Phase transitions in the ${\mathbb{Z}}_p$ and U(1) clock models

Quantum phase transitions are studied in the nonchiral $p$-clock chain, and a new explicitly U(1)-symmetric clock model, by monitoring the ground-state fidelity susceptibility. For $p\ge 5$, the self-dual ${\mathbb{Z}}_p$-symmetric chain displays a double-hump structure in the fidelity susceptibility with both peak positions and heights scaling logarithmically to their corresponding thermodynamic values. This scaling is precisely as expected for two Beresinskii-Kosterlitz-Thouless (BKT) transitions located symmetrically about the self-dual point, and so confirms numerically the theoretical scenario that sets $p=5$ as the lowest $p$ supporting BKT transitions in ${\mathbb{Z}}_p$-symmetric clock models. For our U(1)-symmetric, non-self-dual minimal modification of the $p$-clock model we find that the phase diagram depends strongly on the parity of $p$ and only one BKT transition survives for $p\ge 5$. Using asymptotic calculus we map the self-dual clock model exactly, in the large $p$ limit, to the quantum $O\left(2\right)$ rotor chain. Finally, using bond-algebraic dualities we estimate the critical BKT transition temperatures of the classical planar $p$-clock models defined on square lattices, in the limit of extreme spatial anisotropy. Our values agree remarkably well with those determined via classical Monte Carlo for isotropic lattices. This work highlights the power of the fidelity susceptibility as a tool for diagnosing the BKT transitions even when only discrete symmetries are present.

Figure 1

The classical, planar ($2d$) ${\mathbb{Z}}_p$ clock model with $p=12$.

Figure 2

Energy gaps for low-energy levels as a function of $\lambda$ for $p=3$ and measured with respect to the ground energy level $E_0$. For $\lambda >1$ the ground state is $p$-fold degenerate in the thermodynamic limit. For $\lambda =0$ the ground state is unique. However, the first excited state is $L$-fold degenerate. A qualitatively similar picture is obtained for the $p=2$ and $p=4$ cases as well, with a well pronounced single minimum in the excited-states energy gap near $\lambda =1$.

Figure 3

Energy gaps for $p=5$ and $L=8$ showing clearly a more complex structure than the $p=3$ case shown in Fig. 2. There are a few level crossings between excited states. For large $\lambda$, the ground state becomes fivefold degenerate.

Figure 4

FS ${\chi }_L$ for $p=3$ and system sizes $L=72,96,120$, and 144. One can observe clearly a single peak in the FS. The peak position moves towards the self-dual point $\lambda =1$, and the height of the peak increases with increasing system size. Similarly, for $p=2$ and $p=4$ there is a single peak moving towards the self-dual point $\lambda =1$ with increasing system size.

Figure 5

Log-log plot of the FS peak height versus system size $L$ for quantum $p$-clock chains with $p=2,3$, and 4. The height of the peaks increases as $\sim L$ (two parallel continuous lines connecting little circles and diamonds) for $p=2$ and $p=4$, while it scales as $\sim L^{1.4}$ (continuous line connecting little triangles) for $p=3$. These results are in perfect agreement with Eq. (25).

Figure 6

FS ${\chi }_L$ for $p=5$ and system sizes $L\le 144$. A double-hump structure in the FS curves is clearly visible, with peaks becoming narrower and higher with increasing system size. The peak positions ${\lambda }_{-}$ and ${\lambda }_{+}$ shift towards each other. They move from the gapped regions towards their thermodynamic values ${\lambda }_{-}^{*}$ and ${\lambda }_{+}^{*}$ at the boundaries of the critical phase, with increasing system size.

Figure 7

Scaling of the FS peak positions with system size for $p=5$ and system sizes of up to $L=168$ sites. The numerical constants $c_1$ and $c_2$ are positive. The extrapolated values of the peak positions are (a) ${\lambda }_{-}^{*}\simeq 0.966$ and (b) ${\lambda }_{+}^{*}\simeq 1.035$. For large $L$, the fits follow accurately the BKT scaling law Eq. (21), confirming the BKT nature of the underlying QPTs.

Figure 8

Evolution of the product of the two peak positions ${\lambda }_{-}^m{\lambda }_{+}^m$ towards unity with increasing system size for $p=5$.

Figure 9

The FS ${\chi }_L$ for $p=6$ and system sizes $L\le 144$. The peak positions are farther away from each other when compared to the $p=5$ case shown in Fig. 6. Similar double-hump structure in the FS is observed for other values of $p>6$ as well, with peak positions even farther away from each other. The gapless parameter region is expected to increases as $p$ increases [47].

Figure 10

Scaling of the FS peak positions with system size for $p=6$ and system sizes up to $L=168$ sites; $c_1$ and $c_2$ are positive numerical constants. The extrapolated values of the peak positions are (a) ${\lambda }_{-}^{*}\simeq 0.782$ and (b) ${\lambda }_{+}^{*}\simeq 1.285$ (${\lambda }_{-}^{*}{\lambda }_{+}^{*}=1.005$) and are consistent with the transition points obtained by the level-spectroscopy method [26].

Figure 11

Scaling of the FS peak heights with system size for $p=5,6$, and 7. For the system sizes shown, $(L\ge 72)$, the peak heights follow, in excellent agreement, the leading behavior in Eq. (21), thus confirming the BKT nature of the QPTs.

Figure 12

Quantum phase diagram for (a) ${\mathbb{Z}}_p$ self-dual and (b) U(1)-symmetric $p$-clock chains. Gapless regions are denoted by a thick cyan line. With increasing $p\to \infty$ an spontaneously ${\mathbb{Z}}_p$ symmetry broken region (with $p$-fold degenerate ground states) shrinks towards $h=0$. With increasing $p$ a Ferri phase for even $p$, as well as Néel to gapless transition points for odd $p$, shift towards $h=0$ in panel (b).

Figure 13

Behavior of FS for the U(1) $p=5$ clock model for various system sizes.

Figure 14

Scalings of the FS peak heights with system size for U(1) $p$-clock chains at the critical boundary point between the gapless regime and paramagnetic disordered phase for $h>0$, confirm the BKT nature of the QPTs for $p=5,6$, and 7.

Figure 15

(a) Allowed $n_j$ values for even $p$. Arcs with arrows at the ends connect $n_j$ related by a single action of $H_1$. We have shaded quantum numbers $|n_j|=\mathcal{O}\left(p\right)$, which can be safely truncated when considering ground-state (low-energy) physics for $p\to \infty$ as proved in the text. (b) When quantum numbers $n_j$ are defined on a segment, instead of a circle, the system displays an explicit $O\left(2\right)$ symmetry for any $p>2$ [for $p=3$ the model becomes the spin-1 chain given in Eq. (38)].

Figure 16

Von Neumann entropy $S_{\mathrm{vN}}$ as a function of the partition size $L_A$ for system size $L=36$.