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Assessing the expressive power of a gauge-invariant PEPS ansatz across a topological phase transition

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Etsuko Itou, Tsuyoshi Okubo

High Energy Physics - Lattice · Submitted 10 Nov 2026 · 75 references

Pith's verdict: A well-benchmarked demonstration that the eight-parameter F=2 GGFPEPS captures the Z2 gauge theory ground state across the topological transition; the energy claims hold up, the topological entanglement entropy claim is oversold near lambda_c.

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CitationEtsuko Itou, Tsuyoshi Okubo. "Assessing the expressive power of a gauge-invariant PEPS ansatz across a topological phase transition." arXiv:2611.13852 (2026). https://pith.science/paper/GEODTXNK
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The paper

The paper claims

The eight-parameter gauged Gaussian fermionic PEPS ansatz, evaluated with deterministic Grassmann tensor contractions, reproduces the Z2 gauge theory critical point at lambda_c around 0.573 and captures the topological entanglement entropy across the transition.

The reading

This work studies a toy gauge theory in two spatial dimensions, the pure Z2 gauge theory, which has a topological phase transition between a deconfined phase (toric code) and a confined phase. The authors use a compact wavefunction ansatz called the gauged Gaussian fermionic projected entangled pair state, where the gauge fields are encoded using virtual fermions on each link. Earlier work tuned this ansatz with variational Monte Carlo, but statistical noise made it hard to pin down the optimal parameters, especially near the transition. The authors replace the stochastic evaluation with a Grassmann tensor network, which contracts the whole state exactly and deterministically. They renormalize to reach lattices of linear size up to 34 and use the corner transfer matrix method for the infinite volume. With only eight complex variational parameters, the optimized state matches exact diagonalization on small lattices across all couplings, matches perturbation theory in both the strong and weak coupling regimes, and yields susceptibility peaks whose critical coupling, in the range 0.568 to 0.592, is consistent with the quantum Monte Carlo value 0.573. The topological entanglement entropy drops sharply near that point. They also show the optimized state has lower energy than any state in the one-parameter plaquette product family used to build the ansatz, so near the transition the ansatz represents physics beyond that simplified family. The main caveat is right at the critical point. The optimization landscape has several local minima with very similar energies; five random seeds converge to noticeably different states, with the topological entanglement entropy ranging from about 0.60 to 0.74 of the toric code value. The truncation error is also largest there.
Extended reading notes

Core claim

The F=2 GGFPEPS ansatz with only eight complex variational parameters accurately describes the pure Z2 gauge theory including the region of the topological phase transition, not only in the energy but also in the topological entanglement entropy (b5), and the critical coupling obtained from susceptibility peaks agrees quantitatively with the continuous-time QMC value lambda_c = 0.573 (b6). Near the transition point the ansatz represents a state outside the one-parameter plaquette product family (b7, b59). If correct, the paper establishes that this small, explicitly gauge-invariant fermionic tensor network captures the full ground state physics of the model, including the hard transition region, when evaluated deterministically.

Load-bearing premise

The numerical optimization and truncation are reliable in the critical region. Near lambda_c the singular values decay slowly, so truncation to Dcut=32 changes Etot by about 5e-4 (Table 2), and five random seeds converge to different physical states with Etot between 1.05918 and 1.06669 and normalized TEE between 0.604 and 0.741 at lambda=0.6, L=34 (Table 1, Sec 6.3.1). The conclusion that the eight-parameter ansatz captures the transition, and that its optimized state sits outside the plaquette product family, rests on these optimized parameters being representative of what the ansatz can achieve. If the optimizer only samples unrepresentative local minima near the transition, the expressive-power conclusion would be incomplete.

Original abstract

We investigate the expressive power of the gauged Gaussian fermionic projected entangled pair state (GGFPEPS) ansatz, which is a manifestly gauge-invariant tensor network ansatz for $(2+1)$-dimensional lattice gauge theories. In the GGFPEPS ansatz, the gauge field on each link is represented by coupling its configuration to the fermion number of virtual fermions placed at both ends of the link. This ansatz has been studied by variational Monte Carlo (VMC) methods in previous work~\cite{Emonts:2022yom}. However, the statistical error of VMC makes a precise determination of the variational parameters difficult, so that the expressive power of the ansatz could not be examined accurately. In this work, we reformulate the computation as a Grassmann tensor network (TN), in which the whole network is contracted deterministically, and introduce the higher-order tensor renormalization group (HOTRG), which allows us to perform calculations in large volumes up to $L=34$. We also evaluate the infinite volume system using the corner transfer matrix renormalization group (CTMRG). We find that the ansatz with only eight complex variational parameters accurately describes the pure $\mathbb{Z}_2$ gauge theory including the region of the topological phase transition, not only in the energy but also in the topological entanglement entropy. The critical coupling obtained from the peaks of their susceptibilities agrees quantitatively with the one predicted by continuous-time quantum Monte Carlo simulations~\cite{wu2012phase}. Furthermore, near the transition point, we show that the ansatz represents a state outside the plaquette product family, which is a one-parameter family of states assumed in the construction of the ansatz as a simple interpolation between the strong and weak coupling limits.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-11: 1 request, 7,310 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.22 · randomized assignment 0.01 · states quantitative result 0.89 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We investigate the expressive power of the gauged Gaussian fermionic projected entangled pair state (GGFPEPS) ansatz, which is a manifestly gauge-invariant tensor network ansatz for $(2+1)$-dimensional lattice gauge theories. In the GGFPEPS ansatz, the gauge field on each link is represented by coupling its configuration to the fermion number of virtual fermions placed at both ends of the link. This ansatz has been studied by variational Monte Carlo (VMC) methods in previous work~\cite{Emonts:2022yom}. However, the statistical error of VMC makes a precise determination of the variational parameters difficult, so that the expressive power of the ansatz could not be examined accurately. In this work, we reformulate the computation as a Grassmann tensor network (TN), in which the whole network is contracted deterministically, and introduce the higher-order tensor renormalization group (HOTRG), which allows us to perform calculations in large volumes up to $L=34$. We also evaluate the infinite volume system using the corner transfer matrix renormalization group (CTMRG). We find that the ansatz with only eight complex variational parameters accurately describes the pure $\mathbb{Z}_2$ gauge theory including the region of the topological phase transition, not only in the energy but also in the topological entanglement entropy. The critical coupling obtained from the peaks of their susceptibilities agrees quantitatively with the one predicted by continuous-time quantum Monte Carlo simulations~\cite{wu2012phase}. Furthermore, near the transition point, we show that the ansatz represents a state outside the plaquette product family, which is a one-parameter family of states assumed in the construction of the ansatz as a simple interpolation between the strong and weak coupling limits.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate the expressive power of the gauged Gaussian fermionic projected entangled pair state (GGFPEPS) ansatz, which is a manifestly gauge-invariant tensor network ansatz for $(2+1)$-dimensional lattice gauge theories. contribution 0.75 0.49 0.03 asserted 0.93
s1 In the GGFPEPS ansatz, the gauge field on each link is represented by coupling its configuration to the fermion number of virtual fermions placed at both ends of the link. background 0.51 0.20 0.02 asserted 1.00
s2 This ansatz has been studied by variational Monte Carlo (VMC) methods in previous work~\cite{Emonts:2022yom}. background 1.00 0.04 0.01 asserted 1.00
s3 However, the statistical error of VMC makes a precise determination of the variational parameters difficult, so that the expressive power of the ansatz could not be examined accurately. gap 0.98 0.13 0.02 hedged possibility 0.62
s4 In this work, we reformulate the computation as a Grassmann tensor network (TN), in which the whole network is contracted deterministically, and introduce the higher-order tensor renormalization group (HOTRG), which allows us to perform calculations in large volumes up to $L=34$. method 0.90 0.95 0.04 asserted 1.00
s5 We also evaluate the infinite volume system using the corner transfer matrix renormalization group (CTMRG). method 0.97 0.94 0.03 asserted 0.98
s6 We find that the ansatz with only eight complex variational parameters accurately describes the pure $\mathbb{Z}_2$ gauge theory including the region of the topological phase transition, not only in the energy but also in the topological entanglement entropy. result 1.00 0.96 0.84 asserted 1.00
s7 The critical coupling obtained from the peaks of their susceptibilities agrees quantitatively with the one predicted by continuous-time quantum Monte Carlo simulations~\cite{wu2012phase}. result 0.99 0.81 0.10 asserted 1.00
s8 Furthermore, near the transition point, we show that the ansatz represents a state outside the plaquette product family, which is a one-parameter family of states assumed in the construction of the ansatz as a simple interpolation between the strong and weak coupling limits. result 0.96 0.95 0.16 asserted 1.00

What kind of paper the classifier reports this is: simulation study 0.94 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
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  • references and citing sentences
  • relations to other papers
levels run
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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. (Left) Definition of the plaquette operator. (Right) The Gauss law operator V (x). Let us now consider Z2 gauge theory with N = 2. In the p-basis, we choose P = P † = σz, Q = Q † = σx. (2.5) The eigenstates can then be taken as those of σz and σx, with eigenvalues (−1)p and (−1)q , respectively. Since P and Q are Hermitian, the link orientation does not matter for N = 2, and the Hamiltonian becomes H = λ ∑ l [1 − Pl ] + 1 λ ∑ p [1 − (Ql1Ql2Ql3Ql4 )] . (2.6) – 4 –
Figure 3. Virtual and physical legs of the GGFPEPS ansatz. Note that the legs remaining in the state |ψ⟩ are physical legs, described in the q-basis as in Eq. (3.3). Therefore, the operators for physical observables below must also be written as tensors in the q-basis. 3.2 Gauge invariance We now show that the state in Eq. (3.1) is invariant under the Gauss law operator, namely, that it satisfies Eq. (2.8). We write Eq. (3.1) as |ψ⟩ = ⟨Ω| ∏ x,i w † (x, i)UG ∏ x A(x)|Ω⟩|ψE⟩, ≡ O|ψE⟩. (3.8) Since |ψE⟩ = ⊗l |p = 0⟩ is a product of eigenstates of Pl with eigenvalue +1 on each link, V (x)
Figure 4. Construction of a plaquette in the F = 2 ansatz. The two modes on each link are shown in red and blue. The red modes form the loop around the central plaquette, and the blue modes belong to the neighboring plaquettes. the other (blue) is used for the loops around the neighboring plaquettes. The neighboring plaquettes can be excited independently of the central one. This is the essential point of the F = 2 ansatz. Let us write down this construction explicitly. The C4-invariant form of the matrix Tαβ in the A(x) tensor with eight complex parameters is given by T =   0…
Figure 5. Evaluation of a local observable Oˆ. (Left) Decomposition of the L×L = (2+2k )×(2+2k ) lattice. The 2 × 2 part (blue square) is treated as the local cell, and the remaining part (red) as the environment tensor. (Right) To evaluate ⟨Oˆ⟩, the local operator Oˆ is sandwiched between the GGFPEPS ansatz and its conjugate. remaining region to reduce the computational cost. The remaining region is treated as the environment tensor, in which both the virtual fermions and the physical legs (qx, qy) are contracted. A local observable Oˆ is inserted in the 2 × 2 local cell, and its expectation…
Figure 6. (Left) Subsystems A, B, and C used for the TEE. (Right) Evaluation of a component of the reduced density matrix. arrangement fits within a 2 × 2 lattice, we place it inside the local cell. We first outline how to calculate the EE for each subsystem. Writing the density matrix of the full system as ρ = |ψ⟩⟨ψ|/⟨ψ|ψ⟩, we define the reduced density matrix for the subsystem A by ρA = TrA¯ρ. (4.7) Using the product basis |q⟩A of the links in the subsystem A, its matrix elements can be written as (ρA)q ′q = A⟨q ′ |ρA|q⟩A = ⟨ψ| ( |q⟩⟨q ′ | ) A ⊗ 1A¯|ψ⟩ ⟨ψ|ψ⟩ . (4.8) Thus, the (
Figure 7. The bond R cut in the y-direction coarse-graining (dashed line). The two links crossing the cut carry the virtual legs rtop and rbot, and the three plaquettes p1, p2, and p3 touch these links, with p2 shared by the two. The link m between the two sites is contracted. For F = 1, the occupation r ∈ {0, 1} of the virtual mode on a link equals the electric flux p of that link. Since the ket leg r and the bra leg r ′ have the same fermion parity, Eq. (5.6), which means r = r ′ for F = 1, the bond R is effectively labeled by the pair (rtop, rbot), which takes 2 2 = 4 values. The…
Figure 8. Singular values si/s1 in the y-direction coarse-graining step for L = 4 with the F = 2 ansatz at λ = 0.1 (left), 0.6 (middle), and 4.0 (right). The results for the 10 random seeds are overlaid
Figure 9. Energy density Etot for L = 2, 3, and 4 with the F = 2 ansatz, compared with ED (solid curves). The right panel is an enlarged plot of the region near λc
Figure 10. TEE normalized by − ln 2 for L = 2, 3, and 4 with the F = 2 ansatz, compared with ED (solid curves). The right panel is an enlarged plot of the region near λc. show a nearly flat plateau, Stopo/(− ln 2) ≃ 1, in the small λ region, a rapid decrease around the maximum of Etot, and a value consistent with zero in the strong coupling region. The – 26 –
Figure 11. TEE normalized by − ln 2 and the winding ’t Hooft loop ⟨W⟩ for L = 4 with the F = 2 ansatz. The solid curve is the ED result for the TEE. The right panel is an enlarged plot of the region near λc. the transition is not of first order but of second order or a crossover. 6.3 Large volume calculations: L = 6 to 34 lattices 6.3.1 Seed dependence and truncation error Let us see what kind of state is obtained by the optimization of the F = 2 GGFPEPS in large volume calculations
Figure 12. shows the optimized parameters for L = 34 at λ = 0.1 (left), 0.6 (middle), and 4.0 (right), where the results for the five seeds are presented. -1.5 -1 -0.5 0 0.5 1 1.5 Re y1 Re z1 Re y2 Re z2 Re a Re b Re c Re d Im y1 Im z1 Im y2 Im z2 Im a Im b Im c Im d Real part Imaginary part λ = 0.10 seed1 seed2 seed3 seed4 seed5 -1.5 -1 -0.5 0 0.5 1 1.5 Re y1 Re z1 Re y2 Re z2 Re a Re b Re c Re d Im y1 Im z1 Im y2 Im z2 Im a Im b Im c Im d Real part Imaginary part λ = 0.60 seed1 seed2 seed3 seed4 seed5 -3 -2 -1 0 1 2 3 Re y1 Re z1 Re y2 Re z2 Re a Re b Re c Re d Im y1 Im z1 Im y2 Im z2 Im a…
Figure 13. Energy density Etot for L = 6–34 with the F = 2 ansatz, compared with the strong (magenta dotted curve) and weak coupling expansions (cyan dashed curve). The right panel is an enlarged plot of the region near λc. The size of errors is comparable to the symbol size. also find that no volume dependence is visible at these volumes, even near λc. 6.3.3 Winding ’t Hooft loop Next, we show the results for the winding ’t Hooft loop, Eq. (2.19)
Figure 14. Winding ’t Hooft loop ⟨W⟩ for L = 6–34 with the F = 2 ansatz. The magenta dotted line indicates the critical point λc = 0.573 obtained by the continuous-time QMC [2]. leading order of the weak coupling expansion for L > 2, as shown in Sec. 2.3.2. For L = 4 in Sec. 6.2, ⟨W⟩ increased in the region where the TEE changes, but this is because e −αL is still large for such a small L. Note that ⟨W⟩ is exactly zero in our calculations below the onset because of the property of the isometry described in Sec. 4.2. In perturbation theory, however, ⟨W⟩ in this region would scale as O(λ 2L),…
Figure 15. (Left) L dependence of ln⟨W⟩ for λ ≥ 1.5. The lines are the fits ln⟨W⟩ = c0 + c1L. (Right) The slope −c1 as a function of λ, compared with the strong coupling expansion α (SC) = 1/16λ 4 (magenta dotted line). in the strong coupling region. The lines present the fit functions, ln⟨W⟩ = c0 + c1L, for each λ ≥ 1.5. In this range of λ, our data can be well fitted by a linear function of L and ⟨W⟩ converges to zero as L → ∞. The linearity in L is consistent with the cluster decomposition ⟨W⟩ ≃ ⟨P⟩ L. The right panel of
Figure 16. TEE normalized by − ln 2 for L = 6–34 with the F = 2 ansatz. The magenta dotted line indicates the critical point λc = 0.573 obtained by the continuous-time QMC [2]. The right panel is an enlarged plot of the region near λc. The left panel shows that the point where the TEE starts to decrease rapidly is located almost at λc. The right panel, an enlarged plot near λc, shows that the TEE has no volume dependence even near λc for these sizes, in contrast to the small volumes L = 2, 3, and 4 in Sec. 6.2. The change of the TEE is relatively smooth and no discontinuity is seen, which is…
Figure 17. shows the results for Etot. We fit the data of Etot by a smooth function and evaluate the susceptibility, defined as the second derivative χE(λ) = −∂ 2Etot/∂λ2 , from the fit function. Since no volume dependence is seen in the data, all the data for L = 6 to 34 are used in the fit, where each data point (L, λ) is weighted by its error. To evaluate the
Figure 18. TEE for L = 18 and 34 with the additional data points near λc (top) and its sus￾ceptibility χS evaluated by the finite difference (bottom). The horizontal error bars indicate the half width of the interval, and the vertical ones are propagated from the errors of the TEE. The magenta dotted line indicates λc = 0.573 obtained by the continuous-time QMC [2]. λ and drops rapidly around λc, it is difficult to fit it by a smooth function. We therefore define the susceptibility as the first derivative, χS(λ) = 1 ln 2 ∂Stopo ∂λ , (6.1) and evaluate it by the finite difference. Namely, for…
Figure 19. Comparison with the plaquette product family. (Left) Etot for L = 6–34 compared with the lower bound within the family, mint Efam(t; λ) (purple solid curve), and the weak coupling expansion (cyan dash-dotted curve). (Right) α(λ) normalized by α (SC) = 1/16λ 4 for L = 6–34, compared with the prediction of the family (purple solid curve), which diverges at λ = 1/2. The magenta dotted line indicates λc = 0.573 obtained by the continuous-time QMC [2], and the gray dotted line λ = 1/2. near λc. Since a variational energy within the family cannot be lower than mint Efam, this shows that…
Figure 20. Diagram of the CTMRG environment for calculating the expectation value ⟨Oˆ⟩ of a local observable. We approximate the infinitely large environment by a network consisting of corner (Ci) and edge (Exi, Eyi) tensors. The accuracy of the approximation is controlled by the maximum bond dimension Dcut of the corner and edge tensors. – 35 –
Figure 21. (Left) Energy density Etot and its electric and magnetic contributions, Ee and Em, in the infinite volume limit using the F = 2 ansatz, compared with the strong coupling expansion (magenta dotted curve) and the weak coupling expansion (cyan dashed curve). (Right) The TEE normalized by (− ln 2). The magenta vertical dotted line indicates the critical point λc = 0.573 obtained by the continuous-time QMC [2]. To further investigate the nature of the phase transition, we examine the susceptibilities χE of Etot and χS of the TEE by the finite difference method, as shown in
Figure 22. (Left) Energy density Etot (top) and its susceptibility χE (bottom) near λc in the infinite volume limit using the F = 2 ansatz. Etot is compared with the strong coupling expansion (magenta dotted curve) and the weak coupling expansion (cyan dashed curve). (Right) The TEE normalized by (− ln 2) (top) and its susceptibility χS (bottom) near λc. The susceptibilities are obtained by the finite difference method. The magenta vertical dotted line indicates the critical point λc = 0.573 obtained by the continuous-time QMC [2]. test the ansatz systematically. We have introduced the HOTRG…
Figure 23. Singular values si/s1 in the x-direction (left) and y-direction (right) coarse-graining steps for L = 4 with the F = 1 ansatz at λ = 0.1. The results for the five seeds are overlaid. For the F = 1 ansatz, the bond dimension of R is only 4, since each leg of Tii has the effective dimension 2 2F −1 = 2. As discussed in Sec. 5.2.2, the leading singular values are expected to be twofold degenerate for F = 1, in contrast to the eightfold degeneracy for F = 2. This is indeed seen in
Figure 24. shows the results for a wide range of λ from the strong to weak coupling regions. Here, the energy density Etot and its electric and magnetic parts, Ee and Em, are plotted for the F = 1 and F = 2 ansatzes and compared with ED (solid curves). For λ ≲ 0.3 and λ ≳ 1.5, there is almost no difference between F = 1 and 2, and both 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 ED, Etot ED, Ee ED, Em F=2, Etot F=2, Ee F=2, Em F=1, Etot F=1, Ee F=1, Em Energy density λ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 0.6 0.7 0.8 0.9 1 ED, Etot ED, Ee ED, Em F=2, Etot F=2, Ee F=2, Em F=1, Etot F=1, Ee F=1, Em Energy…

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