Kitaev Chains with Long-Range Pairing

We propose and analyze a generalization of the Kitaev chain for fermions with long-range $p$-wave pairing, which decays with distance as a power law with exponent $\alpha$. Using the integrability of the model, we demonstrate the existence of two types of gapped regimes, where correlation functions decay exponentially at short range and algebraically at long range ($\alpha >1$) or purely algebraically ($\alpha <1$). Most interestingly, along the critical lines, long-range pairing is found to break conformal symmetry for sufficiently small $\alpha$. This is accompanied by a violation of the area law for the entanglement entropy in large parts of the phase diagram in the presence of a gap and can be detected via the dynamics of entanglement following a quench. Some of these features may be relevant for current experiments with cold atomic ions.

Figure 1

(a) Effective central charge $c_{\mathrm{eff}}$ obtained by fitting $S(L/2)$. Two gapless conformal field theories with $c=1/2$ are visible for $\mu =1$ ($\alpha >3/2$) and $\mu =-1$ ($\alpha >2$). White vertical dotted lines: gapless lines with broken conformal symmetry. Horizontal dashed line separates two regions: correlation functions display a hybrid exponential-algebraic ($\alpha >1$) and purely algebraic decay ($\alpha <1$). (b) Time evolution of $S(L/2)$ after a quench from a product state with $\mu \gg 1$ to $\mu =1$: $\alpha >1$, $S(L/2)$ grows linearly, $\alpha <1$, $S(L/2)$ grows logarithmically. (c) $g_2(R)$ correlation function for $\mu =2$ and $\alpha =10$ (squares), showing exponential behavior and $\alpha =7$ (circles), showing an exponential with an algebraic tail even in the gapped region.

Figure 2

(a) Long-distance behavior of $g_2(R)$ for $\alpha =7$ and $\mu =2$ in log-log scale, displaying algebraic decay. Continuous line: analytic prediction $\sim 1/R^{14}$. [Figure 1 shows the same data set on a log plot.] (Inset) Purely algebraic decay for $\alpha \le 1$. Here, $\mu =2$ and $\alpha =0.5$. (b) $g_2(R)$ for $\mu =1$ and $\alpha =0.5$, displaying algebraic decay with oscillating behavior.

Figure 3

Exponents (a) $\gamma$ and (b) $\delta$ of the algebraic decay of $g_1(R)$ and $g_2(R)$ vs $\alpha$, obtained by fitting with power-law functions, namely, $g_1(R)\sim R^{-\gamma }$ and $g_2(R)\sim R^{-\delta }$. The equations of the two straight lines in (a) are $2\alpha -1$ and $\alpha +1$.

Figure 4

Open chain. (a) Wave function $|\mathrm{\Psi }(j){|}^2$ for $\mu =0.5$. (b) Mass gap $M(L\to \infty )$ vs $\alpha$.