Causality and quantum criticality in long-range lattice models

Long-range quantum lattice systems often exhibit drastically different behavior than their short-range counterparts. In particular, because they do not satisfy the conditions for the Lieb-Robinson theorem, they need not have an emergent relativistic structure in the form of a light cone. Adopting a field-theoretic approach, we study the one-dimensional transverse-field Ising model with long-range interactions, and a fermionic model with long-range hopping and pairing terms, explore their critical and near-critical behavior, and characterize their response to local perturbations. We deduce the dynamic critical exponent, up to the two-loop order within the renormalization group theory, which we then use to characterize the emergent causal behavior. We show that beyond a critical value of the power-law exponent of the long-range couplings, the dynamics effectively becomes relativistic. Various other critical exponents describing correlations in the ground state, as well as deviations from a linear causal cone, are deduced for a wide range of the power-law exponent.

Figure 1

Contour plot of the (unnormalized) response function $D_{2/3}(t,R)$. The response function is computed via the scaling function $g_{2/3}$. The causal region exhibits a nonlinear behavior with a fingerlike pattern extending beyond the linear light-cone structure of short-range interactions. The numerical evaluation of the response function with the lattice dispersion relation and momentum cutoff produces an almost identical plot. The (red) solid line represents the curve $t\sim R^{\sigma /2}$ with $\sigma =\frac{2}{3}$.

Figure 2

(a) The scaling function $f_{\sigma }$ characterizing the response function as a function of $s=(t-R)/t^{1/(1-\sigma )}$ for $\sigma =\frac{5}{2}$. The function $f_{\sigma }$ approaches a constant inside the linear light cone $\left(s>0\right)$, but decays rapidly outside the light cone $\left(s<0\right)$. (b) The first local maximum of the response function (solid curve) is bent inside the light cone (dashed line). $\xi$ characterizes the deviation from the linear light cone. For long times, $\xi /t\to 0$, and the causal region becomes a sharp linear cone.

Figure 3

The norm of the (unnormalized) response function $|D_1(t,R)|$ away from criticality evaluated with the full dispersion relation ${\omega }_1\left(q\right)=\sqrt{1+2q-q^2/\pi }$ for $q\in (0,\pi )$ (with $\varrho =1$ and the $q$ dependence obtained from the polylog function of the lattice dispersion relation at $\sigma =1$). The analytical expression in the text produces a very similar plot sufficiently away from the vertical line $\left(R=0\right)$. The response function exhibits a linear causal behavior.

Figure 4

The norm of the (unnormalized) response function $|D_{\sigma }(t,R)|$ evaluated at $t=R^{\alpha }$ for $\sigma =\frac{1}{3}$ away from criticality. The three plots correspond to different choices of $\alpha =\sigma /2,\sigma$, and 1 (with $\sigma =\frac{1}{3}$). The dispersion relation is taken as ${\omega }_{1/3}\left(q\right)=\sqrt{1+\frac{2}{\sqrt{3}}\mathrm{\Gamma }\left(\frac{-4}{3}\right)q^{1/3}+\frac{1}{2}\zeta \left(\frac{-2}{3}\right)q^2}$ for $q\in (0,\pi )$ (with $\varrho =1$ and the $q$ dependence obtained from the polylog function of the lattice dispersion relation at $\sigma =\frac{1}{3}$). For the exponent $\alpha =\sigma /2$, we find a simple power-law decay of the response function with $R$, suggesting $z=\sigma /2$.

Figure 5

The one- and two-loop diagrams contributing to the two-point function.

Figure 6

The critical exponents $\eta$ and $z$ as a function of $\sigma$. These exponents assume their short-range values $\eta ={\eta }_{\mathrm{SR}}=\frac{1}{4}$ and $z=1$ for $\sigma >\frac{7}{4}$. For $\sigma <\frac{2}{3}$, they stick to their mean-field values $\eta =1-\sigma /2$ and $z=\sigma /2$, and, for $\frac{2}{3}<\sigma <\frac{7}{4}$, their values are computed up to the two-loop order (the dashed lines show the mean-field exponents). The inset shows $\mathrm{\Delta }z=z-\sigma /2$ and $\mathrm{\Delta }\eta =\eta -(1-\sigma /2)$, the difference of the critical exponents from their mean-field values, up to the two-loop order.