Superuniversality of dimerized $\mathrm{SU}(N+M)$ spin chains

We explore the physics of the quantum Hall effect using the Haldane mapping of dimerized $\mathrm{SU}(N+M)$ spin chains, the large $N$ expansion, and the density matrix renormalization group technique. We show that while the transition is first order for $N+M>2$, the system at zero temperature nevertheless displays a continuously diverging length scale $\xi$ (correlation length). The numerical results for $(N,M)=(3,1),(2,2),(5,1)$, and $(7,1)$ indicate that $\xi$ is a directly observable physical quantity, namely the spatial width of the edge states. We relate the physical observables of the quantum spin chain to those of the quantum Hall system (and, hence, the $\vartheta$ vacuum concept in quantum field theory). Our numerical investigations provide strong evidence for the conjecture of superuniversality which says that the dimerized spin chain quite generally displays all the basic features of the quantum Hall effect, independent of the specific values of $N$ and $M$. For the cases at hand we show that the singularity structure of the quantum Hall plateau transitions involves a universal function with two scale parameters that may in general depend on $N$ and $M$. This includes not only the Hall conductance but also the ground state energy as well as the correlation length $\xi$ with varying values of $\vartheta \sim \pi$.

Figure 1

Interacting dimerizd spin chain.

Figure 2

Sketch of $\left|\varepsilon \right|=m\left(L\right)$ (solid red line) with $m\left(L\right)$ given by Eqs. (13) and (14). This line defines the correlation length $\xi \left(\varepsilon \right)$ which diverges continuously as $\varepsilon \to 0$, see text.

Figure 3

DMRG results for the dimensionless ground state energy (${\mathcal{E}}_g$) vs dimerization parameter ($\varepsilon$) for different $N,M$ values and system sizes ($L$). The dashed lines represent the large $N$ saddle point result of Eq. (2).

Figure 4

DMRG data of $log[1-\mathbf{P}(j\left)\right]$ vs $j^2$, see text. The solid lines represent the optimal fitting based on Eq. (20).

Figure 5

DMRG results for the Hall conductance (${\tilde{\sigma }}_H$) vs dimerization parameter ($\varepsilon$) for different $N,M$ values and system sizes ($L$).

Figure 6

DMRG results for the Hall conductance (${\tilde{\sigma }}_H$) vs system size ($L$) for $N=M=2$ and different values of the dimerization parameter ($\varepsilon$). The solid lines with $0.3\lesssim {\tilde{\sigma }}_H\lesssim 0.7$ are the best fit based on the function ${\tilde{\sigma }}_H=\frac{1}{2}\pm {\mathrm{\Gamma }}_{1}exp\{{\beta }_{1}L+2.1lnL\}$ with positive coefficients ${\mathrm{\Gamma }}_1\propto \left|\varepsilon \right|$ and ${\beta }_1$. Those with ${\tilde{\sigma }}_H\lesssim 0.3$ and ${\tilde{\sigma }}_H\gtrsim 0.7$ have been obtained using ${\tilde{\sigma }}_H={\mathrm{\Gamma }}_{2}exp\{-{\beta }_2{L}^2\}$ and ${\tilde{\sigma }}_H=1-{\mathrm{\Gamma }}_{2}exp\{-{\beta }_2{L}^2\}$, respectively, with positive ${\mathrm{\Gamma }}_2$ and ${\beta }_2$.

Figure 7

DMRG results for the functions $F_{\text{Energy}}\left(L\right)$ and $F_{\text{Edge}}\left(\xi \right)$ for different $N,M$ values, see text. The solid lines are optimal fittings based on Eqs. (26) and (28).

Figure 8

DMRG results for the functions $F_{\text{Hall}}\left(L\right), F_{\text{Energy}}\left(L\right)$, and $F_{\text{Edge}}\left(\xi \right)$ with $N=M=2$, see text. The solid lines are optimal fittings based on Eqs. (26) and (28).

Figure 9

Collapse of the nine different data sets of Figs. 7 and 8 after rescaling. The solid line represents the function $F\left(X\right)$, Eq. (28), with $X=L$ or $\xi$.