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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S={1 \over 2}} form with a singlet Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=0} ground state and triplet Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=1} excitations called magnons (some people call them triplons). Some examples of this system are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle TlCuCl_3} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle BaCuSi_2O_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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle TlCuCl_3} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle BaCuSi_2O_6} . The lattice of magnetic ions can be regarded as a set of dimers carrying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S={1 \over 2}} each. We assume the Hamiltonian is in the form .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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^z_{m,i},\quad\quad\quad (1) }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_0} is the intra-dimer exchange coupling which is positive because this is antiferromagnetic system. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_{mnij}} denotes the spin-spin interaction coupling, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_B} is the usual Bohr magneton, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} denotes an external magnetic field in z-direction. For the indexes, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i,j} are number dimers, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=0} and a triply degenerate excited state of spin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=1} and energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_0} (Fig. 1c). In the quasiparticle language, the triplet states can be identified with the presence of triplons which are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_{1,2,...}>0} .The interdimer couplings Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_{1,2,...}} can be constructed by summing over spin-spin interactions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_{mnij}} (Fig. 1b). In the case of dimers forming a square lattice, the energy of a triplon with spin projection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^z=0, \pm 1} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(\mathbf k) = J_0 + J_1[\cos{(k_x a)} + \cos{(k_y a)}]- g \mu_B H S^z, \quad\quad\quad (2) }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf k =(k_x,k_y)} is the wavevector of particle, a is the lattice constant, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta$} (Fig. 1c). The external magnetic field reduces the gap between singlet and triplet states according to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta (h)= \Delta - g\mu_B H.\quad\quad\quad (3) }