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arXiv:cond-mat/0404324 · Physical Review Letters

Magnetic-Field-Induced Condensation of Triplons in Han Purple Pigment B a C u S i 2 O 6

Marcelo Jaime, Víctor Félix Correa, N. Harrison, Cristian D. Batista, Naoki Kawashima, Yasuo Kazuma, G. A. Jorge, Raivo Stern, Ivo Heinmaa, Sergei A. Zvyagin, Y. Sasago, K. Uchinokura

Published Aug 18, 2004• 292 citations
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Abstract

Besides being an ancient pigment, ${\mathrm{B}\mathrm{a}\mathrm{C}\mathrm{u}\mathrm{S}\mathrm{i}}_{2}{\mathrm{O}}_{6}$ is a quasi-2D magnetic insulator with a gapped spin dimer ground state. The application of strong magnetic fields closes this gap, creating a gas of bosonic spin triplet excitations. The topology of the spin lattice makes ${\mathrm{B}\mathrm{a}\mathrm{C}\mathrm{u}\mathrm{S}\mathrm{i}}_{2}{\mathrm{O}}_{6}$ an ideal candidate for studying the Bose-Einstein condensation of triplet excitations as a function of the external magnetic field, which acts as a chemical potential. In agreement with quantum Monte Carlo numerical simulations, we observe a distinct lambda anomaly in the specific heat together with a maximum in the magnetic susceptibility upon cooling down to liquid helium temperatures.

DOI
10.1103/physrevlett.93.087203

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Besides being an ancient pigment, BaCuSi2O6{\mathrm{B}\mathrm{a}\mathrm{C}\mathrm{u}\mathrm{S}\mathrm{i}}_{2}{\mathrm{O}}_{6}BaCuSi2​O6​ is a quasi-2D magnetic insulator with a gapped spin dimer ground state. The application of strong magnetic fields closes this gap, creating a gas of bosonic spin triplet excitations. The topology of the spin lattice makes BaCuSi2O6{\mathrm{B}\mathrm{a}\mathrm{C}\mathrm{u}\mathrm{S}\mathrm{i}}_{2}{\mathrm{O}}_{6}BaCuSi2​O6​ an ideal candidate for studying the Bose-Einstein condensation of triplet excitations as a function of the external magnetic field, which acts as a chemical potential. In agreement with quantum Monte Carlo numerical simulations, we observe a distinct lambda anomaly in the specific heat together with a maximum in the magnetic susceptibility upon cooling down to liquid helium temperatures.
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 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.
Sufficiently dimerized quantum antiferromagnets display elementary S =1 excitations, triplon quasiparticles, protected by a gap at low energies. At higher energies, the triplons may decay into two or more triplons. A strong enough magnetic field induces Bose-Einstein condensation of triplons. For both phenomena the compound IPA-CuCl 3 is an excellent model system. Nevertheless no quantitative model was determined so far despite numerous studies. Recent theoretical progress allows us to analyse data of inelastic neutron scattering (INS) and of magnetic susceptibility to determine the four magnetic couplings J 1 ≈− 2.3 meV, J 2 ≈1.2 meV, J 3 ≈2.9 meV and J 4 ≈− 0.3 meV. These couplings determine IPA-CuCl 3 as system of coupled asymmetric S =1/2 Heisenberg ladders quantitatively. The magnetic field dependence of the lowest modes in the condensed phase as well as the temperature dependence of the gap without magnetic field corroborate this microscopic model.
Motivated by recent experiments on Ba3Cr2O8{\text{Ba}}_{3}{\text{Cr}}_{2}{\text{O}}_{8}Ba3​Cr2​O8​, a new spin-dimer compound with spin-1/2 moments of Cr5+{\text{Cr}}^{5+}Cr5+ ions, we theoretically investigate the field-induced magnetic ordering in this material in view of the Bose-Einstein condensation (BEC) of triplet excitations (triplons). We apply the self-consistent Hartree-Fock-Popov (HFP) approach to a microscopic Hamiltonian, using the realistic triplon dispersion measured in an inelastic neutron-scattering experiment. In particular, we ask to what extent the BEC of dilute triplons near the critical field can explain the magnetic ordering in this material. For example, we investigate the temperature range where the BEC picture of triplons can be applied via the HFP approach. We also determine the temperature regime where a quadratic approximation of the triplon dispersion works. It is found that the strength of the effective repulsive interaction between triplons is much weaker in Ba3Cr2O8{\text{Ba}}_{3}{\text{Cr}}_{2}{\text{O}}_{8}Ba3​Cr2​O8​ than in the canonical spin-dimer compound TlCuCl3{\text{TlCuCl}}_{3}TlCuCl3​. Small effective repulsive interaction in combination with the narrow band of triplons leads to higher density of triplons ncr{n}_{\text{cr}}ncr​ at the critical point. The combined effect points to a bigger HFP correction UncrU{n}_{\text{cr}}Uncr​ in Ba3Cr2O8{\text{Ba}}_{3}{\text{Cr}}_{2}{\text{O}}_{8}Ba3​Cr2​O8​ than in TlCuCl3{\text{TlCuCl}}_{3}TlCuCl3​. Nonetheless, the HFP approach provides a reasonable explanation of the transverse magnetization and the specific-heat data of Ba3Cr2O8{\text{Ba}}_{3}{\text{Cr}}_{2}{\text{O}}_{8}Ba3​Cr2​O8​.
Structural and magnetic properties of a quasi-one-dimensional spin-12\frac{1}{2}21​ compound NaVOPO4{\mathrm{NaVOPO}}_{4}NaVOPO4​ are explored by x-ray diffraction, magnetic susceptibility, high-field magnetization, specific heat, electron spin resonance, and 31P^{31}\mathrm{P}31P nuclear magnetic resonance measurements, as well as complementary ab initio calculations. Whereas magnetic susceptibility of NaVOPO4{\mathrm{NaVOPO}}_{4}NaVOPO4​ may be compatible with the gapless uniform spin chain model, detailed examination of the crystal structure reveals a weak alternation of the exchange couplings with the alternation ratio $\ensuremath{\alpha}\ensuremath{\simeq}0.98$ and the ensuing zero-field spin gap ${\mathrm{\ensuremath{\Delta}}}_{0}/{k}_{\mathrm{B}}\ensuremath{\simeq}2.4\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ directly probed by field-dependent magnetization measurements. No long-range order is observed down to 50 mK in zero field. However, applied fields above the critical field ${H}_{c1}\ensuremath{\simeq}1.6\phantom{\rule{0.28em}{0ex}}\mathrm{T}$ give rise to a magnetic ordering transition with the phase boundary ${T}_{N}\ensuremath{\propto}{(H\ensuremath{-}{H}_{c1})}^{\frac{1}{\ensuremath{\phi}}}$, where $\ensuremath{\phi}\ensuremath{\simeq}1.8$ is close to the value expected for Bose-Einstein condensation of triplons. With its weak alternation of the exchange couplings and small spin gap, NaVOPO4{\mathrm{NaVOPO}}_{4}NaVOPO4​ lies close to the quantum critical point.
Dimerized quantum magnets provide a unique possibility to investigate the Bose-Einstein condensation of magnetic excitations in crystalline systems at low temperature. Here, we model the low-temperature magnetic properties of the recently synthesized spin S=1S=1S=1 dimer system K2Ni(MoO4)2{\mathrm{K}}_{2}\mathrm{Ni}{({\mathrm{MoO}}_{4})}_{2}K2​Ni(MoO4​)2​ and propose it as a candidate material for triplon and quintuplon condensation. Based on a first-principles analysis of its electronic structure, we derive an effective spin dimer model that we first solve within a mean-field approximation to refine its parameters in comparison to experiment. Finally, the model is solved by employing a numerically exact quantum Monte Carlo technique which leads to magnetic properties in good agreement with experimental magnetization and thermodynamic results. We discuss the emergent spin model of K2Ni(MoO4)2{\mathrm{K}}_{2}\mathrm{Ni}{({\mathrm{MoO}}_{4})}_{2}K2​Ni(MoO4​)2​ in view of the condensation of magnetic excitations in a broad parameter regime. Finally, we comment on a geometrical peculiarity of the proposed model and discuss how it could host a supersolid phase upon structural distortions.