Scaling of entanglement support for Matrix Product States

Abstract

The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground state properties of a system is limited by the size $\chi$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S\sim 1/6\log \chi$ with high precision. This result can be understood as the emergence of an effective finite correlation length ${\xi }_{\chi }$ ruling of all the scaling properties in the system. We produce five extra pieces of evidence for this finite-$\chi$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, and the scaling of the entanglement entropy for a finite block of spins. All our computations are consistent with a scaling relation of the form ${\xi }_{\chi }\sim {\chi }^{\kappa }$, with $\kappa =2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\kappa \sim 1.37$. We also show how finite-$\chi$ scaling allow to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density matrix renormalization group results.

A new section comparing with previous results. Published version (small differences due to proof corrections)

References in corpus (7)

• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• From density-matrix renormalization group to matrix product states
• New Trends in Density Matrix Renormalization
• Long-distance entanglement in spin systems
• Simulation of time evolution with the MERA
• Local Measures of Entanglement and Critical Exponents at Quantum Phase Transitions
• Rapidly-converging methods for the location of quantum critical points from finite-size data

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