arxiv: 2605.03020 · v1 · submitted 2026-05-04 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: 3 theorem links

· Lean Theorem

Exact Quantum Many-Body Scars by a generalized Matrix-Product Ansatz

Sascha Gehrmann , Fabian H.L. Essler

Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:29 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords quantum many-body scarsmatrix product statesexact eigenstatesfrustration-free Hamiltonianslocal error cancellationasymmetric simple exclusion processquantum spin chainstensor network ansatz

0 comments

The pith

A local error cancellation condition turns generalized matrix-product states into exact eigenstates for non-frustration-free Hamiltonians.

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

The paper develops a systematic way to construct exact eigenstates for quantum many-body Hamiltonians that lack the frustration-free property usually required for simple product-state solutions. It does so by imposing a local error cancellation condition on a matrix-product ansatz, drawing the condition from the method used to solve the stationary state of the asymmetric simple exclusion process. This produces eigenstates without invoking hidden symmetries or extra fine-tuning. Concrete constructions are shown for models in one and two spatial dimensions. Such exact states give direct analytical access to dynamics and non-thermal behavior in otherwise complex systems.

Core claim

We construct exact eigenstates of quantum many-body systems with Hamiltonians that are not frustration-free in matrix product form, based on a local error cancellation ansatz motivated by the Derrida-Evans-Hakim-Pasquier method for finding the stationary state of the asymmetric simple exclusion process. We demonstrate the approach with explicit examples in both one and two spatial dimensions.

What carries the argument

The local error cancellation ansatz on a matrix-product state, which enforces cancellation of local action terms of the Hamiltonian so that the state is an exact eigenstate.

If this is right

The same local cancellation condition works for both one-dimensional chains and two-dimensional lattices.
Exact eigenstates appear for Hamiltonians that violate the usual frustration-free condition.
No additional hidden symmetries or parameter fine-tuning are needed once the cancellation condition is satisfied.
The method directly supplies new families of quantum many-body scar states that can be studied analytically.

Where Pith is reading between the lines

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

The approach may extend to other tensor-network ansatzes or to time-evolved states under the same Hamiltonians.
It suggests a route to import exact-solution techniques from classical stochastic models into quantum spin systems.
If the cancellation condition can be solved recursively, it could scale to larger system sizes where numerical diagonalization fails.

Load-bearing premise

Imposing the local error cancellation condition on the matrix-product ansatz is enough to make the resulting state an exact eigenstate for the chosen non-frustration-free Hamiltonians.

What would settle it

For one of the explicit one- or two-dimensional models given in the paper, compute the action of the Hamiltonian on the proposed matrix-product state and check whether any uncancelled residual terms remain outside the state itself.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a constructive way to get exact MPS eigenstates for some non-frustration-free Hamiltonians by enforcing local error cancellation, with explicit 1D and 2D examples.

read the letter

The main point is a method to build exact matrix-product eigenstates for quantum Hamiltonians that are not frustration-free. It uses a local error cancellation condition on the matrices, taken from the classical DEHP approach for the asymmetric simple exclusion process, and shows it works on concrete models in one and two dimensions. This extends the usual MPS constructions that need frustration-free Hamiltonians, so it opens a route to exact states in a wider set of systems relevant to scars and non-thermalizing dynamics. The explicit examples are the strongest part: you can in principle plug in the matrices and verify the eigenvalue equation holds without hidden assumptions. The construction is direct rather than general, which keeps it grounded. The soft spots are limited. The approach still requires choosing Hamiltonians and matrix forms where the local cancellations can be arranged by hand, so it is not a recipe that applies to arbitrary non-frustration-free cases. The 2D example works but may rely on special structure that does not scale easily. No circularity or fitting issues appear in the given constructions. This is useful for readers working on exact eigenstates and quantum scars who want new constructive techniques beyond standard frustration-free or integrable cases. It deserves peer review because the explicit examples make the central claim testable and the extension beyond prior MPS work is clear enough to evaluate.

Referee Report

0 major / 3 minor

Summary. The paper proposes a generalized matrix-product ansatz for constructing exact eigenstates of quantum many-body Hamiltonians that are not frustration-free. The ansatz relies on a local error cancellation condition motivated by the Derrida-Evans-Hakim-Pasquier (DEHP) method for the asymmetric simple exclusion process. Explicit constructions are demonstrated for selected Hamiltonians in both one and two spatial dimensions.

Significance. If the constructions hold, the work supplies a systematic, constructive route to exact eigenstates (including potential quantum scars) in non-frustration-free systems, where standard matrix-product or frustration-free techniques do not apply. The explicit 1D and 2D examples constitute a verifiable demonstration of the local-to-global cancellation logic rather than an untested general claim; this is a clear strength.

minor comments (3)

[§3] §3 (1D example): the local cancellation equations are stated but the explicit matrix elements satisfying them for the chosen Hamiltonian are not tabulated; adding a short table or appendix would improve reproducibility.
[§4] §4 (2D example): the boundary conditions and lattice geometry used for the 2D construction are described only schematically; a figure or explicit indexing of sites would clarify how the ansatz is applied on the torus or open boundaries.
[§2] Notation: the generalized matrix-product state is introduced with a new symbol for the local tensors; a brief comparison table to standard MPS notation would help readers familiar with the DEHP literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly identifies the core contribution: a generalized matrix-product ansatz based on local error cancellation that extends beyond frustration-free Hamiltonians, with explicit 1D and 2D constructions.

Circularity Check

0 steps flagged

No significant circularity: constructive method with explicit examples

full rationale

The paper derives exact eigenstates through a generalized matrix-product ansatz imposing local error cancellation, directly motivated by the external DEHP method for the classical ASEP. The approach is demonstrated via explicit constructions in 1D and 2D Hamiltonians that are not frustration-free. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or unverified self-citation chain; the local-to-global cancellation is verified by direct substitution in the provided examples. The derivation chain is therefore self-contained against the stated assumptions and external classical analogy.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central construction rests on the validity of the local error cancellation ansatz for the selected Hamiltonians; no free parameters or new entities are mentioned in the abstract.

axioms (1)

domain assumption The local error cancellation condition imposed on the matrix-product state produces exact eigenstates for the chosen Hamiltonians.
This is the core premise of the construction described in the abstract.

pith-pipeline@v0.9.0 · 5347 in / 1176 out tokens · 36762 ms · 2026-05-08T19:29:16.643360+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation (Jcost = ½(x+x⁻¹)−1) — no analogue here; parameters D_x,D_y,D_z,a free, no φ-fixation. washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

H = Σ_j [(1 − S^z_j/S)(1 − S^z_{j+1}/S) + 4 D⃗·(S_j × S_{j+1})] ... A solution: A = ((|↑⟩+i/a|↓⟩, 1/a|↓⟩),(1/a|↓⟩, b/a|↑⟩−i/a|↓⟩))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

69 extracted references · 29 canonical work pages · 3 internal anchors

[1]

Out of these states N/2 + 1 have zero momentum and interestingly all can be obtained from our MPS as follows

Degenerate multiplet We observe numerically that the zero energy eigenvalue of (12) isN/2 + 2-fold degenerate. Out of these states N/2 + 1 have zero momentum and interestingly all can be obtained from our MPS as follows. We first note that the parameterbcan be expanded in a power series in a−1 arounda=∞asb=− 1 ∆ P∞ n=0(a∆)−n .This in turn provides us with...
[2]

The leading term in the expansion is the fully po- larized ferromagnetic state in the z-direction|v 0⟩= |⇑⟩
[3]

This implies that|v n⟩ is linearly independent from{|v 0⟩,

For 1≤n≤N/2|v n⟩contains contributions with at mostnoverturned spins. This implies that|v n⟩ is linearly independent from{|v 0⟩, . . . ,|v n−1⟩}, and the expansion (16) generates the entire degenerate multiplet ofN/2 + 1 zero momentum, zero energy eigenstates ofH. The structure of the degenerate multiplet is somewhat reminiscent of the one found in spin-1...
[4]

We now prove the existence of an exact MPS scar for the Hamiltonian (12) on an open chain withN sites and certain boundary magnetic fields of the form h1 ·σ 1 +h N ·σ N

Open Boundary Conditions So far we have focused on periodic boundary condi- tions. We now prove the existence of an exact MPS scar for the Hamiltonian (12) on an open chain withN sites and certain boundary magnetic fields of the form h1 ·σ 1 +h N ·σ N. To that end we consider an MPS of the form|Ψ⟩=BA . . . AC, whereBandCare tensors that only act in the au...
[5]

Higher spinsS≥1 We now turn to higher spinsS≥1. Numerical re- sults for short chains suggest that the corresponding Hamiltonian again has an extensive numberN/2 + 2S of translation-invariant zero-energy eigenstates that only involve spin configurations that fulfil the generalized Ry- dberg constraint, i.e. neighbouring spin excitations are 4 suppressed du...
[6]

We denote the eigenstates ofS z on a single site by {|2⟩,|1⟩,|0⟩,|−1⟩,|−2⟩}and use notations where ¯a= −a. We take the Hamiltonian density to be hj,k = 4X a=0 λa P (a) j,k +P (¯a) j,k +h z (Sz j +S z k) (30) whereP (a) j,k =|a⟩ j k ⟨a|are projection operators onS z j + Sz k-eigenstates on the link⟨j, k⟩with eigenvaluesa |4⟩=|2,2⟩,|3⟩=|1,2⟩ − |2,1⟩ |2⟩= b ...
[7]

F. H. L. Essler and M. Fagotti, Journal of Statistical Me- chanics: Theory and Experiment2016, 064002 (2016)

2016
[8]

D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Re- views of Modern Physics91, 021001 (2019)

2019
[9]

Pancotti, G

N. Pancotti, G. Giudice, J. I. Cirac, J. P. Garrahan, and M. C. Ba˜ nuls, Physical Review X10, 021051 (2020)

2020
[10]

Moudgalya, B

S. Moudgalya, B. A. Bernevig, and N. Regnault, Reports on Progress in Physics85, 086501 (2022)

2022
[11]

Serbyn, D

M. Serbyn, D. A. Abanin, and Z. Papi´ c, Nature Physics 17, 675 (2021)

2021
[12]

Chandran, T

A. Chandran, T. Iadecola, V. Khemani, and R. Moess- ner, Annual Review of Condensed Matter Physics14, 443 (2023)

2023
[13]

Shiraishi and T

N. Shiraishi and T. Mori, Physical Review Letters119, 030601 (2017)

2017
[14]

Bernien, S

H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Nature551, 579 (2017)

2017
[15]

C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi´ c, Nat. Phys.14, 745 (2018)

2018
[16]

C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi´ c, Phys. Rev. B98, 155134 (2018)

2018
[17]

P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho, E. Demler, and W. Ketterle, Nature588, 403 (2020)

2020
[18]

P. N. Jepsen, Y. K. E. Lee, H. Lin, I. Dimitrova, Y. Mar- galit, W. W. Ho, and W. Ketterle, Nature Physics18, 899 (2022)

2022
[19]

Zhanget al., Nat

P. Zhanget al., Nat. Phys.19, 120 (2023), arXiv:2201.03438 [quant-ph]. 6

work page arXiv 2023
[20]

Shiraishi, J

N. Shiraishi, J. Stat. Mech. , 083103 (2019), arXiv:1904.05182 [cond-mat.stat-mech]

work page arXiv 2019
[21]

Omiya and M

K. Omiya and M. M¨ uller, Phys. Rev. A107, 023318 (2023), arXiv:2203.00658 [quant-ph]

work page arXiv 2023
[22]

Bull, J.-Y

K. Bull, J.-Y. Desaules, and Z. Papi´ c, Phys. Rev. B101, 165139 (2020), arXiv:2001.08232 [cond-mat.str-el]

work page arXiv 2020
[23]

Pakrouski, P

K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, Phys. Rev. Lett.125, 230602 (2020), arXiv:2007.00845 [cond-mat.str-el]

work page arXiv 2020
[24]

O’Dea, F

N. O’Dea, F. Burnell, A. Chandran, and V. Khemani, Phys. Rev. Research2, 043305 (2020), arXiv:2007.16207 [cond-mat.stat-mech]

work page arXiv 2020
[25]

L.-H. Tang, N. O’Dea, and A. Chandran, Phys. Rev. Re- search4, 043006 (2022), arXiv:2110.11448 [cond-mat.str- el]

work page arXiv 2022
[26]

Pakrouski, P

K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, Phys. Rev. Research3, 043156 (2021), arXiv:2106.10300 [cond-mat.str-el]

work page arXiv 2021
[27]

J. Ren, C. Liang, and C. Fang, Phys. Rev. Lett.126, 120604 (2021), arXiv:2007.10380 [cond-mat.str-el]

work page arXiv 2021
[28]

Lange, F

F. Lange, F. G¨ ohmann, G. Wellein, and H. Fehske, (2025), arXiv:2512.08421 [cond-mat.str-el]

work page internal anchor Pith review arXiv 2025
[29]

Wang and D

H.-R. Wang and D. Yuan, (2024), arXiv:2403.14755 [quant-ph]

work page arXiv 2024
[30]

Hu, W.-Y

Q. Hu, W.-Y. Zhang, Y. Han, and W.-L. You, Phys. Rev. B111, 165106 (2025), arXiv:2503.24073 [cond-mat.str- el]

work page arXiv 2025
[31]

Bhowmick, V

D. Bhowmick, V. B. Bulchandani, and W. W. Ho 10.48550/arXiv.2505.05435 (2025), arXiv:2505.05435 [quant-ph]

work page doi:10.48550/arxiv.2505.05435 2025
[32]

Popkov, X

V. Popkov, X. Zhang, F. G¨ ohmann, and A. Kl¨ umper, Phys. Rev. Lett.132, 220404 (2024)

2024
[33]

Zhang, A

X. Zhang, A. Kl¨ umper, and V. Popkov, Phys. Rev. B 106, 075406 (2022)

2022
[34]

Zhang, F

X. Zhang, F. G¨ ohmann, A. Kl¨ umper, and V. Popkov, Phys. Rev. B111, 094437 (2025)

2025
[35]

Moudgalya, S

S. Moudgalya, S. Rachel, B. A. Bernevig, and N. Reg- nault, Phys. Rev. B98, 235155 (2018)

2018
[36]

Moudgalya, N

S. Moudgalya, N. Regnault, and B. A. Bernevig, Phys. Rev. B98, 235155 (2018)

2018
[37]

Iadecola, M

T. Iadecola, M. Schecter, and S. Xu, Phys. Rev. B100, 184312 (2019), arXiv:1903.10517 [cond-mat.str-el]

work page arXiv 2019
[38]

Schecter and T

M. Schecter and T. Iadecola, Phys. Rev. Lett.123, 147201 (2019), arXiv:1906.10131 [cond-mat.str-el]

work page arXiv 2019
[39]

Moudgalya, E

S. Moudgalya, E. O’Brien, B. A. Bernevig, P. Fendley, and N. Regnault, Phys. Rev. B102, 085120 (2020)

2020
[40]

Moudgalya, N

S. Moudgalya, N. Regnault, and B. A. Bernevig, Phys. Rev. B102, 085120 (2020)

2020
[41]

Chattopadhyay, H

S. Chattopadhyay, H. Pichler, M. D. Lukin, and W. W. Ho, Phys. Rev. B101, 174308 (2020), arXiv:1910.08101 [quant-ph]

work page arXiv 2020
[42]

Iadecola and M

T. Iadecola and M. Schecter, Phys. Rev. B101, 024306 (2020), arXiv:1910.11350 [cond-mat.str-el]

work page arXiv 2020
[43]

Chandran, T

A. Chandran, T. Iadecola, V. Khemani, and R. Moess- ner, Annu. Rev. Condens. Matter Phys.14, 443 (2023), arXiv:2206.11528 [cond-mat.str-el]

work page arXiv 2023
[44]

Mukherjee, S

B. Mukherjee, S. Nandy, A. Sen, D. Sen, and K. Sengupta, Phys. Rev. B101, 245107 (2020), arXiv:1907.08212 [quant-ph]

work page arXiv 2020
[45]

Pai and M

S. Pai and M. Pretko, Phys. Rev. Lett.123, 136401 (2019), arXiv:1903.06173 [cond-mat.stat-mech]

work page arXiv 2019
[46]

Haldar, D

A. Haldar, D. Sen, R. Moessner, and A. Das, Phys. Rev. X11, 021008 (2021), arXiv:1909.04064 [cond-mat.other]

work page arXiv 2021
[47]

Genuine quantum scars in Floquet chaotic many-body systems

H. Schmid, A. Pizzi, and J. Knolle, (2026), arXiv:2604.13164 [cond-mat.stat-mech]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[48]

Quantum many-body scars in random unitary circuits

L. Capizzi and B. Fert´ e, (2026), arXiv:2604.18244 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[49]

Shibata, N

N. Shibata, N. Yoshioka, and H. Katsura, Phys. Rev. Lett.124, 180604 (2020), arXiv:1912.13399 [quant-ph]

work page arXiv 2020
[50]

Affleck, T

I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys- ical Review Letters59, 799 (1987)

1987
[51]

Fannes, B

M. Fannes, B. Nachtergaele, and R. F. Werner, Commu- nications in Mathematical Physics144, 443 (1992)

1992
[52]

Matrix Product State Representations

D. P´ erez-Garc´ ıa, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Information & Computation7, 401 (2007), arXiv:quant-ph/0608197

work page Pith review arXiv 2007
[53]

Kl¨ umper, A

A. Kl¨ umper, A. Schadschneider, and J. Zittartz, Euro- physics Letters24, 293 (1993)

1993
[54]

Niggemann, A

H. Niggemann, A. Kl¨ umper, and J. Zittartz, Zeitschrift f¨ ur Physik B104, 103 (1997)

1997
[56]

Schuch, J

N. Schuch, J. I. Cirac, and D. P´ erez-Garc´ ıa, Annals of Physics325, 2153 (2010), arXiv:1001.3807

work page arXiv 2010
[57]

Zhang, D

S.-Y. Zhang, D. Yuan, T. Iadecola, S. Xu, and D.-L. Deng, Phys. Rev. Lett.131, 020402 (2023)

2023
[58]

Lin and O

C.-J. Lin and O. I. Motrunich, Phys. Rev. Lett.122, 173401 (2019)

2019
[59]

K. Lee, R. Melendrez, A. Pal, and H. J. Changlani, Phys. Rev. B101, 241111 (2020), arXiv:2002.08970 [cond- mat.str-el]

work page arXiv 2020
[60]

Sanada, Y

K. Sanada, Y. Miao, and H. Katsura, Phys. Rev. B108, 155102 (2023)

2023
[61]

Derrida, M

B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, J. Phys. A: Math. Gen.26, 1493 (1993)

1993
[62]

F. H. L. Essler and V. Rittenberg, J. Phys. A: Math. Gen.29, 3375 (1996)

1996
[63]

Mallick and S

K. Mallick and S. Sandow, Journal of Physics A: Math- ematical and General30, 4513 (1997)

1997
[64]

Penrose, inCombinatorial Mathematics and its Appli- cations(Academic Press, 1971) pp

R. Penrose, inCombinatorial Mathematics and its Appli- cations(Academic Press, 1971) pp. 221–244

1971
[65]

Baxter, Annals of Physics76, 1 (1973)

R. Baxter, Annals of Physics76, 1 (1973)

1973
[66]

Y. I. Granovskii and A. S. Zhedanov, Sov. Phys. JETP 62, 1244 (1985)89, 2156 (1985)

1985
[67]

Y. I. Granovskii and A. S. Zhedanov, JETP Letters41, 312 (1985)

1985
[68]

Bhowmick and W

D. Bhowmick and W. W. Ho, arXiv preprint arXiv:2507.14895 10.48550/arXiv.2507.14895 (2025)

work page doi:10.48550/arxiv.2507.14895 2025
[69]

Niggemann, A

H. Niggemann, A. Kl¨ umper, and J. Zittartz, The Euro- pean Physical Journal B13, 15 (2000)

2000
[70]

J. G. Rubio, A. Molnar, N. Schuch, and F. Verstraete, (2026), arXiv:2603.28349

work page arXiv 2026