Conformally invariant free-parafermionic quantum chains with multispin interactions

We calculate the spectral properties of two related families of non-Hermitian free-particle quantum chains with $N$-multispin interactions ($N=2,3,...$). The first family have a $Z\left(N\right)$ symmetry and are described by free parafermions. The second have a $U\left(1\right)$ symmetry and are generalizations of $XX$ quantum chains described by free fermions. The eigenspectra of both free-particle families are formed by the combination of the same pseudo-energies. The models have a multicritical point with dynamical critical exponent $z=1$. The finite-size behavior of their eigenspectra, as well as the entanglement properties of their ground-state wave function, indicate the models are conformally invariant. The models with open and periodic boundary conditions show quite distinct physics due to their non-Hermiticity. The models defined with open boundaries have a single conformal invariant phase, while the $XX$ multispin models show multiple phases with distinct conformal central charges in the periodic case.

Figure 1

Representation in the complex plane of the eigenenergies (4) with $\overline{M}=3$, for the $Z\left(N\right)$ models with $N=3$ (a), (b) and $N=5$ (c), (d). The circles have the radius ${\varepsilon }_i$, and the possible values (open circles) are the intercepts with the $Z\left(N\right)$ circles. Each circle contributes with one and only one of the possible $N$ intercepts (black circles).

Figure 2

Representation of configurations that are present in the $N=\mathrm{th}$ree-multispin $XX$ quantum chains (29) but not present in the multispin $Z\left(3\right)$ quantum chain (4), in the case $\overline{M}=2$. The circles have the radius ${\varepsilon }_i$, and the possible values (open circles) are the intercepts with the $Z\left(N\right)$ circles. Distinct from the configurations in Fig. 1, each circle may have multiple contributions (black circles), and do not satisfy the $Z\left(N\right)$ circle exclusion constraint.

Figure 3

Representation in the complex plane of the eigenenergies for the $Z\left(N\right)$ multispin model with $N=3$. The configuration of the ground-state energy is (a), and (b) and (c) are the configurations that produce the lowest gaps (43).

Figure 4

Representation in the complex plane of the energies of some excited states for the $N=3XX$ multispin model.

Figure 5

Representation in the complex plane of the ground-state eigenenergy of the $N=5$ multispin $XX$ model.

Figure 6

Dispersion relation $\mathrm{\Lambda }\left(k\right)$, given in (66), for the multispin $XX$ quantum chain with $N=3$ and some values of $\lambda$. The Fermi points are the ones where $\mathrm{\Lambda }\left(k_F\right)=0$.

Figure 7

Dispersion relation $\mathrm{\Lambda }\left(k\right)$, given in (66), for the multispin $XX$ quantum chain with $N=4$, and some values of $\lambda$. The Fermi points are the ones where $\mathrm{\Lambda }\left(k_F\right)=0$.

Figure 8

Von Neumann entanglement entropy $S_1(L,\ell )$, as a function of $ln[\frac{L}{\pi }sin\left(\frac{\ell \pi }{L}\right)]/3$, for some values of $\lambda$. The data are for the $XX$ multispin quantum chain with $N=3, L=600$ sites and PBC.

Figure 9

Rényi entanglement entropies $S_2(L,\ell )$ and $S_3(L,\ell )$, as a function of $ln[\frac{L}{\pi }sin\left(\frac{\ell \pi }{L}\right)]$. The data are for the $XX$ multispin quantum chain with $N=3, L=600$ sites and PBC.

Figure 10

Estimated values of the central charge $c$ as a function of $\lambda$ for the multispin $XX$ quantum chain with $N=3$ and PBC.

Figure 11

Estimated values of the central charge $c$ as a function of $\lambda$ for the multispin $XX$ quantum chains with $N=3$ and $N=6$, with PBC.

Figure 12

Estimated values of the central charge $c$ as a function of $\lambda$ for the multispin $XX$ quantum chains with $N=5$ and $N=7$ with PBC.