A Review of Bose-Einstein Condensation in Certain Quantum Magnets Containing Cu and Ni

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1 A Review of Bose-Einstein Condensation in Certain Quantum Magnets Containing Cu and Ni Vivien S. Zapf Abstract We summarize a set of recent experimental and theoretical work on the phenomenon of Bose-Einstein Condensation (BEC) of magnetic degrees of freedom in quantum magnets. We will focus on two examples: BaCuSi 2 O 6, which contains planes of Cu S = 1/2 dimers, and NiCl 2-4SC(NH 2 ) 2 (DTN), an organic magnet with Ni S = 1 weakly-coupled chains. In both compounds, the BEC corresponds to a field-induced antiferromagnetic ordered state that persists over a magnetic field range H c1 to H c2. At the quantum critical point at H c1, the predicted power-law behavior of the critical fields and the magnetization for 3-D BEC is observed at low temperatures. In addition, BaCuSi 2 O 6 shows a dimensional crossover from a 3-D BEC to 2-D behavior below 1 K. 1 Introduction In the past decade there has been a surge of interest in the topic of quantum magnetism in condensed-matter systems. Quantum spin systems come in a wide variety of structures including reduced-dimensional ladders, chains or planes, dimers, frustrated spins, and single-molecule magnets. These materials all share the common trait that quantum effects, such as spin fluctuations and quantized spin levels, play a significant role in shaping the ground state and the physical properties of the system. The resultant behavior can be complex and challenge our understanding. In particular, there have been a number of attempts recently to observe Bose- Einstein Condensation (BEC) in quantum magnets. BEC was first observed in dilute gases of 87 Rb atoms [1], leading to a nobel prize being awarded in 21. It turns out that a form of BEC can also be observed in quantum magnets, e.g. crystalline lattices containing spins. BEC in quantum magnets was first predicted to occur in 1991 by V.S. Zapf National High Magnetic Field Laboratory, Los Alamos National Laboratory, vzapf@lanl.gov B. Barbara et al. (eds.), Quantum Magnetism. c Springer Science + Business Media B.V

2 24 V.S. Zapf Ian Affleck [2]. In the past few years, several reports of BEC in real spin systems have been published including TlCuCl 3 [3, 4], BaCu 2 SiO 6 [5, 6], CsCuCl 4 [7, 8], and NiCl 2-4SC(NH 2 ) 2 [9, 1]. In these compounds, the BEC phase transition occurs at a critical magnetic field at which XY antiferromagnetic order is induced. The quantum phase transition into the magnetically ordered state belongs to the same universality class as Bose-Einstein Condensation, and the bosons can be identified via a mapping from the spin degrees of freedom. All quantum magnets investigated for BEC to date contain either Cu 2+ (S = 1/2) or Ni 2+ (S = 1). In this paper we give a review of some recent experimental data on a material containing Cu, BaCuSi 2 O 6 and an organic crystal containing Ni, NiCl 2-4SC(NH 2 ) 2 (DTN). We describe the details of these systems and how BEC can be derived from the Hamiltonian and observed via thermodynamic measurements. Both BaCuSi 2 O 6 and DTN have body-centered tetragonal crystal structures (see Fig. 1) with antiferromagnetic exchange coupling. In BaCuSi 2 O 6, the strongest antiferromagnetic coupling (J = 51 K) occurs within the Cu dimers, such that the two S = 1/2 spins in each dimer form a S = singlet groundstate separated by a gap J from the S = 1 excited triplet (see Figs. 2 and 6). The dimers are arranged in square planes with an intraplane coupling between dimers of J = 6.7 K. The planes in turn are staggered along the c-axis, which results in frustrated antiferromagnetic coupling between planes of J f = 1.3 K. Thus J >> J >> J f [12, 5]. DTN, by contrast, is a chain compound, consisting of Ni S = 1 spins strongly coupled along Ni-Cl-Cl-Ni bonds in the c-axis with J c = 2.2 K. The interchain coupling J a =.18 K is an order of magnitude weaker than the intrachain coupling and no diagonal couplings analogous to J f in BaCuSi 2 O 6 have been observed in experiment. The exchange couplings were determined from inelastic neutron diffraction Cu J' J a J f J Ni Cl J c BaCuSi 2 O 6 c b NiCl 2-4SC(NH 2 ) 2 (DTN) Fig. 1 Tetragonal crystal structures of BaCuSi 2 O 6 (left) showing the Cu S = 1/2 dimers arranged in staggered planes, and DTN (right) showing the Ni and Cl atoms. The lines indicate the antiferromagnetic coupling strengths J (intradimer), J (intraplane), and J f (interplane) for BaCuSi 2 O 6 and J c and J a for DTN [9, 11] a

3 Bose-Einstein Condensation in Quantum Magnets 241 measurements [9], refined via Quantum Monte Carlo calculations and Electron Spin Resonance experiments [1]. The important feature of both DTN and BaCuSi 2 O 6 is that the zero field ground state is nonmagnetic and is separated by a gap from a magnetic excited state (see Fig. 2). In BaCuSi 2 O 6 the gap between the singlet ground state (S = ) and the excited triplet states (S = 1) is created by the intradimer coupling J. InDTN,there is no S = singlet only a S = 1 triplet. A uniaxial anisotropy D splits this S = 1 triplet into a S z = ground state and a S z = ±1 excited doublet. In both compounds, the excited S z =+1 state can be suppressed with magnetic field until it becomes degenerate with the S = ors z = ground state. The critical region where these two states overlap is where the antiferromagnetic order/bose-einstein Condensation occurs. The S z = ±1 spin levels are dispersed by the antiferromagnetic coupling, e.g. the spins can raise or lower their energy depending on their orientation with respect to their neighbors. In Fig. 2, the S z = ±1 energy levels are shown schematically as broad bands where the upper and lower edges of the bands correspond to the k = FM wave vector and the k =(π,π,π) AFM wave vector, respectively. Thus, the region of overlap between the ground state and the field-suppressed S z =+1 excited state occurs over a broad range of fields between H c1 and H c2. In both systems, 3-D XY antiferromagnetic order is observed between H c1 and H c2, (23.5 T to 49 T for BaCuSi 2 O 6 and 2.1 T to 12.6 T for DTN) [5, 13]. The magnetic order occurs in a dome-shaped region of the phase diagram with a maximum Néel temperature T N of 3.8 K for BaCuSi 2 O 6 and 1.2 K for DTN. The temperature-field phase diagrams are shown in Fig. 3. In BaCuSi 2 O 6 the groundstate is nonmagnetic for H < H c1 (S = ). In DTN, the spins form a disordered S z = spin liquid for H < H c1. When the field is increased to H c1,thespinsorder antiferromagnetically in the plane perpendicular to the applied field and then cant along the applied field direction as the field is increased from H c1 to H c2. Finally, the spins polarize along the field direction above H c2. The longitudinal magnetization therefore increases monotonically between H c1 and H c2, as shown in Fig. 3. It should be noted that for DTN, this phase diagram is only valid for fields along the tetragonal c-axis. For fields perpendicular to c, the spin levels within the triplet mix with one another leading to paramagnetic behavior for all fields. For BaCuSi 2 O 6, there is no uniaxial anisotropy created by the crystal structure. Thus, BEC should occur for all field directions with a small anisotropy due to an anisotropic g factor. The data discussed in this review was taken for H c. 2 Boson Mapping The bosons are created via a mapping from the spin levels. In the spin language, the Hamiltonians H DT N for DTN and H B for BaCuSi 2 O 6 are given by: H DTN = D(S z )2 +,α J a S S +eα + J c S S +eγ

4 242 V.S. Zapf DTN E S z = -1 S z = 1 H c1 = 2.2 T S z = H c2 = 12.6 T H E BaCuSi 2 O 6 S z = -1 S z = S z = 1 H c1 = 23.5 T H c2 = 49 T S = H Fig. 2 Energy level diagrams of BaCuSi 2 O 6 and DTN as a function of magnetic field parallel to the crystallographic c-axis. The broad colored bands for the S z = ±1 levels schematically indicate the width antiferromagnetic dispersion due to the coupling J a and J c for DTN, and J and J f for BaCuSi 2 O 6 H B = JS,1 S,2 +,α J S S +eα + J f S S +eγ,γ Here e α is the unit vector along a and b. e γ is the vector for coupling along the c-axis. For DTN, e γ = e c. However, for BaCuSi 2 O 6, the coupling occurs between dimers on staggered planes (see Fig. 1) such that e γ = e c ± 1/2e a ± 1/2e b. In both Hamiltonians, the first term creates the zero-field splitting between the ground state and the excited state. The indices 1, 2 in this term refer to the two Cu spins within the dimerofbacusi 2 O 6. The second term of the Hamiltonians expresses the AFM coupling within the plane, and the third term couples along the c-axis. For BaCuSi 2 O 6 it is implied that the couplings J and J f are summed over both spins within each dimer. When a magnetic field H is turned on, both Hamiltonians acquire an extra Zeeman term gµ B H S z. These Hamiltonians can be transformed into boson language via the identification S + = b,whereb is the boson creation operator. Then

5 Bose-Einstein Condensation in Quantum Magnets 243 T N (K) T N (K) data QMC MCE Specific Heat QMC H c1 H H (T) c DTN H c BaCuSi 2 O 6 H c AFM/BEC data QMC Torque Specific Heat MCE QMC 2 H c1 H (T) H c2 Fig. 3 Temperature T - Magnetic field H phase diagram of DTN (top) and BaCuSi 2 O 6 (bottom). Antiferromagnetic order/bose-einstein Condensation occurs under the dome-shaped region with the spin configuration indicated by arrows in the bottom figure. The Néel temperatures were determined from magnetocaloric effect (MCE) and specific heat data for DTN (top) [9] and torque, specific heat and MCE data for BaCuSi 2 O 6 (bottom) [6]. Quantum Monte Carlo (QMC) calculations of the phase diagrams are shown for comparison [5, 1]. The magnetization vs field at 16 mk (DTN) [13] and.5 K (BaCuSi 2 O 6 ) [5] is overlayed onto the phase diagram together with the predicted magnetization determined from QMC calculations [5, 1] M/M sat M/M sat H ef f = t,α (b +e α b + b b +e α )+t (b +e γ b + b b +e γ ) + V,α n n +eα +V n n +eγ + µ n The definitions of t, t, V, V,andµ in terms of the parameters of the spin-language Hamiltonian are summarized in Table 1. The Hamiltonian consists of kinetic energy terms (t and t ), potential energy terms (V and V ), and a chemical potential µ. The kinetic energy terms allow the bosons to hop on the lattice in the square a-b plane (t), and along the c-axis t, subect to the hard-core constraint of one boson per lattice site. The bosons repel each other proportional to V and V in and out of the plane, respectively. The repulsion is also proportional to the number of bosons, and therefore to the S z =+1 component of the spins. Finally, there is a chemical potential µ that is linear in the magnetic field and controls the number of bosons.

6 244 V.S. Zapf In this two-level boson-mapping model, [2, 14, 3] the mapping from spins to bosons treats the S = (ors z = for DTN) state as an unoccupied bosonic state, and the S z =+1 state as occupied bosonic state. The magnetic field acts as the chemical potential in this system, tuning the number of bosons, e.g. the weight of the S z =+1 component of the ground state. Condensation occurs as the number of bosons is tuned from zero to nonzero at H c1. As the magnetic field is further increased, the bosons become less dilute, and eventually near H c2, where the ground state is mostly S z =+1, a reverse mapping is necessary to create a dilute Bose gas, with S z =+1 being the unoccupied state and S = the occupied state. A second BEC transition then occurs across H c2. This two-level model described above, while didactic, is not exactly accurate in DTN because the third spin level S z = 1 is low enough in energy that it needs to be taken into account. In comparing the phase diagrams of BaCuSi 2 O 6 and DTN in Fig. 3, it is clear that the region of AFM order in BaCuSi 2 O 6 is more symmetric in field about the midpoint between H c1 and H c2. This reflects a particle-hole symmetry of the bosons in this compound. In DTN, by contrast, the upper S z = 1 level is low enough in energy that it breaks particle-hole symmetry of the bosons, and distorts the phase diagram in the T H plane. Therefore in DTN, the idea of two spin levels corresponding to occupied and unoccupied bosons is not a complete picture. Two theoretical papers have presented models of Bose-Einstein Condensation for DTN that take all three spin levels (S z =, S z =+1, S z = 1) into account. Wang and Wang [15] treat each spin level as a different type of boson and Ng et al. [16] interpret the S z = 1 level as an energetically unfavorable double occupancy state. In any case, the concept of bosons that condense is still valid in a three-level model and the universality class of Bose-Einstein Condensation is still applicable at the quantum phase transition. A key condition that separates bosonic systems that condense from those that don t is boson number conservation. The boson number must be set by some external constraint or else the bosons will merely be excitations of the system and vanish as the temperature is lowered to zero, as is the case e.g. for phonons. In DTN and BaCuSi 2 O 6, the number conservation is created by the tetragonal crystal structures, which provides an approximate uniaxial symmetry of the spin environment about the direction of the applied field. In the effective Hamiltonian it can be seen that every creation operator b is multiplied by a destruction operator b. This is an indication that the Hamiltonian obeys the uniaxial symmetry. If the Hamiltonian were Table 1 Relation between the parameters of the Hamiltonian in the boson-picture and the Hamiltonian in the spin picture DTN BaCuSi 2 O 6 t J a J t J c J f V J a /2 J /2 V J c /2 J f /2 µ D gµ B H J gµ B H

7 Bose-Einstein Condensation in Quantum Magnets 245 rotated by an angle φ in the plane perpendicular to the field, then b b e iφ and b be iφ, such that (b e iφ )(be iφ )=b b and the Hamiltonian is independent of φ. The fact that every creation operator is paired with a destruction operator also ensures that the boson number is conserved. Since b b is the number operator, a Hamiltonian containing only b b terms will commute with the number operator. Thus the uniaxial symmetry of the Hamiltonian creates a number conservation law for the bosons. We should mention that there are several caveats to the idea of BEC in quantum spin systems. First of all, the uniaxial spin symmetry of the Hamiltonian is approximate. The square lattice of the crystal does introduce a small anisotropy in the a-b plane and dipole-dipole interactions and Dzyaloshinskii-Moriya interactions could also create anisotropies. However, these effects have been shown to occur at lower energy than the mk temperatures measured scales (possibly) and can thus be neglected at the temperatures of tens to hundreds of mk at which these quantum magnets are studied [8, 1]. Another caveat is that the uniaxial symmetry is only obeyed in equilibrium and the conservation of the boson number is therefore only obeyed on average. On short time scales, thermal and quantum fluctuations can distort the symmetry of the lattice thereby produce fluctuations in the boson number. Thus, the effects of Bose-Einstein Condensation are studied through thermodynamic measurements in equilibrium. Nonequilibrium effects such as supercurrents are not robust. The boson number fluctuations create relaxation mechanisms for supercurrent excitations and thus supercurrents have finite lifetimes in quantum magnets. Nevertheless, the Bose-Einstein condensation picture in quantum magnets is valid for the temperatures at which these compounds are studied, and more importantly it provides a way of understanding the observed thermodynamic behavior near the quantum phase transition. The thermal phase transitions in these system belong to the d = 3, z = 1 universality class of an XY antiferromagnet, where d is the spatial dimension and z is the dynamical exponent. However, the field-induced quantum phase transition belongs to the d = 3, z = 2 universality class, and the challenge, as with all quantum phase transitions, is to find a classical phase transition to map it onto, allowing us to create a physical picture of what is happening. Bose-Einstein Condensation provides an intuitive way to describe the quantum phase transition in this system. 3 Experimental Investigation into Bose-Einstein Condensation One experimental approach to identifying Bose-Einstein Condensation is to measure the power-law temperature dependencies of the critical fields and the magnetization near T =. The mean-field theory for Bose-Einstein condensation has a critical dimension d + z 4. Near the quantum critical point (QCP), z = 2forthese systems, thus the condition d 2 must be satisfied for the theory to hold. The theory then predicts the following:

8 246 V.S. Zapf H c H c (T = ) T α (1) M z (H c ) ρ T α (2) where H c is H c1 or H c2, α = d/2 and the longitudinal magnetization M z is proportional to the boson density ρ. Thus, α = 3/2 for a 3-D BEC and α = 1fora 2-D BEC [2, 3, 14]. In contrast, the prediction for an Ising magnet is α = 2. These power laws are valid in the dilute boson limit, which is satisfied near H c1 and H c2. Experimentally, these power laws have proven difficult to measure since they are a low-temperature approximation to the boson distribution function and only valid as T. In addition, the power-law exponent α is very sensitive to the fitting range used and to the extrapolated value of the critical field H c [8]. To solve these problems, a method has been developed to determine the critical field H c1 independent of the exponent α, and to then extrapolate the exponent α in the limit as T [6]. Using this approach, an exponent within experimental error of α = 3/2 was found [6, 9] for BaCuSi 2 O 6 down to 1 K, and DTN down to 1 mk. For BaCuSi 2 O 6, H c1 (T ) was determined using torque, magnetocaloric effect and specific heat data [6]. The magnetization was extracted from torque measurements. For DTN, magnetocaloric effect and specific heat data was used to measure H c1 (T ) and the magnetization was measured using a VSM [9]. 4 Frustration and Dimensional Reduction Recent experiments have found that the exponent α = 3/2 for BaCuSi 2 O 6 is only valid down to 1 K. Magnetic torque data taken to down to 35 mk showed that the critical field vs temperature H c1 (T ) and the magnetization M z become linear with temperature (α = 1) as shown in Fig. 4 [11]. This data suggests a dimensional reduction at the quantum critical point with 2-D behavior occurring below T 1K in the vicinity of H c T c (K) BaCuSi 2 O 6 H c H (T) Fig. 4 Critical temperature vs field for BaCuSi 2 O 6 determined from magnetic torque data, showing linear behavior between 35 mk and 1 K [11]

9 Bose-Einstein Condensation in Quantum Magnets 247 Fig. 5 Body-centered tetragonal structure of NiCl 2-4SC(NH 2 ) 2 (DTN) showing the two interlocking Ni sublattices (red and blue atoms). The two sublattices are decoupled due to frustration, but this doesn t lead to a dimensional reduction. The significant AFM couplings along the c and a axes are shown as thick lines (J c = 2.2 K,J a =.18 K). The thin lines along (1,1,1) indicate the frustrated couplings, which are either very small or absent [9] This dimensional reduction at the quantum critical point can be understood by considering the body-centered tetragonal structure of the BaCuSi 2 O 6 (see Fig. 1). The square a-b planes are arranged in a staggered structure such that each spin has an even number of nearest neighbors on the next plane with coupling J f along (1,1,1). Since the AFM wave vector is commensurate with the lattice, this results in frustration that suppresses the coupling along the c-axis. The c-axis boson hopping term, which is derived from the c-axis AFM coupling J f, therefore goes to zero, restricting boson motion to the 2-D planes. One might expect phase fluctuations at the quantum critical point to restore 3-D behavior. However, the magnitude of these fluctuations in quadratic in the boson number ρ, and thus goes to zero as the critical field H c1 is approached [17]. This confluence of effects leading to 2-D behavior at the QCP in BaCuSi 2 O 6 make it a unique system. 2-D quantum critical behavior has been predicted to occur in many systems near quantum critical points but has never unambiguously observed. DTN also has body-centered structure with a commensurate AFM wave vector. However, frustration does not lead to dimensional reduction in this system. The reason is that the dominant AFM coupling occurs along the c-axis, J c with a weaker coupling J a in the tetragonal a-b plane. As shown in Fig. 5, the frustration occurs between spins at the edge of the unit cell, and those in the center. This decouples the two interlocking lattices (shown as red and blue in Fig. 5), giving rise to a two-fold degenerate ground state. However, each lattice is still three-dimensional so no dimensional reduction would be expected near the quantum critical points. Inelastic neutron scattering measurements at zero field [9] do not show any diagonal couplings to within experimental error, confirming that the frustrated couplings are either suppressed to a small value, or entirely absent. Due to the prevalence of quantum fluctuations in these systems, particularly in DTN, Quantum Monte Carlo (QMC) simulations are necessary to calculate the phase diagram and all finite-temperature behavior. QMC calculations have been performed for both BaCuSi 2 O 6 and DTN and the resulting predictions for the phase

10 248 V.S. Zapf S z = -1 S z = S z = 1 T (K) 3 2 DTN BaCuSi 2 O 6 S z = -1 1 S = 1 z H c1 H H c2 c1 H c2 H (T) Fig. 6 Energy levels (lines) and Néel temperatures T N (points) for DTN and BaCuSi 2 O 6,shown on the same scale for comparison. AFM/BEC occurs under the dome-shaped regions [9, 6] diagrams are shown in Fig. 3, together with the experimental data [5, 6, 9, 1]. For both systems, the QMC calculations fit the experimental phase diagrams very well. In addition, the longitudinal magnetization is also shown together with the QMC prediction. Much further work is needed in both compounds. For example, the 2-D behavior in BaCuSi 2 O 6 should be observable as a broad Shastry-Sutherland crossover in the specific heat at H c1. Neutron diffraction data examining the low-energy spin excitations in these systems will also shed light on the symmetry of the spin configurations. Finally, in DTN, a study is underway to investigate magnetostriction in the soft organic lattice, and its effects on the phase diagram and energy levels. In summary, the quantum magnets DTN and BaCuSi 2 O 6 have different spin level configurations, antiferromagnetic couplings, and energy scales (see Fig. 6). However, the underlying physics of field-induced antiferromagnetism corresponding to Bose-Einstein Condensation is largely the same and both systems show the critical exponent α = 3/2 at the quantum critical point. The most significant difference between the two compounds is that in BaCuSi 2 O 6 a frustration-induced decoupling along the c-axis results in a dimensional reduction at the quantum critical point H c1, leading to 2-D behavior with α = 1belowT = 1K. References 1. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269, 198 (1995) 2. I. Affleck, Phys. Rev. B 43, 3215 (1991) 3. T. Nikuni, M. Oshikawa, A. Oosawa, H. Tanaka, Phys. Rev. Lett. 84, 5868 (2)

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