Qubit regularization of the $O(3)$ sigma model

We construct a qubit regularization of the $O(3)$ nonlinear sigma model in two and three spatial dimensions using a quantum Hamiltonian with two qubits per lattice site. Using a worldline formulation and worm algorithms, we show that in two spatial dimensions our model has a quantum critical point where the well-known scale-invariant physics of the three-dimensional Wilson-Fisher fixed point is reproduced. In three spatial dimensions, we recover mean-field critical exponents at a similar quantum critical point. These results show that our qubit Hamiltonian is in the same universality class as the traditional classical lattice model close to the critical points. Simple modifications to our model also allow us to study the physics of traditional lattice models with $O(2)$ and $Z_2$ symmetries close to the corresponding critical points.

Figure 1

Schematic of how qubit regularizations fit into the usual picture of Wilson’s renormalization group ideas. The two lines shown as Qubit Regularization 1 and Qubit Regularization 2 show a set of qubit Hamiltonians where one parameter is varied. These are not RG flow lines, which are shown with arrows.

Figure 2

The zero temperature phase diagram of our qubit Hamiltonian in $d=2$, 3.

Figure 3

Illustration of a worldline configuration (left) and a worm (or defect) configuration (right) in one spatial dimension. Positive time direction is upwards. The worm is shown as the thick solid line with a head (open circle) and a tail (filled square). The lines without a head or a tail show particle worldlines, while sites with a Fock vacuum (singlets) are shown as filled circles. Particle with charge $m=\pm 1$ are shown as oriented worldlines while those with charge $m=0$ are unoriented.

Figure 4

Possible local moves for the worm head. The configurations shown follow the same convention as Fig. 3. Starting from the configuration shown in the center, the worm head can move to one of the four positions $A$, $B$, $C$, or $D$. We choose the probability of each move to satisfy detailed balance.

Figure 5

Critical scaling in the $O(3)$ qubit model. Plots of ${\rho }_s{L}^{d-1}$ and $\chi /L^{2-\eta }$ as a function of $(J/J_c-1)L^{1/\nu }$ for the relativistic limit and as a function of $(1-\lambda /{\lambda }_c)L^{1/\nu }$ for the Hamiltonian limit in $d=2$, 3 dimensions. The black line shows a combined fit in each case assuming that $f(x)$ and $g(x)$ in Eq. (2) can be approximated by a polynomial up to fourth order.

Figure 6

Critical scaling in the $O(2)$ qubit model. Plots of ${\rho }_s{L}^{d-1}$ and $\chi /L^{2-\eta }$ as a function of $(J/J_c-1)L^{1/\nu }$ for the relativistic limit and as a function of $(1-\lambda /{\lambda }_c)L^{1/\nu }$ for the Hamiltonian limit in $d=2$, 3 dimensions. The black line shows a combined fit in each case assuming that $f(x)$ and $g(x)$ in Eq. (22) can be approximated by a polynomial up to fourth order.

Figure 7

Critical scaling in the $Z_2$ qubit model. Plots of $\chi /L^{2-\eta }$ as a function of $(J/J_c-1)L^{1/\nu }$ for the relativistic limit and as a function of $(1-\lambda /{\lambda }_c)L^{1/\nu }$ for the Hamiltonian limit in $d=2$, 3 dimensions.

Figure 8

Continuum time extrapolation of monomer density $v$ and the $O(3)$ charge $⟨Q⟩$ for the $d=1$ data shown in the middle row ($L=4$) of Table 3.