Phy5670/Bose-Einstein Condensation in Spin-gaped Systems

Introduction

Bose-Einstein theory describes the behaviour of integer spin objects (bosons). This theory predicted the so-called Bose-Einstein Condensation (BEC) phenomenon. Bose Einstein condensates is one of exotic ground states in strongly correlated systems. At first, this condensation concept was applied to dilute gases of bosons which are weakly interacting. Those gases were confined in an external potential and cooled to temperatures very near to absolute zero. These cooling bosonic atoms then fall (or "condensate") into the lowest accessible quantum state, resulting in a new form of matter. One example of these gases is helium-3.

Not long after the aplications of Bose and Einstein statistics to photons and atoms, Bloch applied the same concept to excitations in solid. He explained that the state of misaligned spins in a ferromagnet can be regarded as magnons, quasiparticles with integer spin and bosonic statistics. In 1965 paper, Matsubara and Matsuda pointed out the correpondences between a quantum ferromagnet and a lattice Bose gas .

The similarity between the Bose gases and magnons suggests that magnons can undergo a process like Bose-Einstein condensation. However, in this case we are only considering simple spin systems, if we want to assume more realistic cases, such factors like anisotropies could restrict the usefulness of BEC concept.

Nevertheless, the analogy between bosons and spins has been very useful in antiferromagnetic systems which closely spaced pairs of spins $S={1 \over 2}$ form with a singlet $S=0$ ground state and triplet $S=1$ excitations called magnons (some people call them triplons). Some examples of this system are $TlCuCl_{3}$ and $BaCuSi_{2}O_{6}\$$.

Here I present an overview of BEC in antiferromagnetic systems.

Bosons in Magnets

In this part we will explain the basics of magnon BEC in real dimerized antiferromagnets, such as $TlCuCl_{3}$ and $BaCuSi_{2}O_{6}$. The lattice of magnetic ions can be regarded as a set of dimers carrying $S={1 \over 2}$ each. We assume the Hamiltonian is in the form .

$\mathbf {H} =\sum _{i}J_{0}\,\mathbf {S} _{1,i}\cdot \mathbf {S} _{2,i}+\sum _{\langle mnij\rangle }J_{mnij}\,\mathbf {S} _{m,i}\cdot \mathbf {S} _{n,j}-g\mu _{B}H\sum _{\langle mi\rangle }S_{m,i}^{z},\quad \quad \quad (1)$

where $J_{0}$ is the intra-dimer exchange coupling which is positive because this is antiferromagnetic system. $J_{mnij}$ denotes the spin-spin interaction coupling, $\mu _{B}$ is the usual Bohr magneton, and $H$ denotes an external magnetic field in z-direction. For the indexes, $i,j$ are number dimers, and $m,n=1,2$ denote their magnetic sites.

Here, the intra-dimer exchange is the strongest interaction. The system is antiferromagnetic which means that its isolated dimer has a ground state with total spin $S=0$ and a triply degenerate excited state of spin $S=1$ and energy $J_{0}$ (Fig. 1c). In the quasiparticle language, the triplet states can be identified with the presence of triplons which are $S=1$ bosonic particles, and the singlet states are states with the absence of triplons (Fig. 1b). With the assumption that inter-dimer interactions are weak, non-magnetic singlets ground state is disordered down to zero kelvin temperature with no long-range magnetic ordering. The triplon interacting with each other through weak interdimer couplings $J_{1,2,...}>0$.The interdimer couplings $J_{1,2,...}$ can be constructed by summing over spin-spin interactions $J_{mnij}$ (Fig. 1b). In the case of dimers forming a square lattice, the energy of a triplon with spin projection $S^{z}=0,\pm 1$ is

$\varepsilon (\mathbf {k} )=J_{0}+J_{1}[\cos {(k_{x}a)}+\cos {(k_{y}a)}]-g\mu _{B}HS^{z},\quad \quad \quad (2)$

where $\mathbf {k} =(k_{x},k_{y})$ is the wavevector of particle, a is the lattice constant, and $D=4J_{1}$ is the bandwidth (Fig 1.c). The dispersion relation of the triplons and singlet-triplet correlations can be measured directly by inelastic neutron scattering

With the assumption that the system is isotropic, the spin singlet ground state is separated from the first excited triplet by a gap $\Delta \$$ (Fig. 1c). The external magnetic field reduces the gap between singlet and triplet states according to

$\Delta (h)=\Delta -g\mu _{B}H.\quad \quad \quad (3)$

When $\Delta (h)=0$, the system undergoes the antiferromagnetic ordering in which can be viewed as triplons condensation . We can see clearly that the external magnetic field controls the density of triplons. From figure 1c , the excitation energy of triplons with $S^{z}=1$ is lowered and eventually crosses zero as the magnetic field increases. There are two critical magnetic fields $H_{c1}$ and $H_{c2}$ in the phase diagrams (Fig 1d). We take consideration at $T=0K$, below $H_{c1}$ the system is in disordered phase, the magnetization $m_{z}(H)$ is zero and only singlets exist. Between $H_{c1}$ and $H_{c2}$ , the magnetization starts increasing and the triplon band fills up (Fig. 1c). Above $H_{c2}$, the triplon band is full and the magnetisation saturates.

The bosonic nature of triplons is guaranted by the simple fact that spin operators of two different dimers commute. However, the bosonic representation of dimers require a hard-core constraint to exclude states with more than quasi-particle per dimer.