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Tensor-Network QFT

Tensor networks organize states or partition functions by restricting how correlations pass between local tensors. This chapter distinguishes the resulting representation from the algorithms used to contract and optimize it, then relates all of those approximations to continuum observables and real-time dynamics.

Enter this chapter

Section titled “Enter this chapter”

Begin with Entanglement Structure and Tensor-Network Ansätze to choose the state or partition function and the network geometry that represents it.

Topics

Section titled “Topics”
  1. Entanglement Structure and Tensor-Network Ansätze — Connects network geometry to retained correlations.
  2. Matrix Product States, Finite Entanglement, and Continuum Limits — Separates finite bond dimension from lattice and volume limits.
  3. PEPS and Higher-Dimensional Field Theories — Controls higher-dimensional states and approximate contraction.
  4. Tensor Renormalization of Euclidean Path Integrals — Coarse-grains a Euclidean partition function.
  5. MERA, cMERA, and Renormalization Geometry — Organizes an ansatz by scale without assuming an emergent geometry.
  6. Symmetric and Gauge-Invariant Tensor Networks — Imposes charges and Gauss constraints directly on tensors.
  7. Real-Time Tensor-Network Dynamics — Tracks entanglement growth and time-evolution errors.
  8. Contraction, Truncation, and Continuum Error Certification — Tests each approximation axis in the final observable.

How the ideas fit together

Section titled “How the ideas fit together”

Increasing bond dimension enlarges the represented state space, but it does not automatically improve an independently approximate contraction or find the best state in that space. Exact symmetry constraints can remove forbidden states without controlling these errors. Near criticality, finite entanglement itself introduces a correlation-length scale, which must be distinguished from finite volume and lattice spacing. Consequently, one discarded-weight number cannot certify a continuum QFT result or its real-time accuracy. Cirac et al. 2021, §§ II–III and VI.

Where to go next

Section titled “Where to go next”

Continue to Quantum Simulation of QFT to connect a continuum question to a finite encoded calculation.

References

Section titled “References”
  • Cirac, J. Ignacio, David Pérez-García, Norbert Schuch, and Frank Verstraete. “Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, Theorems.” Reviews of Modern Physics 93 (2021): 045003. DOI.