Entanglement in quantum critical phenomena

TL;DR

The results establish a precise connection between concepts of quantum information, condensed matter physics, and quantum field theory, by showing that the behavior of critical entanglement in spin systems is analogous to that of entropy in conformal field theories.

Abstract

Quantum phase transitions occur at zero temperature and involve the appearance of long-range correlations. These correlations are not due to thermal fluctuations but to the intricate structure of a strongly entangled ground state of the system. We present a microscopic computation of the scaling properties of the ground-state entanglement in several 1D spin chain models both near and at the quantum critical regimes. We quantify entanglement by using the entropy of the ground state when the system is traced down to $L$ spins. This entropy is seen to scale logarithmically with $L$, with a coefficient that corresponds to the central charge associated to the conformal theory that describes the universal properties of the quantum phase transition. Thus we show that entanglement, a key concept of quantum information science, obeys universal scaling laws as dictated by the representations of the conformal group and its classification motivated by string theory. This connection unveils a monotonicity law for ground-state entanglement along the renormalization group flow. We also identify a majorization rule possibly associated to conformal invariance and apply the present results to interpret the breakdown of density matrix renormalization group techniques near a critical point.

Figures (2)

• Figure 1: Non-critical $S_L$ for the Ising model, $H_{XY}(\gamma=1)$ in Eq. (\ref{['eq:XY']}), as a function of the size $L$ of the spin block (left axis) and parameter $a$ (right axis). The finite correlation length governing non-critical phenomena translates into a finite entanglement length, that is, a finite value of $L$ for which adding new spins to a block does not increase its entanglement with the rest of the chain. Such an entanglement length (which scales as the correlation length) diverges only at the critical point $a=1$. Scaling arguments also imply that the entropy surface is given by $\log[Lf(L|1-a|)]$. For any given $a$, the saturation value for the entropy is given by Eq. (\ref{['eq:isingKi']}).
• Figure 2: Numerical calculation of critical entanglement $S_L$ for 1D spin chains. Points ($+$) come from an infinite Ising chain with critical magnetic field, $H_{XY}(a=1,\gamma=1)$ in Eq. (\ref{['eq:XY']}), and corresponds to $S_L^{Ising} \approx \log_2(L)/6+k_2$. Curve ($\times$) comes from an infinite XX spin chain without magnetic field, $H_{XY}(a=\infty, \gamma=0)$ in Eq. (\ref{['eq:XY']}), and corresponds to $S^{XX}_L \approx \log_2(L)/3+k_1$ . Thus the growth in $L$ of entanglement in the XX model, asymptotically described by a free boson, is twice that in the Ising model, corresponding to a free fermion, $S^{XX}_{L+1}-S^{XX}_L \approx 2(S_{L+1}^{Ising}-S_L^{Ising})$. Finally, curve ($*$) comes from a XXX chain of $20$ spins without magnetic field, $H_{XXZ}(\Delta=1, \lambda=0)$ in Eq. (\ref{['eq:XXZ']}). These finite-chain results, obtained using the Bethe ansatz, combine the logarithmic behavior ($L=1,\cdots, 5$) of a free boson field theory with a finite-size saturation effect ($L=6,\cdots, 10$). We have added the lines ${c+\bar{c}\over 6} [\log_2(L)+\pi]$ both for bosons and fermions to highlight their remarkable agreement with the numerical diagonalization.