Supersymmetric spin chains with nonmonotonic dispersion relation: Criticality and entanglement entropy

We study the critical behavior and the ground-state entanglement of a large class of $\mathrm{su}\left(1\right|1)$ supersymmetric spin chains with a general (not necessarily monotonic) dispersion relation. We show that this class includes several relevant models, with both short- and long-range interactions of a simple form. We determine the low temperature behavior of the free energy per spin, and deduce that the models considered have a critical phase in the same universality class as a $(1+1)$-dimensional conformal field theory (CFT) with central charge equal to the number of connected components of the Fermi sea. We also study the Rényi entanglement entropy of the ground state, deriving its asymptotic behavior as the block size tends to infinity. In particular, we show that this entropy exhibits the logarithmic growth characteristic of $(1+1)$-dimensional CFTs and one-dimensional (fermionic) critical lattice models, with a central charge consistent with the low-temperature behavior of the free energy. Our results confirm the widely believed conjecture that the critical behavior of fermionic lattice models is completely determined by the topology of their Fermi surface.

Figure 1

Dispersion relation of the $\mathrm{su}\left(1\right|1)$ chain (3) with power-law interaction $h_N\left(x\right)=x^{-\nu }$ for several values of the exponent $\nu$ between $3/2$ and 10.

Figure 2

Comparison of the interaction strength (13) of the $\mathrm{su}\left(1\right|1)$ HS chain (red, upper) with the simple inverse-square law $h_N\left(x\right)=1/x^2$ (blue, lower) for $N=500$. Inset: same plot for the range $100\le x\le 250$.

Figure 3

Dispersion relation of the spin chain (3) with interaction (23) for $J=0.9$. Inset: plot of the function $\phi \left(p\right)$ in Eq. (24).

Figure 4

Integration path ${\gamma }_{\varepsilon ,\delta }$ in Eq. (38).

Figure 5

Plot of the constant term ${\tilde{C}}_{\alpha }$ in Eq. (58).

Figure 6

Relative error $r_L\equiv S_{\mathrm{app}}/S-1$ of the approximation $S_{\mathrm{app}}$ in the right-hand side of Eq. (63) to the von Neumann ground-state entanglement entropy of the chain (22) with $J=1/2$ and $\mu =17/4$ as a function of the block length $L$. Inset: dispersion relation of the chain (22) with the latter values of $J$ and $\mu$. [The interval $(\mathcal{E}\left(\pi \right),{\mathcal{E}}_{\text{max}})$ is in this case is $(4,9/2)$, $m=1$, $p_0\simeq 1.71777$, and $p_1\simeq 2.59356$.]

Figure 7

Dispersion relation $\mathcal{E}\left(p\right)$ with a single maximum in $(0,\pi )$.