Magnetostriction in an array of spin chains under a magnetic field

We consider an array of $XX$ spin-$1∕2$ chains coupled to acoustic phonons and placed in a magnetic field. Treating the phonons in the mean field approximation, we show that this system presents a first-order transition as a function of the magnetic field between a partially magnetized distorted state and the fully polarized undistorted state at low temperature. This behavior results from the magnetostriction of the coupled chain system. A dip in the elastic constant of the material near the saturation field along with an anomaly in the magnetic susceptibility is predicted. We also predict the contraction of the material as the magnetic field is reduced (positive magnetostriction) and the reciprocal effect, i.e., a decrease of magnetization under applied pressure. At higher temperature, the first-order transition is replaced by a crossover. However, the anomalies in the susceptibilities in the system near the saturation field are still present. We discuss the relevance of our analysis in relation to recent experiments on spin-$1∕2$ chain and ladder materials in high magnetic fields.

Figure 1

The ground state energy as a function of $\delta$ for different values of the magnetic field $h$. (a) For low field, there is a single minimum at $\delta <0$. (b) For a higher field $h>h_{c_{<}}$, a second minimum, at a higher value of the energy is present. This second minimum corresponds to a metastable state. (c) For a critical value of the magnetic field, the two minima become degenerate. (d) Above the critical field, the absolute minimum of the ground state energy is for $\delta =0$. A metastable minimum exists for $\delta <0$. (e) For $h=h_{c_{>}}$, the minimum at $\delta <0$ becomes unstable. (f) Above $h_{c_{>}}$, the only minimum is for $\delta =0$. In the range of $h_{c_{<}}<\delta <h_{c_{>}}$, hysteresis in $\delta$ as $h$ is varied is expected.

Figure 2

Striction for $T=0,P=0,K=1,J=1$. This is obtained from the solution of Eq. (20). It does not correspond to the thermodynamic striction which presents a jump for $h=h_c<h_{c_{>}}$.

Figure 3

Magnetization for $T=0,P=0,K=1,J=1$. This is obtained from the solution of Eq. (20) and this is not the thermodynamic magnetization.

Figure 4

Hysteresis cycle for $T=0$ and $K=J=1$, $P=0$. The thermodynamic transition happens for $h=h_c\simeq 1.03J$; however, the distorted state is metastable for $h_c<h<h_{c_{>}}\simeq 1.045J$ and the undistorted state is metastable in the region $h_{c_{<}}=J<h<h_c$. As the field $h$ is increased from 0 to $h_{c_{>}}$, the system remains on the metastable branch, and the magnetization jumps to $1∕2$ at $h=h_{c_{>}}$. As the magnetic field is decreased, the magnetization drops from $1∕2$ at $h=h_{c_{<}}$. This produces a hysteresis cycle in the magnetization. A similar hysteresis cycle is expected for the lattice distortion.

Figure 5

The effective elastic constant as a function of the magnetic field. Below the saturation field $h_c$, the elastic constant is very sharply reduced. For $h=h_c$, the effective elastic constant jumps to $K$. The curve is drawn for $J=K=1$.

Figure 6

The Joule coefficient $\Lambda ={(\partial M∕\partial p)}_h=-{(\partial \delta ∕\partial h)}_p$ as a function of the magnetic field. We see that applying pressure results in a decrease of the magnetization, which indicates a positive magnetostriction. The plot is done for $J=K=1$.

Figure 7

The magnetic susceptibility as a function of the magnetic field. It diverges at the critical field where the transition takes place. The plot is done for $J=K=1$.

Figure 8

The evolution of the free energy as a function of temperature for $K=J=1$ and $h=1.04$. At zero temperature, the double minima are clearly visible. The absolute minimum corresponds to $\delta =0$ indicating a nonmagnetized phase, without deformation. For not too large positive temperature, the two minima are still visible, but the barrier between them becomes smaller. Also, one notices that the minimum of the free energy corresponds to a solution with $\delta <0$, indicating a contraction of the lattice as $T$ increases, i.e., a negative dilatation coefficient. As temperature is further increased, i.e., for $T=0.03J$, there is only a single minimum indicating the disappearance of the first-order transition.

Figure 9

The striction as a function of the magnetic field for various temperatures and $K=J=1$. For $T=0.01$, the discontinuity becomes smaller than for $T=0$ but it is still present. For $T=0.03$ the discontinuity disappears, and is replaced by a rapid but continuous change of $\delta$. For $T=0.1$, the change becomes even more gradual. One can notice that $\delta$ reaches zero at higher field when the temperature is increased.

Figure 10

The magnetization as a function of the magnetic field for various temperatures and $K=J=1$. For $T=0.01$, the discontinuity is smaller, but it is still present. For $T=0.03$ the discontinuity disappears, and is replaced by a rapid but continuous change of $M$. As temperature is further increased, the change of $M$ becomes more gradual. One can notice that as $T$ is increased, the saturation field becomes higher.

Figure 11

Derivative of the lattice parameter with respect to $h$ and varying the temperature. The plot is done for $J=K=1$. Below the transition temperature, the discontinuity is visible. Above the transition temperature, the discontinuity is replaced by a maximum. As the temperature increases, this maximum is lowered, and the region of anomaly is broadened.

Figure 12

The magnetic susceptibility vs $h$ for varying $T$. At low $T=0.01$, the magnetic susceptibility possesses a discontinuity. It is zero in the fully polarized phase, and becomes very large in the partially polarized state as the Fermi wave vector of magnetic excitations is nearing the band edge when the magnetization is approaching saturation. For $T=0.03$, the discontinuity is replaced by a peak. As temperature is further increased, the height of the peak diminishes, while its breadth increases.

Figure 13

The effective elastic constant as a function of the magnetic field and different temperatures. The curve is drawn for $J=K=1$. We notice that in the fully polarized state at low temperature, the elastic constant quickly returns to its value in the absence of interaction with magnetic excitations. A jump of elastic constant is observed as the magnetic field is crossing the saturation value. At higher temperature, the jump is replaced by a minimum in the elastic constant. The minimal value of the elastic constant is an increasing function of $T$. We also note that the width of the region where the elastic constant decreases is larger as $T$ increases.

Figure 14

The specific heat for $T=0.01,0.03,0.05$ and $P=0$. We have $K=J=1$. The specific heat becomes very large near the transition. Above the transition temperature, a peak can be observed in the specific heat. The height of this peak diminishes as $T$ increases, whereas its breadth increases.

Figure 15

Dilatation coefficient for $T=0.01,0.03,0.05$ for $P=0$ and $K=J=1$ as a function of the magnetic field. The dilatation coefficient vanishes in the fully polarized phase, and is negative in the partially polarized phase. We notice that the dilatation coefficient decreases strongly as temperature is increased. The behavior of the dilatation coefficient is nonmonotonic as a function of the magnetic field.