Decomposition of entanglement entropy in lattice gauge theory (1109.0036v2)

Abstract: We consider entanglement entropy between regions of space in lattice gauge theory. The Hilbert space corresponding to a region of space includes edge states that transform nontrivially under gauge transformations. By decomposing the edge states in irreducible representations of the gauge group, the entropy of an arbitrary state is expressed as the sum of three positive terms: a term associated with the classical Shannon entropy of the distribution of boundary representations, a term that appears only for non-Abelian gauge theories and depends on the dimension of the boundary representations, and a term representing nonlocal correlations. The first two terms are the entropy of the edge states, and depend only on observables measurable at the boundary. These results are applied to several examples of lattice gauge theory states, including the ground state in the strong coupling expansion of Kogut and Susskind. In all these examples we find that the entropy of the edge states is the dominant contribution to the entanglement entropy.

• The paper decomposes entanglement entropy into three distinct positive components related to boundary observables and nonlocal correlations.
• It introduces a method that separates the Shannon entropy of boundary representations and the log-dimension term unique to non-Abelian gauge theories.
• These insights facilitate numerical simulations and potential extensions to quantum gravity by clarifying the role of edge state contributions.

Decomposition of Entanglement Entropy in Lattice Gauge Theory

The paper by William Donnelly provides a comprehensive analysis of entanglement entropy in the framework of lattice gauge theory, focusing on the critical interplay between spatial regions and gauge degrees of freedom. The paper delineates a method to express the entropy of arbitrary states by segregating contributions from distinct components tied to boundary observables and nonlocal correlations.

Summary of Key Results

The principal achievement of this research is the decomposition of the entanglement entropy into three positive-definite components:

1. Shannon Entropy of Boundary Representations: This term relates to the classical distribution of irreducible representations on the boundary of a region. It is significant in capturing the classical randomness associated with the various representations that the gauge field might assume at the boundary.
2. Log-Dimension of Boundary Representations: Appearing only in non-Abelian gauge theories, this term incorporates the entropic contribution linked to the dimensionality of representations found on the boundary. The dimension of these representations provides insight into the degrees of freedom that contribute to the entropy in a quantum manner.
3. Nonlocal Correlation Contributions: Capturing the essence of correlations that extend beyond local boundary effects, this term represents the nonlocal quantum correlations that exist in the bulk of the field.

This decomposition is particularly applicable to states in lattice gauge theory, such as those in the strong coupling expansion or other model-specific states. The entropy dominance by edge state contributions illustrates the significance of boundary observables in gauge theories, providing enhanced insights into quantum field behavior in discrete settings.

Implications for Lattice Gauge Theories and Quantum Gravity

This work resonates profoundly within the lattice gauge theory framework and potential extensions to loop quantum gravity. The paper adherently uses lattice structure to simulate scenarios where edge states significantly influence the entanglement entropy. This approach circumvents difficulties encountered in the continuum such as the ultraviolet divergence issue in defining entanglement entropy, making the findings more tractable in computational and theoretical explorations.

Applying to loop quantum gravity, these findings might generalize to superpositions of lattices that embody quantized spacetimes, suggesting analogous roles for gauge structures in the emergent space. Additionally, the relation of SU(2) edge states to those in loop quantum gravity offers potential calculations pivotal to the behavior of entropy in quantum gravitational systems, such as black hole entropy.

Theoretical and Practical Consequences

The paper's implication extends into multiple theoretical insights:

• Phase Transitions and Topological Phases: The entropic measures proposed may serve as diagnostic tools for identifying phase transitions in quantum field theories. Moreover, in nontrivial topological phases, the unique terms of this decomposition could relate profoundly to constructs such as topological entanglement entropy, already linked to conducting calculations in condensed matter systems.
• Numerical Simulations and Computational Physics: The clear articulation of entropy components tied to boundary representations aids in simulations, where the Maxwellian aspect of the Shannon entropy offers a practical route for exploration. Particularly, the paper’s approach affirms the feasibleness of computing complex systems’ entanglement in lattice gauge theories.

In conclusion, this paper extends the foundational understanding of quantum state entropy in lattice gauge theories and suggests intriguing parallels and extensions to quantum gravity research. The delineation of entropy through novel decomposition contributes to potentially significant future advancements, both in simulating field theories and in understanding the fabric of quantum gravitation narratives.