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An experimental condensed matter physicist whose measurements of field-induced ordering in $\mathrm{TlCuC}{\mathrm{l}}_3$ motivated a magnon Bose-Einstein condensation interpretation. The subsequent collaborative model reproduced the unusual temperature dependence of magnetization, linking an experimentally measured bulk response to a dilute gas of magnetic excitations.^[ref]

An experimental condensed matter physicist associated with measurements of field-induced antiferromagnetism in $\mathrm{TlCuC}{\mathrm{l}}_3$. With Akira Oosawa, Tetsuro Nikuni and Masaki Oshikawa, he connected this ordering to Bose-Einstein condensation of magnons: varying the magnetic field controls a gas of magnetic excitations in a solid.^[ref]

Masaki Oshikawa coauthored a theory interpreting field-induced Néel order in thallium copper chloride as Bose-Einstein condensation of dilute magnons. An applied magnetic field changes the energy cost of these magnetic excitations, allowing their population to become large enough for collective ordering. The 2000 work compared a dilute-boson mean-field theory with the measured temperature dependence of magnetization. This provides a connection between an insulating magnet and the collective quantum mechanics of a gas of bosons, without requiring the magnetic material to become a fluid.^[ref]

For a single quantum state, the equilibrium occupations areBoth reduce to $e^{(\mu -ϵ)/(k_{B}T)}$ when $ϵ-\mu \gg k_{B}T$.

For one three-dimensional quadratic branch, the density of states per unit volume isWith ${\lambda }_T=\sqrt{2\pi {\hslash }^2/(m{k}_{B}T)}$, changing integration variable to $x=ϵ/(k_{B}T)$ givesA degeneracy factor multiplies both densities if additional equivalent branches are present. The first expression is the noncondensed density: a macroscopically occupied ground state must be counted separately.

Bose-Einstein condensation occurs when the excited states can no longer hold all particles and the ground-state occupation becomes macroscopic. At fixed total density,For field-induced triplon condensation, write $\Delta =\nu n-\eta (H-H_g)$, so $ϵ(k)={\hslash }^2{k}^2/(2m)+\Delta$. At fixed reference $n$, the three requested dispersions have gaps $\nu n+\eta H_g$, $\nu n$ and zero respectively. Their curvature is unchanged. Because the actual thermodynamic chemical potential is zero, the Bose distribution requires $\Delta \ge 0$. At $H_c=H_g+\nu n/\eta$, the minimum reaches zero and the noncondensed population saturates, permitting macroscopic occupation.

For the triplon condensation field scaling, the critical density at onset is $n_T=\zeta (3/2)/{\lambda }_T^3$. ThereforeThe low-temperature phase boundary rises from $(T,H)=(0,H_g)$; the condensed phase lies on the higher-field side.

For $H>H_c$, simply continuing the gap below zero would give unphysical Bose occupations. For the condensate density in a shifted-parabola triplon model, added condensate density keeps the minimum at zero: $\nu (n_0+n_T)=\eta (H-H_g)$. At fixed temperature this giveswith $n_0=0$ below onset. At zero temperature it becomes $n_0=\eta (H-H_g)/\nu$ within that model.

This prefactor is conditional on continuing the stated single-density mean-field shift into the condensed phase. A microscopic interacting Bose condensate has collective Bogoliubov excitations $E_k=\sqrt{{ϵ}_k^0({ϵ}_k^0+2g{n}_0)}$, with ${ϵ}_k^0={\hslash }^2{k}^2/(2m)$, which are linear at small $k$. The normal-state shift alone does not fix the condensed interaction closure or identify $g$ with $\nu$. The gap closes and the condensate density grows linearly with excess field in the simple mean-field description; the detailed collective dispersion needs that additional interaction model. A dilute-boson treatment of field-induced magnetic ordering makes this normal/condensed distinction explicitly.^[ref]

A theoretical condensed matter physicist who coauthored a dilute-magnon description of field-induced ordering in $\mathrm{TlCuC}{\mathrm{l}}_3$. Its mean-field theory connected the measured magnetization to Bose-Einstein condensation. The magnetic field permits the excitation population to vary, providing a grand canonical ensemble setting for condensation physics.^[ref]