Theory of Finite-Entanglement Scaling at One-Dimensional Quantum Critical Points

Studies of entanglement in many-particle systems suggest that most quantum critical ground states have infinitely more entanglement than noncritical states. Standard algorithms for one-dimensional systems construct model states with limited entanglement, which are a worse approximation to quantum critical states than to others. We give a quantitative theory of previously observed scaling behavior resulting from finite entanglement at quantum criticality. Finite-entanglement scaling in one-dimensional systems is governed not by the scaling dimension of an operator but by the "central charge" of the critical point. An important ingredient is the universal distribution of density-matrix eigenvalues at a critical point [P. Calabrese and A. Lefevre, Phys. Rev. A 78, 032329 (2008)]. The parameter-free theory is checked against numerical scaling at several quantum critical points.

Figure 1

Scaling of the entropy with matrix dimension $\chi$. The entropy is calculated numerically using the iTEBD algorithm for several critical points in one dimension: the $XX$ and Heisenberg ($XXX$) models, the transverse Ising model, and the $s=1$ biquadratic Heisenberg model. It can be seen that in each case $S\propto \log \chi$. The lines for models with the same central charge $c$ have the same slope, although other density-matrix-renormalization-group-observed properties differ, such as the marginal operator present in the $XXX$ model [24] but not the $XX$ model.

Figure 2

Scaling of the parameter $b$. $b=-\log {\lambda }_{\max }$ is calculated numerically for different models, where ${\lambda }_{\max }$ is the largest eigenvalue of the Schmidt decomposition. For all models, $b\propto \log \chi$ for sufficiently large matrix dimension $\chi$. A comparison to Fig. 1 shows that the relationship $S=2b$ (single-copy entanglement) is approximately satisfied.

Figure 3

Energy vs correlation length. The energy calculated numerically scales as $1/{\xi }^2$ with correlation length $\xi$. The same scaling also appears (inset) when the state in question is obtained as the ground state of a slightly off-critical Hamiltonian. The correlation length in a matrix product state can be obtained from the transfer operator [25].

Figure 4

$\kappa$ for different values of the central charge. $\kappa$ is calculated numerically using different definitions of the entropy [definitions from Eqs. (2) and (5) nearly agree, while the single-copy entanglement $S=2b$ gives different values as shown], and according to the parameter-free asymptotic theoretical result in Eq. (10). The values of $\kappa$ are, within the numerical accuracy, identical for different models with the same central charge $c$.