arXiv:2607.06028

Bose Einstein Condensation of Magnons in BaCuSi$_{2}$O$_{6}$: An experimental perspective

Marcelo Jaime, Franziska Weickert

Published Jul 7, 2026• 0 citations

Abstract

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 BaCuSi$_{2}$O$_{6}$, and its structure is a layered cyclosilicate in which magnetic Cu$^{2+}$ 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, BaCuSi$_{2}$O$_{6}$ 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, BaCuSi$_{2}$O$_{6}$ serves as an extraordinary playground for testing predictions of quantum many-body physics.

DOI: 10.48550/arxiv.2607.06028

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arXiv:2607.06028

Bose Einstein Condensation of Magnons in BaCuSi $_{2}$ O $_{6}$ : An experimental perspectiveBose Einstein Condensation of Magnons in BaCuSi$_{2}$O$_{6}$: An experimental perspective

Marcelo Jaime, Franziska Weickert

Published Jul 7, 2026• 0 citations

arXiv HTML PDF

Abstract

_{2}

_{6}

, and its structure is a layered cyclosilicate in which magnetic Cu

^{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, BaCuSi

_{2}

_{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, BaCuSi

_{2}

_{6}

serves as an extraordinary playground for testing predictions of quantum many-body physics.

DOI: 10.48550/arxiv.2607.06028

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S. Krämer, Raivo Stern, M. Horvatić +3 more

We present a $^{63,65}\mathrm{Cu}$ and $^{29}\mathrm{Si}$ NMR study of the quasi-2D coupled spin 1/2 dimer compound ${\mathrm{BaCuSi}}_{2}{\mathrm{O}}_{6}$ in the magnetic field range $13--26\phantom{\rule{0.3em}{0ex}}\mathrm{T}$ and at temperatures as low as $50\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$ . NMR data in the gapped phase reveal that below $90\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ different intradimer exchange couplings and different gaps $({\ensuremath{\Delta}}_{\mathrm{B}}∕{\ensuremath{\Delta}}_{\mathrm{A}}=1.16)$ exist in every second plane along the $c$ axis, in addition to a planar incommensurate (IC) modulation. $^{29}\mathrm{Si}$ spectra in the field induced magnetic ordered phase reveal that close to the quantum critical point at ${H}_{\mathrm{c}1}=23.35\phantom{\rule{0.3em}{0ex}}\mathrm{T}$ the average boson density $\overline{n}$ of the Bose-Einstein condensate is strongly modulated along the $c$ 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.We present a $^{63,65}\mathrm{Cu}$ and $^{29}\mathrm{Si}$ NMR study of the quasi-2D coupled spin 1/2 dimer compound ${\mathrm{BaCuSi}}_{2}{\mathrm{O}}_{6}$ in the magnetic field range $13--26\phantom{\rule{0.3em}{0ex}}\mathrm{T}$ and at temperatures as low as $50\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$. NMR data in the gapped phase reveal that below $90\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ different intradimer exchange couplings and different gaps $({\ensuremath{\Delta}}_{\mathrm{B}}∕{\ensuremath{\Delta}}_{\mathrm{A}}=1.16)$ exist in every second plane along the $c$ axis, in addition to a planar incommensurate (IC) modulation. $^{29}\mathrm{Si}$ spectra in the field induced magnetic ordered phase reveal that close to the quantum critical point at ${H}_{\mathrm{c}1}=23.35\phantom{\rule{0.3em}{0ex}}\mathrm{T}$ the average boson density $\overline{n}$ of the Bose-Einstein condensate is strongly modulated along the $c$ 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.

arXiv:0704.0888Physical Review BSep 18, 200733 citations

Bose Einstein Condensation of Magnons in BaCuSi

_{2}

_{6}

: An experimental perspective

_{2}

_{6}

, and its structure is a layered cyclosilicate in which magnetic Cu

^{2+}

_{2}

_{6}

_{2}

_{6}

serves as an extraordinary playground for testing predictions of quantum many-body physics.

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.

We synthesized single crystals of composition

{\mathrm{Ba}}_{2}{\mathrm{CuSi}}_{2}{\mathrm{O}}_{6}{\mathrm{Cl}}_{2}

and investigated their quantum magnetic properties. The crystal structure is closely related to that of the quasi-two-dimensional (2D) dimerized magnet

{\mathrm{BaCuSi}}_{2}{\mathrm{O}}_{6}

also known as Han purple.

{\mathrm{Ba}}_{2}{\mathrm{CuSi}}_{2}{\mathrm{O}}_{6}{\mathrm{Cl}}_{2}

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

g

factor except for a small jump anomaly for a magnetic field perpendicular to the

c

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

{\mathrm{Ba}}_{2}{\mathrm{CuSi}}_{2}{\mathrm{O}}_{6}{\mathrm{Cl}}_{2}

is described as the quasi-2D Bose-Einstein condensation of triplets.

The material BaCuSi

{}_{2}

{}_{6}

has a fascinating dual status as an ancient pigment (the original Terracotta Army was colored with Han purple) and as a quasi-two-dimensional quantum magnet at the modern frontier of research into unconventional quantum many-body phenomena. The HFM/EXED facility at the Helmholtz-Zentrum Berlin neutron source allowed the authors to investigate the field-driven quantum phase transition to magnetic order above 23 T by neutron diffraction and spectroscopy. They report their search for fingerprints of the exotic critical behavior predicted to appear in the two-dimensional regime.

In order to understand the nature of the two-dimensional Bose-Einstein condensed (BEC) phase in BaCuSi

{}_{2}

{}_{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.

We present a

^{63,65}\mathrm{Cu}

and

^{29}\mathrm{Si}

NMR study of the quasi-2D coupled spin 1/2 dimer compound

{\mathrm{BaCuSi}}_{2}{\mathrm{O}}_{6}

in the magnetic field range

13--26\phantom{\rule{0.3em}{0ex}}\mathrm{T}

and at temperatures as low as

50\phantom{\rule{0.3em}{0ex}}\mathrm{mK}

. NMR data in the gapped phase reveal that below

90\phantom{\rule{0.3em}{0ex}}\mathrm{K}

different intradimer exchange couplings and different gaps $({\ensuremath{\Delta}}_{\mathrm{B}}∕{\ensuremath{\Delta}}_{\mathrm{A}}=1.16)$ exist in every second plane along the

c

axis, in addition to a planar incommensurate (IC) modulation.

^{29}\mathrm{Si}

spectra in the field induced magnetic ordered phase reveal that close to the quantum critical point at

{H}_{\mathrm{c}1}=23.35\phantom{\rule{0.3em}{0ex}}\mathrm{T}

the average boson density

\overline{n}

of the Bose-Einstein condensate is strongly modulated along the

c

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.