BEC phase boundary in

Science and Technology of Advanced Materials, 2007

We report specific heat, magnetocaloric effect and magnetization measurements on single crystals of the frustrated quasi-2D spin À 1 2 antiferromagnet Cs 2 CuCl 4 in the external magnetic field 0pBp12 T along a-axis and in the temperature range 0:03 KpTp6 K. Decreasing the applied magnetic field B from high fields leads to the closure of the field induced gap in the magnon spectrum at a critical field B c ' 8:44 T and a long-range incommensurate state below B c . In the vicinity of B c , the phase transition boundary is well described by the power law T N $ðB c À BÞ 1=f with the measured critical exponent f ' 1:5. These findings provide experimental evidence that the scaling law of the transition temperature T N can be described by the universality class of 3D Bose-Einstein condensation (BEC) of magnons. r

JETP Letters, 2015

We discuss a possible origin of the experimentally observed nonlinear contribution to the shift ∆Tc = Tc − T 0 c of the critical temperature Tc in an atomic Bose-Einstein condensate (BEC) with respect to the critical temperature T 0 c of an ideal gas. We found that accounting for a nonlinear (quadratic) Zeeman effect (with applied magnetic field closely matching a Feshbach resonance field B0) in the mean-field approximation results in a rather significant renormalization of the field-free nonlinear contribution b2, namely ∆Tc/T 0 c ≃ b * 2 (a/λT ) 2 (where a is the s-wave scattering length, λT is the thermal wavelength at T 0 c ) with b * 2 = γ 2 b2 and γ = γ(B0). In particular, we predict b * 2 ≃ 42.3 for the B0 ≃ 403G resonance observed in the 39 K BEC. PACS: 67.85.Hj, 67.85.Jk Studies of Bose-Einstein condensates (BECs) continue to be an important subject in modern physics (see, e.g., Refs.[1, 2, 3, 4] and further references therein). Atomic BECs are produced in the laboratory in lasercooled, magnetically-trapped ultra-cold bosonic clouds of different atomic species (including 87 Rb [5, 9], 7 Li [6], 23 N a [7], 1 H [8], 4 He [10], 41 K [11], 133 Cs [12], 174 Y b [13] and 52 Cr [14], among others). Also, a discussion of a relativistic BEC has appeared in Ref.[15] and BECs of photons are most recently under investigation [16]. In addition, BECs are successfully utilized in cosmology and astrophysics [17] as they have been shown to constrain quantum gravity models [18].

Physical Review A, 2011

We investigate effects of thermal and quantum fluctuations on the phase diagram of a spin-1 87 Rb Bose-Einstein condensate (BEC) under a quadratic Zeeman effect. Due to the large ratio of spinindependent to spin-dependent interactions of 87 Rb atoms, the effect of noncondensed atoms on the condensate is much more significant than that in scalar BECs. We find that the condensate and spontaneous magnetization emerge at different temperatures when the ground state is in the brokenaxisymmetry phase. In this phase, a magnetized condensate induces spin coherence of noncondensed atoms in different magnetic sublevels, resulting in temperature-dependent magnetization of the noncondensate. We also examine the effect of quantum fluctuations on the order parameter at absolute zero, and find that the ground-state phase diagram is significantly altered by quantum depletion.

Physical Review B, 2005

Physical Review A, 1999

ABSTRACT Three hyperfine states of Bose-condensed sodium atoms, recently optically trapped, can be described as a spin-1 Bose gas. We study the behaviour of this system in a magnetic field, and construct the phase diagram, where the temperature of the Bose condensation $T_{BEC}$ increases with magnetic field. In particular the system is ferromagnetic below $T_{BEC}$ and the magnetization is proportional to the condensate fraction in a vanishing magnetic field. Second derivatives of the magnetisation with regard to temperature or magnetic field are discontinuous along the phase boundary. Comment: 5 pages, 5 figures included, to appear in Phys. Rev. A

Journal of Experimental and Theoretical Physics, 1999

A phase diagram for a 2D metal with variable carrier density has been derived. It consists of a normal phase, where the order parameter is absent; a so-called "abnormal normal" phase where this parameter is also absent but the mean number of composite bosons (bound pairs) exceeds the mean number of free fermions; a pseudogap phase where the absolute value of the order parameter gradually increases but its phase is a random value, and finally a superconducting (here Berezinskiȋ-Kosterlitz-Thouless) phase. The characteristic transition temperatures between these phases are found. The chemical potential and paramagnetic susceptibility behavior as functions of the fermion density and the temperature are also studied. An attempt is made to qualitatively compare the resulting phase diagram with the features of underdoped high-T c superconducting compounds above their critical temperature.

Physical Review A, 2012

We study experimentally the equilibrium phase diagram of a spin 1 Bose-Einstein condensate with antiferromagnetic interactions, in a regime where spin and spatial degrees of freedom are decoupled. For a given total magnetization mz, we observe for low magnetic fields an "antiferromagnetic" phase where atoms condense in the m = ±1 Zeeman states, and occupation of the m = 0 state is suppressed. Conversely, for large enough magnetic fields, a phase transition to a "broken axisymmetry" phase takes place: The m = 0 component becomes populated and rises sharply above a critical field Bc(mz). This behavior results from the competition between antiferromagnetic spin-dependent interactions (dominant at low fields) and the quadratic Zeeman energy (dominant at large fields). We compare the measured Bc as well as the global shape of the phase diagram with mean-field theory, and find good quantitative agreement.

Physical Review B, 2009

By performing heat capacity, magnetocaloric effect, torque magnetometry and force magnetometry measurements up to 33 T, we have mapped out the T-H phase diagram of the S = 1/2 spin dimer compound Ba3Cr2O8. We found evidence for field-induced magnetic order between Hc1 = 12.52(2) T and Hc2 = 23.65(5) T, with the maximum transition temperature Tc ∼ 2.7 K at H ∼ 18 T. The lower transition can likely be described by Bose-Einstein condensation of triplons theory, and this is consistent with the absence of any magnetization plateaus in our magnetic torque and force measurements. In contrast, the nature of the upper phase transition appears to be quite different as our measurements suggest that this transition is actually first order.

Nature Physics, 2008

The elementary excitations in antiferromagnets are magnons, quasiparticles with integer spin and Bose statistics. In an experiment their density is controlled efficiently by an applied magnetic field and can be made finite to cause the formation of a Bose-Einstein condensate (BEC). Studies of magnon condensation in a growing number of magnetic materials provide a unique window into an exciting world of quantum phase transitions (QPT) and exotic quantum states.

The European Physical Journal D, 2008

The quartic confining potential has emerged as a key ingredient to obtain fast rotating vortices in BEC as well as observation of quantum phase transitions in optical lattices. We calculate the critical temperature Tc of bosons at which normal to BEC transition occurs for the quartic confining potential. Further more, we evaluate the effect of finite particle number on Tc and find that ∆Tc/Tc is larger in quartic potential as compared to quadratic potential for number of particles < 10 5 . Interestingly, the situation is reversed if the number of particles is 10 5 .

Physical Review B, 2008

At zero temperature and strong applied magnetic fields the ground sate of an anisotropic antiferromagnet is a saturated paramagnet with fully aligned spins. We study the quantum phase transition as the field is reduced below an upper critical $H_{c2}$ and the system enters a XY-antiferromagnetic phase. Using a bond operator representation we consider a model spin-1 Heisenberg antiferromagnetic with single-ion anisotropy in hyper-cubic lattices under strong magnetic fields. We show that the transition at $H_{c2}$ can be interpreted as a Bose-Einstein condensation (BEC) of magnons. The theoretical results are used to analyze our magnetization versus field data in the organic compound $NiCl_2$-$4SC(NH_2)_2$ (DTN) at very low temperatures. This is the ideal BEC system to study this transition since $H_{c2}$ is sufficiently low to be reached with static magnetic fields (as opposed to pulsed fields). The scaling of the magnetization as a function of field and temperature close to $H_{c2}$ shows excellent agreement with the theoretical predictions. It allows to obtain the quantum critical exponents and confirm the BEC nature of the transition at $H_{c2}$.

Physical Review Letters, 2014

Han purple (BaCuSi 2 O 6 ) is not only an ancient pigment, but also a valuable model material for studying Bose-Einstein condensation of magnons in high magnetic fields. Using precise low-temperature structural data and extensive density-functional calculations, we elucidate magnetic couplings in this compound. The resulting magnetic model comprises two types of nonequivalent spin dimers, in excellent agreement with the 63;65 Cu nuclear magnetic resonance data. We further argue that leading interdimer couplings connect the upper site of one dimer to the bottom site of the contiguous dimer, and not the upper-to-upper and bottomto-bottom sites, as assumed previously. This finding is verified by inelastic neutron scattering data and implies the lack of frustration between the layers of spin dimers in BaCuSi 2 O 6 , thus challenging existing theories of the two-dimensional-like Bose-Einstein condensation of magnons in this compound.

We discuss the effect of interatomic interactions on the condensation temperature $T_c$ of a laboratory atomic Bose-Einstein condensate under the influence of an external trapping magnetic field. We predict that accounting for hyperfine interactions mediated Zeeman term in the mean-field approximation produces, in the case of the $403 \, G$ Feshbach resonance in the $|F,m_F&gt; = |1,1&gt;$ hyperfine state of a $^{39}K$ condensate, with $F$ the total spin of the atom, an experimentally observed (and not yet explained) shift in the condensation temperature $\Delta T_{c}/T_{c}^{0}=b^{*}_0+b^{*}_1 (a/\lambda_{T}) + b^{*}_2 (a/\lambda_{T})^2$ with $b^{*}_0 \simeq 0.0002$, $b^{*}_1 \simeq -3.4$ and $b^{*}_2 \simeq 47$, where $a$ is the s-wave scattering length, and $\lambda_T$ is the thermal wavelength at $T_{c}^{0}$. Generic expressions for the coefficients $b^*_0$, $b^*_1$ and $b^*_2$ are also obtained, which can be used to predict the temperature shift for other Feshbach resonances of ...

Physical Review B, 2012

In this paper we investigate the quantum phase transition from magnetic Bose glass to magnetic Bose-Einstein condensation induced by a magnetic field in NiCl2·4SC(NH2)2 (dichloro-tetrakis-thiourea-Nickel, or DTN), doped with Br (Br-DTN) or site diluted. Quantum Monte Carlo simulations for the quantum phase transition of the model Hamiltonian for Br-DTN, as well as for site-diluted DTN, are consistent with conventional scaling at the quantum critical point and with a critical exponent z verifying the prediction z = d; moreover the correlation length exponent is found to be ν = 0.75(10), and the order parameter exponent to be β = 0.95(10). We investigate the low-temperature thermodynamics at the quantum critical field of Br-DTN both numerically and experimentally, and extract the power-law behavior of the magnetization and of the specific heat. Our results for the exponents of the power laws, as well as previous results for the scaling of the critical temperature to magnetic ordering with the applied field, are incompatible with the conventional crossover-scaling Ansatz proposed by Fisher et al., [Phys. Rev. B 40, 546 (1989)], but they can all be reconciled within a phenomenological Ansatz in the presence of a dangerously irrelevant operator.

2004

Classic and recent results for the critical behaviour of ideal Bose gas at constant volume and constant pressure and for various spatial dimensionalities d>0 are reviewed. New results about the critical properties in a close vicinity of the $\lambda-$point are presented.