NMR evidence for a strong modulation of the Bose-Einstein Condensate in BaCuSi$_2$O$_6$

TL;DR

This study uses NMR to reveal strong modulation of the Bose-Einstein condensate in BaCuSi$_2$O$_6$, showing significant density variations along the c-axis and in each plane, which impacts understanding of its quantum critical behavior.

Contribution

It provides the first detailed NMR evidence of strong spatial modulation of the Bose-Einstein condensate in BaCuSi$_2$O$_6$, highlighting inhomogeneities in the density near the quantum critical point.

Findings

Strong c-axis modulation of boson density near critical field

Existence of different intra-dimer exchange couplings and gaps

Incommensurate density modulation within planes

Abstract

We present a $^{63,65}$Cu and $^{29}$Si NMR study of the quasi-2D coupled spin 1/2 dimer compound BaCuSi$_2$O$_6$ 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 ($\Delta_{\rm{B}}/\Delta_{\rm{A}}$ = 1.16) exist in every second plane along the c-axis, in addition to a planar incommensurate (IC) modulation. $^{29}$Si spectra in the field induced magnetic ordered phase reveal that close to the quantum critical point at $H_{\rm{c1}}$ = 23.35 T the average boson density $\bar{n}$ of the Bose-Einstein condensate is strongly modulated along the c-axis with a density ratio for every second plane $\bar{n}_{\rm{A}}/\bar{n}_{\rm{B}} \simeq 5$. An IC modulation of the local density is also present in each plane. This adds new constraints for the understanding of the 2D value $\phi$ = 1 of the critical exponent describing the phase boundary.

Figures

Equations

$$K_{zz}^{\alpha }(T)-K_{zz}^{\alpha }(0)=A_{zz}^{\alpha }{m}_z^{\text{d}}({\Delta }_{\alpha },H,T)/H,$$

Full text

The interest in Bose-Einstein condensation (BEC) has been considerably renewed since it was shown to occur in cold atomic gases Anderson . In condensed matter, a formal analog of the BEC can also be obtained in antiferromagnetic (AF) quantum spin systems Affleck91 ; Giamarchi99 ; Nikuni00 ; Tanaka01 under an applied magnetic field. Many of these systems have a collective singlet ground state, separated by an energy gap $\Delta$ from a band of triplet excitations. Applying a magnetic field ($H$) lowers the energy of the $M_{z}=-1$ sub-band and leads to a quantum phase transition between a gapped non magnetic phase and a field induced magnetic ordered (FIMO) phase at the critical field $H_{\rm{c1}}$ corresponding to $\Delta_{\rm{min}}$-$g\mu_{\rm{B}}H_{\rm{c1}}=0$, where $\Delta_{\rm{min}}$ is the minimum gap value corresponding to some q vector qmin Affleck91 ; Giamarchi99 ; Nikuni00 ; Tanaka01 . 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 Sirker04 ; Clemancey06 ; Miyahara06 . From that point of view, BaCuSi2O6  Jaime04 seems at the moment the most promising candidate for the observation of a true BEC quantum critical point (QCP) Sebastian06a . In addition, this system exhibits an unusual dimensionality reduction at the QCP, which was attributed to frustration between adjacent planes in the nominally body-centered tetragonal structure Sebastian06 . The material also exhibits a weak orthorhombic distortion at $\simeq$90 K which is accompanied by an in-plane IC lattice modulation Samulon06 . This structural phase transition affects the triplon dispersion, and the possibility of a modulation of the amplitude of the BEC along the c-axis has been speculated based on low field inelastic neutron data Ruegg06 .

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 alternating 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 $T_{\rm{BEC}}$ with $H-H_{\rm{c1}}$ as expected for a 2D BEC. Our main finding is that the average boson density $\overline{n}$ in the BEC is strongly modulated along the $c$-axis in a ratio of the order of 1:5 for every second plane, whereas its local value $n$(R) is IC modulated within each plane.

NMR measurements have been obtained on $\sim$10 mg single crystals of BaCuSi2O6 using a home-made spectrometer and applying an external magnetic field $H$ along the $c$ axis. The gapped phase was studied using a superconducting 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 temperature 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 summing 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 surrounded 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 interaction. For 29Si nuclei both the transferred hyperfine interaction through oxygen atoms with a single dimer and the direct dipolar interaction are important. According to the room temperature structure $I4_{1}/acd$ Sparta04 , 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 Samulon06 . 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 $\nu=\nu_{0}=$ 29$\gamma H$ defined by the Si gyromagnetic ratio ${}^{29}\gamma$.

On the 63,65Cu NMR spectra recorded at 3 K and 13.2 T (Fig. 2), however, one can distinguish two different Cu sites, denoted A and B. That is, each of the 6 lines of Cu spectrum (for 2 copper isotopes $\times$ 3 transitions of a spin 3/2 nucleus) is split into two, which is particularly obvious on the lowest frequency “satellite” 63Cu line. The whole spectra can be nicely fitted with the following parameters: ${}^{63}\nu_{\rm{Q}}^{\rm{A}(\rm{B})}$ = 14.85 (14.14) MHz, $\eta$ = 0, and $K^{\rm{A}(\rm{B})}_{zz}$ = 1.80 (1.93) %, where $\nu_{\rm{Q}}$ is the quadrupolar frequency and $\eta$ the asymmetry parameter. The $K_{zz}$ is the hyperfine shift, expected to be purely orbital since the susceptibility has fully vanished. On increasing $T$ 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/$T_{2}$. Since the satellite NMR lines at 3 K (the lowest temperature) are narrow, the modulation of $\nu_{\rm{Q}}$ 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 distribution of local hyperfine fields: $\delta h_{z}(\textbf{R})=A_{zz}(\textbf{R})m_{z}(\textbf{R})$ in which $A$(R) is the hyperfine coupling tensor and $m_{z}(\textbf{R})$ the longitudinal magnetization at site R. Since $\nu_{\rm{Q}}$(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 $m_{z}$ in the plane.

Keeping constant the $\nu_{\rm{Q}}$ parameters obtained at 3 K, one can analyze the $T$ dependence of the shift $K_{zz}^{\alpha}(T)$ of each component $\alpha=$ A or B according to the formula

[TABLE]

where $m_{z}^{\text{d}}$ is the magnetization of a non-interacting dimer, $m_{z}^{\rm{d}}=g_{c}\mu_{\rm{B}}/(e^{(\Delta_{\alpha}-g_{c}\mu_{\rm{B}}H)/k_{\rm{B}}T}+1)$ in the given $T$ range, $g_{c}=$ 2.3 Zvyagin06 , and $K_{zz}^{\alpha}$ is determined from the average line position, i.e., the first moment. The best fit was obtained for $\Delta_{\rm{A(B)}}$ = 3.7 (4.3) meV and $A_{cc}^{\rm{A(B)}}$ = -16.4 T/$\mu_{\rm{B}}$. We assumed that $A^{\rm{A}}_{cc}$ = $A^{\rm{B}}_{cc}$, but the values of $\Delta$ depend only weakly with this quantity. The values are slightly higher than those determined by neutron inelastic scattering for Qmin = [$\pi,\pi$] Ruegg06 , which is normal considering our approximate description. However, the ratio $\Delta_{\rm{B}}$/$\Delta_{\rm{A}}$ = 1.16 is in excellent agreement with the neutron result 1.15. Considering the fact that there is no disorder in the system (as Cu lines at low $T$ are narrow), 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. Looking 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 $\Delta_{\rm{A}}$ and $\Delta_{\rm{B}}$ 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 structurefactor , with intra-dimer exchange in the ratio $J_{\rm{B}}/J_{\rm{A}}\cong\Delta_{\rm{B}}/\Delta_{\rm{A}}=1.16$

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 systems. As soon as a finite density of bosons $n$ is present ($H>H_{\rm{c1}}$ = $\Delta_{\rm{min}}$/$g\mu_{\rm{B}}$), a transverse staggered magnetization $m_{\perp}$ ($\perp$ to $H$) appears. Its amplitude and direction correspond respectively to the amplitude and phase of the order parameter. At the same time, the longitudinal magnetization $m_{z}$ is proportional to the number of bosons at a given temperature and field, this latter playing the role of the chemical potential. Due to the appearance of a static $\textbf{m}_{\perp}$, 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 modulation, they should give a single line for $H$ $\|$ $c$. Inside the condensate the NMR lines of this pair of Si sites will split by $\pm^{29}\gamma|A_{z\perp}|m_{\perp}$ because their $A_{z\perp}$ couplings are of opposite sign. Obviously, observing a splitting of lines requires the existence of off-diagonal terms in the hyperfine 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 spectra 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 $\nu_{0}$, which corresponds to a negligible boson density, suddenly 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 increases, but at a much lower rate. Both peculiar broadenings 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 total 4 Si atoms (denoted by $k$ = 0,1,2,3) situated around in a nearly symmetrical square coordination. The absolute 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 ${}^{29}\nu_{k}(\textbf{R})=\nu_{0}+\nu_{1}(\textbf{R})+\nu_{2,k}(\textbf{R})$, where $\nu_{1}(\textbf{R})=$ {}^{29}\gamma$$A_{zz}g\mu_{\rm{B}}n(R) and $\nu_{2,k}(\textbf{R})=$ {}^{29}\gamma$$A_{z\perp}m_{\perp}(\textbf{R})\cos(\phi-k\pi/2). Note that $\nu_{2,k}$ only exists when the bosons are condensed, that is when there is a transverse magnetization $m_{\perp}$ pointing in the direction $\phi$. In a uniform condensate $m_{\perp}$ is proportional to $\sqrt{n}$ 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 $m_{\perp}(\textbf{R})\propto\sqrt{n(\textbf{R})}$. The line shape is the histogram of the distribution of ${}^{29}\nu_{k}(\textbf{R})$, 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 $\overline{n}(H,T)$, the field $H_{\rm{c1}}(T)$ corresponding to to the BEC phase boundary, and the field dependence of the BEC order parameter (for $T$ close to zero). The average number of bosons $\overline{n}$ per dimer is directly proportional to the first moment $M_{1}$ (i.e., the average position) of the line: $M_{1}=\int_{-\infty}^{\infty}(\nu-\nu_{0})f(\nu){\rm d}\nu=$ {}^{29}\gamma$$A_{zz}g\mu_{\rm{B}}\overline{n}(H,T), where the line shape $f(\nu)$ is supposed to be normalized. The second moment (i.e., the square of the width) of the line $M_{2}=\int_{-\infty}^{\infty}(\nu-\nu_{0}-M_{1})^{2}f(\nu){\rm d}\nu$ has two origins: the broadening due to the IC distribution of ($n$(R)-$\overline{n}$), and that due to the onset of $m_{\perp}\propto\sqrt{n(\textbf{R})}$ in the condensate. When increasing $H$ at $T\simeq 0$, 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 $M_{1}$ ($\overline{n}$) and $M_{2}$ ($m_{\perp}$) vary linearly with the field and the extrapolation of $M_{2}$ to zero allows the determination of $H_{\rm{c1}}$ at 50 mK. For higher temperatures a thermal population of bosons $\overline{n}$ exists and increases with $H$ before entering the BEC phase. As a result both $M_{1}$ and $M_{2}$ increase non-linearly with $H$, as shown in the upper inset of Fig. 3b. However, the increase of $M_{2}(H)$ shows two clearly separated regimes and allows the determination of $H_{\rm{c1}}(T)$ as the point where the rate of change of $M_{2}(H)$ strongly increases due to the appearance of $m_{\perp}$. Applying this criterion to all temperatures, we were able to determine the field dependence of $T_{\rm{BEC}}$ (lower inset of Fig. 3b) and define precisely the QCP at $H_{\rm c1}$ = 23.35 T. In agreement with the torque measurements Sebastian06 , we find a linear field dependence. This is the signature of a 2D BEC QCP, where $T_{\rm{c}}\propto(H-H_{\rm{c1}})^{\phi}$ with $\phi=2/d$ and $d$ = 2 Batista .

This analysis, however, does not take into account the specificity of the line shapes, which are related to the existence 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 exhibiting 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 without any splitting, whereas the other immediately heavily broadens in some sort of triangular line shape. The field dependence of $M_{1}$ 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 corresponding 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 $\Delta_{\rm{A}}<g\mu_{\rm{B}}H<\Delta_{\rm{B}}$. Although the observed density of boson is finite in the B planes, it is strongly reduced, giving rise to a strong commensurate modulation of $\overline{n}$ along the c-axis. According to Sebastian06 , 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 account the IC modulation of the boson density.

In conclusion, this NMR study of the 2D weakly coupled dimers BaCuSi2O6 reveals that the microscopic nature of the BEC in this system is much more complicated than first expected. Two types of planes are clearly evidenced, 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 $\simeq$ 5. This provides new constraints for the understanding of the quasi-2D character of the BEC close to the QCP.

• Specific Support Action (contract RITA-CT-2003-505474), the Estonian Science Foundation (grant 6852) and the NSF (grant DMR-0134613).