Pith. sign in

REVIEW 3 major objections 4 minor 71 references

A tensor-network renormalization group that keeps lattice reflection, rotation, and PT symmetries explicit at every coarse-graining step can locate both phase transitions of the nearest-neighbor hard-square lattice gas with relative errors

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-02 17:22 UTC pith:KA5SM7YR

load-bearing objection A genuinely new symmetry-preserving TNRG scheme with a clean derivation and strong positive-z results, but the negative-z accuracy rests on per-χ tuning of ε_inv and shouldn't be sold as intrinsic. the 3 major comments →

arxiv 2603.25492 v2 pith:KA5SM7YR submitted 2026-03-26 cond-mat.stat-mech hep-thphysics.comp-phquant-ph

Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model

Xinliang Lyu This is my paper
classification cond-mat.stat-mech hep-thphysics.comp-phquant-ph
keywords hard-square lattice gastensor-network renormalization grouplattice symmetryPT symmetryspontaneous symmetry breakingloop optimizationYang-Lee edge singularitycritical exponents
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper shows how to build a tensor-network renormalization group (TNRG) that keeps the lattice-reflection, lattice-rotation, and parity-time (PT) symmetries of a statistical model explicit at every coarse-graining step, rather than letting numerical approximations break them. Using the nearest-neighbor hard-square lattice gas as a benchmark, it defines what these symmetries mean for a coarse-grained tensor network and constructs a scheme that both preserves and imposes them through a symmetric singular-value decomposition and a loop-optimization entanglement filter. The payoff is concrete: the critical activities of both the Ising-type ordering transition at positive activity and the Yang-Lee edge singularity at negative activity are estimated with relative errors as small as 7e-7 and 4e-8, roughly two orders of magnitude better than plain TRG at the same bond dimension, and scaling dimensions stop drifting with RG step. The central insight is that the sign information carried by bond matrices—which ordinary tensor renormalization absorbs away or ignores—is exactly what must be tracked to keep PT and lattice symmetries alive through coarse graining.

Core claim

The paper's central claim is that lattice-reflection, lattice-rotation, and PT symmetries can be incorporated into a two-dimensional tensor-network renormalization group in a way that is both exact and computationally useful. The key is to work with a coarse-grained tensor network of the form in Eq. (16), in which two tensors A and B sit on alternating sublattices joined by diagonal ±1 bond matrices σ. Lattice symmetries are expressed in a weak form, where B is determined by rotations and reflections of A, and a strong form, where A itself carries a diagonal SWAP-gauge matrix g. The symmetric SVD splitting replaces ordinary SVD by an eigendecomposition of the tensor treated as a matrix acros

What carries the argument

The central object is the symmetric SVD splitting: the 4-leg tensor A is viewed as a symmetric matrix across the diagonal l1 axis and decomposed by truncated eigendecomposition; the signs of the eigenvalues become a new diagonal bond matrix σ′, while their absolute values are split evenly into two 3-leg tensors vΛ. Because the signs are carried by σ′ rather than absorbed into orthogonal matrices, the PT symmetry (reality of all tensors) and the lattice reflection and rotation symmetries remain explicit. The loop optimization then varies vΛ to remove the corner-double-line entanglement while keeping A, σ, and σ′ fixed; the weak form of lattice symmetry survives, and the strong form is re-impo

Load-bearing premise

The load-bearing premise is that the loop-optimization update, in which an inverse of a certain environment matrix is used to improve the split tensors, can be made to reliably remove redundant short-range entanglement without losing critical information; the paper itself cautions that this update is rudimentary, does not guarantee improved fidelity, and works for the negative-activity transition only with carefully hand-tuned regulator values.

What would settle it

Run the published implementation at the negative-activity critical point with bond dimension χ=20 but replace the tuned regulator ϵ_inv=6×10^-12 by 1×10^-8, keeping everything else fixed. If the estimated z_c^- does not degrade from 4×10^-8 toward the plain-TRG error of 2×10^-5, the paper's stated sensitivity to ϵ_inv—and its implication that the reported accuracy depends on hand-tuning—is falsified. If it does degrade, that confirms the load-bearing role of the loop-optimization regulator.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Any model whose tensor network can be cast in the two-sublattice form with real tensors and diagonal ±1 bond matrices can use this RG without modifying the core algorithm; the paper singles out hard-core lattice gases with longer exclusion ranges as the immediate next step.
  • The sign-tracking bond matrix makes the method applicable to non-unitary critical points such as the Yang-Lee edge, where the partition function can become negative; this extends TNRG beyond the usual positive Boltzmann weight setting.
  • Because the strong form of lattice symmetry is imposed at the end of every RG step, RG-relevant perturbations from algorithm artifacts are eliminated, so the spontaneous-symmetry-breaking fixed points become strictly stable and critical tensors can be found reliably by a bisection method.
  • The truncation error at criticality stops growing and converges below 1e-6 rather than growing to 1e-2, so scaling dimensions extracted from near-fixed-point tensors are stable with respect to RG step rather than drifting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same diagonal sign-and-gauge bookkeeping could encode other discrete symmetries, such as glide or point-group operations, by choosing different diagonal matrices on the bonds, making symmetry-preserving TNRG available for models whose order parameters break those symmetries.
  • Editorial inference: because the method works at the Yang-Lee edge despite the failure of the standard entanglement-entropy argument, it suggests that entanglement filtering in TNRG can be effective at non-unitary fixed points for reasons unrelated to area-law entanglement; testing it at another non-unitary fixed point would show whether this is generic.
  • Editorial inference: the hand-tuned regulator for the negative-activity transition means the method is not yet turnkey; a principled way to set or eliminate that regulator, for instance from the spectrum of the environment tensor, is the difference between a demonstration and a robust numerical tool.
  • Editorial inference: the weak-form lattice symmetry requirement may be enough for other tensor-network algorithms that update tensors sequentially, pointing toward a route for making entanglement filtering symmetry-aware beyond the specific TRG-based scheme proposed here.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a tensor-network renormalization group (TNRG) scheme that explicitly defines and imposes lattice-reflection, lattice-rotation, and PT symmetries on coarse-grained tensor networks, using a symmetric SVD splitting plus loop optimization. It benchmarks the method on the nearest-neighbor hard-square lattice gas, reporting highly accurate estimates of the two critical activities (relative errors down to 7e-7 for z_c^+ at chi=12 and 4e-8 for z_c^- at chi=20) and scaling dimensions consistent with the 2D Ising and Yang-Lee edge universality classes. The paper also provides numerical experiments showing that the degeneracy-index RG flow at the symmetry-broken fixed point is more stable when the relevant symmetry is preserved.

Significance. If the claims hold, this is the first EF-enhanced TNRG that systematically incorporates lattice rotation and PT symmetries, going beyond the well-studied on-site global symmetries. The symmetric SVD splitting and the weak/strong symmetry definitions are clean and potentially reusable for other lattice models. The numerical gains over plain TRG at equal bond dimension are large and consistent for the positive-activity transition. The availability of reproducible Python code strengthens the paper. However, the negative-activity results rely on a hand-tuned regulator in the loop-optimization step, which is a load-bearing caveat for the central claim of accurate estimation at both transitions.

major comments (3)
  1. [Sec. IV B, Appendix A, Table III] The reported accuracy for z_c^- is produced by per-bond-dimension tuning of epsilon_inv in Eq. (A7). Table III lists epsilon_inv = 1e-8, 1e-10, 5e-11, 6e-12 for chi = 10,12,16,20, and Sec. V admits performance is 'very sensitive' to this regulator. Since no a priori selection rule is given, the 4e-8 error at chi=20 cannot be cleanly attributed to the symmetry-preserving RG map; it may be a favorable hyperparameter choice. The non-monotonic errors in Table II (2e-6, 1e-5, 4e-6, 4e-8) reinforce this concern. Please provide a principled criterion for epsilon_inv or a sensitivity analysis showing a plateau around the chosen values.
  2. [Appendix A, Eqs. (A4)-(A7); Sec. V] The loop-optimization update rule does not guarantee an increase of fidelity because Q and Upsilon in Eq. (A3) depend on v-tilde_Lambda; the convex-combination trick in Eq. (A5) only prevents a decrease for a fixed environment, not under the true nonlinear map. This is acknowledged in Sec. V ('does not guarantee an increase of the fidelity'). Since the loop optimization is the essential EF step that tames RG errors at criticality (Figs. 6 and 8), the paper should present stronger convergence diagnostics and, in particular, show that the negative-z results are not artifacts of a favorable initialization and regulator. A robustness test across random initializations or a demonstration that the fidelity actually increases for the reported runs would address this.
  3. [Sec. III D and Eq. (28)] The loop approximation introduces an uncontrolled error: the optimized v-tilde_Lambda no longer satisfies the strong lattice symmetry of Eq. (22), and the paper does not bound the deviation. The claim that this is the only approximation in the first half of the RG step is formally true but does not quantify how much the loop optimization distorts the critical tensor. Given that the accuracy gain over TRG is the main evidence for the method, a quantitative comparison of the optimized tensor with the SVD initialization (e.g., fidelity curves versus RG step) would make the claim more convincing.
minor comments (4)
  1. [Tables I-II] The errors for the proposed method are not monotonic in chi (e.g., z_c^+ error worsens from chi=12 to chi=18 before improving at chi=20; z_c^- error worsens from chi=10 to chi=12). The text says 'there is a clear trend of improvement' but the non-monotonicity deserves a comment, especially in relation to the regulator sensitivity.
  2. [Figs. 7 and 9] The scaling-dimension estimates are only presented graphically. Since these are central quantitative outputs, a table with numerical values and their convergence in RG step would be more useful to readers and reviewers.
  3. [Throughout] The notation 'TNRG' is used inconsistently; sometimes it refers to the general method and other times to the specific TRG-based scheme. This can confuse readers comparing with the literature.
  4. [Sec. V] The statement that 'no EF-enhanced TNRG scheme is suitable for studying a phase transition where some lattice symmetries are spontaneously broken' should be moderated; Ref. [17] treats lattice-reflection symmetry with EF, and the distinction is more about rotation/reflection combinations and PT symmetry. A sentence clarifying the incremental contribution would help.

Circularity Check

0 steps flagged

No significant circularity: the critical parameters and scaling dimensions are algorithm outputs benchmarked against external high-precision data; the main caveats are minor self-citations and the per-χ tuning of the ε_inv regulator, neither of which feeds the target values back into the derivation.

full rationale

The paper's central claims are numerical estimates of z+_c, z−_c, and scaling dimensions for the hard-square lattice gas. These are produced by running the proposed symmetry-preserving TNRG map, locating phase boundaries via the degeneracy index X, and extracting transfer-matrix spectra near the estimated fixed point. The benchmark values used for comparison (z+_c from Ref. [39], z−_c from Ref. [40], and exact 2D Ising/Yang-Lee CFT dimensions) are external to the algorithm and are not used to set any RG-map parameter. The symmetry conditions in Eqs. (17)-(20) are imposed as design constraints on the coarse-grained ansatz Eq. (16); they are not derived from the critical data, so no self-definitional circularity is present. The loop-optimization fidelity objective in Eq. (A2) and the FET-style update in Eq. (A4) define an approximation problem whose solution is the optimized tensor, not a restatement of the desired critical activity. The paper honestly flags that the EF updating rule 'does not guarantee an increase of the fidelity' and that the regulated inverse is 'very sensitive to the choice of ε_inv' (Table III). This is a robustness limitation, and it does create some risk that the negative-z results are partly dependent on favorable regulator choices. However, there is no statement or equation showing that ε_inv was chosen to reproduce the benchmark z−_c; the reported estimates remain outputs of an RG calculation rather than fitted inputs. Self-citations appear for established methodology (bisection in Ref. [7], the SWAP-gauge and FET update in Ref. [17], and related prior work). These are used as algorithmic ingredients, and Theorem 1 is proven in the text rather than imported as an unexamined uniqueness claim. Therefore the derivation chain does not reduce to its inputs by construction; any concern about the EF regulator belongs to numerical robustness, not circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

No new physical entities are introduced. The sign-carrying bond matrix sigma' and SWAP-gauge matrix g are algorithmic gauge degrees of freedom, not new particles or forces. The main input pulled from outside is the external benchmark data for the two critical points, plus the standard linear-algebra theorems used in the SVD-splitting step.

free parameters (2)
  • epsilon_inv regulator for Moore-Penrose inverse = Table III: 1e-8 (chi=10), 1e-10 (chi=12), 5e-11 (chi=16), 6e-12 (chi=20)
    Used in regulated inverse Eq. (A7) for negative-activity transition; paper says effectiveness is 'very sensitive' to its choice, so it is a hand-tuned parameter per bond dimension.
  • min_iterations for loop optimization = 10
    Paper states 'A minimum of 10 iterations works well in our numerical calculations' (Appendix A); a hand-chosen algorithmic hyperparameter.
axioms (4)
  • standard math Spectral theorem for real symmetric matrices; eigendecomposition of a real symmetric matrix yields real eigenvalues and orthonormal eigenvectors.
    Used to justify the truncated eigendecomposition in Eq. (21a) and the symmetric SVD splitting, including the sign-carrying bond matrix sigma'.
  • domain assumption The 1NN hard-square lattice gas has two phase transitions belonging to the 2D Ising universality class (at z+_c = 3.79625517...) and the Yang-Lee edge singularity (at z-_c = -0.11933888188...), with those reference values from transfer-matrix calculations.
    External benchmarks used to validate the method in Sec. IV; not derived in this paper.
  • domain assumption The coarse-grained tensor network can always be represented in the form of Eq. (16) with a diagonal +/-1 bond matrix sigma and tensors A, B satisfying the weak lattice symmetries Eq. (18).
    The proposed RG map is built on this ansatz; after loop optimization the strong form is lost and preservation of the weak form is asserted.
  • ad hoc to paper The FET-style updating rule Eq. (A4), with Moore-Penrose or regulated inverse, filters CDL tensors without damaging the critical fixed-point tensor.
    The paper states the scheme is 'still rudimentary' and requires hand-tuning (Table III); there is no proof that fidelity increases.

pith-pipeline@v1.3.0-alltime-deepseek · 24111 in / 15808 out tokens · 150660 ms · 2026-08-02T17:22:05.266913+00:00 · methodology

0 comments
read the original abstract

The tensor-network renormalization group (TNRG) is an accurate numerical real-space renormalization group method for studying phase transitions in both quantum and classical systems. Continuous phase transitions, as an important class of phase transitions, are usually accompanied by spontaneous breaking of various symmetries. However, the understanding of symmetries in the TNRG is well-established mainly for global on-site symmetries like U(1) and SU(2). In this paper, we demonstrate how to incorporate lattice symmetries (including reflection and rotation) and the PT symmetry in the TNRG in two dimensions (2D) through a case study of the hard-square lattice gas with nearest-neighbor exclusion. This model is chosen because it is well-understood and has two continuous phase transitions whose spontaneously-broken symmetries are lattice and PT symmetries. Specifically, we write down proper definitions of these symmetries in a coarse-grained tensor network and propose a TNRG scheme that incorporates these symmetries. We demonstrate the validity of the proposed method by estimating the critical parameters and the scaling dimensions of the two phase transitions of the model. The technical development in this paper has made the 2D TNRG a more well-rounded numerical method.

Figures

Figures reproduced from arXiv: 2603.25492 by Xinliang Lyu.

Figure 1
Figure 1. Figure 1: (a) Black dots represent sites of the square lattice, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: SSB of lattice symmetries for positive-z transition. Number 0, 1 on the site means number of particle. When the activity z = +∞, all particles occupy either the shaded (left) or unshaded (right) sites. These two configurations can be mapped to each other through any of the following symmetry transformations: 1) translation by one lattice constant in any direction, 2) 90◦ lattice rotation around the central… view at source ↗
Figure 3
Figure 3. Figure 3: The four largest eigenvalues of the transfer matrix [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The RG flows of the degeneracy index X near the positive-z transition point z + c generated by (a) the HOTRG and (b) the TRG at bond dimension χ = 10. The solid line in the middle is the RG flow closest to the transition point z + c ≈ 3.80. Two lines below have z < z+ c and belong to the trivial phase, while two lines above have z > z+ c and belong to the SSB phase. For the flows generated by the HOTRG sta… view at source ↗
Figure 5
Figure 5. Figure 5: The RG flows of the degeneracy index X near the negative-z transition point z − c generated by the TRG at bond dimension χ = 10. The solid line in the middle is the RG flow closest to the repulsive-core singularity z − c ≈ −0.119. Two lines below have z > z− c and belong to the trivial phase, while two lines above have z < z− c and belong to the SSB phase. (a) The RG flow starts with a complex tensor-netwo… view at source ↗
Figure 6
Figure 6. Figure 6: Flows of the RG errors of the TRG and the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Estimates of the scaling dimensions at the positive [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Estimates of the scaling dimensions at the negative [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

71 extracted references · 3 linked inside Pith

  1. [1]

    Notice that the above weak form in Eq

    Weak form of the lattice symmetries The weak form is expressed in terms of how the tensorB can be determined by several lattice symmetry operations of the tensorA: B A A= = A= A= y x .(18) The first two equal signs are lattice reflections along the x and y axes, while the second two are 90 ◦ lattice rotations counterclockwise and clockwise. Notice that th...

  2. [2]

    Strong form of the lattice symmetries The strong form of the two lattice symmetries is repre- sented by the following symmetry of the tensor A itself, which involves a diagonal SW AP-gauge matrixg[17]: AA =A= y x g g g g (20a) for lattice reflections along thexandyaxes, A A == A g g g g (20b) for 90 ◦ lattice rotation counterclockwise and clockwise, and =...

  3. [3]

    shooting method

    In this paper, we regard two TRG trans- formations as a single RG step withb= 2: A, B, σ TRG − − − →A′, B′, σ′ TRG − − − →A′′, B′′, σ′′.(26) We call A, B, σoriginal tensors, A′′, B′′, σ′ coarse-grained tensors, andA ′, B′, σ′ intermediate tensors. Remark.The proposed symmetric TRG is equivalent to the usual TRG if the machine precision in numerical cal- c...

  4. [4]

    Levin and C

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

    2007

  5. [5]

    Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang, Coarse-graining renormalization by higher-order singular value decomposition, Phys. Rev. B86, 045139 (2012)

    2012

  6. [6]

    Evenbly, Algorithms for tensor network renormaliza- tion, Phys

    G. Evenbly, Algorithms for tensor network renormaliza- tion, Phys. Rev. B95, 045117 (2017)

    2017

  7. [7]

    Okunishi, T

    K. Okunishi, T. Nishino, and H. Ueda, Developments in the tensor network—from statistical mechanics to quan- tum entanglement, Journal of the Physical Society of Japan91, 062001 (2022)

    2022

  8. [8]

    Gu and X.-G

    Z.-C. Gu and X.-G. Wen, Tensor-entanglement-filtering renormalization approach and symmetry-protected topo- logical order, Phys. Rev. B80, 155131 (2009)

    2009

  9. [9]

    Evenbly and G

    G. Evenbly and G. Vidal, Local scale transformations on the lattice with tensor network renormalization, Phys. Rev. Lett.116, 040401 (2016)

    2016

  10. [10]

    X. Lyu, R. G. Xu, and N. Kawashima, Scaling dimensions from linearized tensor renormalization group transforma- tions, Phys. Rev. Res.3, 023048 (2021)

    2021

  11. [11]

    Guo and T.-C

    W. Guo and T.-C. Wei, Tensor network methods for extracting conformal field theory data from fixed-point tensors and defect coarse graining, Phys. Rev. E109, 034111 (2024)

    2024

  12. [12]

    Kennedy and S

    T. Kennedy and S. Rychkov, Tensor rg approach to high- temperature fixed point, Journal of Statistical Physics 187, 10.1007/s10955-022-02924-4 (2022)

    doi:10.1007/s10955-022-02924-4 2022

  13. [13]

    Kennedy and S

    T. Kennedy and S. Rychkov, Tensor renormalization group at low temperatures: Discontinuity fixed point, Annales Henri Poincar´ e25, 773–841 (2023)

    2023

  14. [14]

    Ebel, 3d tensor renormalisation group at high temper- atures, Annales Henri Poincar´ e26, 1291–1351 (2024)

    N. Ebel, 3d tensor renormalisation group at high temper- atures, Annales Henri Poincar´ e26, 1291–1351 (2024)

    2024

  15. [15]

    N. Ebel, T. Kennedy, and S. Rychkov, Tensor renor- malization group meets computer assistance (2025), arXiv:2506.03247 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2025

  16. [16]

    Singh, R

    S. Singh, R. N. C. Pfeifer, and G. Vidal, Tensor network decompositions in the presence of a global symmetry, Phys. Rev. A82, 050301 (2010)

    2010

  17. [17]

    Singh, R

    S. Singh, R. N. C. Pfeifer, and G. Vidal, Tensor network states and algorithms in the presence of a global u(1) symmetry, Phys. Rev. B83, 115125 (2011)

    2011

  18. [18]

    Singh and G

    S. Singh and G. Vidal, Tensor network states and algo- rithms in the presence of a global su(2) symmetry, Phys. Rev. B86, 195114 (2012). 20

    2012

  19. [19]

    Lyu and N

    X. Lyu and N. Kawashima, Three-dimensional real-space renormalization group with well-controlled approxima- tions, Phys. Rev. E111, 054140 (2025)

    2025

  20. [20]

    Lyu and N

    X. Lyu and N. Kawashima, Lattice-reflection symme- try in tensor-network renormalization group with entan- glement filtering in two and three dimensions (2025), arXiv:2510.19428 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2025

  21. [21]

    N. Ebel, T. Kennedy, and S. Rychkov, Rotations, nega- tive eigenvalues, and newton method in tensor network renormalization group, Phys. Rev. X15, 031047 (2025)

    2025

  22. [22]

    D. Dhar, R. Rajesh, and A. Kumar, Phase transitions in systems of particles with only hard-core interactions, Europhysics Letters143, 61001 (2023)

    2023

  23. [23]

    Summer School on Collective Behaviour in Quantum Matter

    D. Dhar, Phase transitions in hard core models V (2018), A lecture during “Summer School on Collective Behaviour in Quantum Matter” at ICTP

    2018

  24. [24]

    H. C. M. Fernandes, J. J. Arenzon, and Y. Levin, Monte carlo simulations of two-dimensional hard core lattice gases, The Journal of Chemical Physics126, 114508 (2007)

    2007

  25. [25]

    Ramola and D

    K. Ramola and D. Dhar, High-activity perturbation ex- pansion for the hard square lattice gas, Phys. Rev. E86, 031135 (2012)

    2012

  26. [26]

    Ramola,Onset of Order in Lattice Systems: Kitaev Model and Hard Squares, PhD dissertation, Tata Institute of Fundamental Research, Mumbai, India (2012)

    K. Ramola,Onset of Order in Lattice Systems: Kitaev Model and Hard Squares, PhD dissertation, Tata Institute of Fundamental Research, Mumbai, India (2012)

    2012

  27. [27]

    Ramola, K

    K. Ramola, K. Damle, and D. Dhar, Columnar order and ashkin-teller criticality in mixtures of hard squares and dimers, Phys. Rev. Lett.114, 190601 (2015)

    2015

  28. [28]

    S. S. Akimenko, V. A. Gorbunov, A. V. Myshlyavtsev, and P. V. Stishenko, Tensor renormalization group study of hard-disk models on a triangular lattice, Phys. Rev. E 100, 022108 (2019)

    2019

  29. [29]

    S. S. Akimenko, Tensor network construction for lattice gas models: Hard-core and triangular lattice models, Phys. Rev. E107, 054116 (2023)

    2023

  30. [30]

    Evenbly and G

    G. Evenbly and G. Vidal, Tensor network renormalization, Phys. Rev. Lett.115, 180405 (2015)

    2015

  31. [31]

    Yang, Z.-C

    S. Yang, Z.-C. Gu, and X.-G. Wen, Loop optimization for tensor network renormalization, Phys. Rev. Lett.118, 110504 (2017)

    2017

  32. [32]

    M. Bal, M. Mari¨ en, J. Haegeman, and F. Verstraete, Renormalization group flows of hamiltonians using tensor networks, Phys. Rev. Lett.118, 250602 (2017)

    2017

  33. [33]

    Evenbly, Gauge fixing, canonical forms, and optimal truncations in tensor networks with closed loops, Phys

    G. Evenbly, Gauge fixing, canonical forms, and optimal truncations in tensor networks with closed loops, Phys. Rev. B98, 085155 (2018)

    2018

  34. [34]

    Hauru, C

    M. Hauru, C. Delcamp, and S. Mizera, Renormalization of tensor networks using graph-independent local trunca- tions, Phys. Rev. B97, 045111 (2018)

    2018

  35. [35]

    Harada, Entanglement branching operator, Phys

    K. Harada, Entanglement branching operator, Phys. Rev. B97, 045124 (2018)

    2018

  36. [36]

    Lee and N

    H.-Y. Lee and N. Kawashima, Tensor-ring decomposition with index-splitting, Journal of the Physical Society of Japan89, 054003 (2020)

    2020

  37. [37]

    Homma, T

    K. Homma, T. Okubo, and N. Kawashima, Nuclear norm regularized loop optimization for tensor network, Phys. Rev. Res.6, 043102 (2024)

    2024

  38. [38]

    P. N. Meisinger and M. C. Ogilvie, PT symmetry in classical and quantum statistical mechanics, Philosophi- cal Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences371, 20120058 (2013)

    2013

  39. [39]

    C. M. Bender and S. Boettcher, Real Spectra in Non- Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett.80, 5243 (1998)

    1998

  40. [40]

    C. M. Bender and D. W. Hook, PT -symmetric quantum mechanics, Rev. Mod. Phys.96, 045002 (2024)

    2024

  41. [41]

    Todo and M

    S. Todo and M. Suzuki, Transfer-matrix method for hard-square lattice gas, International Journal of Mod- ern Physics C07, 811 (1996)

    1996

  42. [42]

    Guo and H

    W. Guo and H. W. J. Bl¨ ote, Finite-size analysis of the hard-square lattice gas, Phys. Rev. E66, 046140 (2002)

    2002

  43. [43]

    Todo, Transfer-matrix study of negative-fugacity sin- gularity of hard-core lattice gas, International Journal of Modern Physics C10, 517 (1999)

    S. Todo, Transfer-matrix study of negative-fugacity sin- gularity of hard-core lattice gas, International Journal of Modern Physics C10, 517 (1999)

    1999

  44. [44]

    R. J. Baxter, Planar lattice gases with nearest-neighbor exclusion, Annals of Combinatorics3, 191–203 (1999)

    1999

  45. [45]

    C. N. Yang and T. D. Lee, Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation, Phys. Rev.87, 404 (1952)

    1952

  46. [46]

    T. D. Lee and C. N. Yang, Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model, Phys. Rev.87, 410 (1952)

    1952

  47. [47]

    Groeneveld, Two theorems on classical many-particle systems, Physics Letters3, 50 (1962)

    J. Groeneveld, Two theorems on classical many-particle systems, Physics Letters3, 50 (1962)

    1962

  48. [48]

    Chan, Series expansions from the corner transfer matrix renormalization group method: the hard-squares model, Journal of Physics A: Mathematical and Theoreti- cal45, 085001 (2012)

    Y.-b. Chan, Series expansions from the corner transfer matrix renormalization group method: the hard-squares model, Journal of Physics A: Mathematical and Theoreti- cal45, 085001 (2012)

    2012

  49. [49]

    I. Jensen, Comment on ‘Series expansions from the corner transfer matrix renormalization group method: the hard- squares model’, Journal of Physics A: Mathematical and Theoretical45, 508001 (2012)

    2012

  50. [50]

    Poland, On the universality of the nonphase transition singularity in hard-particle systems, Journal of Statistical Physics35, 341–353 (1984)

    D. Poland, On the universality of the nonphase transition singularity in hard-particle systems, Journal of Statistical Physics35, 341–353 (1984)

    1984

  51. [51]

    Baram and M

    A. Baram and M. Luban, Universality of the cluster inte- grals of repulsive systems, Phys. Rev. A36, 760 (1987)

    1987

  52. [52]

    R. J. Baxter, Hard hexagons: exact solution, Journal of Physics A: Mathematical and General13, L61 (1980)

    1980

  53. [53]

    P. J. Kortman and R. B. Griffiths, Density of zeros on the lee-yang circle for two ising ferromagnets, Phys. Rev. Lett.27, 1439 (1971)

    1971

  54. [54]

    M. E. Fisher, Yang-Lee Edge Singularity and ϕ3 Field Theory, Phys. Rev. Lett.40, 1610 (1978)

    1978

  55. [55]

    Lai and M

    S. Lai and M. E. Fisher, The universal repulsive-core singularity and yang–lee edge criticality, The Journal of Chemical Physics103, 8144 (1995)

    1995

  56. [56]

    Assis, J

    M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard, and B. M. McCoy, Integrability versus non-integrability: hard hexagons and hard squares compared, Journal of Physics A: Mathematical and Theoretical47, 445001 (2014)

    2014

  57. [57]

    J. L. Cardy, Conformal invariance and the yang-lee edge singularity in two dimensions, Phys. Rev. Lett.54, 1354 (1985)

    1985

  58. [58]

    Cardy, The Yang–Lee Edge Singularity and Related Problems, in50 Years of the Renormalization Group (WORLD SCIENTIFIC, 2024) p

    J. Cardy, The Yang–Lee Edge Singularity and Related Problems, in50 Years of the Renormalization Group (WORLD SCIENTIFIC, 2024) p. 281–302

    2024

  59. [59]

    Kardar, Statistical fields, inStatistical Physics of Fields(Cambridge University Press, 2007) Chap

    M. Kardar, Statistical fields, inStatistical Physics of Fields(Cambridge University Press, 2007) Chap. 2, p. 19–34

    2007

  60. [60]

    Dorey, C

    P. Dorey, C. Dunning, and R. Tateo, FromPT -symmetric quantum mechanics to conformal field theory, Pramana 73, 217–239 (2009)

    2009

  61. [61]

    J. D. Biamonte, S. R. Clark, and D. Jaksch, Categorical tensor network states, AIP Advances1, 042172 (2011)

    2011

  62. [62]

    Lyu, A TNRG study of the 1NN hard-square lattice gas model, github.com/brucelyu/lattice-gas-TNRG (2026)

    X. Lyu, A TNRG study of the 1NN hard-square lattice gas model, github.com/brucelyu/lattice-gas-TNRG (2026). 21

    2026

  63. [63]

    N. Ebel, T. Kennedy, and S. Rychkov, Transfer matrix and lattice dilatation operator for high-quality fixed points in tensor network renormalization group, Phys. Rev. B112, 104424 (2025)

    2025

  64. [64]

    Holzhey, F

    C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nuclear Physics B424, 443 (1994)

    1994

  65. [65]

    Vidal, J

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entan- glement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003)

    2003

  66. [66]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and con- formal field theory, Journal of Physics A: Mathematical and Theoretical42, 504005 (2009)

    2009

  67. [67]

    Lyu and N

    X. Lyu and N. Kawashima, Essential difference between 2d and 3d from the perspective of real-space renormalization group, Journal of Statistical Physics193, 10.1007/s10955- 026-03587-1 (2026)

    doi:10.1007/s10955- 2026

  68. [68]

    Shimizu and K

    H. Shimizu and K. Kawabata, Complex entanglement entropy for complex conformal field theory, Phys. Rev. B 112, 085112 (2025)

    2025

  69. [69]

    Evenbly, Example codes of the TNR, tensors.net/p-tnr (2019)

    G. Evenbly, Example codes of the TNR, tensors.net/p-tnr (2019)

    2019

  70. [70]

    Ghojogh, F

    B. Ghojogh, F. Karray, and M. Crowley, Eigenvalue and generalized eigenvalue problems: Tutorial (2023), arXiv:1903.11240 [stat.ML]

    Pith/arXiv arXiv 2023

  71. [71]

    Strang,Introduction to linear algebra, 6th ed

    G. Strang,Introduction to linear algebra, 6th ed. (Wellesley-Cambridge Press, Wellesley, MA, 2023)

    2023