Quantum phase transition and criticality in quasi-one-dimensional spinless Dirac fermions

We study the quantum criticality of spinless fermions on a quasi-one-dimensional $\pi$-flux square lattice in cylinder geometry, by using the infinite density matrix renormalization group and Abelian bosonization. For a series of cylinder circumferences $L_y=4n+2=2,6,...$ with a periodic boundary condition, there are quantum phase transitions from gapped Dirac fermion states to charge density wave (CDW) states. We find that the quantum phase transitions for such circumferences are continuous and belong to the ($1+1$)-dimensional Ising universality class. On the other hand, when $L_y=4n=4,8,...$, there are gapless Dirac fermions at the noninteracting point and the phase transition to the CDW state is Gaussian. Both of these criticalities are described in a unified way by bosonization. We clarify their intimate relationship and demonstrate that a central charge $c=1/2$ Ising transition line arises as a critical state of an emergent Majorana fermion from the $c=2$ Gaussian transition point.

Figure 1

A schematic phase diagram including both insulating and (semi)metallic states. Generally, the blue and green phase transition lines and the red transition point would be characterized by different criticalities.

Figure 2

(a) An $L_y=4$ $\pi$-flux square lattice with the periodic boundary condition. The hopping on the black bonds is $-t$ and that on the red bonds is $+t$, which gives a $\pi$ flux for each square plaquette. (b) Schematic picture of the staggered CDW order. The blue circles represent the fermion particle density.

Figure 3

Single-particle dispersion relations (a) for $L_y=8$ and (b) for $L_y=10$ under the periodic boundary condition in the $y$ direction.

Figure 4

The CDW order parameter $\mathrm{\Delta }$ as a function of the interaction $V$ calculated by iDMRG with the periodic boundary condition for the $y$ direction. (a) $L_y=4n+2=6,10,14$ with $\chi =1000$ (red), 1600 (blue). For $L_y=2$, $\chi =100$ (red), 200 (blue). (b) $L_y=4n=4,8,12$ with $\chi =1000$ (red), 1600 (blue). Note that the data for $L_y=2,4$ with the different values of $\chi$ almost coincide in the present scale of the figures.

Figure 5

The scaling plots of the CDW order parameter for $L_y=2$. The correlation length ${\xi }_{\chi }$ is denoted as $\xi$ for simplicity. (a) The scaling plot Eq. (7), and the black line is $\mathrm{\Delta }\sim {\xi }^{-\beta /\nu }$ with $\beta =0.125$, $\nu =1$. (b) The scaling plot Eq. (8), and the black line is ${\partial }_g\mathrm{\Delta }/\mathrm{\Delta }\sim {\xi }^{1/\nu }$ with $\nu =1$. The $g$ derivative is approximated by ${\partial }_g\mathrm{\Delta }\left(V\right)=V_c[\mathrm{\Delta }(V+\delta V)-\mathrm{\Delta }(V-\delta V)]/2\delta V$ with $\delta V=0.0001$. The bond dimension is used up to $\chi \le 200$.

Figure 6

The scaling plots of the CDW order parameter for $L_y=6$. (a) The scaling plot Eq. (7) and (b) Eq. (8). The black lines are the same as in Fig. 5, while the $g$ derivative is approximated with $\delta V=0.001$. The bond dimension is used up to $\chi \le 2800$.

Figure 7

The scaling plots of the CDW order parameter for $L_y=10$. (a) The scaling plot Eq. (7) and (b) Eq. (8). The black lines are the same as in Fig. 5, while the $g$ derivative is approximated with $\delta V=0.005$. The bond dimension is used up to $\chi \le 2400$.

Figure 8

The scaling plot of the CDW order parameter $\mathrm{\Delta }$ for (a) $L_y=2$ and (b) $L_y=6$. The critical exponents used are those for the ($1+1$)D Ising universality class $\beta =0.125$, $\nu =1$.

Figure 9

The entanglement entropy $S$ for (a) $L_y=2$ and (b) $L_y=6$. The black lines are $S=(c/6)ln\xi +S_0$ with the central charge $c=1/2$.

Figure 10

(a) Schematic global phase diagram in the $V\text{−}{t}_{\perp }$ plane in ($1+1$)D. The red point at the origin is the $c=2$ Gaussian transition point, and the green curve is the $c=1/2$ Ising transition line separating the band insulator and CDW state. The arrows correspond to the renormalization group flow in the effective low-energy model. (b) Expected phase diagram in ($2+1$)D. $t^{\prime }$ is an additional hopping which induces a band gap.