Scaling of entanglement support for Matrix Product States (0712.1976v2)

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.

• The paper establishes scaling relations showing how the ability of Matrix Product States (MPS) to capture entanglement entropy and correlation length in critical systems is limited by the finite matrix size (χ).
• Numerical evidence, particularly for the quantum Ising model, demonstrates that entanglement entropy scales logarithmically with matrix size, S ~ (1/6)log χ, linking it to a finite correlation length scaling as ξ ~ χ².
• The findings reveal that the scaling exponent varies with the universality class (e.g., different for Ising vs. Heisenberg models), providing insight into critical phenomena and guiding efficient numerical simulations.

Overview of "Scaling of Entanglement Support for Matrix Product States"

Introduction and Context

The paper "Scaling of Entanglement Support for Matrix Product States" by L. Tagliacozzo et al. addresses the capacity of Matrix Product States (MPS) to accurately describe infinite-size, translational-invariant critical spin chains, particularly focusing on the quantum Ising and Heisenberg models. This paper is situated within the broader context of using approximations to understand quantum systems, as exact solutions are often infeasible.

Entanglement and Matrix Size Limitations

The primary limitation in using MPS is the finite size of matrices, denoted by $\chi$. This constraint translates into a bounded ability to capture the entanglement entropy of half-chains in critical systems—an issue that becomes evident at quantum critical points where entanglement diverges. The authors find that for the Ising model, the half-chain entanglement entropy scales as $S \sim \frac{1}{6} \log \chi$ with high precision. This result is interpreted as the manifestation of an effective finite correlation length $\xi_\chi \sim \chi^\kappa$, where $\kappa = 2$ for the Ising model.

Numerical Evidence and Theoretical Insights

The paper presents a suite of numerical evidence supporting the finite-$\chi$ scaling hypothesis. These include:

• Entanglement Entropy: The scaling of the entanglement entropy is confirmed across different models, with a particular focus on the Ising model showing a fit to $S_\chi = \frac{1}{6} \log \chi$.
• Correlation Length: The computed correlation lengths for finite $\chi$ align with the scaling law $\xi_\chi \sim \chi^\kappa$, reinforcing the derived entanglement entropy results.
• Shift of Critical Point: The shift in the critical point with finite $\chi$ is shown to follow the scaling relations, offering another layer of validation.
• Magnetization and Block Entropy: These observables further verify the finite-$\chi$ scaling, demonstrating consistent behavior with predicted power laws and a saturation behavior for block entropies.

Universality and Model-Dependency

For the Heisenberg model, similar scaling behaviors are observed, but with a different $\kappa$ value, approximately 1.37. These findings suggest that while the concept of finite-$\chi$ scaling is robust, the specific scaling exponent $\kappa$ is model-dependent and corresponds to the universality class of the phase transition.

Methodological Approach

The authors employ the infinite Time Evolving Block Decimation (iTEBD) algorithm to perform the calculations over infinite systems, ensuring the observations are free from finite-size effects. This algorithm efficiently handles infinite systems by exploiting translational invariance.

Implications and Future Directions

The established connection between entanglement entropy and correlation length through finite-$\chi$ scaling offers deeper insights into the universality classes of quantum phase transitions. Practically, it enables more efficient numerical simulations of critical systems by determining the minimal $\chi$ necessary for accurate description. Future work could extend these insights to other tensor network states beyond MPS, such as Projected Entangled Pair States (PEPS), potentially broadening the applications to higher-dimensional systems.

Conclusion

This paper contributes to the understanding of how matrix size constraints in MPS representation affect the accurate depiction of entangled quantum systems, particularly at critical points. The defined scaling relations provide a framework for leveraging MPS in the paper of quantum phase transitions, thereby enhancing the toolbox available for theoretical and computational physicists in the field of quantum many-body systems.