEC transitions. For this purpose, we use field-theoretical (FT) renormalization-group (RG) methods and perform mean-field and numerical calculations. The FT RG analysis is based on the Landau-Ginzburg-Wilson ${\mathrm{\ensuremath{\Phi}}}^{4}$ theory with two complex scalar fields which has the same symmetry as the bosonic system. In particular, for identical bosons with exchange Z2{\mathbb{Z}}_{2}Z2​ symmetry, coupled by effective density-density interactions, the global symmetry is ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\bigotimes}\mathrm{U}(1)$. At the BEC transition, it may break into ${\mathbb{Z}}_{2,e}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when both components condense simultaneously, or to $\mathrm{U}(1)\ensuremath{\bigotimes}{\mathbb{Z}}_{2}$ when only one component condenses. This implies different universality classes for the corresponding critical behaviors. Numerical simulations of the two-component Bose-Hubbard model in the hard-core limit support the RG prediction: when both components condense simultaneously, the critical behavior is controlled by a decoupled XY fixed point, with unusual slowly decaying scaling corrections arising from the onsite interspecies interaction.