Quantum phase transitions in multileg spin ladders with ring exchange

Four-spin exchange interaction has been raising intriguing questions regarding the exotic phase transitions it induces in two-dimensional quantum spin systems. In this context, we investigate the effects of a cyclic four-spin exchange in the quasi-1D limit by considering a general $N$-leg spin ladder. We show by means of a low-energy approach that depending on its sign, this ring exchange interaction can engender either a staggered or a uniform dimerization from the conventional phases of spin ladders. The resulting quantum phase transition is found to be described by the SU(2)${}_N$ conformal field theory. This result, as well as the fractional value of the central charge at the transition, is further confirmed by a large-scale numerical study performed by means of exact diagonalization and density matrix renormalization group approaches for $N\le 4$.

Figure 1

$N$-leg spin ladder with leg coupling $J_{\parallel }$, rung coupling $J_{\perp }$, and a ring-exchange interaction $K$.

Figure 2

$N=2$ and $J_{\perp }=1$: crossing point with PBC (circles) and TBC (squares) vs system size (see text for definitions). The most precise extrapolations are obtained using TBC data and lead to a critical point at $K=0.26$ (dashed line).

Figure 3

$N=2$ and $J_{\perp }=1$. Finite-size scaling of various quantities at the critical point $K=0.26$: (a) ground-state energy per rung $e_0$, (b) velocity $v$, and (c) scaling dimension $x$ of the SU(2)${}_2$ WNZW primary field (see text for details). Data are compatible with CFT behavior and extrapolations obtain $x=0.375$ and the central charge $c=1.52$.

Figure 4

$N=2$ and $J_{\perp }=1$: dimerization at the center for $N=2$ for various values of $K$. The critical point corresponds to a power-law decay (see fit, dashed line) for $K=0.255$ (black).

Figure 5

$N=2$ and $J_{\perp }=1$: von Neumann entropy vs conformal distance $d\left(x\right|L)$ at the critical point for $K=0.255$.

Figure 6

$N=2$ and $J_{\perp }=-1$: crossing point with PBC (circles) and TBC (squares) vs system size (see text for definitions). The most precise extrapolations are obtained using TBC data and lead to a critical point at $K=-0.28$ (dashed line).

Figure 7

$N=2$ and $J_{\perp }=-1$. Finite-size scaling of various quantities at the critical point $K=-0.28$: (a) ground-state energy per rung $e_0$, (b) velocity $v$, and (c) scaling dimension $x$ of the SU(2)${}_2$ WNZW primary field (see text for details). Data are compatible with CFT behavior and extrapolations obtain $x=0.376$ and the central charge $c=1.55$.

Figure 8

$N=2$ and $J_{\perp }=-1$: dimerization at the center for various $K$. Critical point is found at $K/J_{\parallel }=-0.275$ with a power-law behavior, possibly with logarithmic corrections (see text).

Figure 9

$N=2$ and $J_{\perp }=-1$: von Neumann entropy vs conformal distance $d\left(x\right|L)$ at the critical point $K=-0.275$. Oscillations prevent from a reliable fit of the central charge.

Figure 10

$N=2$ and $J_{\perp }=-1$: same data as in Fig. 9 after removing the oscillations due to the bond modulations using $S_{vN}+\langle S_i·S_{i+1}\rangle$, see Eq. (29). The fit of the central charge (dashed line) then gives $c=1.53$.

Figure 11

$N=3$ and $J_{\perp }=1$: dimerization at the center for various $K$. Power-law decay allows to locate the critical point for $K/J_{\parallel }=0.37$.

Figure 12

$N=3$ and $J_{\perp }=1$: von Neumann entropy vs conformal distance $d\left(x\right|L)$ at the critical point for various sizes. Again, oscillations were removed (see text).

Figure 13

$N=3$ and $J_{\perp }=1$. Finite-size scaling of various quantities at the critical point $K=0.37$: (a) ground-state energy per rung $e_0$, (b) velocity $v$, and (c) scaling dimension $x$ of the primary CFT field (see text for details).

Figure 14

$N=3$ and $J_{\perp }=-1$. Finite-size scaling of various quantities at the critical point $K=-0.28$: (a) ground-state energy per rung $e_0$, (b) velocity $v$, and (c) scaling dimension $x$ of the primary CFT field (see text for details). Inset: scaling of the level crossing vs system size.

Figure 15

$N=3$ and $J_{\perp }=-1$: dimerization at the center for various $K$. The critical point corresponds to a power-law decay for $K/J_{\parallel }=-0.275$.

Figure 16

$N=3$ and $J_{\perp }=-1$: von Neumann entropy plus bond energy vs conformal distance $d\left(x\right|L)$ at the critical point $K=-0.275$ for $L=128$.

Figure 17

$N=4$ and $J_{\perp }=1$: crossing point with PBC and TBC vs system size (see text for definitions). Extrapolations are difficult but point towards a critical value between $0.3<K_c<0.5$.

Figure 18

$N=4$ and $J_{\perp }=1$: dimerization at the center for various $K$. Critical point corresponds to a power-law decay for $K/J_{\parallel }\simeq 0.4$

Figure 19

$N=4$ and $J_{\perp }=1$: von Neumann entropy vs conformal distance $d\left(x\right|L)$ at the critical point $K=0.4$ for various sizes.

Figure 20

$N=4$ and $J_{\perp }=1$. Finite-size scaling of various quantities at the critical point $K=0.4$: (a) ground-state energy per rung $e_0$, (b) velocity $v$, and (c) scaling dimension $x$ of the primary CFT field (see text for details).

Figure 21

$N=4$ and $J_{\perp }=-1$: crossing point with PBC and TBC vs system size (see text for definitions). Both PBC and TBC allow for an accurate extrapolation to $K_c=-0.28$.

Figure 22

$N=4$ and $J_{\perp }=-1$. Finite-size scaling of various quantities at the critical point $K=-0.28$: (a) ground-state energy per rung $e_0$, (b) velocity $v$, and (c) scaling dimension $x$ of the primary CFT field.

Figure 23

$N=4$ and $J_{\perp }=-1$: dimerization at the center for various $K$. Critical point corresponds to $K/J_{\parallel }=-0.272$ with a power-law decay with exponent $x=1/4$ up to logarithmic corrections.

Figure 24

$N=4$ and $J_{\perp }=-1$: von Neumann entropy plus bond energy vs conformal distance $d\left(x\right|L)$ at the critical point $K=-0.272$ for various sizes.