Tensor-Network QFT

Tensor networks become QFT methods only when their geometry and every cutoff are tied to a specified observable. Use an MPS for a one-dimensional Hamiltonian state, a PEPS for a higher-dimensional state, Euclidean tensor renormalization for a partition function, MERA or cMERA for an explicitly scale-organized ansatz, and symmetric tensors when a charge or Gauss constraint must hold by construction. The distinction between state geometry, virtual bonds, symmetry structure, and contraction is reviewed in Cirac et al. 2021, §§ II–III and VI. In every route, continuum evidence requires separate control of representation, contraction, optimization, local Hilbert space, volume, lattice spacing, and—when relevant—time evolution.

Enter this chapter

The shortest route depends on the mathematical object being approximated:

Start with Entanglement Structure and Tensor-Network Ansätze to distinguish a state network from a partition-function network and to identify physical and virtual indices.
Continue to Matrix Product States, Finite Entanglement, and Continuum Limits for one spatial dimension, transfer spectra, and joint finite-size–finite-entanglement scaling.
Use PEPS and Higher-Dimensional Field Theories when the state lives in two or more spatial dimensions and contraction itself becomes an approximation.
Use Tensor Renormalization of Euclidean Path Integrals when the starting object is a Euclidean lattice partition function rather than a wavefunction.
Use MERA, cMERA, and Renormalization Geometry when scale transformations or scaling operators are central. The geometric interpretation is conditional, not an automatic holographic statement.
Use Symmetric and Gauge-Invariant Tensor Networks when global charges or local Gauss laws must survive every truncation.
Use Real-Time Tensor-Network Dynamics for quenches, response functions, or scattering-time correlators, with a convergence-defined maximum time.
Finish with Contraction, Truncation, and Continuum Error Certification before attaching a continuum or precision claim to any tensor result.

One object, several independent approximations

For a regulated Hamiltonian $H(a,L,d_{\rm loc})$ , a tensor-state calculation replaces the exact ground state by

|\psi_0\rangle \longrightarrow |\psi(\mathcal G,\chi;\theta)\rangle,

where $\mathcal G$ is the network geometry, $\chi$ denotes one or more virtual-bond dimensions, and $\theta$ denotes optimized tensor entries. This introduces an ansatz error and an optimization error. In PEPS or MERA, evaluating the ansatz can additionally introduce an environment or contraction error. The local cutoff $d_{\rm loc}$ , size $L$ , and lattice spacing $a$ remain physical regulator axes; increasing $\chi$ does not remove any of them.

For a Euclidean theory, the exact finite-lattice identity instead has the form

Z(a,L,d_{\rm loc}) =\operatorname{tTr}\!\prod_x T_x,

after an exact local character, quadrature, or discrete-variable decomposition. Truncating bonds during blocking changes the contraction of $Z$ ; it is not a variational approximation to a Hamiltonian state unless an additional transfer-matrix argument establishes that connection.

Choosing the evidence route

Target	Natural starting route	First nontrivial diagnostic	Control that is often confused with it
Ground-state gap or correlator in one dimension	MPS	Transfer spectrum and energy residual	Finite volume or local field cutoff
Higher-dimensional Hamiltonian state	PEPS	Environment-dimension and contraction-family scan	PEPS bond dimension
Free energy or Euclidean insertion	Tensor renormalization	Blocking-normalization and discarded-spectrum scan	Monte Carlo sampling uncertainty
Scaling operator	Scale-invariant MERA	Eigenoperator of the scaling channel	A continuum extrapolation in lattice spacing
Exact charge or Gauss sector	Symmetric tensor	Local intertwiner or Gauss-law residual	A penalty term that only suppresses leakage
Real-time response	MPS/PEPS evolution	Step-size, projection, and bond-growth window	Smoothness at late times

The decisive question is never merely “which tensor network?” It is “which finite-regulator quantity is computed exactly, which operation is approximate, and which independent variation could falsify the claimed observable?” The chapter’s final page gives a common regulator and error record for answering those questions.

Entanglement measures and information-theoretic interpretations belong to the quantum-information volume; dated PEPS, MERA/cMERA, and real-time capability comparisons belong to the Research methods dossier rather than to these durable method pages.

A compact chapter benchmark

The critical transverse-field Ising chain provides a useful common fixture because several independent checks coexist. At criticality its continuum limit has central charge $c=1/2$ ; finite chains can be diagonalized exactly, an MPS supplies a transfer correlation length $\xi_\chi$ , and the Euclidean two-dimensional Ising partition function can be tensorized. A credible study does not force these data into one fit. It first verifies finite- $L$ energies, then finite-entanglement scaling at fixed lattice model, then Euclidean contraction and normalization, and only then compares dimensionless continuum quantities.

This is a benchmark of the control chain, not evidence that the same bond dimension or contraction scheme suffices for an interacting gauge theory.

Review the chapter

For a target mass ratio $M_2/M_1$ , write the minimum list of axes that must be varied in an MPS calculation with a bosonic local cutoff.
Explain why agreement between two PEPS optimizers that use the same approximate environment is correlated evidence.
State one exact finite-regulator identity available to a Euclidean tensor calculation and one additional inference needed before calling its result continuum QFT.

Answers

At minimum vary bond dimension, optimization tolerance or initialization, local Hilbert cutoff, volume, and lattice spacing; also vary any environment cutoff used to evaluate the ratio and match the operators defining both masses.
Both optimizers may minimize the same biased approximate objective. Changing initialization probes optimization, but it does not probe the shared environment contraction; an independent environment or contraction family is required.
The equality between the finite-lattice partition sum and the untruncated local tensor contraction is exact. Continuum language additionally requires a tuned line of constant physics, volume control, operator normalization, bond/contraction convergence, and a justified $a\to0$ extrapolation.

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.