Bose-Einstein condensation in quantum magnets

Vivien Zapf, Marcelo Jaime, and C. D. Batista

Rev. Mod. Phys. 86, 563 – Published 15 May 2014; Erratum Rev. Mod. Phys. 86, 1453 (2014)

Abstract

This article reviews experimental and theoretical work on Bose-Einstein condensation in quantum magnets. These magnets are natural realizations of gases of interacting bosons whose relevant parameters such as dimensionality, lattice geometry, amount of disorder, nature of the interactions, and particle concentration can vary widely between different compounds. The particle concentration can be easily tuned by applying an external magnetic field which plays the role of a chemical potential. This rich spectrum of realizations offers a unique possibility for studying the different physical behaviors that emerge in interacting Bose gases from the interplay between their relevant parameters. The plethora of other bosonic phases that can emerge in quantum magnets, of which the Bose-Einstein condensate is the most basic ground state, is reviewed. The compounds discussed in this review have been intensively studied in the last two decades and have led to important contributions in the area of quantum magnetism. In spite of their apparent simplicity, these systems often exhibit surprising behaviors. The possibility of using controlled theoretical approaches has triggered the discovery of unusual effects induced by frustration, dimensionality, or disorder.

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Received 4 October 2012

DOI:https://doi.org/10.1103/RevModPhys.86.563

Erratum

Erratum: Bose-Einstein condensation in quantum magnets [Rev. Mod. Phys. 86, 563 (2014)]

Vivien Zapf, Marcelo Jaime, and C. D. Batista

Rev. Mod. Phys. 86, 1453 (2014)

Authors & Affiliations

Vivien Zapf^* and Marcelo Jaime^†

National High Magnetic Field Laboratory, Materials Physics and Applications Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

C. D. Batista^‡

Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

^*vzapf@lanl.gov
^†mjaime@lanl.gov
^‡cdb@lanl.gov

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Vol. 86, Iss. 2 — April - June 2014

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Figure 1
$(H, T)$ phase diagrams of (a) $S = 1 / 2$ XXZ and (b) weakly coupled dimer models.
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Figure 2
(a) Antiferromagnetic $X X Z$ model on a body-centered tetragonal lattice. (b) Blocking of interlayer hopping for bosons in the condensate of each layer. (c) Lowest-order diagram that generates an effective coherent interlayer hopping.
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Figure 3
Pair of weakly coupled dimers.
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Figure 4
Diagrammatic representation of Eq. (64). The ladder diagrams give the dominant contribution in an expansion in $a ρ^{1 / d}$ , where $a$ is the scattering length of the interaction potential $V_{q}$ .
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Figure 5
$(T, H)$ phase diagram for several quantum magnets studied in the context of BEC.
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Figure 6
(a) Cartoon of the spin levels of ${TlCuCl}_{3}$ showing the zero-field gap due to the interdimer interactions $J_{interdimer}$ . The levels are dispersed, forming bands that evolve in magnetic fields due to the Zeeman interaction. Long-range order occurs in the region $H_{c 1} < H < H_{c 2}$ . (b) Phase diagram of ${TlCuCl}_{3}$ from elastic neutron diffraction (open symbols) and magnetization measurements (solid symbols). The complete phase diagram extends to approximately 100 T, and the antiferromagnetically ordered phase occurs underneath and to the right of the data points. Adapted from [265].
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Figure 7
(a) Cartoon of the spin levels of DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] showing the $S^{z} = \pm 1$ excited doublet and $S^{z} = 0$ ground state, separated by a zero-field gap $D$ due to single-ion anisotropy. As with Fig. 6, the spin levels are shown as bands due to magnetic dispersion. (b) Phase diagram from specific heat (solid symbols) and magnetocaloric effect (open symbols) showing a dome-shaped region of quasi- $X Y$ antiferromagnetic order ( $X Y$ AFM), the low-field quantum paramagnet (QPM), and the high-field spin saturated phase (SP). Adapted from [312].
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Figure 8
The $S = 1$ antiferromagnetically coupled dimer compound ${Ba}_{3} {Mn}_{2} O_{8}$ forms an $S = 0$ ground state, an $S = 1$ excited triplet separated by the intradimer antiferromagnetic coupling $J_{0}$ , and an $S = 2$ quintuplet separated by $3 J_{0}$ from the ground state. (a) Evolution of the spin levels with applied magnetic field. (b) Schematic of the $S = 1$ dimers in the lattice. (c) Magnetization $M$ vs magnetic field $H$ at various temperatures and (d) temperature–magnetic-field phase diagram determined from specific heat and magnetocaloric effect (MCE) showing two regions of BEC, one where the lowest triplet level crosses the ground state, and one where the lowest quintuplet level crosses. The magnetization $M$ is overlaid. From [218].
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Figure 9
(a),(b) Frustrated triangular arrangement of $S = 1 / 2$ Cu spins in ${Cs}_{2} {CuCl}_{4}$ with couplings $J$ and $J^{'}$ in the $b − c$ plane and $J^{''}$ between planes. Dzyaloshinskii-Moriya (DM) couplings $D_{a}$ are also shown pointing into (dots) and out of (crosses) the $b c$ plane. (c) Temperature–magnetic-field phase diagram for magnetic fields along the $a$ axis showing a conelike magnetic state below the critical field 8.5 T, and the opening of a gap above 8.5 T ([41]; [198]).
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Figure 10
Conceptual $(T, H)$ phase diagram for a quantum magnet with a field-induced QCP belonging to the BEC universality class. The region of magnetic ordering is delimited by the critical fields $H_{c 1}$ and $H_{c 2}$ and the maximum ordering temperature $T_{cmax}$ in the $(T, H)$ plane. The energy scale $T_{s b}$ is the lower limit for observing a BEC QCP, and is given by uniaxial-symmetry-breaking terms below which discrete symmetry (DS) becomes important to the QCP. $T_{U}$ is the maximum temperature expected for universal properties that are derived from low-temperature approximations to the boson distribution function, such as $T^{3 / 2}$ scaling of thermodynamic properties for a $d = 3$ BEC QCP.
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Figure 11
(a) The temperature dependence of the magnetic susceptibilities of ${KCuCl}_{3}$ and ${TlCuCl}_{3}$ measured at $H = 5 kOe$ . $χ_{1}$ is the susceptibility for $H$ perpendicular to the cleavage plane, while $χ_{2}$ and $χ_{3}$ are the maximum and minimum susceptibilities when $H$ is applied in the cleavage plane, respectively. From [266]. (b) Magnetic susceptibility vs temperature $χ (T)$ of ${Sr}_{3} {Cr}_{2} O_{8}$ measured in a magnetic field of 10 kG showing fits with Eq. (72) for $H ∥ c$ and $H ⊥ c$ . The $g$ factors are fixed in the fits to the values obtained by ESR $g_{c} = 1.938$ and $g_{a b} = 1.950$ . The fitting indicates 0.8% free $S = 1 / 2$ moments, with the rest being dimerized, and the estimated exchange constants are $Δ = J = 61.5 (3) K$ (intradimer) and $J^{'} = 12 (2) K$ (interdimer). From [5].
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Figure 12
Magnetization (left vertical axis) versus magnetic field $M (H)$ measured at low temperatures for (a) DTN $[{NiCl}_{2} − 4 SC ({NH}_{2})_{2}]$ and (b) ${BaCuSi}_{2} O_{6}$ . Critical fields $H_{c 1}$ and $H_{c 2}$ delimit the region of magnetic-field-induced long-range order. A temperature (right vertical axis) vs magnetic field $T_{c} (H)$ phase diagram is superimposed on the magnetization data for both materials. Adapted from [95], and [321].
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Figure 13
Magnetization vs temperature $M (T)$ measured at constant magnetic fields for (a) ${TlCuCl}_{3}$ ([174]), (b) DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] ([185]), and (c) ${Pb}_{2} V_{3} O_{9}$ ([290]). A clear minimum in $M (T)$ is observed in all three compounds, as expected for a BEC.
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Figure 14
(a) Magnetic contribution to the specific heat $C$ vs temperature $T$ measured in ${TlCuCl}_{3}$ . From [177]. (b) Total $C (T) / T$ for ${BaCuSi}_{2} O_{6}$ for a number of magnetic fields $H < H_{c 2}$ . These data show a remarkable symmetry for magnetic fields around $H (T_{c \max})$ due to particle-hole symmetry. From [95].
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Figure 15
(a) High-temperature specific heat vs temperature $C (T)$ measured in ${Cs}_{2} {CuCl}_{4}$ from which the spin energy gap is estimated. (b) Magnetic contribution to the specific heat measured for a number of applied magnetic fields showing a $λ$ -like 3D ordering peak. From [198].
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Figure 16
Specific heat $C$ vs magnetic field $H$ for DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] obtained using an ac technique. $C (H)$ measured at several temperatures show sharp anomalies at the critical magnetic fields $H_{c 1} (T)$ and $H_{c 2} (T)$ . The contribution from the sample holder was not subtracted. The light shaded region is the $X Y$ AFM phase in the $(H, T)$ phase diagram of DTN From [121].
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Figure 17
(a) Typical $T (H)$ traces at a field-induced disorder-order transition. From top to bottom, curves are shown for the adiabatic ( $τ_{1} ≫ τ_{exp}$ ), semiadiabatic ( $τ_{1} \sim τ_{exp}$ ), quasi-isothermal ( $τ_{1} ≪ τ_{exp}$ ), and completely thermalized ( $κ \to \infty$ ), where $τ_{1} = C_{sample} / κ$ . Note that the size of the jumps are not to scale—the jump in the adiabatic case would be largest. (b) $T (H)$ traces at a field-induced ordering transition in a system that also has spin levels that cross, e.g., the quantum magnets with BEC QCPs.
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Figure 18
(a) Magnetocaloric effect for DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] in the adiabatic limit, near $H_{c 1}$ (unpublished). (b) Magnetocaloric effect in the equilibrium limit near $H_{c 1}$ (in which the data were incorrectly interpreted as showing irreversible behavior). The different limits are achieved by varying the sample size and the magnetic field sweep rate from [312].
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Figure 19
Magnetocaloric effect and specific heat vs temperature were used to build the $(H, T)$ phase diagram for ${Ba}_{3} {Mn}_{2} O_{8}$ , shown here for $H ∥ c$ . The magnetocaloric effect was measured under semiadiabatic conditions and shows an anomaly near the spin level crossing, and another set of anomalies at $H_{c 1}$ and $H_{c 2}$ . Adapted from [217].
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Figure 20
(a) Phase diagram of ${Sr}_{3} {Cr}_{2} O_{8}$ near $H_{c 1}$ , with the entropy extracted from Eq. (75) shown as a contour plot. (b) Phase diagram and magnetocaloric effect for ${Sr}_{3} {Cr}_{2} O_{8}$ in the equilibrium limit for $T < 4 K$ and the nearly adiabatic limit for $T > 4 K$ from [5].
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Figure 21
Thermal conductivity $κ$ vs temperature $T$ along the crystallographic $a$ and $c$ axes measured in DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] and ${Ba}_{3} {Mn}_{2} O_{8}$ for different magnetic field values ([111]; [121]).
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Figure 22
Specific heat $C_{p}$ vs temperature $T$ and magnetic field $H$ of ${Ba}_{3} {Mn}_{2} O_{8}$ showing the ordered transitions between $H_{c 3}$ and $H_{c 4}$ in this material. From [218].
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Figure 23
Asymmetry between physical properties of DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] at $H_{c 1}$ and $H_{c 2}$ . Here $χ$ is the magnetic susceptibility, $C_{p}$ is the specific heat, $d L / d H$ is the magnetostriction, $d P / d t$ is the magnetoelectric current due to changes in the electric polarization of the sample, and $T_{N}$ is the Néel temperature determined from the magnetocaloric effect (open symbols) and the specific heat (solid symbols) ([312]; [311]; Yu et al. [310]; [121]).
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Figure 24
Thermal conductivity $κ$ , along the $c$ axis as function of $H ∥ c$ for DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ]. $κ$ is measured at constant temperatures between $T = 0.3$ and 0.81 K, showing a clear enhancement at the critical fields $H_{c 1}$ and $H_{c 2}$ From [121].
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Figure 25
(a) Magnetostriction $Δ L / L$ measured along the $c$ and $a$ crystallographic axes vs magnetic field $H$ applied along the $c$ axis. (b) Experimental (open circles) and numerically simulated (line) magnetostriction $Δ L / L$ of DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] along the $c$ axis vs magnetic field ([310]; [293]).
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Figure 26
(a) Field dependence of the relative change of the sound velocity and (b) the sound attenuation of the acoustic $c 33$ mode in DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] at temperatures below the maximum Néel temperature $T_{N}$ . The magnetic field was applied along the (001) axis. The ultrasonic frequency was 78 MHz. The insets show the sound velocity and attenuation in the vicinity of $H_{c}$ in an enlarged scale. (c) Field dependence of the relative change of the sound velocity and (d) the sound attenuation of the acoustic $c 33$ mode in DTN for temperatures above the maximum $T_{N}$ Froom [38].
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Figure 27
Windowing analysis of the critical field $H_{c} (T)$ in ${BaCuSi}_{2} O_{6}$ . (a) Circles represent estimates of $H_{c 1}$ obtained from fitting experimental points on the phase boundary of ${BaCuSi}_{2} O_{6}$ in a window of increasing size $t_{w} = T / T_{c max}$ , for different fixed values of $ν = 1 / ϕ$ . $t_{w}$ labels the highest reduced temperature of the fit window. The dotted lines show the linear convergence of $H_{c 1}$ values at $t_{w} = 0$ . Triangles represent estimates of $H_{c 1}$ similarly obtained from quantum Monte Carlo simulations for corresponding fixed values of $ν$ , and similar convergence is observed. (b) Circles represent estimates of $ν$ from fitting experimental points on the phase boundary in a window of increasing size $t_{w}$ , with $H_{c 1} = 23.52 T$ determined from (a). The error bars are due to experimental uncertainty in determining values of $H_{c}$ . Triangles represent estimates of $ν$ from a similar fit to QMC simulation data. The dotted line is a guide to the eye, illustrating the approach of the QMC data to the mean-field value as $t_{w} \to 0$ . (c) The same as (a) with temperatures now reaching 30 mK in a dilution refrigerator. A rapid change is found in the estimate for $H_{c 1}$ for $t_{w} < 0.6$ . (d) The same as (b), but again to 30 mK. The change in $H_{c 1}$ accordingly affects the zero-temperature limit for $ν$ , indicating a crossover from 3D to 2D universality as the temperature is dropped. (e) $T$ vs $H - H_{c 1}$ phase boundary plotted for the two different values of $H_{c 1}$ in (c) showing $ϕ = 1$ ( $H_{c 1} = 23.17 T$ ) and $ϕ = 3 / 2$ ( $H_{c 1} = 23.52 T$ ) behaviors ([233]; [231]).
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Figure 28
Critical field $H_{c 1}$ vs $T^{1.5}$ for DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] down to 1 mK, determined from ac susceptibility data. The observed linear behavior of this function is consistent with the expected BEC power-law exponent $ϕ = 3 / 2$ . Deviations at low temperatures are eventually expected due to uniaxial-symmetry-breaking terms. The inset shows the curve $H_{c 1} (T^{2})$ showing that the exponent $ϕ = 2$ expected for an Ising QCP in dimension $3 + 1$ is inconsistent with the experimental data From [305].
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Figure 29
(a) Critical field $H_{c} (T)$ vs magnetization at the critical field $M_{c} (T) / M_{sat}$ for DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ]. (b) Scaled magnetization vs temperature below and above the upper critical field showing the approach toward the $T^{3 / 2}$ dependence expected for a BEC QCP in spatial dimension $d = 3$ ([200]; [182]).
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Figure 30
(a) DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ] magnetization $M$ , vs $T^{3 / 2}$ for magnetic field values between 1 and 3 T. (b) Measured specific heat vs temperature compared against the results of QMC simulations From [292].
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Figure 31
(a) Measured magnetization $M$ , vs temperature $T$ compared against the results of QMC simulations and analytical results for DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ]. This comparison shows that the experimental data follow the expected behavior for $T \to 0$ . Inset: $T^{- 1}$ behavior of the magnetic Grüneisen parameter. (b) Coefficient of volumetric thermal expansion $α$ divided by temperature as a function of $T$ for different magnetic fields. Inset: $T^{- 1}$ behavior of the thermal Grüneisen parameter From [292].
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Figure 32
Magnetic field vs pressure phase diagram for ${TlCuCl}_{3}$ at $T = 0 K$ . A square root phase boundary separates the spin-liquid (SL) phase from the AFM phase. (a)–(e) Dispersion relation of triplet modes in for different temperatures and pressures. (a) Degenerate triplet modes at $H = 0$ (anisotropies are neglected). (b) Zeeman-split triplet modes. (c) Spin dynamics in the field-induced ordered phase. (d) Degenerate and linear triplet modes close the spin energy gap at the critical pressure $P_{c}$ . (e) Spin dynamics in the pressure-induced ordered phase. Two degenerate transverse Goldstone modes (solid line) and a longitudinal amplitude mode (dashed line) are expected From [207].
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Figure 33
${TlCuCl}_{3}$ elastic neutron diffraction study. (a) $θ − 2 θ$ scans for (1, 0, 3) reflection measured at $μ_{0} H = 0$ and 12 T at 1.9 K. (b) Spin structure in the ordered phase. The external field is applied along the $b$ axis. The double chains located at the corner and the center of the chemical unit cell in the $b − c$ plane are represented by solid and dashed lines, respectively. The shaded area is the chemical unit cell in the $a − c$ plane. (c) Temperature dependence of the magnetic peak intensity at (1, 0, 3) in various magnetic fields. The square of the transverse magnetization per site $m_{⊥}^{2}$ is shown on the right $y$ axis. (d) Magnetic field dependence of the magnetic peak intensity for (1, 0, 3) at various temperatures. The square of the transverse magnetization per site $m_{⊥}^{2}$ is shown on the right $y$ axis. From [265].
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Figure 34
(a)–(c) Measured neutron intensity vs energy transfer in the $a − c$ plane of ${TlCuCl}_{3}$ for (1.35,0,0), (top) and (0,0,3.15) (bottom) (r.l.u.). (d) Observed energy dispersion of the magnetic excitation modes in ${TlCuCl}_{3}$ at $T = 1.5 K$ . Full circles from the relevant directions of reciprocal space are arranged in a reduced scheme representation, with $A = E = (1 / 2, 0, 0)$ , $B = (0, 1, 0)$ , $D = (0, 0, 0)$ (r.l.u.). Zone centering corresponds to $C = (0, 0, 1)$ for $Δ q = (h, 0, l)$ , $C = (0, 0, 0)$ for $Δ q = (0, k, 0)$ . From [35].
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Figure 35
$T = 1.7 K$ inelastic neutron scattering data for ${Ba}_{3} {Mn}_{2} O_{8}$ , which have six degenerate minima (only one shown) in the dispersion relation. (a)–(d) Scattering intensity (contour plot) and $ℏ ω$ (points) vs $Q$ plotted in reciprocal lattice units (r.l.u.). Solid lines are fits to a calculated dispersion relation. (e)–(h) Integrated scattering intensity from Gaussian fits to constant $Q$ scans. Solid lines are fits. (i) Path through the $(h h l)$ plane shown in (a)–(h). From [253].
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Figure 36
Energy-dependent scattering intensity of ${Ba}_{3} {Mn}_{2} O_{8}$ for $0.9 < Q < 1.0 Å^{- 1}$ for (from top to bottom): $T = 1.4$ , 8, 16, 24, 40, and 160 K. Data for different temperatures are vertically offset. Solid lines are fits to Lorentzian functions convolved with the energy-dependent resolution of the instrument. The inset illustrates scattering intensity of the $T = 12 K$ measurement plotted on a logarithmic intensity scale. The dotted line corresponds to the spectral contribution of the lower-energy singlet-triplet mode, and the dashed line corresponds to the higher-energy singlet-quintet mode. From [254].
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Figure 37
${TlCuCl}_{3}$ inelastic neutron diffraction study. (a) Intensity vs energy transfer showing the emergence of three spin levels with different energies ([68]). (b) Spin levels as a function of external magnetic field $H$ . (c) Contour plot of the neutron scattering intensity vs momentum transfer, in the spin gapped state at $H = 0$ (top), and inside the ordered phase at $H = 14 T$ showing the linear dispersion of the Goldstone mode. From [206].
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Figure 38
(a) Energy-level diagram for an $S = 1$ spin triplet with a Hamiltonian $H = g μ_{B} H_{z} S^{z}$ that obeys uniaxial symmetry. The transitions $α$ and $β$ obey the $Δ S^{z} = \pm 1$ selection rules and would be observed in ESR. (b) $H$ contains additional terms $D (S^{z})^{2} + E [(S_{x})^{2} - (S_{y})^{2}]$ . The $E$ term breaks the continuous uniaxial symmetry and disallows BEC for $k_{B} T < E$ . The $D$ term breaks uniaxial symmetry only if $H$ is not parallel to the $z$ axis. The eigenstates of the new Hamiltonian now contain mixtures of the different $S^{z}$ states; thus the energies are no longer linear in $H$ and follow $E_{\pm} = D \pm \sqrt{(2 g μ_{B} H)^{2} + E^{2} / 4}$ .
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Figure 39
ESR measurements on ${BaCuSi}_{2} O_{6}$ single crystals at $T = 0.6 K$ . The frequencies of intratriplet splittings of the Cu dimer $S = 1$ triplet are plotted as a function of $H | | \hat{c}$ , labeled R1 and R2, showing linear evolution with a very small offset from zero, from which the maximum size of $D$ can be estimated. There is no evidence for a nonzero $E$ in these data. The inset shows a cartoon of the triplet levels evolving in magnetic field. From [234].
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Figure 40
Electron spin resonance data at 1.4 K for single crystals of DTN [ ${NiCl}_{2} − 4 SC ({NH}_{2})_{2}$ ]. (a) Top: resonant absorption of energy at a fixed frequency as a function of magnetic field. Bottom: magnetic field dependence of the resonant absorption lines labeled A–F. (b) Schematic of the five absorption lines A–F, which include not only resonant transitions between spin levels (A and B), but also single- and two-magnon bound states (C, E, F). The expected dispersion of the spin levels are shown, which illustrated the fact that ESR measures ferromagnetic (uniform field) excitations, which correspond to the top (energy maximum) of the dispersion in DTN because the interactions are antiferromagnetic. From [321].
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Figure 41
(a) ${}^{63, 65}{Cu}$ NMR spectra of ${BaCuSi}_{2} O_{6}$ single crystals in the gapped phase $H < H_{c 1}$ . The $T$ dependence of the high-frequency satellite reveals two different Cu sites associated with two different types of planes in the crystalline structure. Inset: field-sweep spectrum that reveals the incommensurate nature of the line shape for each of the two sites. Shading separates the contribution of the ${}^{65}{Cu}$ high-frequency satellite from the rest of the spectrum. The analysis of the latter part confirms that the observed line shape has a purely magnetic origin. (b) ${}^{29}{Si}$ NMR at 50 mK near $H_{c 1}$ . A decomposition of the line shape into A and B as shown yields the different magnetizations of the two types of planes A and B. This corresponds to a ratio of the average boson density $n_{A} / n_{B} \sim 5$ . From [125].
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Figure 42
(a) Schematic (not crystallographic) representation of frustration in the four-leg spin tube of $Sul − {Cu}_{2} {Cl}_{4}$ . From [317]. (b) Incommensurate helical structure induced by magnetic fields in $Sul − {Cu}_{2} {Cl}_{4}$ with wave vector $k = (- 0.22, 0, 0.48)$ . From [69].
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Figure 43
Phase diagram of $(HPIP)_{2} {− CuBr}_{4}$ showing regions of spin-Luttinger-liquid behavior and 3D BEC ordering at the top. The magnetocaloric effect and the magnetization derived from the magnetocaloric effect are shown in the contour plot. The line at the bottom of the plot the $T_{N}$ points is a theoretical prediction. From [270].
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