Spin locking and freezing phenomena in the antiferromagnetic Heisenberg model on the three-leg ladder

The antiferromagnetic Heisenberg model on the three-leg ladder is studied using the generalized Jordan-Wigner transformation in dimensions higher than 1, and the bond-mean-field theory. The magnetic susceptibility and other thermodynamic quantities are analyzed as a function of the rung-to-leg coupling ratio $\alpha$ and temperature $T$. We fit the experimental susceptibility data of the three-leg material ${\mathrm{Sr}}_2{\mathrm{Cu}}_3{\mathrm{O}}_5$ of Azuma and co-workers with good agreement. One of the main findings of this work is the proposal that close to two-thirds of the spin degrees of freedom on each of the rungs of the ladder lock at low $T$ for small $\alpha$, then collectively almost $2∕3$ of the spin degrees of freedom on all the rungs freeze completely at low $T$ for $\alpha$ greater than a threshold value. The approach developed here can be used to study the three-leg ladder for all values of $\alpha$, and is thus suitable for the description of the crossover regime between the weak- and strong-coupling regimes.

Figure 1

Sites within the shaded area enter in the summation in the expression of the phase ${\Phi }_{i,1}$ of $S_{i,1}$ in the JW transformation (2) for the three-leg ladder. The phases for sites $S_{i,2}$ and $S_{i,3}$ are obtained by adding to ${\Phi }_{i,1}$, $\pi n_{i,1}$, and $\pi (n_{i,1}+n_{i,2})$, respectively.

Figure 2

The mean-field parameters $Q$, $Q^{\prime }$, and $P$ are plotted as functions of $\alpha =J_{\perp }∕J$ at zero temperature.

Figure 3

The parameters $Q$, $Q^{\prime }$, and $P$ are plotted as a function of temperature $(T∕J)$ for four values of $\alpha$. For $\alpha =0$, $P=0$ and $Q=Q^{\prime }$ at any temperatures.

Figure 4

The quantity $Q-Q^{\prime }$ is plotted as a function of temperature for three values of $\alpha$. $Q-Q^{\prime }$ becomes significantly greater than zero at low temperature.

Figure 5

The excitation spectra $\mid E_j\left(k\right)\mid$, with $j=1,2,3$, are plotted as a function of $k$ for $\alpha =1$ in (a) and 3 in (b).

Figure 6

The energy gap in $E_2$ or $E_3$ is drawn versus $\alpha$.

Figure 7

The effective coupling constant $J_{\mathit{eff}}$ is plotted as a function of $\alpha$.

Figure 8

The susceptibility is drawn versus temperature for various values of $\alpha$: (a) $0<\alpha <0.4$, (b) $0<\alpha <10$.

Figure 9

(Color online) Low-$T$ schematic representation of spin locking (a) and freezing (b). (a) $\alpha <{\alpha }_{\mathit{th}}$: there is no preferred location for the “singlets;” each “singlet” can jump back and forth between the two top and two bottom chains. We say that the singlets form a fluid state. (b) $\alpha >{\alpha }_{\mathit{th}}$: the “singlets” occupy either the two top (like here) or the two bottom chains; the three-leg ladder in this case behaves as a combination of a single chain and a two-leg ladder. A dashed ellipse symbolizes a pair of spins locked into a singlet.

Figure 10

The susceptibility is drawn versus temperature for values of $\alpha$ near the threshold value ${\alpha }_{\mathit{tr}}\approx 2.75$.

Figure 11

The temperature $T_{\mathit{max}}$, at which the maximum (the first of the maxima for $\alpha >{\alpha }_{\mathit{tr}}$) of $\chi \left(T\right)$ takes place, is plotted as a function of $\alpha$.

Figure 12

The susceptibility $\chi \left(T\right)$ is drawn versus temperature for $\alpha =10$. It is compared with the contribution ${\chi }_{\mathit{eff}}$ from the $E_1$ term in Eq. (15). For $T<2J$, ${\chi }_{\mathit{eff}}$ coincides exactly with the total susceptibility $\chi \left(T\right)$.

Figure 13

The experimental magnetic susceptibility of ${\mathrm{Sr}}_2{\mathrm{Cu}}_3{\mathrm{O}}_5$ from Ref. 13 is shown as circles. The solid line is our theoretical fit obtained using the coupling constants $J∕k_B=1700\mathrm{K}$ and $\alpha =J_{\perp }∕J=0.4$. For comparison the dotted line is the susceptibility calculated for the one-dimensional $(J_{\perp }∕J=0)$ case with $J∕k_B=1700\mathrm{K}$.

Figure 14

(a) The entropy is displayed as a function of temperature for several values of $\alpha$. (b) The entropy for $\alpha =10$ is compared to the entropy contribution of the lower-energy band $E_1$. Both entropies are equal for $T<2J$.

Figure 15

The specific heat is drawn as a function of $T$: (a) for $0⩽\alpha ⩽1$, (b) for greater values of $\alpha$ together with the $\alpha =0$ curve.

Figure 16

The specific heat for $\alpha =10$ is compared to the contribution from the lower-energy band $E_1\left(k\right)$. The maximum at the higher temperature of $C\left(T\right)$ is a consequence of the bands $E_2$ and $E_3$.

Figure 17

The WS ratio $R_W$ is plotted as a function of $T$ for various values of the rung-to-leg couplings ratio $\alpha$.

Figure 18

The ratio $R_W$ is compared to the ratio $R_W^{\mathit{eff}}$ of the Heisenberg chain with the effective coupling $J_{\mathit{eff}}$.