Introduction to the spin-Peierls transition in MgCr 2 O 4

5.1.1 Basic theory of the spin-Peierls transition

The minimal Hamiltonian of a spin-lattice coupled system can be written as [199, 201]:

H =X

〈i j〉

·

J(1−αρi j)Si·Sj+K 2ρ²_{i j}

¸

, (5.1)

whereρi j is the change in the bond length between the spinsSiandSj,αis the first derivative of the exchange coupling strength overρi j, andKis the elastic constant. Assuming a uniform

distortion, the equilibrium value ofρi j can be solved by: spin-lattice coupled Hamiltonian becomes purely spin-dependent [202]:

H =X

〈i j〉

J[S_i·S_j−b(S_i·S_j)²] , (5.3)

whereb =Jα²/2K is a dimensionless parameter that describes the strength of the spin-lattice coupling. The existence of the quartic coupling termb(S_i·S_j)²modifies the original Heisenberg exchange couplings and favors collinear structures as the ground state [201].

The real situation might be more complex than eqn. (5.1). Firstly, theρi jparameters, which describe the change in the bond length, are not independent from each other. When a magnetic ion at the siterimoves, all the neighbouring bonds will be elongated or shortened with it. One remedy is to treat the site displacementδr_i instead of the bond changeρi j as the independent variable [203–206]. With a minimization process similar to that of eqns.

(5.1-5.3), an additional spin quartic term appear, and the effective spin-only Hamiltonian can be expressed as:

where ˆei jis the unit vector along the undistorted bondi j. For the pyrochlore lattice, Monte Carlo simulations [206] for the Hamiltonian (5.4) have revealed lattice-mediated magnetic order withq=(110) and (0.5 0.5 0.5), where the latterq-vector corresponds to one of the magnetic domains observed experimentally in ZnCr2O4[207].

The second short-coming of the model (5.3) concerns the over-simplified lattice dynamics.

The number of phonon modes scales with the total number of atoms, and the softened mode, which determines the distorted structure, is not necessarily uniform across the lattice. One step beyond this over-simplification scenario is to allow anti-symmetric distortions over the neighbouring tetrahedra. References [199, 208] discuss such a possibility based on symmetry arguments. By absorbing the spin-interactions into the coefficients, the Landau free energy of the lattice system can be expressed phenomenologically. Rich phases with long-range ordered magnetic structures can be identified in this way, which include also the coplanar magnetic structures that cannot be derived from the simplified model of (5.3).

The existence of the effective quartic interactions has a direct impact on the spin dynamics.

As a first approximation, the Hamiltonian (5.3) for a distorted state can be re-interpreted with

5.1. Introduction to the spin-Peierls transition in MgCr2O4

Figure 5.1 –(a)Candidate spin-Peierls long-range ordered structure on the pyrochlore lattice.

The solid (dashed) lines show the shortened (elongated) bonds with stronger (weaker) anti-ferromagnetic couplingJ+δJ(J−δJ).(b)Density-of-states (DOS) of the spin wave excitations calculated for the magnetic structure shown in panel (a) withS=3/2. The dashed line shows the calculation results with a uniform nearest-neighbour couplingJ, where the high DOS near E=0 originates from the ground state degeneracy. The solid line shows the calculation results withδJ=0.2J, where the zero-energy degenerate states are raised to 8SδJ.

staggered couplings:

H =X

〈i j〉

(J±δJ)S_i·S_j, (5.5)

where the shortened (elongated) bonds receive stronger (weaker) anti-ferromagnetic couplings ofJ+δJ(J−δJ). Fig. 5.1 shows the calculated density-of-states (DOS) for the colinearly ordered pyrochlore with ferromagnetic spin chains along the [110] and [110] directions [199]. With δJ=0, that is without the spin-lattice couplings, a high DOS can be observed nearE=0, which is a result of the large ground state degeneracy. However, with finiteδJ, these degenerate states move to higher energies nearE=8SδJand form resonance-like excitations. Similar to the weather-vane modes on the kagomé lattice [209], these resonance-like excitations involve local fluctuations of ferromagnetic strings in real space [199], where the shortest closed loops are the hexagons in the kagomé layers shown in Fig. 1.9 [210].

5.1.2 Experimental studies of MgCr₂O₄

MgCr₂O₄belongs to the chromite familyACr₂X₄, which is well-known for its strong spin-lattice coupling [211]. The reason for its high sensitivity to spin-lattice distortions is the competition between direct exchange and superexchange interactions. Taking MgCr2O4as an example, its Cr³⁺–O²⁻–Cr³⁺superexchange path has an angle of∼95^◦in theF d3mphase, which is ferromagnetic according to the Goodenough-Kanamori rule. As a contrast, the direct exchange interaction, which heavily depends on the Cr³⁺–Cr³⁺distance, is anti-ferromagnetic due to the half-filledt_2gorbitals of the Cr³⁺ions. Therefore, a small lattice distortion is able to introduce a relatively large variation of the coupling strength inACr₂X₄, which finally results in its strong

a b

Figure 5.2 –(a)Magnetic excitations of MgCr2O4in the (hkl) plane withl=0, 1, and 2 at E=4.5, 9, 13.5, and 18.0 meV measured by inelastic neutron scattering atT=8 K below the spin-Peierls transition. (b)The calculated intensity distribution using the effective model of hexagons (4.5 meV) and heptamers (9.0, 13.5, and 18.0 meV). This figure is adapted from Ref. [217].

spin-lattice coupling.

Among the ACr₂X₄compounds, MgCr₂O₄and ZnCr₂O₄have the most similar properties.

At room temperature, close lattice parameters ofa=8.328 Å for MgCr2O4and 8.335 Å for ZnCr₂O₄are observed [212, 213]. Dc-susceptibility measurements have revealed very similar Curie-Weiss behaviour with parameters ofµeff=3.97µB andΘCW = −433 K for MgCr₂O₄and µeff=3.82µB andΘCW = −400 K for ZnCr2O4[212]. Even the spin-Peierls transitions in these two compounds occur at about the same temperature of∼12.7 K. Across the transition, the symmetry of both MgCr₂O₄and ZnCr₂O₄is lowered from cubic to tetragonal or orthogonal [213] with a shortenedc-axis. And the same dominant propagation vectors ofq=(0.5 0.5 0) and (1 0 0.5) for the magnetic structure have been observed by neutron diffraction [207, 214, 215]. Similarities between MgCr₂O₄and ZnCr₂O₄can also be found from the ultra-high-field magnetization measurements [216], which indicate comparable spin-lattice coupling strength in these two compounds [201].

Of special interest are the unconventional spin correlations in these two compounds. Above the spin-Peierls transition, neutron diffuse scattering data reveals that the spin fluctuations can be described by anti-ferromagnetic hexagons [218, 219]. Below the transition, these zero-energy modes rise in energy, forming resonance-like excitations nearE∼4.5 meV [200].

Besides the mode at 4.5 meV, three other resonance-like modes were recently identified at E=9, 13.5, and 18 meV [210, 217]. The measured intensity distribution presented in Fig. 5.2 can be effectively described by the excitations of hexamers and heptamers. Existence of the resonance-like excitations is very unusual for long-range ordered systems and calls for further