Fixed-Point Tensor Network Representations

• Fixed-point tensor network representations are tensors invariant under specific maps, capturing macroscopic observables while discarding microscopic details.
• They serve as canonical transfer-matrix fixed points in MPS/PEPS, enabling extraction of key metrics like correlation lengths, scaling spectra, and entanglement structures.
• Applications span numerical renormalization algorithms, topological state sums, and conformal field theory, providing a unified framework for phase and criticality analysis.

Fixed-point tensor network representations are tensors, or sets of tensors, that are invariant under a transfer, contraction, or coarse-graining map up to gauge transformations and normalization. In practice they arise as canonical transfer-matrix fixed points of matrix product states, as boundary states for projected entangled-pair states, as infrared representatives of tensor-network renormalization flows, and as exact fixed-point descriptions of topological and conformal field theories. Across these settings, the fixed-point object encodes macroscopic data—correlation length, free-energy density, scaling spectra, entanglement structure, superselection sectors, or boundary conformal data—while discarding nonuniversal microscopic details (Ran et al., 2017, Vanthilt et al., 8 Apr 2026, Luo et al., 2016).

1. Formal definition and basic fixed-point equations

A fixed-point tensor network representation is defined relative to a map acting on tensors or boundary states. For a transfer operator $E$, fixed-point means that dominant left and right eigenvectors are preserved up to normalization,

$$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$

For a coarse-graining map $\mathcal{R}$ acting on local tensors, fixed-point means

$\mathcal{R}(\{T\}) \simeq \{T\}$

up to gauge and normalization. In tensor-network renormalization this is often written as

$R(T^*) = \lambda T^*,$

or, more explicitly, $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$, where the equality is understood as equality of partition functions and correlators on all scales rather than literal componentwise identity (Ran et al., 2017, Ueda, 2024).

For a translation-invariant MPS with local tensors $\{A^s\}$, the transfer operator is

$E = \sum_s A^s \otimes \overline{A^s},$

and the correlation length follows from the dominant and subleading eigenvalues,

$\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$

In canonical form one chooses $\lambda_0=1$, so the dominant fixed-point eigenvectors are identities. In the left- and right-canonical gauges,

$$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$0

and the singular-value spectra directly encode bipartite entanglement (Ran et al., 2017).

The same logic extends to higher-dimensional tensor networks. For an infinite PEPS, the fixed-point boundary state is the dominant eigenvector of a row- or column-transfer MPO. For classical partition-function tensor networks on two-dimensional lattices, the fixed-point is the tensor to which repeated coarse-graining flows; trivial/product-like fixed points occur in gapped phases, while nontrivial scale-invariant fixed points characterize criticality. A central theme across algorithms is therefore that the physically relevant tensor is not the microscopic tensor itself, but its representative in the quotient by gauge freedom and normalization after repeated renormalization (Ran et al., 2017, Ueda, 2024).

2. Boundary fixed points in variational, transfer-matrix, and environment algorithms

In one-dimensional and quasi-one-dimensional settings, fixed-point tensor network representations appear most directly as boundary or environment states. For an infinite PEPS on a square lattice, contraction in one direction is reduced to the dominant-eigenvector problem

$$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$1

where $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$2 is a row-MPO built from a horizontal line of double tensors. The boundary MPS $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$3 is the fixed-point boundary state used to contract the two-dimensional network (Ran et al., 2017).

This viewpoint unifies several standard algorithms. Imaginary-time TEBD and iTEBD iteratively apply local gates and truncate, converging to a stationary MPS up to gauge. DMRG and iDMRG solve effective eigenvalue problems for central tensors together with left and right environments satisfying fixed-point equations. Variational boundary-state methods for PEPS likewise replace the full environment by the dominant eigenvector of a transfer operator, obtained by power iteration, iTEBD, or VMPS/DMRG. In each case, convergence is diagnosed by stabilization of the energy per site, spectra such as $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$4 and $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$5, transfer-matrix eigenvalues, and the induced correlation length $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$6 (Ran et al., 2017).

Corner transfer matrix renormalization group makes the fixed-point structure especially explicit. The boundary environment is represented by corner tensors $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$7 and edge tensors $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$8, updated by absorbing a row or column of double tensors and truncating enlarged bonds with an isometry $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$9 computed from corner environments. The iteration defines a map

$\mathcal{R}$0

and convergence to $\mathcal{R}$1 gives the CTMRG fixed point. These fixed points determine environments, free energies, and corner spectra, and they improve on strictly local truncation schemes by incorporating nonlocal boundary dependence (Ran et al., 2017).

From a multilinear-algebra perspective, canonicalization and super-orthogonalization are the procedures that remove gauge freedom and expose fixed-point structure. SVD and QR underlie local truncation and canonical form, Tucker/HOSVD generalizes this logic to higher-order tensors, and rank-1 decompositions provide zero-loop approximations in which fixed-point equations become explicitly self-consistent (Ran et al., 2017).

3. Renormalization-group fixed points of classical tensor networks

For classical partition functions, fixed-point tensor network representations are the stable objects of a renormalization flow. In square-lattice TRG, a rank-4 tensor $\mathcal{R}$2 is split by SVD in two diagonal channels and recombined into a renormalized tensor $\mathcal{R}$3. Iteration defines a coarse-graining map $\mathcal{R}$4, and fixed points $\mathcal{R}$5 characterize phases: trivial fixed points for gapped disordered phases, and nontrivial fixed points for ordered or topological phases. At criticality, scale invariance implies a nontrivial fixed point, but plain TRG converges slowly because its truncation environment is local (Ran et al., 2017).

TNR-based schemes reformulate this flow by explicitly removing short-range entanglement. In TNRKit.jl, the coarse-graining map $\mathcal{R}$6 includes TRG, HOTRG, LoopTNR, and nuclear-norm-regularized LoopTNR, and fixed-point tensors are defined by

$\mathcal{R}$7

The scalar $\mathcal{R}$8 carries the local intensive free-energy density accumulated in a step, while physical observables are obtained after normalization or by ratios. TRG has cost $\mathcal{R}$9 in two dimensions and HOTRG has cost $\mathcal{R}(\{T\}) \simeq \{T\}$0; both can produce accurate thermodynamics, but both retain residual short-range structures at criticality. LoopTNR introduces entanglement filtering and local optimization on a $\mathcal{R}(\{T\}) \simeq \{T\}$1 cell, removing corner-double-line loops and yielding stable scale-invariant fixed-point tensors across many RG steps (Vanthilt et al., 8 Apr 2026).

The practical distinction between gapped and critical fixed points is reflected in the transfer spectrum. In a gapped phase the dominant fixed point is unique and $\mathcal{R}(\{T\}) \simeq \{T\}$2 is finite; at criticality subleading eigenvalues approach the dominant one and $\mathcal{R}(\{T\}) \simeq \{T\}$3 diverges. TNRKit uses stability of transfer or scaling spectra, invariance of extracted scaling dimensions and central charge under further coarse-graining and shape changes, small truncation errors, and symmetry consistency across charge blocks as fixed-point diagnostics (Vanthilt et al., 8 Apr 2026).

Energy-based finite-size scaling gives an additional RG interpretation. Perturbing a critical fixed-point tensor by local operators,

$\mathcal{R}(\{T\}) \simeq \{T\}$4

one obtains running couplings with beta functions

$\mathcal{R}(\{T\}) \simeq \{T\}$5

Finite-size scaling of free energy, energy density, and transfer spectra then identifies whether a numerical flow has reached a fixed point and relates the linearized RG data to CFT scaling dimensions and OPE coefficients (Ueda, 2024).

4. Exact topological fixed points and categorical structure

A distinct but closely related notion of fixed-point tensor network representation appears in topological phases. In two dimensions, fixed-point tensor network states on a trivalent lattice are exactly invariant under local dual Pachner moves, and the consistency conditions for that invariance reproduce the data of a unitary fusion category $\mathcal{R}(\{T\}) \simeq \{T\}$6. The local tensors are built from quantum dimensions and $\mathcal{R}(\{T\}) \simeq \{T\}$7-symbols, and the fixed-point condition under $\mathcal{R}(\{T\}) \simeq \{T\}$8–$\mathcal{R}(\{T\}) \simeq \{T\}$9 and $R(T^*) = \lambda T^*,$0–$R(T^*) = \lambda T^*,$1–$R(T^*) = \lambda T^*,$2 recouplings is equivalent to the orthogonality and pentagon identities. In this sense, exact RG invariance is identical to categorical coherence (Luo et al., 2016).

This algebraic structure organizes long-range entanglement. Loop or MPO operators obey pulling-through constraints governed by the same $R(T^*) = \lambda T^*,$3-symbols, and the resulting tube algebra has central idempotents projecting onto superselection sectors. Those sectors are classified by the Drinfeld center $R(T^*) = \lambda T^*,$4, which is the anyon theory of the doubled topological order. The topological entanglement entropy is

$R(T^*) = \lambda T^*,$5

with $R(T^*) = \lambda T^*,$6 the total quantum dimension. The same categorical input also defines the three-dimensional Turaev–Viro state sum, so the boundary fixed-point PEPS/string-net state and the bulk TQFT are two realizations of the same data (Luo et al., 2016).

A Euclidean path-integral version of this statement is furnished by diagrammatic “liquids,” where fixed-point tensor networks are built as discrete spacetime path integrals on triangulations or more general cellulations. Here fixed point means zero correlation length, rank-1 annular transfer, and invariance under local topology-preserving moves. In $R(T^*) = \lambda T^*,$7 dimensions this framework recovers Turaev–Viro/string-net models and Kitaev quantum doubles, including weak-Hopf-algebra generalizations, by assigning local tensors to tetrahedra, faces, and edges together with branching structures, edge weights, and cancellation moves. The local tensor equations are combinatorial versions of Pachner moves, and commuting-projector Hamiltonians appear as exact Euclidean tensor-network path integrals after Trotterization (Bauer et al., 2020).

Symmetry enrichment fits naturally into the same fixed-point language. A finite onsite unitary symmetry $R(T^*) = \lambda T^*,$8 is incorporated by passing from an ordinary UFC to a $R(T^*) = \lambda T^*,$9-graded UFC, imposing flatness constraints on group labels, and modifying the local tensor so that global $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$0 acts by right multiplication on plaquette variables and conjugation on link variables. This yields symmetry-enriched topological fixed points and, in the bulk, a homotopy QFT with target $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$1 (Luo et al., 2016).

5. Fixed-point tensors as carriers of conformal data

At criticality, fixed-point tensor network representations become discrete realizations of CFT data. One route is purely numerical: transfer operators built from a fixed-point tensor yield scaling dimensions, spins, and central charge. Another route is structural: the entries of the fixed-point tensor themselves can be identified with CFT correlators.

A four-leg critical fixed-point tensor $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$2 can be identified with a four-point function. After gauge fixing by diagonalizing the transfer matrix on a cylinder so that each leg is labeled by an orthonormal basis of CFT states, the tensor elements satisfy

$T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$3

For the canonical square geometry used in the explicit construction, the normalized tensor is the four-point correlator at the points $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$4, so the cross ratio is $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$5. The associated three-leg tensor $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$6 is a three-point function, and it yields OPE coefficients directly. In the Ising case, measured fixed-point tensor elements agree with exact CFT values for channels such as $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$7 and $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$8, while symmetry-forbidden channels are numerically zero (Ueda et al., 2023).

The same conclusion appears from an RG-flow perspective. At a nontrivial fixed point, tensor elements align with four-point functions of primary operators, and the conformal-block decomposition of those four-point functions organizes the tensor’s block structure. In this description virtual indices decompose into symmetry sectors, and the nonzero pattern of $T^\star \simeq \lambda\, G\!\cdot\! R[T^\star]$9 follows CFT fusion rules. Finite-size scaling of free energy and transfer spectra yields

$\{A^s\}$0

linking the fixed-point tensor directly to $\{A^s\}$1, $\{A^s\}$2, and the RG exponents $\{A^s\}$3 (Ueda, 2024).

Exact fixed-point tensors are available for rational CFTs. In the boundary-changing-operator construction, the local rank-3 tensor is

$\{A^s\}$4

where $\{A^s\}$5 label conformal boundary conditions, $\{A^s\}$6 label primary families and descendants, $\{A^s\}$7 are boundary structure constants, and $\{A^s\}$8 are conformal blocks. The tensor satisfies crossing identities and real-space RG consistency conditions exactly, reproduces the RCFT partition function on triangulated surfaces, and yields transfer-matrix spectra whose eigenvalues obey

$\{A^s\}$9

Explicit constructions for Ising, Yang–Lee, and tricritical Ising recover the expected central charges and scaling dimensions with modest descendant cutoffs (Cheng et al., 2023).

Linearized TRG furnishes a related fixed-point formalism. A fixed point $E = \sum_s A^s \otimes \overline{A^s},$0 satisfies $E = \sum_s A^s \otimes \overline{A^s},$1, and the Jacobian $E = \sum_s A^s \otimes \overline{A^s},$2 of the RG map at $E = \sum_s A^s \otimes \overline{A^s},$3 acts as a scaling operator,

$E = \sum_s A^s \otimes \overline{A^s},$4

In the Ising study based on GILT+HOTRG and minimal canonical form, this approach recovers $E = \sum_s A^s \otimes \overline{A^s},$5, $E = \sum_s A^s \otimes \overline{A^s},$6, and OPE coefficients from overlaps of fixed-point tensors or defect tensors; the minimal canonical form further improves stability of the RG flow (Guo et al., 2023).

High-quality numerical fixed points sharpen transfer-matrix methods still further. For the 2D Ising model, direct and crossed transfer matrices built from a Newton-improved Gilt-TNR fixed point extract scaling dimensions and spins modulo integers, while the lattice dilatation operator (LDO) reaches higher levels. For the rotating map, quasiprimary Jacobian eigenvalues obey

$E = \sum_s A^s \otimes \overline{A^s},$7

so the phase yields spin modulo $E = \sum_s A^s \otimes \overline{A^s},$8. The transfer-matrix method achieves good agreement up to $E = \sum_s A^s \otimes \overline{A^s},$9, while the LDO yields good agreement up to $\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$0 (Ebel et al., 2024).

Fixed-point tensors are not restricted to rational theories. For the compactified boson at generic radius $\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$1, local rank-3 and rank-4 tensors are built from Dirichlet and Neumann boundary data, with leg charges determined by boundary-condition differences and winding or momentum sectors. The closed-string spectrum is extracted from cylinder transfer matrices through

$\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$2

The resulting fixed-point tensors reproduce the $\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$3 spectrum at irrational radii, exhibit T-duality, and realize an exactly marginal deformation at the single-tensor level by tuning the coefficient of the level-$\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$4 state $\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$5 (Cheng et al., 3 Jul 2026).

6. Direct fixed-point solvers, rigorous fixed points, and mixed-state extensions

A recent line of work treats the fixed-point equation itself as the primary computational object. For a normalized RG map $\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$6, one solves

$\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$7

by Newton’s method. The main obstruction is that, for a nonrotating critical map, the stress-tensor directions have marginal eigenvalue $\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$8, so the Jacobian is singular. This is resolved by modifying the RG map with a $\xi = -\frac{1}{\ln |\lambda_1/\lambda_0|}.$9 rotation,

$\lambda_0=1$0

which changes quasiprimary eigenvalues to

$\lambda_0=1$1

For the stress tensor with $\lambda_0=1$2 and $\lambda_0=1$3, the eigenvalue becomes $\lambda_0=1$4, isolating the fixed point. Using Gilt-TNR at bond dimension $\lambda_0=1$5, this method finds Ising and 3-state Potts fixed-point tensors with residuals $\lambda_0=1$6 and $\lambda_0=1$7, respectively (Ebel et al., 2024).

Rigorous RG analysis gives complementary fixed-point results in infinite-dimensional tensor spaces without truncation. In the high-temperature phase, there is a trivial fixed-point tensor $\lambda_0=1$8 with only

$\lambda_0=1$9

nonzero. The normalized RG map satisfies

$$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$00

and for tensors sufficiently close to $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$01 the deviation contracts as

$$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$02

This super-exponential contraction proves stability of the high-temperature fixed point and also clarifies why the naive contraction-only RG is not contractive: corner-double-line tensors generate marginal directions that must be removed by a disentangling step (Kennedy et al., 2021).

At low temperature, the rigorous fixed-point structure is qualitatively different. The tensor is decomposed as

$$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$03

where $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$04 and $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$05 are the two pure-phase fixed points and $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$06 is exponentially small at large $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$07. The RG map induces

$$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$08

with

$$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$09

The equal-weight direct sum is therefore an unstable discontinuity fixed point, while the two pure phases are stable fixed points. This yields a tensor-RG proof of a first-order transition, analyticity of the free energy away from the coexistence surface, and a discontinuity in the magnetization at the critical field (Kennedy et al., 2022).

Fixed-point tensor network representations also extend beyond pure states. A general protocol constructs fixed-point PEPS or MPS for intrinsically mixed-state topological orders by anyon condensation in Choi states. In the doubled space, local decoherence channels become commuting projectors, and the mixed-state tensor network is assembled from doubled tensors $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$10, on-site basis changes $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$11, and connector tensors $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$12 that enforce condensation constraints. For $$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$13 channels one has, for example,

$$E\,|r\rangle = \lambda_0 |r\rangle,\qquad \langle l|\,E = \lambda_0 \langle l| .$$14

The resulting density-matrix tensor network is an exact RG fixed point, has zero correlation length away from operator insertions, and realizes intrinsically mixed-state topological phases with transparent anyons or chiral features that do not have pure-state commuting-projector counterparts (Aldossari et al., 30 Jul 2025).

Taken together, these developments show that fixed-point tensor network representations are not a single construction but a unifying language. The same basic notion—tensorial invariance under renormalization, transfer, or recoupling—organizes canonical infinite-MPS forms, PEPS boundary environments, scale-invariant classical partition-function tensors, exact topological state sums, rational and irrational CFT fixed points, rigorous high- and low-temperature phases, and intrinsically mixed-state topological orders.