NMR evidence for a strong modulation of the Bose-Einstein Condensate in
  BaCuSi$_2$O$_6$
NMR evidence for a strong modulation of the Bose-Einstein Condensate in BaCuSi2O6
S. Krämer,1 R. Stern,2 M. Horvatić,1 C. Berthier,1 T. Kimura,3 and I. R. Fisher4
1Grenoble High Magnetic Field Laboratory (GHMFL) - CNRS, BP 166, 38042 Grenoble Cedex 09, France
2National Institute of Chemical Physics and Biophysics, 12618,Tallinn, Estonia
3Los Alamos National Laboratory, Los Alamos NM 87545, USA
4Geballe Laboratory for Advanced Materials and Department of Applied Physics, Stanford University, Stanford CA 94305, USA
(Dated: October 30, 2018)
We present a 63,65Cu and 29Si NMR study of the quasi-2D coupled spin 1/2 dimer compound
BaCuSi2O6 in the magnetic field range 13-26 T and at temperatures as low as 50 mK. NMR data in
the gapped phase reveal that below 90 K different intra-dimer exchange couplings and different gaps
(∆B/∆A = 1.16) exist in every second plane along the c-axis, in addition to a planar incommensurate
(IC) modulation. 29Si spectra in the field induced magnetic ordered phase reveal that close to the
quantum critical point at Hc1 = 23.35 T the average boson density n of the Bose-Einstein condensate
is strongly modulated along the c-axis with a density ratio for every second plane nA/nB ≃ 5. An
IC modulation of the local density is also present in each plane. This adds new constraints for the
understanding of the 2D value φ = 1 of the critical exponent describing the phase boundary.
PACS numbers: 75.10.Jm,75.40.Cx,75.30.Gw
The interest in Bose-Einstein condensation (BEC) has
been considerably renewed since it was shown to occur
in cold atomic gases [1]. In condensed matter, a formal
analog of the BEC can also be obtained in antiferromag-
netic (AF) quantum spin systems [2, 3, 4, 5] under an
applied magnetic field. Many of these systems have a
collective singlet ground state, separated by an energy
gap ∆ from a band of triplet excitations. Applying a
magnetic field (H) lowers the energy of the Mz = −1
sub-band and leads to a quantum phase transition be-
tween a gapped non magnetic phase and a field induced
magnetic ordered (FIMO) phase at the critical field Hc1
corresponding to ∆min-gµBHc1 = 0, where ∆min is the
minimum gap value corresponding to some q vector qmin
[2, 3, 4, 5]. This phase transition can be described as a
BEC of hard core bosons for which the field plays the role
of the chemical potential, provided the U(1) symmetry is
conserved. Quite often, however, anisotropic interactions
can change the universality class of the transition and
open a gap [6, 7, 8]. From that point of view, BaCuSi2O6
[9] seems at the moment the most promising candidate
for the observation of a true BEC quantum critical point
(QCP) [10]. In addition, this system exhibits an un-
usual dimensionality reduction at the QCP, which was
attributed to frustration between adjacent planes in the
nominally body-centered tetragonal structure [11]. The
material also exhibits a weak orthorhombic distortion at
≃90 K which is accompanied by an in-plane IC lattice
modulation [12]. This structural phase transition affects
the triplon dispersion, and the possibility of a modula-
tion of the amplitude of the BEC along the c-axis has
been speculated based on low field inelastic neutron data
[13].
In order to get a microscopic insight of this system, we
performed 29Si and 63,65Cu NMR in BaCuSi2O6 single
crystals. Our data in the gapped phase reveal that the
structural phase transition which occurs around 90 K not
only introduces an IC distortion within the planes, but
also leads to the existence of two types of planes alter-
nating along the c-axis. From one plane to the other, the
intra-dimer exchange coupling and the energy gap for the
triplet states differs by 16 %. Exploring the vicinity of
the QCP in the temperature (T ) range 50-720 mK, we
confirm the linear dependence of TBEC with H −Hc1 as
expected for a 2D BEC. Our main finding is that the av-
erage boson density n in the BEC is strongly modulated
along the c-axis in a ratio of the order of 1:5 for every sec-
ond plane, whereas its local value n(R) is IC modulated
within each plane.
NMR measurements have been obtained on ∼10 mg
single crystals of BaCuSi2O6 using a home-made spec-
trometer and applying an external magnetic fieldH along
the c axis. The gapped phase was studied using a su-
perconducting magnet in the field range 13-15 T and
the temperature range 3-100 K. The investigation of the
FIMO phase was conducted in a 20 MW resistive magnet
at the GHMFL in the field range 22-25 T and the tem-
perature range 50-720 mK. Except for a few field sweeps
in the gapped phase, the spectra were obtained at fixed
fields by sweeping the frequency in regular steps and sum-
ming the Fourier transforms of the recorded echoes.
Before discussing the microscopic nature of the QCP,
let us first consider the NMR data in the gapped phase.
The system consists of S = 1/2 Cu spin dimers parallel
to the c axis and arranged (at room temperature) on a
square lattice in the ab plane. Each Cu dimer is sur-
rounded by four Si atoms, lying approximately in the
equatorial plane. For Cu nuclei, the interaction with the
electronic spins is dominated by the on-site hyperfine in-
teraction. For 29Si nuclei both the transferred hyperfine
interaction through oxygen atoms with a single dimer and
the direct dipolar interaction are important. According
http://arxiv.org/abs/0704.0888v1
125.0 125.2 125.4 125.6 125.8
0 20 40 60 80 100
 [ MHz]
T [ K ]
29Si NMR
H = 14.79 T
H || c-axis
I x 0.5
I x 0.25
T [ K ]
FIG. 1: (Color online) Evolution of the normalized 29Si NMR
spectra as a function of T in the gapped phase. Below 90 K
the line splits into two components, each of them correspond-
ing to an IC pattern. Inset: T dependence of the 1st moment
(i.e., the average position) for i) the total spectra (squares)
and ii) the individual components before they overlap (up
and down triangles). The solid and dashed lines are fits for
non-interacting dimers.
to the room temperature structure I41/acd [14], there
should be only one single Cu and two nearly equivalent
Si sites for NMR when H‖c. As far as 29Si is concerned,
one actually observes a single line above 90 K, as can be
seen in Fig. 1. However, below 90 K, the line splits into
two components, each of them corresponding to an IC
pattern, that is an infinite number of inequivalent sites.
This corresponds to the IC structural phase transition
discovered by X-ray measurements [12]. At 3 K, when
T is much smaller than the gap, the spin polarization is
zero and one observes again a single unshifted line, at the
frequency ν = ν0 =
29γH defined by the Si gyromagnetic
ratio 29γ.
On the 63,65Cu NMR spectra recorded at 3 K and
13.2 T (Fig. 2), however, one can distinguish two dif-
ferent Cu sites, denoted A and B. That is, each of the
6 lines of Cu spectrum (for 2 copper isotopes × 3 tran-
sitions of a spin 3/2 nucleus) is split into two, which
is particularly obvious on the lowest frequency “satel-
lite” 63Cu line. The whole spectra can be nicely fitted
with the following parameters: 63ν
Q = 14.85 (14.14)
MHz, η = 0, and K
zz = 1.80 (1.93) %, where νQ is the
quadrupolar frequency and η the asymmetry parameter.
The Kzz is the hyperfine shift, expected to be purely
orbital since the susceptibility has fully vanished. On in-
creasing T the highest frequency 65Cu “satellite” lines of
sites A and B become well separated and both exhibit
a line shape typical of an IC modulation of the nuclear
spin-Hamiltonian. Although the apparent intensities of
lines A and B look different, they correspond to the same
number of nuclei after corrections due to different spin-
spin relaxation rate 1/T2. Since the satellite NMR lines
135 140 145 150 155 160 165 170 175
 = 3.7 meV
 = 4.3 meV
63,65Cu NMR
 [ MHz ]
H = 13.205 T
15.0 14.5 14.0 13.5 13.0
upper sat.
T = 8.9 K
167 MHz
H [ T ]
FIG. 2: (Color online) 63,65Cu NMR spectra of BaCuSi2O6
in the gapped phase, well below the critical field. The T de-
pendence of the high-frequency “satellite” line clearly reveals
two different copper sites. From their shifts, the two corre-
sponding gap values have been determined. Inset: field sweep
spectrum that reveals the IC nature of the line shape for each
of the two sites. Shading separates the contribution of the
65Cu high-frequency satellite from the rest of the spectrum.
The analysis of the latter part confirms that the observed line
shape has a pure magnetic origin.
at 3 K (the lowest temperature) are narrow, the modu-
lation of νQ is negligible, meaning that the IC lineshapes
visible at higher temperature are purely magnetic. This
is confirmed by the analysis of the spectrum shown in the
inset of Fig. 2, which shows that at 8.9 K the broadening
of the “central” line is the same as that observed on the
“satellites”. Such a broadening results from a distribu-
tion of local hyperfine fields: δhz(R) = Azz(R)mz(R) in
which A(R) is the hyperfine coupling tensor and mz(R)
the longitudinal magnetization at site R. Since νQ(R) is
not modulated by the distortion, one expects that the
modulation of A(R) is negligible too, A(R) =A. This
means that the NMR lineshape directly reflects the IC
modulation of mz in the plane.
Keeping constant the νQ parameters obtained at 3 K,
one can analyze the T dependence of the shift Kαzz(T ) of
each component α = A or B according to the formula
Kαzz(T )−Kαzz(0) = Aαzzmdz(∆α, H, T )/H, (1)
where mdz is the magnetization of a non-interacting
dimer, mdz = gcµB/(e
(∆α−gcµBH)/kBT + 1) in the given
T range, gc = 2.3 [15], and K
zz is determined from the
average line position, i.e., the first moment. The best fit
was obtained for ∆A(B) = 3.7 (4.3) meV and A
-16.4 T/µB. We assumed that A
cc = A
cc, but the values
of ∆ depend only weakly with this quantity. The values
are slightly higher than those determined by neutron in-
elastic scattering for Qmin = [π, π] [13], which is normal
considering our approximate description. However, the
ratio ∆B/∆A = 1.16 is in excellent agreement with the
neutron result 1.15. Considering the fact that there is
FIG. 3: (Color online) Evolution of the normalized 29Si spec-
trum as a function of H at fixed T . The colored spectra
correspond to the BEC. a) T = 50 mK: Instead of a simple
splitting of the line as expected for a standard BEC, a com-
plex pattern appears, typical of an IC distribution of the local
hyperfine field. Inset: H dependence of the 1st (M1, squares)
and 2nd (M2, circles) moment of the spectra. M1 is propor-
tional to mz and M2 to the square of the order parameter. b)
T = 720 mK: The non zero magnetization outside the BEC
leads to an IC pattern for fields H ≤ Hc1(T ), where Hc1(T )
is determined from the H dependence of M2, as shown in the
inset. Lower inset: Tc is linear in H −Hc1, as expected for a
2D BEC QCP.
no disorder in the system (as Cu lines at low T are nar-
row), and that X-rays did not detect any commensurate
peak corresponding to a doubling of the unit cell in the
ab plane, our NMR data can only be explained if there
are two types of planes with different gap values. Look-
ing back at the 29Si spectra in Fig. 1, one also observes
just below 90 K two well separated components, both of
them exhibiting an IC pattern. They indeed correspond
to the two types of planes, as the T dependence of their
positions can be well fit using values close to ∆A and
∆B determined from Cu NMR (inset to Fig. 1). This
means that the 90 K structural phase transition not only
corresponds to the onset of an IC distortion in the ab
plane, but also leads simultaneously to an alternation of
different planes along the c-axis [16], with intra-dimer
exchange in the ratio JB/JA ∼= ∆B/∆A = 1.16
Let us now recall what is expected from a microscopic
point of view in the vicinity of the QCP corresponding to
the onset of a homogeneous BEC for coupled dimer sys-
tems. As soon as a finite density of bosons n is present
(H > Hc1 = ∆min/gµB), a transverse staggered magne-
tization m⊥ (⊥ to H) appears. Its amplitude and direc-
tion correspond respectively to the amplitude and phase
of the order parameter. At the same time, the longitu-
dinal magnetization mz is proportional to the number of
bosons at a given temperature and field, this latter play-
ing the role of the chemical potential. Due to the appear-
ance of a static m⊥, the degeneracy between sites which
were equivalent outside the condensate will be lifted and
their corresponding NMR lines will be split into two. To
be more specific, we consider a pair of Si sites situated
in the ab plane on opposite sides of a Cu dimer. Outside
the condensate, and in the absence of the IC modula-
tion, they should give a single line for H ‖ c. Inside
the condensate the NMR lines of this pair of Si sites will
split by ±29γ|Az⊥|m⊥ because their Az⊥ couplings are
of opposite sign. Obviously, observing a splitting of lines
requires the existence of off-diagonal terms in the hyper-
fine tensor. Such terms are always present due to the
direct dipole interaction between an electronic and the
nuclear spin, which can be easily calculated.
Instead of this expected simple line splitting, the spec-
tra of Fig. 3a reveal a quite complex modification of the
line-shape when entering the condensate. The narrow
single line, observed at 23.41 T at the frequency ν0,
which corresponds to a negligible boson density, sud-
denly changes into a composite line-shape including a
narrow and a broad component. The spread-out of
the broad component increases very quickly with the
field. The width of the narrow component also in-
creases, but at a much lower rate. Both peculiar broad-
enings are related to the IC modulation of the boson
density n(R) due to the structural modulation. To be
more precise, a copper dimer at position R has in to-
tal 4 Si atoms (denoted by k = 0,1,2,3) situated around
in a nearly symmetrical square coordination. The ab-
solute values of the corresponding hyperfine couplings
will thus be nearly identical, and we will also neglect
their dependence on R. These 4 Si sites will give rise
to four NMR lines at the frequencies 29νk(R) = ν0 +
ν1(R) + ν2,k(R), where ν1(R) =
29γAzzgµBn(R) and
ν2,k(R) =
29γAz⊥m⊥(R) cos(φ − kπ/2). Note that ν2,k
only exists when the bosons are condensed, that is when
there is a transverse magnetizationm⊥ pointing in the di-
rection φ. In a uniform condensate m⊥ is proportional to√
n near the QCP, since the mean field behavior is valid
in both, 2D and 3D. We assume that only the amplitude
of the order parameter is spatially modulated, and that
m⊥(R) ∝
n(R). The line shape is the histogram of the
distribution of 29νk(R), convoluted by some broadening
due to nuclei – nuclei interaction.
Three quantities can be derived from the analysis of
NMR lines at fixed T values and variable H : the average
boson density n(H,T ), the field Hc1(T ) corresponding to
to the BEC phase boundary, and the field dependence of
the BEC order parameter (for T close to zero). The aver-
age number of bosons n per dimer is directly proportional
to the first moment M1 (i.e., the average position) of the
line: M1 =
(ν − ν0)f(ν)dν = 29γAzzgµBn(H,T ),
where the line shape f(ν) is supposed to be normalized.
The second moment (i.e., the square of the width) of the
line M2 =
(ν − ν0 − M1)2f(ν)dν has two origins:
the broadening due to the IC distribution of (n(R)-n),
and that due to the onset of m⊥ ∝
n(R) in the con-
densate. When increasing H at T ≃ 0, the condensation
occurs as soon as bosons populate the dimer plane. This
is observed in the inset of Fig. 3a at T = 50 mK. Both
M1 (n) and M2 (m⊥) vary linearly with the field and the
extrapolation of M2 to zero allows the determination of
Hc1 at 50 mK. For higher temperatures a thermal pop-
ulation of bosons n exists and increases with H before
entering the BEC phase. As a result both M1 and M2
increase non-linearly with H , as shown in the upper in-
set of Fig. 3b. However, the increase of M2(H) shows
two clearly separated regimes and allows the determina-
tion of Hc1(T ) as the point where the rate of change of
M2(H) strongly increases due to the appearance of m⊥.
Applying this criterion to all temperatures, we were able
to determine the field dependence of TBEC (lower inset of
Fig. 3b) and define precisely the QCP at Hc1 = 23.35 T.
In agreement with the torque measurements [11], we find
a linear field dependence. This is the signature of a 2D
BEC QCP, where Tc ∝ (H − Hc1)φ with φ = 2/d and
d = 2 [17].
This analysis, however, does not take into account the
specificity of the line shapes, which are related to the ex-
istence of two types of planes with different energy gaps.
A careful examination of the spectra clearly reveals that
they correspond to the superposition of two lines exhibit-
ing different field dependence at fixed T value. For sake
of simplicity, we have made a decomposition only for the
spectra at 50 mK, as shown in the inset to Fig. 4. Clearly,
one of the components remains relatively narrow with-
out any splitting, whereas the other immediately heav-
ily broadens in some sort of triangular line shape. The
field dependence of M1 of the two components, shown in
Fig. 4, reveals that they differ by a factor of 5. This is
attributed to the difference by a factor of 5 in the cor-
responding average populations of bosons. If there were
no hopping of bosons between A and B planes, the B
planes should be empty for the range of field such that
∆A < gµBH < ∆B. Although the observed density of
boson is finite in the B planes, it is strongly reduced,
giving rise to a strong commensurate modulation of n
along the c-axis. According to [11], the hopping along
the c-axis of bosons in the condensate is forbidden by
the frustration, and can only occur as a correlated jump
of a pair. However, this argument does not take into
23.5 24.0 24.5 25.0 25.5
T = 50 mK
total line  
H [ T ]
0.0 0.1 0.2
20H = 23.59 T
(  - 29  H ) [ MHz ]
FIG. 4: (Color online) Using a simple decomposition of the
spectra into two components as shown in the inset, we deter-
mined the 1st moments of the 29Si lines corresponding to the
different types of planes A and B. From the slopes of their
field dependence, the ratio of the average boson density is
found equal to nA/nB ≃ 5.
account the IC modulation of the boson density.
In conclusion, this NMR study of the 2D weakly cou-
pled dimers BaCuSi2O6 reveals that the microscopic na-
ture of the BEC in this system is much more complicated
than first expected. Two types of planes are clearly evi-
denced, with different intra-dimer J couplings and a gap
ratio of 1.16. Close to the QCP we observed that the
density of bosons, which is IC modulated within each
plane, is reduced in every second plane along the c-axis
by a factor of ≃ 5. This provides new constraints for
the understanding of the quasi-2D character of the BEC
close to the QCP.
We thank S.E. Sebastian, C.D. Batista and T. Gia-
marchi for discussions. Part of this work has been sup-
ported by the European Commission through the Euro-
MagNET network (contract RII3-CT-2004-506239), the
Transnational Access - Specific Support Action (contract
RITA-CT-2003-505474), the Estonian Science Founda-
tion (grant 6852) and the NSF (grant DMR-0134613).
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http://arxiv.org/abs/cond-mat/0610861
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http://arxiv.org/abs/cond-mat/0608703
