Quantum phase transitions of the asymmetric three-leg spin tube

We investigate quantum phase transitions of the $S=\frac{1}{2}$ three-leg antiferromagnetic spin tube with asymmetric interchain (rung) exchange interactions. On the basis of the electron tube system, we propose a useful effective theory to give the global phase diagram of the asymmetric spin tube. In addition, using other effective theories we raise the reliability of the phase diagram. The density-matrix renormalization-group and the numerical diagonalization analyses show that the finite spin gap appears in a narrow region around the rung-symmetric line, in contrast to a recent paper by Nishimoto and Arikawa, [Phys. Rev. B 78, 054421 (2008)]. The numerical calculations indicate that this global phase diagram obtained by use of the effective theories is qualitatively correct. In the gapless phase on the phase diagram, the numerical data are fitted by a finite-size scaling in the $c=1$ conformal field theory. We argue that all the phase transitions between the gapful and gapless phases belong to the Berezinskii-Kosterlitz-Thouless universality class.

Figure 1

Structure of the three-leg asymmetric spin tube (1).

Figure 2

Phase diagram obtained from the effective Hubbard model (2). In the strong-coupling limit $\left(U/t,U/s⪢1\right)$, the horizontal axis ${\beta }^2$ and vertical one ${\left(s/t\right)}^2$ can be regarded as $\alpha$ and $J_r/J_1$, respectively. The effective theory claims that the phases (II) and (IV) are gapless and the phase (III) is gapful. The phase (I) is possibly gapless.

Figure 3

(a) Scaled gap calculated by DMRG for $J_1=0.5$. (b) Difference of the scaled gaps between two systems with sizes $L_1$ and $L_2$.

Figure 4

Ground-state phase diagram of the asymmetric three-leg spin tube (1), derived from the numerical analysis. The phases (I)–(IV) correspond to those in Fig. 2. The cross points are determined by the DMRG. The three kinds of lines are done by the numerical diagonalization. The circle points are obtained from the level-spectroscopy method in Sec. 4C.

Figure 5

Central charge $c$ and critical exponents $L{\Delta }_t/\pi v_s$ and $L{\Delta }_s/\pi v_s$ including finite-size corrections for $J_1=0.3$. These are estimated from the numerical diagonalization up to $L=10$, based on the size dependence of low-lying energy spectra predicted by the CFT.

Figure 6

Critical points ${\alpha }_c$ evaluated by the level-spectroscopy analysis for $J_1=0.2$, assuming the size correction is proportional to $1/L^2$.

Figure 7

Critical value of $J_1$ for the regular triangle spin tube $\left(\alpha =1\right)$ evaluated by the level-spectroscopy analysis.