Entanglement renormalization for quantum fields (1102.5524v2)

Abstract: It is shown how to construct renormalization group flows of quantum field theories in real space, as opposed to the usual Wilsonian approach in momentum space. This is achieved by generalizing the multiscale entanglement renormalization ansatz to continuum theories. The variational class of wavefunctions arising from this RG flow are translation invariant and exhibit an entropy-area law. We illustrate the construction for a free non-relativistic boson model, and argue that the full power of the construction should emerge in the case of interacting theories.

• The paper presents cMERA as a novel formulation for real-space RG flows in quantum field theories, enabling variational optimization without discretization.
• It demonstrates the method’s effectiveness on both relativistic and non-relativistic systems while preserving translation invariance and adhering to an entropy-area law.
• The approach overcomes lattice limitations such as the fermion doubling problem, paving the way for exploring interacting quantum fields and critical phenomena.

Entanglement Renormalization for Quantum Fields

The paper "Entanglement Renormalization for Quantum Fields" by Haegeman, Osborne, Verschelde, and Verstraete presents a formulation of renormalization group (RG) flows for quantum field theories within a real-space framework, utilizing the multiscale entanglement renormalization ansatz (MERA). The authors propose a generalization of MERA to continuum theories, referred to as cMERA (continuous MERA), thus extending their applicability beyond lattice to continuous quantum systems.

The paper effectively demonstrates how entanglement renormalization can be applied to both relativistic and non-relativistic quantum field theories. This stands in contrast to traditional RG methods, which operate predominantly in momentum space and often necessitate perturbative approaches for interacting theories. The cMERA approach, on the other hand, offers a real-space renormalization process conducive to handling interacting quantum fields without the need for discretization.

Key Methodological Contributions

1. Translation Invariance: The cMERA construction results in wavefunctions that are translation invariant, adhering to an entropy-area law which is consistent with expectations for critical theories in lower dimensions.
2. General Construction for Bosons: The analysis provides a theoretical framework utilizing a free non-relativistic boson model to illustrate cMERA's construction, showcasing the potential for addressing interacting theories.
3. Continuum Approach: cMERA circumvents the fermion doubling problem inherent in lattice frameworks, thus facilitating the paper of problems directly within continuous space—an advantage for critical systems and certain interacting theories that are less tractable in a lattice setting.

Comparisons and Implications

The paper contrasts cMERA with Wilsonian momentum-shell RG, emphasizing its variational flexibility and real-space applicability. One of the primary distinctions is cMERA's capability to variationally optimize RG flows by disentangling modes via Hamiltonian dynamics, rather than merely integrating out high-frequency modes as in momentum-shell RG. Additionally, cMERA incorporates both ultraviolet and infrared cutoffs explicitly, controlling the creation and entanglement of modes through smoothed operators, which differ from discretization-based cutoffs seen in lattice methods.

By conceptualizing quantum field renormalization in the framework of tensor networks, cMERA provides a compelling real-space analog to illuminate RG processes. This approach is instrumental in exploring algebraically decaying correlations characteristic of critical theories, and supporting computations of properties such as scaling exponents.

Future Perspectives

The potential implications of cMERA are significant across numerous domains of theoretical physics. It holds promise for expanding our understanding of strongly interacting quantum fields, addressing phenomena such as topological effects, confinement, and symmetry breaking. Furthermore, the paper posits cMERA as a physical realization of the holographic principle, drawing possible connections to the AdS/CFT correspondence, a cornerstone in the understanding of gravitational and quantum field interactions.

The rigorous mathematical groundwork laid by cMERA invites further research to validate this framework's efficacy for more complex systems. While the current work provides proof-of-concept analytical results, future experimental validation and computational implementations will be critical to its advancement.

In sum, the paper provides a sophisticated, technical treatment of quantum field theory through real-space renormalization, offering a robust platform for capturing the emergent properties of complex quantum systems. As such, it marks an important step in the application of tensor network methods to continuum quantum fields, with broad potential utility across theoretical and applied physics research landscapes.