Pith. sign in

REVIEW 4 major objections 7 minor 2 cited by

Two-color lattice QCD in $(1+1)$ dimensions with Grassmann tensor renormalization group

T0 review · 4 major / 7 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that at $m \approx -0.49084721$ the Wilson-fermion two-color lattice QCD is critical with $c \approx 1.017$ and $\Delta_i \approx 1/2$, and that Grassmann TRG represents staggered and Wilson fermions with equal bond…

desk verdict The Wilson-fermion Grassmann representation and compression scheme are real steps forward, but the claimed c=1 critical point rests on a single tuned extraction with no convergence checks and a radius assignment that looks off. read the letter →

arxiv 2501.18918 v1 pith:7LUQ5YO5 submitted 2025-01-31 hep-lat

Kwok Ho Pai , Shinichiro Akiyama , Synge Todo This is my paper · ORCID
classification hep-lat
keywords Grassmanntensorrenormalizationgrouptwo-colorlatticeQCD1+1dimensionsstaggeredfermionWilsonconformalfieldtheorydatafinitedensitysignproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that the Grassmann tensor renormalization group can handle a non-Abelian gauge theory coupled to fermions in $(1+1)$ dimensions without Monte Carlo sampling, which would remove the sign problem at finite density. It constructs Grassmann tensor network representations for both staggered and Wilson fermions using the same initial bond dimension, and adds a compression step that reduces the initial bond dimension of the gauge tensors. Using the staggered fermion action it computes the number density, chiral condensate, and diquark condensate at finite density, reproducing the Silver-Blaze phenomenon (observables independent of the chemical potential below a threshold) and an intermediate phase seen in mean-field studies. For the Wilson fermion theory at infinite coupling it reports a critical point at $m \approx -0.49084721$ with central charge $c \approx 1.017453$ and the first four scaling dimensions near $1/2$, suggesting a compactified free-boson conformal field theory with radius $R = 1/\sqrt{2}$. If correct, the method identifies a universality class in a non-Abelian fermion-gauge theory purely from tensor-network data.

What carries the argument

The carrying mechanism is the Grassmann tensor network representation of the partition function, in which anticommuting Grassmann variables are encoded in tensor indices so that fermion signs are handled exactly, combined with the bond-weighted TRG coarse-graining algorithm, a renormalization scheme based on weighted singular value decompositions. The initial tensor for the gauge sector is compressed by a truncated singular value decomposition that keeps the smallest bond dimension satisfying a retained-weight criterion. After repeated coarse graining, a transfer matrix is built from renormalized tensors, and the central charge $c$ and scaling dimensions $\Delta_i$ are extracted from its eigenvalues through Eq. (16), using the scale-invariance relation $\lambda_0^{(2q-2)}/Z_q = \lambda_0^{(2q-4)}/Z_{q-1}$ that holds only at criticality.

What would settle it

Run the same transfer-matrix calculation at larger bond dimensions, say $D = 150, 180, 240$, and at additional volumes; if $c$ and the leading $\Delta_i$ drift away from $1$ and $1/2$ as $D$ grows, the reported free-boson fixed point is a truncation artifact. A complementary check is an independent finite-size-scaling collapse of the pseudoscalar condensate at $m = -0.49084721$ using the predicted scaling dimensions.

Watch

Extended reading notes

Core claim

The paper's central claim is that a Grassmann tensor network with the same construction and the same initial bond dimension describes both staggered and one-flavor Wilson fermions in $(1+1)$-dimensional two-color lattice QCD, with gauge degrees of freedom efficiently compressed before renormalization. At infinite gauge coupling, the Wilson theory is reported to have a critical point in the negative mass region: at $m \approx -0.49084721$ on a volume $V = 2^{20}$ with bond dimension $D = 120$, the transfer-matrix analysis gives $c \approx 1.017453$, and the first four scaling dimensions are approximately $1/2$ in the large-volume limit. The paper interprets this as evidence that the conformal field theory describing the criticality is the compactified free boson with radius $R = 1/\sqrt{2}$.

Load-bearing premise

The central critical-point claim stands or falls on whether the tensor-network data at one lattice volume $V=2^{20}$ and one bond dimension $D=120$ have converged to the thermodynamic, infinite-bond-dimension limit; the paper presents no extrapolation in either quantity.

Editorial extensions

If this is right

  • The same Grassmann tensor construction applies to staggered and Wilson fermions with equal initial bond dimension, so a single implementation can compare the two discretizations directly.
  • Finite-density observables such as the number density, chiral condensate, and diquark condensate can be computed without a sign problem, reproducing the Silver-Blaze plateau and a saturated-density phase.
  • The proposed initial compression reduces the bond dimension before renormalization, cutting the cost of finite-coupling simulations and making larger gauge groups more accessible.
  • At $m \approx -0.49084721$, the Wilson-fermion theory flows to a conformal point with $c \approx 1.017$ and $\Delta_i \approx 1/2$, consistent with a compactified free boson at $R = 1/\sqrt{2}$.
  • The transfer-matrix CFT extraction provides a route to universality-class identification in non-Abelian fermion-gauge theories.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the free-boson identification is correct, the fixed point has a continuously varying radius parameter; computing $c$ and $\Delta_i$ at several negative masses would reveal whether the radius changes along a critical line, the signature of a topological phase transition.
  • The paper reports the critical point at a single volume and bond dimension; testing whether $m_c$, $c$, and $\Delta_i$ shift under larger $D$ and $V$ would determine how much of the free-boson match is a truncation artifact.
  • The same transfer-matrix machinery could be applied to the staggered-fermion theory at finite density, potentially locating critical endpoints or quantum phase transitions in the $\mu$-$m$ plane that the current observables only hint at.
  • For finite gauge coupling $\beta$, the initial compression becomes less efficient, so extending the Wilson-fermion analysis beyond $\beta = 0$ will require either larger bond dimensions or a better compression scheme; the construction described in the paper is the natural starting point for that test.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. This proceedings paper (LATTICE2024) develops Grassmann tensor-network representations of (1+1)-dimensional SU(2) lattice gauge theory coupled to staggered or Wilson fermions, and applies the bond-weighted TRG to finite-density observables and to critical phenomena. For the staggered theory the authors compute the number density, chiral condensate, and diquark condensate as functions of the chemical potential, observing Silver-Blaze behavior, an intermediate-density phase, and saturation; these results (Figs. 2-3) are adapted from the companion preprint Ref. [9]. The central new claim is in Sec. 3: in the infinite-coupling Wilson-fermion theory, a critical point at m ≈ -0.49084721 is identified from the pseudoscalar condensate and from transfer-matrix CFT data, with central charge c = 1.017453 at V = 2^20 and four scaling dimensions near 1/2, interpreted as a compactified free boson with radius R = 1/√2.

Significance. The paper offers two potentially useful contributions. First, the unified Grassmann formulation of staggered and Wilson fermions with the same initial bond dimension, together with the β=0 exact gauge integration and the initial compression scheme, is a helpful technical step toward applying tensor networks to non-Abelian fermion-gauge theories. Second, if substantiated, the Wilson-fermion critical point with c ≈ 1 and Δᵢ ≈ 1/2 would demonstrate that GTRG transfer-matrix data can identify universality classes in this setting and would connect the negative-mass regime to the Aoki-type critical point studied in the Schwinger model with Wilson fermions [13]. Credit is due for the finite-density observables at β=0 and β=0.8 (Silver-Blaze plateau, intermediate phase, saturation), which are consistent with expectations. The significance is moderated by three facts: the staggered results are adapted from the companion paper Ref. [9]; the Wilson-fermion section is explicitly preliminary; and the central numerical evidence is reported at a single bond dimension without error estimates. This is a promising proceedings report whose central claim needs additional support.

major comments (4)
  1. [Sec. 3.2, Figs. 5-6] The critical-point identification lacks an independent criticality criterion. The mass m = -0.49084721 is quoted to eight significant digits, but the manuscript never states how this value was obtained: the only described indicator is the broad peak of the pseudoscalar condensate near m = -0.5 in Fig. 5, which is a weak estimator at fixed h = 10^-4, and no m-resolution for that scan is given. If the value was refined by requiring c ≈ 1 and Δ_i ≈ 1/2 from Eq. (16), the reported plateau in Fig. 6 is a consistency condition rather than an independent test, especially because the paper itself notes in Sec. 3.1 that Eq. (16) is only valid at criticality. The central charge is quoted at V = 2^20 while the dimensions are attributed to the thermodynamic limit; please specify the extrapolation procedure. To make the claim load-bearing, the authors should describe the m-scan, determine m_c by an independent method (for example, data collapse of the condensate, the h -> 0 limit, or Binder-type cumulants), and then report c and Δ_i at the fixed m_c with error estimates.
  2. [Sec. 3.2 (radius interpretation)] The stated interpretation is internally inconsistent with the reported spectrum under the standard compactified-boson formula. With Δ_{n,m} = n^2/(2R^2) + m^2 R^2/2, the lowest excited state at R = 1/√2 has Δ = 1/4, which should be the best-resolved level in the transfer-matrix spectrum, whereas the manuscript reports the first four (and Fig. 6 shows a fifth) scaling dimensions near 1/2. A degenerate level at Δ = 1/2 is instead the signature of the self-dual radius R = 1, i.e., the free Dirac fermion; this also matches the four-fold degeneracy at 1/2. If a different normalization of vertex-operator dimensions is intended, it must be stated explicitly. As written, the radius R = 1/√2 and the quoted Δ_i cannot both be correct.
  3. [Sec. 3.1, Eq. (16), Fig. 6] There is no bond-dimension convergence check for the CFT data. All values of c and Δ_i are computed at D = 120, and the scale-invariance relation underlying Eq. (16) holds exactly only for the fixed-point tensor in the infinite-bond-dimension limit; the deviation c = 1.017453 from 1 could therefore be a truncation artifact, and the plateau in Fig. 6 could be a fixed-D truncation plateau rather than evidence of criticality. The authors should report c and Δ_i at larger D (for example 150 and 180) or provide a truncation-error estimate, and should give the numerical values of Δ_1 through Δ_5 at V = 2^20, since the text states four dimensions near 1/2 while the figure caption shows five curves.
  4. [Secs. 2.3-2.5, Figs. 2-3] The finite-density results are presented without error bars, and the finite-β calculations rest on a single gauge-group sample size K = 14 with no K-convergence check and on an unspecified compression ratio r in Eq. (7). Because the initial compression truncates the discretized gauge integral, the systematic error in the number density and the condensates is not controlled; please state the value of r and the resulting compressed bond dimension D' used for Figs. 2c/2d and 3, and provide at least one check in K or r, or an error estimate, for the quoted values of μ_c1 and μ_c2.
minor comments (7)
  1. [Sec. 3.2] The phrase 'strong coupling limit β -> 0' is imprecise: the calculations are performed at β = 0, where the gauge integration is exact, rather than in a limiting sense.
  2. [Sec. 3] Please clarify what is meant by 'four-component Grassmann fields' for the two-dimensional Wilson fermion, since a 2D Dirac spinor has two components; the counting (spin components times colors, or ψ together with ψ̄) should be stated.
  3. [Fig. 5] The m-resolution of the scan in Fig. 5 is not stated; please give the increment in m and the number of data points used to draw the curve.
  4. [Fig. 6] The body text says that the first four scaling dimensions are near 1/2 while the caption lists five curves; please report the numerical values of Δ_1 through Δ_5 at V = 2^20 and comment on the degeneracy pattern.
  5. [Sec. 3.2] Quoting c = 1.017453 with six digits and m_c = -0.49084721 with eight digits overstates the numerical precision of a D = 120 calculation presented without error estimates; the number of digits should reflect the actual resolution.
  6. [Fig. 3 caption] The caption's claim that the curves are 'in the thermodynamic limit' should be qualified, since the TRG result is an approximation to that limit at fixed bond dimension.
  7. [Eq. (6)] Please state explicitly that the SU(2) matrices are sampled with the Haar measure and whether the same random sample set is used for all values of μ in Fig. 3, since correlated samples affect the apparent smoothness of the curves.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Wilson critical-point CFT data are a nontrivial consistency check, not a fitted input; same-team citations are methodological, not load-bearing.

full rationale

The central Wilson-fermion claim does not reduce to its inputs by construction. The paper locates the transition region independently via the pseudoscalar condensate peak near m = -0.5, then evaluates Eq. (16) at m = -0.49084721. Although Eq. (16) is stated to be valid only at criticality, the plateaus in c and the scaling dimensions shown in Fig. 6 are exactly the scale-invariance check needed to justify that application. A single mass parameter cannot simultaneously force c ≈ 1 and five scaling dimensions to ≈ 1/2, so the observed free-boson-like spectrum is a nontrivial numerical result rather than an identity. The lack of bond-dimension extrapolation, error bars, and a systematic mass scan is a robustness or correctness concern, not circularity. The staggered-fermion results are adapted from the authors' companion paper Ref. [9], but that is a separate prior calculation; citing it does not make the present derivation circular. No uniqueness theorem is imported from the authors, and the Grassmann tensor representation is derived from the lattice actions. Overall, no prediction in the paper is equivalent by definition to an input.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

All physical content rests on established Grassmann TRG formalism (Ref [7]), on a numerical approximation of SU(2) integration (Eq. 6), and on the transfer-matrix CFT formula (Eq. 16). Truncation parameters r, K, D, and finite-difference increments are chosen by hand; the critical mass m_c is tuned to the claimed CFT behavior. No new physical entities are postulated.

free parameters (5)
  • Initial compression ratio r = not stated
    Introduced in Eq. (7) to set the truncated-SVD threshold for reducing initial bond dimension from 16K to D'. The paper never gives the r value used, so the effective truncation error is uncontrolled.
  • Gauge-group sample size K = 14
    Used in Eq. (6) to discretize the SU(2) Haar integration at finite beta (Sec. 2.5). K=14 is stated without a convergence check.
  • Bond dimension cutoffs D = 84, 150, 120
    TRG truncation cutoffs chosen for staggered beta=0, staggered finite beta, and Wilson runs (Figs. 2, 3, 5, 6). No D-extrapolation is shown.
  • Critical mass m_c = -0.49084721
    Tuned in Sec. 3.2 so that the transfer-matrix CFT data give c approximately 1 and Delta_i approximately 1/2; quoted to eight significant digits without an error bar.
  • Finite-difference increments Delta_mu, Delta_m, Delta_lambda, Delta_h = 0.04/0.02, 1e-4, 1e-4, 1e-4
    Used in Eqs. (9) and (12) and for the pseudoscalar condensate; step sizes are chosen small but no step-size convergence or discretization bias is reported.
assumptions (4)
  • standard math Grassmann tensor network representation of the staggered fermion action can be built via auxiliary Grassmann integration as in Ref. [7].
    Sec. 2.2 states 'Following the formalism in Ref. [7]' and does not reproduce the derivation; the central construction depends on this background result.
  • domain assumption The finite sum in Eq. (6) with K samples approximates the SU(2) Haar measure integral.
    Sec. 2.2 and 2.5 use K=14; the paper gives no proof or convergence test that this discrete set is accurate for the observables.
  • standard math The scale-invariance relation behind Eq. (16) gives the central charge and scaling dimensions from transfer-matrix eigenvalues.
    Sec. 3.1 asserts the relation citing Refs. [14,15]; it is valid only at criticality, which is exactly what the paper uses m_c to enforce.
  • domain assumption Thermodynamic limit is well approximated by V=2^20 and fixed bond dimension D.
    Sec. 3.2 computes c and Delta_i at V=2^20, D=120 and calls this the thermodynamic limit; no V or D extrapolation is reported.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Two-color lattice QCD in $(1+1)$ dimensions with Grassmann tensor renormalization group." pith.science (2026). https://pith.science/paper/7LUQ5YO5

@misc{pith2026250118918,
  author       = {Pith},
  title        = {Pith review of: Two-color lattice QCD in $(1+1)$ dimensions with Grassmann tensor renormalization group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7LUQ5YO5}},
  note         = {Machine review of arXiv:2501.18918}
}
abstract

The $(1+1)$-dimensional two-color lattice QCD is studied with the Grassmann tensor renormalization group. We construct tensor network representations of theories with the staggered fermion and the Wilson fermion and show that Grassmann tensor networks can describe both cases with the same bond dimension. We also propose an efficient initial tensor compression scheme to gauge degrees of freedom. We compute the number density, chiral condensate, and diquark condensate at finite density, employing the staggered fermions. For the theory with Wilson fermion, a critical point in the negative mass region is identified by inspecting the pseudoscalar condensate and the conformal field theory data.

Figures

Figures reproduced from arXiv: 2501.18918 by the authors.

Figure 1
Figure 1. A truncated SVD is performed to reduce the bond dimension of the horizontal link which connects T𝑛 and T𝑛+1ˆ , from 16𝐾 to a smaller integer 𝐷 ′ . Tensors with a reduced bond dimension are denoted as T ′ . They are evaluated by the forward difference in this study: ⟨𝑛⟩ ≃ 𝑓 (𝜇 + Δ𝜇) − 𝑓 (𝜇) Δ𝜇 , ⟨𝜒𝜒¯ ⟩ ≃ 𝑓 (𝑚 + Δ𝑚) − 𝑓 (𝑚) Δ𝑚 . (9) In addition, we add a diquark source term to the original action 𝑆 as 𝑆 ′ = 𝑆 + 𝜆 2 ∑︁… view at source ↗
Figure 2
Figure 2. ⟨𝑛⟩, ⟨𝜒𝜒¯ ⟩, and ⟨𝜒𝜒⟩ as a function of 𝜇 at different 𝑚 and 𝛽. 𝐷 refers to the bond dimension cutoff in TRG iterations. To evaluate the numerical differences in Eq. (9) and Eq. (12), we set Δ𝜇 = 0.04 for 𝑚 = 0.1, Δ𝜇 = 0.02 for 𝑚 = 1, Δ𝑚 = 10−4 , and 𝜆 = Δ𝜆 = 10−4 . The figures are adapted from Ref. [9]. takes a constant finite value in the Silver-Blaze region and decreases in the intermediate phase. For 𝜇 > 𝜇𝑐2, ⟨𝑛⟩… view at source ↗
Figure 3
Figure 3. The quark number density ⟨𝑛⟩ as a function of chemical potential 𝜇 at 𝑚 = 0.1, 𝛽 = 0, 0.4, 0.8, 1.2, 1.6 in the thermodynamic limit. The bond dimension in the infinite coupling calculation is 𝐷 = 84, and the bond dimension in the finite 𝛽 calculations is 𝐷 = 150. To evaluate the numerical differences in Eq. (9), we set Δ𝜇 = 0.04. The figure is adapted from Ref. [9]. 3. Two-color QCD with Wilson fermion In the follow… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Construction of the transfer matrix for a lattice volume 𝑉 = 2 2𝑞 by contracting four renormalized tensors T (2𝑞−2) . Here, T (0) denotes the initial tensor. 3.2 Numerical results Hereafter, we consider the theory in the strong coupling limit 𝛽 → 0 and set 𝑟 = 1 [PITH…
Figure 5
Figure 5. Figure 5: ⟨𝜓𝛾¯ 5𝜓⟩ as a function of 𝑚 in the infinite coupling limit at a finite ℎ = 10−4 for the lattice vol￾ume𝑉 = 2 20. The bond dimension in the calculation is 𝐷 = 120. To evaluate the numerical difference, we set Δℎ = 10−4 . 6 8 10 12 14 16 18 20 22 log2V 0.5 0.6 0.7 0.8 0.…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Phase diagram of the single-flavor Gross--Neveu--Wilson model from the Grassmann corner transfer matrix renormalization group

    hep-lat 2026-02 conditional novelty 6.0 of 10

    For one-flavor Gross-Neveu-Wilson fermions, the Aoki phase is bounded by c=1/2 Ising critical lines and terminates at strong coupling, while c=1 lines separate topological and trivial insulators.

  2. Tensor renormalization group study of cold and dense QCD in the strong coupling limit

    hep-lat 2026-01 conditional novelty 6.0 of 10

    In strong-coupling lattice QCD at Nτ=8, the chiral and nuclear transition endpoints coincide at m_c≈2.06, and a first-order transition persists at m=2.07 on a 1024^4 zero-temperature lattice.

Reference graph

Works this paper leans on

15 extracted references · 6 canonical work pages · cited by 2 Pith papers

  1. [9]

    K. H. Pai, S. Akiyama and S. Todo,Grassmann tensor renormalization group approach to (1+ 1)-dimensional two-color lattice QCD at finite density, 2410.09485

    arXiv

  2. [13]

    Shimizu and Y

    Y. Shimizu and Y. Kuramashi,Berezinskii-Kosterlitz-Thouless transition in lattice Schwinger model with one flavor of Wilson fermion, Phys. Rev.D97(2018) 034502, [1712.07808]

    arXiv 2018

  3. [1]

    Levin and C

    M. Levin and C. P. Nave,Tensor renormalization group approach to two-dimensional classical lattice models,Phys. Rev. Lett.99(2007) 120601, [cond-mat/0611687]

    arXiv 2007

  4. [2]

    Bloch and R

    J. Bloch and R. Lohmayer,Grassmann higher-order tensor renormalization group approach fortwo-dimensionalstrong-couplingQCD ,Nucl.Phys.B 986(2023)116032,[ 2206.00545]

    arXiv 2023

  5. [3]

    Tensor network representation of non-abelian gauge theory coupled to reduced staggered fermions

    M. Asaduzzaman, S. Catterall, Y. Meurice, R. Sakai and G. C. Toga,Tensor network representation of non-abelian gauge theory coupled to reduced staggered fermions,JHEP 05 (2024) 195, [2312.16167]

    work page Pith review arXiv 2024

  6. [4]

    Adachi, T

    D. Adachi, T. Okubo and S. Todo,Bond-weighted tensor renormalization group,Phys. Rev. B 105 (2022) L060402, [2011.01679]

    arXiv 2022

  7. [5]

    Bond-weighting method for the Grassmann tensor renormalization group

    S. Akiyama,Bond-weighting method for the Grassmann tensor renormalization group, JHEP 11 (2022) 030, [2208.03227]

    work page Pith review arXiv 2022

  8. [6]

    Aoki,New Phase Structure for Lattice QCD with Wilson Fermions,Phys

    S. Aoki,New Phase Structure for Lattice QCD with Wilson Fermions,Phys. Rev. D30 (1984) 2653

    work page 1984

Show all 15 references
  1. [7]

    Akiyama and D

    S. Akiyama and D. Kadoh,More about the Grassmann tensor renormalization group, JHEP 10 (2021) 188, [2005.07570]

    2021 arXiv

  2. [8]

    Fukuma, D

    M. Fukuma, D. Kadoh and N. Matsumoto,Tensor network approach to two-dimensional Yang–Mills theories, PTEP 2021(2021) 123B03, [2107.14149]

    2021 arXiv

  3. [10]

    N. D. Mermin and H. Wagner,Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models,Phys. Rev. Lett.17(1966) 1133

    1966

  4. [11]

    S. R. Coleman,There are no Goldstone bosons in two-dimensions, Commun. Math. Phys.31 (1973) 259

    1973

  5. [12]

    Nishida, K

    Y. Nishida, K. Fukushima and T. Hatsuda,Thermodynamics of strong coupling two color QCD with chiral and diquark condensates, Phys. Rept.398 (2004) 281, [hep-ph/0306066]

    2004 arXiv

  6. [14]

    Gu and X.-G

    Z.-C. Gu and X.-G. Wen,Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order,Phys. Rev. B80 (2009) 155131, [0903.1069]

    2009 arXiv

  7. [15]

    Ueda,Renormalization group flow and fixed-point in tensor network representations, Ph.D

    A. Ueda,Renormalization group flow and fixed-point in tensor network representations, Ph.D. thesis, University of Tokyo, 2024.2401.18068. 9

    2024

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.