Pith. sign in

REVIEW 2 major objections 5 minor 4 cited by

The paper shows that the anti-symmetric combination H_A = H − \tilde{H} of two commuting free-fermion Hamiltonians has a product reference state and few-body eigenstates built from the fermionic modes.

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

2026-08-03 04:59 UTC pith:OOSOSUHJ

load-bearing objection A genuinely new eigenstate construction for disguised free-fermion models, but the FFD edge operators as written anticommute with the wrong end of the chain. the 2 major comments →

arxiv 2602.03431 v1 pith:OOSOSUHJ submitted 2026-02-03 cond-mat.stat-mech nlin.SI

Solving models with generalized free fermions I: Algebras and eigenstates

Kohei Fukai , Bal\'azs Pozsgay , Istv\'an Vona This is my paper
classification cond-mat.stat-mech nlin.SI
keywords graph-Clifford algebradefining representationfree fermions in disguiseanti-symmetric Hamiltonianreference statefew-body eigenstatesfrustration graphtransfer matrix
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 is about spin chains that are secretly free-fermion systems but with highly nonlocal fermions, where no reference state was previously known. The authors prove that for the anti-symmetric combination H_A of two commuting copies of the Hamiltonian, a simple product state (all spins up) is annihilated, and acting on it with the fermionic operators produces eigenstates with exact energies E = −2Σσ_jε_j. This gives the first explicit few-body wave functions for the 'free fermions in disguise' model, and it works model-independently for the underlying graph-Clifford algebra. The key tool is a 'defining representation' that maps algebra elements to spin-chain states, making the reference state appear as the image of the identity.

Core claim

The central claim is that the all-up state |∅⟩ is a null vector of H_A = Σ_j b_j (h_j − \tilde{h}_j) whenever the two generator families satisfy h_j|∅⟩ = \tilde{h}_j|∅⟩, and that the states ∏_{j=1}^n Ψ^{σ_j}_{a_j}|∅⟩ are eigenstates of H_A with energies E = −2Σ_j σ_j ε_j. In the defining representation of the graph-Clifford algebra this follows from identifying H_A with the commutator with H, so the image of any commuting operator is a null vector (Theorem 7). For the FFD model the paper constructs the required edge operators explicitly for M = 6k, 6k+2, and 6k+5, and shows that the second fermion family can be dropped via the relation Ψ_{±k}|∅⟩ = \tilde{Ψ}_{∓k}|∅⟩. The same construction als

What carries the argument

The defining representation of the tensor product A ⊗ A of two graph-Clifford algebras. An operator–state correspondence maps each ordered basis element of A to a computational basis state of a spin chain; left- and right-multiplication by a generator h_k become Pauli strings X_k∏_{j<k}Z_j and X_k∏_{j>k}Z_j (Theorem 4). This representation turns H_A into the commutator with H, so any operator commuting with H becomes a null vector of H_A (Theorem 7). The edge operator χ, defined to anti-commute with a chosen simplicial clique and commute with all other generators, enters the explicit fermion formula Ψ_{±k} = (1/N_k) T(∓u_k) χ T(±u_k).

Load-bearing premise

The paper's construction rests on the unproven-in-this-paper premise that claw-free and even-hole-free frustration graphs always admit the Dirac fermions of eq. (27), and on a deferred relation (Theorem 11) that lets one discard the second fermion family; if either fails, the eigenstates collapse.

What would settle it

Take the FFD model with M=6 and random couplings b_j, construct Ψ_1|∅⟩ using the explicit edge operator and transfer matrix, and compute H_A on this state; if the eigenvalue is not −2ε_1, Theorem 7 is wrong. Alternatively, check Conjecture 1 by numerically testing whether the 2^M states (101) span the Hilbert space for M=12 with several random couplings; one counterexample would disprove the completeness claim.

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

If this is right

  • If correct, the anti-symmetric Hamiltonian H_A in the XY and FFD models has a product reference state and at least one explicit eigenstate for every energy level of the form (66).
  • Acting with all S fermionic modes on the reference state yields a nonzero eigenvector of H itself, with norm squared 2^{-S}, giving a practical route to H eigenstates in models where no reference state was known.
  • In the FFD model the fermionic operators have a product form consisting of 2M localized rotations and one edge operator, so few-particle eigenstates can be constructed efficiently, and each added fermion contributes only O(1) to bipartite entanglement.
  • The paper conjectures that for M = 6k and M = 6k+2 all eigenstates of H_A can be generated by the spectrum-generating fermions plus the auxiliary fermions; it verifies this numerically for M = 6, 8, 12.

Where Pith is reading between the lines

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

  • One could test the same defining-representation mechanism in non-integrable graph-Clifford models: the reference state is an exact null vector of H_A for any coupling choices, and other null vectors might be found from the conserved charges even without free-fermion solvability.
  • If H_A eigenstates can be prepared in quantum circuits as initial states, the area-law entanglement of few-particle states could make them useful for studying quench dynamics and entanglement growth in disguised free-fermion systems.
  • The deferred relation Ψ_{±k}|∅⟩ = \tilde{Ψ}_{∓k}|∅⟩, once proven in the follow-up paper, is likely the bridge between this construction and standard Jordan-Wigner solvability; checking it explicitly in the XY model would be a direct test of that bridge.

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

2 major / 5 minor

Summary. The paper develops an algebraic framework for spin chains with hidden free-fermion structure. It reviews graph-Clifford (quasi-Clifford) algebras, introduces an operator–state correspondence and a ‘defining representation’ of the tensor product A⊗A, and shows that this representation reproduces the Hamiltonian terms of the XY and FFD models up to boundary modifications. The central new result is that in this representation the anti-symmetric Hamiltonian H_A = H − H̃ admits the all-up product state as an exact null vector, and that fermionic operators built from the known transfer-matrix construction of [11,12,16] create explicit few-body eigenstates of H_A. The FFD model is treated in detail: central elements and determinant parities are analyzed, edge operators are proposed, eigenstates are constructed, and a completeness conjecture is formulated. Selected eigenstates for periodic versions are also given.

Significance. If the construction is correct, the paper provides a genuinely new route to reference states and few-body eigenstates in free-fermion models outside the Jordan–Wigner paradigm, where no such reference state was previously known. The defining representation and Theorem 7 are elegant and likely to be useful for later work, and the explicit FFD formulas give concrete states in a model of active interest. The paper also connects physics results to the mathematical graph-Clifford literature. On the other hand, the paper rests substantially on external results from [11,12,16] and on a theorem deferred to a follow-up paper; the FFD edge-operator formulas contain an internal inconsistency as written. These issues are local and repairable, so the overall approach seems sound, but the manuscript in its current form is not fully self-consistent.

major comments (2)
  1. [Section V.C, Eqs. (92)–(95)] The text declares the simplicial clique to be {h_1}, so the edge operator must satisfy {χ,h_1}=0 and [χ,h_k]=0 for k>1. The explicit operators (93)–(95) do not satisfy this: for example, for M=6, χ=i h_2h_3h_4 has commutation pattern (0,0,0,0,0,1), so it commutes with h_1 and anticommutes with h_6. More generally, (93)–(94) have patterns concentrated at the opposite end of the chain (they anticommute with h_{6k} and h_{6k+2}, respectively), and (95) similarly targets h_M. Consequently the fermionic operators Ψ_k built from these χ via (61)/(28) are not the edge operators required for the stated clique, and the derivation of (63)–(64) and the eigenstates (66)–(67) is not self-consistent for the FFD data as written. This can be repaired by choosing the simplicial clique to be {h_M} (the FFD graph is left-right symmetric), but the current text must be corrected or the clique redefined.
  2. [Section V.D, Theorem 11] Theorem 11 states Ψ_{±k}|∅⟩ = Ψ̃_{∓k}|∅⟩ but its proof is omitted and deferred to ‘the second paper of this series’. This theorem is used to conclude that one fermionic family suffices to generate at least one state per level, and it underpins the later completeness discussion leading to Conjecture 1. Since the main eigenstate construction in (66)–(67) does not itself require discarding the Ψ̃ family, the theorem is not needed for the existence of the stated eigenstates. However, as written the reader cannot verify an assertion that is used as a theorem. The authors should either provide a proof, or explicitly label this as a conjecture and avoid using it as an established result in the completeness argument.
minor comments (5)
  1. [Table I] The determinant values in Table I are asserted without proof. The parity of det(A) is what is actually used (through the center triviality argument), and the n_c column is proven in Theorem 10. Please state explicitly that only the parity of the determinant is needed and that it follows from the n_c computation, or provide a separate proof for the determinant values.
  2. [Section III.B, Eqs. (15) and (42)] Equation (42) is said to be ‘almost identical’ to Fendley’s representation (15), with differences at boundary terms. The abstract says the defining representation ‘coincides’ with the FFD Hamiltonian. Please clarify precisely which Hamiltonian (which boundary conditions, which Hilbert-space length) the concrete eigenstates in Section V.D apply to, so that the reader does not confuse the defining-representation model with the original FFD open chain of length L=M+2.
  3. [Eq. (66) and Fig. 4] The notation Ψ^{σ_j}_{a_j} with σ_j=±1 is introduced without a formal definition of Ψ^{+} versus Ψ^{-}. From (64) it is clear that Ψ_k lowers and Ψ_{-k}=Ψ†_k raises, but the σ label in the multi-particle state should be defined explicitly to avoid ambiguity in the energy formula (67).
  4. [Theorem 9] The proof of Theorem 9 uses the degeneracy 2^{M−S} from [11] and assumes the projectors in (69) have the stated trace. For generic couplings with distinct ε_k this is fine, but the argument should mention that accidental degeneracies of single-particle energies are either absent or do not change the result, or else the theorem should be stated under a generic-coupling assumption.
  5. [General] There are a few minor typographical and notation issues, e.g., in the text around (A4)–(A9) the MPO bond-dimension labels are a bit hard to follow. These do not affect the results.

Circularity Check

0 steps flagged

No significant circularity: the reference state and eigenstates follow algebraically from the defining representation and from externally established free-fermion machinery; deferred and self-cited results are not load-bearing for the central claim.

full rationale

The central derivation is not circular. Theorem 7 follows by construction of the left/right regular representation: with (37)-(38), h_j - \tilde{h}_j represents the commutator [h_j, \cdot], so any operator commuting with H maps to a null vector of H_A; in particular |∅⟩ = φ(1) is annihilated by H_A. The eigenstates (66)-(67) then follow from (61)-(64): using the externally cited free-fermion diagonalization [11,12,16], one has [H_A, Ψ_k] = -2ε_k Ψ_k, and since H_A|∅⟩ = 0, products of Ψ's acting on |∅⟩ are eigenvectors with the stated energies. No fitted parameter is renamed as a prediction. The free-fermion machinery is imported from non-self references ([11,12,16]), not from the authors' own prior work. Self-citations ([17,18,22,23,24,38,39,40]) appear for symmetry, correlation-function, circuit, and crosscap extensions, not to define the target eigenstates. Theorem 11 is explicitly deferred to a follow-up paper, but it is used only to discard the second fermionic family; the states built from the Ψ family alone are already eigenstates, so the deferred relation is not load-bearing. A separate correctness caveat, not a circularity: the explicit FFD edge-operator formulas (93)-(95) have commutation patterns concentrated at h_M, while (92) declares the simplicial clique {h_1}; this is an internal inconsistency repairable by choosing the symmetric clique {h_M}, and it does not turn the derivation into an input-output equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The core theorem depends on prior free-fermion solvability results, on a cited simplicial-clique theorem, and on several assertions the paper itself marks as unproved (determinant values, deferred Theorem 11, Conjecture 1). No constants are fitted to make the construction work.

axioms (6)
  • domain assumption Free-fermion diagonalization theorem: for claw-free and even-hole-free frustration graphs with a simplicial clique and edge operator χ, there exist Dirac fermions Ψ_k satisfying (26) and diagonalizing H via (27), imported from [11,12,16].
    Used throughout Section II.D.2 and Section IV.A; not re-derived in this paper, and it underpins the eigenstate construction for H_A.
  • domain assumption Every eigenenergy of H has degeneracy 2^{M−S}; projectors Ψ^{−σ}Ψ^{σ} have trace 2^{M−n}, used in Theorem 9.
    Assumed from [11] in the proof of Theorem 9 (Section IV.A) to compute norm squares of the constructed eigenstates.
  • standard math Theorem 3 from [28]: two graph-Clifford algebras are isomorphic iff the number of generators and the center dimension coincide; Theorem 1: det(A) odd implies trivial center.
    Cited from Khovanova [28] and used in Section III.A to characterize the algebras and their centers.
  • standard math Claw-free and even-hole-free graphs contain a simplicial clique.
    Cited from [34]; used in Section II.D.1 to justify the existence of the building block for the free-fermion construction.
  • ad hoc to paper Conjecture 1: for M=6k and 6k+2 the states (101) span the full Hilbert space in the defining representation; numerically checked for M=6,8,12.
    Section V.E; used for the completeness claim of the eigenstates of H_A, but explicitly not proved.
  • ad hoc to paper Det(A) values for the FFD adjacency matrix follow the Table I pattern; asserted without proof in the text.
    Section V.B ('We do not prove the values for the determinants'); these values determine which system sizes have a faithful defining representation.

pith-pipeline@v1.3.0-alltime-deepseek · 25539 in / 19892 out tokens · 193581 ms · 2026-08-03T04:59:08.157762+00:00 · methodology

0 comments
read the original abstract

We study quantum spin chains solvable via hidden free fermionic structures. We study the algebras behind such models, establishing connections to the mathematical literature of the so-called ``graph-Clifford'' or ``quasi-Clifford'' algebras. We also introduce the ``defining representation'' for such algebras, and show that this representation actually coincides with the terms of the Hamiltonian in two relevant models: the XY model and the ``free fermions in disguise'' model of Fendley. Afterwards we study a particular anti-symmetric combination of commuting Hamiltonians; this is performed in a model independent way. We show that for this combination there exists a reference state, and few body eigenstates can be created by the fermionic operators. Concrete application is presented in the case of the ``free fermions in disguise'' model.

Figures

Figures reproduced from arXiv: 2602.03431 by Bal\'azs Pozsgay, Istv\'an Vona, Kohei Fukai.

Figure 1
Figure 1. Figure 1: Frustration graph for the XY and Ising models with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Frustration graph for the FFD model with [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: The four states associated with the eigenmode with [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Short summary of the notations for the different [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

discussion (0)

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

Forward citations

Cited by 4 Pith papers

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

  1. Correlation and entanglement dynamics of free fermions in disguise

    cond-mat.stat-mech 2026-07 conditional novelty 7.0

    For Fendley's free-fermions-in-disguise chain, post-quench GGE occupations and local energy-density expectations are derived analytically and match TEBD numerics; the quasi-particle entanglement-growth formula is only...

  2. Correlation and entanglement dynamics of free fermions in disguise

    cond-mat.stat-mech 2026-07 conditional novelty 6.0

    Analytic method for GGE quasi-momentum distribution in degenerate free-fermion models plus a conjecture adjusting the quasi-particle entanglement formula, tested numerically with good agreement on local observables bu...

  3. Free fermions in disguise without exponential degeneracies

    cond-mat.stat-mech 2026-06 unverdicted novelty 6.0

    A perturbation of two Ising chains (or interpolation between Jordan-Wigner and Fendley FFD models) yields an FFD-solvable spin chain without exponential degeneracies for generic couplings.

  4. Solving models with generalized free fermions II: Path-product expansion and conserved charges

    cond-mat.stat-mech 2026-05 unverdicted novelty 6.0

    Derives path-product expansion for free-fermion modes and local conserved charges in generalized free-fermion models from Krylov basis generating function.

Reference graph

Works this paper leans on

47 extracted references · 25 linked inside Pith · cited by 3 Pith papers

  1. [1]

    We identify the generatorsh j and ˜hj as hj =Y j−1Xj, ˜hj =X jYj+1.(8) It is easy to see that (4) holds, and that the two sets of generators satisfy the same algebra

    The XY model via the Dzyaloshinskii–Moriya interaction Let us consider a Hamiltonian given solely by the Dzyaloshinskii–Moriya interaction term: HA = MX j=1 (YjXj+1 −X jYj+1).(7) We impose periodic boundary conditions, and for sim- plicity we used here homogeneous couplings. We identify the generatorsh j and ˜hj as hj =Y j−1Xj, ˜hj =X jYj+1.(8) It is easy...

  2. [2]

    The quantum Ising chain We considerM= 2Land define the Hamiltonian act- ing on a spin-1/2 chain of lengthLas H= LX j=1 (XjXj+1 +gZ j+1).(10) We assume periodic boundary conditions, and also ho- mogeneous couplings. The generators can be identified as h2j−1 =X jXj+1, h 2j =Z j+1.(11) Inspecting the commutation relations of the generators we see that they s...

  3. [3]

    Free fermions with all-to-all interactions Let us now consider a family of Majorana fermionsχ a, witha= 1. . . M′. They are defined via the relations χ† a =χ a,{χ a, χb}= 2δ a,b.(12) If we would construct the frustration graph for this family of operators, we would obtain a complete graph withM ′ vertices. However, instead of taking linear combinations of...

  4. [4]

    Let us consider open boundary conditions, and the spin chain of lengthL=M+ 2

    Free fermions in disguise The model in question was introduced by Paul Fendley in the seminal work [11]. Let us consider open boundary conditions, and the spin chain of lengthL=M+ 2. We introduce generators hj =Z j−2Zj−1Xj, ˜hj =X jZj+1Zj+2.(15) The two sets of generators commute with each other, and they satisfy the same algebra. This algebra is given by...

  5. [5]

    Higher charges and transfer matrix Let us now construct the higher conserved charges of the models that we treat. For future use we introduce the notation Hk =b khk.(19) We say thatQ 1 is the Hamiltonian, in these notations Q1 = X k Hk.(20) The next conserved charge is given by Q2 = X j<k Ajk =0 HjHk.(21) Note that the sum runs over those products where t...

  6. [6]

    An even hole (also called chordless cycle) is a cycle in the graph of even length, where only consecutive vertices are connected, but there are no “chords”

    Inversion relation Further useful properties can be derived if the graphG satisfies another special relation, namely that it is even- hole-free. An even hole (also called chordless cycle) is a cycle in the graph of even length, where only consecutive vertices are connected, but there are no “chords”. It was shown in [12] that ifGis claw-free and even- hol...

  7. [7]

    Simplicial cliques One of the key ingredients of the free fermionic solu- tion is a special subgraph of the frustration graph: the simplicial clique. A simplicial clique is a clique (a sub- graph which is a complete graph), such that for every vertex within the clique, the neighbors of the given ver- tex outside the clique also form a clique. It was prove...

  8. [8]

    free fermions in dis- guise

    Construction of the fermionic operators Let us now turn to the diagonalization of the Hamilto- nianH, assuming that the graphGis claw-free and also even-hole-free, and also that a simplicial clique has been identified. The construction will use a certain edge operatorχ. This is a Hermitian operator satisfyingχ 2 = 1. It has to anti-commute with the genera...

  9. [9]

    T. D. Schultz, D. C. Mattis, and E. H. Lieb, Two- dimensional ising model as a soluble problem of many fermions, Rev. Mod. Phys.36, 856 (1964)

    1964

  10. [10]

    E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys.16, 407 (1961)

    1961

  11. [11]

    A. Y. Kitaev, Unpaired majorana fermions in quantum wires, Physics-Uspekhi44, 131 (2001)

    2001

  12. [12]

    Kitaev, Anyons in an exactly solved model and beyond, Ann

    A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys.321, 2 (2006), arXiv:cond- mat/0506438 [cond-mat.mes-hall]

    arXiv 2006

  13. [13]

    Jordan and E

    P. Jordan and E. Wigner, ¨Uber das paulische ¨ aquivalen- zverbot, Z. Physik47, 631 (1928)

    1928

  14. [14]

    Minami, Solvable Hamiltonians and Fermionization Transformations Obtained from Operators Satisfying Specific Commutation Relations, J

    K. Minami, Solvable Hamiltonians and Fermionization Transformations Obtained from Operators Satisfying Specific Commutation Relations, J. Phys. Soc. Japan85, 024003 (2016)

    2016

  15. [15]

    Minami, Infinite number of solvable generalizations of XY-chain, with cluster state, and with central charge c = m/2, Nucl

    K. Minami, Infinite number of solvable generalizations of XY-chain, with cluster state, and with central charge c = m/2, Nucl. Phys. B925, 144 (2017), arXiv:1710.01851 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2017

  16. [16]

    Yanagihara and K

    Y. Yanagihara and K. Minami, Exact solution of a cluster model with next-nearest-neighbor interaction, Progr. of Theor. Exp. Phys.2020, 113A01 (2020), arXiv:2003.00962 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2020

  17. [17]

    Ogura, Y

    M. Ogura, Y. Imamura, N. Kameyama, K. Minami, and M. Sato, Geometric criterion for solvability of lat- tice spin systems, Phys. Rev. B102, 245118 (2020), arXiv:2003.13264 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2020

  18. [18]

    Chapman and S

    A. Chapman and S. T. Flammia, Characterization of solvable spin models via graph invariants, Quantum4, 278 (2020), arXiv:2003.05465 [quant-ph]

    Pith/arXiv arXiv 2020

  19. [20]

    S. J. Elman, A. Chapman, and S. T. Flammia, Free fermions behind the disguise, Commun. Math. Phys.388, 969 (2021), arXiv:2012.07857 [quant-ph]

    Pith/arXiv arXiv 2021

  20. [21]

    F. C. Alcaraz and R. A. Pimenta, Free fermionic and parafermionic quantum spin chains with multi- spin interactions, Phys. Rev. B102, 121101 (2020), arXiv:2005.14622 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2020

  21. [22]

    F. C. Alcaraz and R. A. Pimenta, Integrable quantum spin chains with free fermionic and parafermionic spectrum, Phys. Rev. B102, 235170 (2020), arXiv:2010.01116 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2020

  22. [23]

    Chapman, S

    A. Chapman, S. T. Flammia, and A. J. Koll´ ar, Free- Fermion Subsystem Codes, PRX Quantum3, 030321 (2022), arXiv:2201.07254 [quant-ph]

    Pith/arXiv arXiv 2022

  23. [24]

    Chapman, S

    A. Chapman, S. J. Elman, and R. L. Mann, A Unified Graph-Theoretic Framework for Free-Fermion Solvability, arXiv e-prints , arXiv:2305.15625 (2023), arXiv:2305.15625 [quant-ph]

    Pith/arXiv arXiv 2023

  24. [25]

    Fendley and B

    P. Fendley and B. Pozsgay, Free fermions beyond Jordan and Wigner, SciPost Physics16, 102 (2024), arXiv:2310.19897 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2024

  25. [26]

    Fukai, I

    K. Fukai, I. Vona, and B. Pozsgay, A free fermions in disguise model with claws, arXiv e-prints 20 10.48550/arXiv.2508.05789 (2025), arXiv:2508.05789 [cond-mat.stat-mech]

    Pith/arXiv arXiv doi:10.48550/arxiv.2508.05789 2025

  26. [27]

    Fendley and K

    P. Fendley and K. Schoutens, Cooper pairs and exclusion statistics from coupled free-fermion chains, J. Stat. Mech. 2007, 02017 (2007), arXiv:cond-mat/0612270 [cond- mat.stat-mech]

    Pith/arXiv arXiv 2007

  27. [28]

    de Gier, G

    J. de Gier, G. Z. Feher, B. Nienhuis, and M. Rusaczonek, Integrable supersymmetric chain without particle conser- vation, J. Stat. Mech. , 023104 (2016), arXiv:1510.02520 [cond-mat.quant-gas]

    Pith/arXiv arXiv 2016

  28. [29]

    G. Z. Feher, A. Garbali, J. de Gier, and K. Schoutens, A curious mapping between supersymmetric quantum chains, in2017 MATRIX Annals(Springer International Publishing, Cham, 2019) pp. 167–184, arXiv:1710.02658 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2019

  29. [30]

    Vernier and L

    E. Vernier and L. Piroli, The Hilbert-space structure of free fermions in disguise, JSTAT2026(1), 013101, arXiv:2507.15959 [cond-mat.stat-mech]

    arXiv

  30. [31]

    I. Vona, M. Mesty´ an, and B. Pozsgay, Exact real-time dy- namics with free fermions in disguise, Phys. Rev. B111, 144306 (2025), arXiv:2405.20832 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2025

  31. [32]

    Fukai and B

    K. Fukai and B. Pozsgay, Quantum circuits with free fermions in disguise, J. Phys. A58, 175202 (2025), arXiv:2402.02984 [quant-ph]

    Pith/arXiv arXiv 2025

  32. [33]

    Sz´ asz-Schagrin, D

    D. Sz´ asz-Schagrin, D. Cristani, L. Piroli, and E. Vernier, Construction and simulability of quantum circuits with free fermions in disguise, Quantum Science and Technol- ogy11, 015044 (2026), arXiv:2509.22585 [quant-ph]

    arXiv 2026

  33. [34]

    Fendley, Free parafermions, J

    P. Fendley, Free parafermions, J. Phys. A47, 075001 (2014), arXiv:1310.6049 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2014

  34. [35]

    Takahashi,Thermodynamics of One-Dimensional Solvable Models(Cambridge University Press, 1999)

    M. Takahashi,Thermodynamics of One-Dimensional Solvable Models(Cambridge University Press, 1999)

    1999

  35. [36]

    Khovanova, Clifford Algebras and Graphs, Geombina- toricsXX (2), 56 (2010), arXiv:0810.3322 [math.CO]

    T. Khovanova, Clifford Algebras and Graphs, Geombina- toricsXX (2), 56 (2010), arXiv:0810.3322 [math.CO]

    Pith/arXiv arXiv 2010

  36. [37]

    H. M. Gastineau-Hills,Systems of orthogonal designs and quasi-Clifford algebras, Phd thesis, University of Sydney (1980)

    1980

  37. [38]

    H. M. Gastineau-Hills, Quasi clifford algebras and sys- tems of orthogonal designs, J. Austr. Math. Soc. Series A32, 1–23 (1982)

    1982

  38. [39]

    Huber, Quasi-Clifford to qubit mappings, arXiv e- prints (2025), arXiv:2508.01470 [quant-ph]

    F. Huber, Quasi-Clifford to qubit mappings, arXiv e- prints (2025), arXiv:2508.01470 [quant-ph]

    Pith/arXiv arXiv 2025

  39. [40]

    Zhang, F

    X. Zhang, F. G¨ ohmann, A. Kl¨ umper, and V. Pop- kov, Chiral eigenbases of the XX and XY quan- tum spin chains, Phys. Rev. B111, 094437 (2025), arXiv:2411.17877 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2025

  40. [41]

    L. D. Faddeev, How Algebraic Bethe Ansatz works for integrable model, arXiv e-prints (1996), arXiv:hep- th/9605187 [hep-th]

    arXiv 1996

  41. [42]

    Chudnovsky, A

    M. Chudnovsky, A. Scott, P. Seymour, and S. Spirkl, A note on simplicial cliques, Discrete Mathematics344, 112470 (2021), arXiv:2012.05287 [math.CO]

    Pith/arXiv arXiv 2021

  42. [43]

    Chapman, Product form of the transfer matrix for chordal graphs, (unpublished)

    A. Chapman, Product form of the transfer matrix for chordal graphs, (unpublished)

  43. [44]

    Fendley, Free fermions in disguise, J

    P. Fendley, Free fermions in disguise, J. Phys. A52, 335002 (2019), arXiv:1901.08078 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2019

  44. [45]

    F. C. Alcaraz and R. A. Pimenta, Free-parafermionic z(n) and free-fermionicxyquantum chains, Phys. Rev. E104, 054121 (2021), arXiv:2108.04372 [cond-mat.stat- mech]

    Pith/arXiv arXiv 2021

  45. [46]

    Gombor and B

    T. Gombor and B. Pozsgay, Integrable spin chains and cellular automata with medium-range interaction, Phys. Rev. E104, 054123 (2021), arXiv:2108.02053 [nlin.SI]

    Pith/arXiv arXiv 2021

  46. [47]

    Fukai, (in preparation)

    K. Fukai, (in preparation)

  47. [48]

    Mesty´ an and B

    M. Mesty´ an and B. Pozsgay, Crosscap states with tun- able entanglement as exact eigenstates of local spin chain Hamiltonians, J. Phys. A58, 265001 (2025), arXiv:2503.15640 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2025