Quantized topological invariant of symmetry-projected Gibbs states | Paper Citation Record

Observation 468e6598-be33-4ddb-ab5a-7847543eaabb · outbound

This paper cites Gu and X.-G.

Quantized topological invariant of symmetry-projected Gibbs states Gu and X.-G

Reference 1

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Observation ea74250d-0992-4c48-b2c5-eac135010963 · outbound

This paper cites Entanglement spectrum of a topological phase in one dimension.

Quantized topological invariant of symmetry-projected Gibbs states Entanglement spectrum of a topological phase in one dimension

Reference 2

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Observation 97b17413-32c2-4e00-b089-48a029351a7c · outbound

This paper cites Symmetry protection of topological order in one-dimensional quantum spin systems.

Quantized topological invariant of symmetry-projected Gibbs states Symmetry protection of topological order in one-dimensional quantum spin systems

Reference 3

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Observation a8b3ef31-4e9a-41f2-ad24-c21c161e7788 · outbound

This paper cites Classification of Gapped Symmetric Phases in 1D Spin Systems.

Quantized topological invariant of symmetry-projected Gibbs states Classification of Gapped Symmetric Phases in 1D Spin Systems

Reference 4

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Observation 88fdd37c-48fa-4e8b-91d1-3989f145751c · outbound

This paper cites Chen, Z.-C.

Quantized topological invariant of symmetry-projected Gibbs states Chen, Z.-C

Reference 5

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Observation ca6cf314-d659-4c5f-abac-e464d4989478 · outbound

This paper cites Symmetry protected topological orders and the group cohomology of their symmetry group.

Quantized topological invariant of symmetry-projected Gibbs states Symmetry protected topological orders and the group cohomology of their symmetry group

Reference 6

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Observation 12d8d68f-35bd-447a-9907-083a1fd736b5 · outbound

This paper cites Symmetry protected topological order at nonzero temperature.

Quantized topological invariant of symmetry-projected Gibbs states Symmetry protected topological order at nonzero temperature

Reference 7

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Observation 1457ddfd-b2c8-4003-aaba-a8ddfcdcd6ef · outbound

This paper cites Average Symmetry-Protected Topological Phases.

Quantized topological invariant of symmetry-projected Gibbs states Average Symmetry-Protected Topological Phases

Reference 8

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Observation 9e93a6c7-ffa0-4571-932a-7f6cec91f3e3 · outbound

This paper cites Symmetry Protected Topological Order in Open Quantum Systems.

Quantized topological invariant of symmetry-projected Gibbs states Symmetry Protected Topological Order in Open Quantum Systems

Reference 9

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Observation 23b3c154-70e6-48d9-b3e0-23ba002fd92e · outbound

This paper cites Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States.

Quantized topological invariant of symmetry-projected Gibbs states Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States

Reference 10

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Observation 23fedef4-590c-40c4-a345-cd68fb402816 · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 11

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Observation 2b5ed480-be1a-4b6e-b551-9dbe0b9e6c2d · outbound

This paper cites Negari, L.

Quantized topological invariant of symmetry-projected Gibbs states Negari, L

Reference 12

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Observation 2439015c-edc4-4230-b635-3a72ba393627 · outbound

This paper cites Raussendorf, S.

Quantized topological invariant of symmetry-projected Gibbs states Raussendorf, S

Reference 13

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Observation d4418de8-8244-4b5c-a405-568d2dc5d2fc · outbound

This paper cites an unresolved cited work.

Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 14

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Observation ea11dd38-cab8-42de-b44e-e38eae6068b8 · outbound

This paper cites Pushing the Limits of Monte Carlo Simulations for the 3d Ising Model.

Quantized topological invariant of symmetry-projected Gibbs states Pushing the Limits of Monte Carlo Simulations for the 3d Ising Model

Reference 15

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Observation 6c9b5a61-d702-4560-94aa-5dcd1286b595 · outbound

This paper cites Topological Order at Non-zero Temperature.

Quantized topological invariant of symmetry-projected Gibbs states Topological Order at Non-zero Temperature

Reference 16

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Observation 2d003806-bd03-4795-a3f9-d63e7545212f · outbound

This paper cites Topological order in a 3D toric code at finite temperature.

Quantized topological invariant of symmetry-projected Gibbs states Topological order in a 3D toric code at finite temperature

Reference 17

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Observation d1dd9ceb-0292-4098-a86c-158da183932c · outbound

This paper cites Distinct finite-temperature phase diagrams of non-invertible Kennedy--Tasaki duals.

Quantized topological invariant of symmetry-projected Gibbs states Distinct finite-temperature phase diagrams of non-invertible Kennedy--Tasaki duals

Reference 18

Resolution: verified exact; local_arxiv, observed 2026-08-08T19:23:28.634435Z

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Observation b2ea74e3-1d52-4251-aff9-5ac2a8872579 · outbound

This paper cites an unresolved cited work.

Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 19

Resolution: unresolved; raw_fallback, observed 2026-08-08T19:23:30.852583Z

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Observation 93bfba5f-4277-4530-898c-149a09709ae4 · outbound

This paper cites Precision Islands in the Ising and $O(N)$ Models.

Quantized topological invariant of symmetry-projected Gibbs states Precision Islands in the Ising and $O(N)$ Models

Reference 20

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Observation 55558974-2eab-49be-abec-9e3886fa83c7 · outbound

This paper cites Detection of Symmetry Protected Topological Phases in 1D.

Quantized topological invariant of symmetry-projected Gibbs states Detection of Symmetry Protected Topological Phases in 1D

Reference 21

Resolution: verified exact; local_arxiv, observed 2026-08-08T19:23:28.536588Z

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Observation 9f06760c-7dad-4299-9dfe-d287deed4af6 · outbound

This paper cites Kennedy and H.

Quantized topological invariant of symmetry-projected Gibbs states Kennedy and H

Reference 22

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Observation 79570631-2775-4b2a-a30e-d80d8a76ccec · outbound

This paper cites Non-Invertible Duality Transformation Between SPT and SSB Phases.

Quantized topological invariant of symmetry-projected Gibbs states Non-Invertible Duality Transformation Between SPT and SSB Phases

Reference 23

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Observation e9e5aa15-bd48-46ac-9b94-12b9717f8d67 · outbound

This paper cites Generating Lattice Non-invertible Symmetries.

Quantized topological invariant of symmetry-projected Gibbs states Generating Lattice Non-invertible Symmetries

Reference 24

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Observation 035d21da-5eae-4dc0-aafb-de534f477d4b · outbound

This paper cites Generalized Kramers-Wannier Duality from Bilinear Phase Map.

Quantized topological invariant of symmetry-projected Gibbs states Generalized Kramers-Wannier Duality from Bilinear Phase Map

Reference 25

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Observation ebde71a0-4f79-4ec4-91c1-9f2c642a43c9 · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 26

Resolution: verified exact; arxiv_id, observed 2026-08-08T19:23:28.394385Z

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Observation aa23d960-b7cc-48be-9e58-53f50ae8cf39 · outbound

This paper cites Li and Y.

Quantized topological invariant of symmetry-projected Gibbs states Li and Y

Reference 27

Resolution: verified exact; arxiv_id, observed 2026-08-08T19:23:28.099843Z

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Observation e007d205-0667-43fa-b952-7376ef5fd455 · outbound

This paper cites Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code.

Quantized topological invariant of symmetry-projected Gibbs states Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code

Reference 28

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Observation 5faaca35-b92e-45ed-8b52-2d3e6d8c3ff2 · outbound

This paper cites Intrinsic Mixed-state Topological Order.

Quantized topological invariant of symmetry-projected Gibbs states Intrinsic Mixed-state Topological Order

Reference 29

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Observation 0dc1fe2b-6531-45ef-aa9c-680213fd8c1d · outbound

This paper cites Towards a classification of mixed-state topological orders in two dimensions.

Quantized topological invariant of symmetry-projected Gibbs states Towards a classification of mixed-state topological orders in two dimensions

Reference 30

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Observation b62da3e7-6b61-4443-8b1a-75efb3cf4948 · outbound

This paper cites Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States.

Quantized topological invariant of symmetry-projected Gibbs states Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States

Reference 31

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Observation aa6b2fb3-5828-4464-9023-3b80f586f4d4 · outbound

This paper cites A Noisy Approach to Intrinsically Mixed-State Topological Order.

Quantized topological invariant of symmetry-projected Gibbs states A Noisy Approach to Intrinsically Mixed-State Topological Order

Reference 32

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Observation e193b22b-302a-4a14-9aa3-064a1db31282 · outbound

This paper cites Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems.

Quantized topological invariant of symmetry-projected Gibbs states Holographic View of Mixed-State Symmetry-Protected Topological Phases in Open Quantum Systems

Reference 33

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Observation dc8566fe-d4ef-4c16-b6f0-2dcead1400e4 · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 34

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Observation 0426b21d-6ad1-40da-9a76-3da1e7b6c255 · outbound

This paper cites Topological Holography for Mixed-State Phases and Phase Transitions.

Quantized topological invariant of symmetry-projected Gibbs states Topological Holography for Mixed-State Phases and Phase Transitions

Reference 35

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Observation 7286f5a8-5950-43bb-8c41-234d42ddef58 · outbound

This paper cites SymTFT Approach for Mixed States with Non-Invertible Symmetries.

Quantized topological invariant of symmetry-projected Gibbs states SymTFT Approach for Mixed States with Non-Invertible Symmetries

Reference 36

Resolution: unresolved; no resolver link, observed 2026-08-08T19:23:26.101343Z

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Observation 341233c6-e10a-4596-9bfd-c10c2c343fcf · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 37

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Observation d3ce4b8c-f1ab-437a-9f0a-0c431b5c98d9 · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 38

Resolution: unresolved; no resolver link, observed 2026-08-08T19:23:26.242444Z

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Observation bcf49092-5114-4361-875b-4906fdf412de · outbound

This paper cites Mixed-state phases from local reversibility.

Quantized topological invariant of symmetry-projected Gibbs states Mixed-state phases from local reversibility

Reference 39

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Observation 176b7523-9a0e-4a09-b040-e11a7e36b9b8 · outbound

This paper cites ParaToric 1.0: Continuous-time quantum Monte Carlo for the toric code in a parallel field.

Quantized topological invariant of symmetry-projected Gibbs states ParaToric 1.0: Continuous-time quantum Monte Carlo for the toric code in a parallel field

Reference 40

Resolution: verified exact; local_arxiv, observed 2026-08-08T19:23:27.052289Z

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Observation d2bfb8b7-4352-4608-906d-88cab6703f98 · outbound

This paper cites Aizenman, D.

Quantized topological invariant of symmetry-projected Gibbs states Aizenman, D

Reference 41

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:30.818114Z

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Observation b1fcf9ef-6674-4b01-bf44-3d278863814d · outbound

This paper cites strongly symmetric.

Quantized topological invariant of symmetry-projected Gibbs states strongly symmetric

Reference 42

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:30.802219Z

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Observation dcbe4b67-932b-48e0-bd58-244a6a732b23 · outbound

This paper cites contractible.

Quantized topological invariant of symmetry-projected Gibbs states contractible

Reference 43

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:30.757971Z

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Observation d581cc49-f443-4b14-9742-37257b92ebc6 · outbound

This paper cites Weighting configurations by (−1) wi(C) is, in the Ising dictionary, an antiperiodic twist of the spin model, as follows.

Quantized topological invariant of symmetry-projected Gibbs states Weighting configurations by (−1) wi(C) is, in the Ising dictionary, an antiperiodic twist of the spin model, as follows

Reference 44

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:30.738654Z

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Observation f767e593-5136-4e08-8981-7be42dbd2510 · outbound

This paper cites Along every scan B :=βµ= 4 is fixed, and the sampled exponent is −βH(s) +BG.

Quantized topological invariant of symmetry-projected Gibbs states Along every scan B :=βµ= 4 is fixed, and the sampled exponent is −βH(s) +BG

Reference 45

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:30.708261Z

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Observation 39547733-2fe2-4c15-8a58-b602256f62ae · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 46

Resolution: unresolved; raw_fallback, observed 2026-08-08T19:23:30.628238Z

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Observation fc47afef-553e-4429-b0ad-25dbce2b1da0 · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 47

Resolution: unresolved; raw_fallback, observed 2026-08-08T19:23:30.502411Z

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Observation 5f7b0066-69e4-41ea-9c6f-55db9b17f287 · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 48

Resolution: unresolved; raw_fallback, observed 2026-08-08T19:23:30.293096Z

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Observation 6ea38d71-661d-4666-b08c-fcbaffb63e2e · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 49

Resolution: unresolved; raw_fallback, observed 2026-08-08T19:23:30.188469Z

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Observation 5f542930-fe31-41ad-97eb-265d1a8097bd · outbound

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Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 50

Resolution: unresolved; raw_fallback, observed 2026-08-08T19:23:30.172649Z

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Observation c8279d3d-fe9f-4c84-971a-d0a6ba6395e7 · outbound

This paper cites (S27) gives IL( 1 2 , T) =−IL( 1 2 , T) = 0.(S33) This holds at every temperature and finiteL, without a thermodynamic limit or Hypothesis (H).

Quantized topological invariant of symmetry-projected Gibbs states (S27) gives IL( 1 2 , T) =−IL( 1 2 , T) = 0.(S33) This holds at every temperature and finiteL, without a thermodynamic limit or Hypothesis (H)

Reference 51

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:30.156026Z

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Observation 6879eff5-8f6c-46db-bffc-1ab67444023a · outbound

This paper cites We first considers ′ > 1 2 at finiteL.

Quantized topological invariant of symmetry-projected Gibbs states We first considers ′ > 1 2 at finiteL

Reference 52

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:30.139348Z

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Observation 7eb7f375-2a9e-4213-aa0e-902bfe828883 · outbound

This paper cites Conjugation byW, which preservesP loc, supplies the same (−1)Σ·γ prefactor as in Eq.

Quantized topological invariant of symmetry-projected Gibbs states Conjugation byW, which preservesP loc, supplies the same (−1)Σ·γ prefactor as in Eq

Reference 53

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:30.120683Z

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Observation 91daa530-1cc0-4cd5-8f37-38e3a93a1d64 · outbound

This paper cites an unresolved cited work.

Quantized topological invariant of symmetry-projected Gibbs states Unresolved cited work

Reference 54

Resolution: unresolved; raw_fallback, observed 2026-08-08T19:23:29.986523Z

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Observation 34c655b2-3c1d-4508-9b5b-a078ffdf4ceb · outbound

This paper cites The two winding sectorsm= + (even) andm=−(odd) are simulated independently, the latter seeded with a single winding line.

Quantized topological invariant of symmetry-projected Gibbs states The two winding sectorsm= + (even) andm=−(odd) are simulated independently, the latter seeded with a single winding line

Reference 55

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:29.969619Z

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Observation e1595a13-0570-4e1e-8a4b-8066812fa3fd · outbound

This paper cites Fora̸= 0, define F A,m sp (a;L) :=−ln[Z (as=a) A,m /Z(as=0) A,m ], τA,m(a;L) :=L −2F A,m sp (a;L).

Quantized topological invariant of symmetry-projected Gibbs states Fora̸= 0, define F A,m sp (a;L) :=−ln[Z (as=a) A,m /Z(as=0) A,m ], τA,m(a;L) :=L −2F A,m sp (a;L)

Reference 56

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:29.952787Z

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Observation c2b14ae9-d1d5-40b1-8a07-0fbd4e734a65 · outbound

This paper cites S2 B–S4 A, where the same observable is com- puted independently by extended-ensemble Ising Monte Carlo.

Quantized topological invariant of symmetry-projected Gibbs states S2 B–S4 A, where the same observable is com- puted independently by extended-ensemble Ising Monte Carlo

Reference 57

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:29.925680Z

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Observation 150fa089-f676-42c9-93c0-5681a574c3cd · outbound

This paper cites S5, not replicas of the density matrix.

Quantized topological invariant of symmetry-projected Gibbs states S5, not replicas of the density matrix

Reference 58

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:29.893421Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation f50469d4-3cf6-4092-a0ee-fd5d36294299 · outbound

This paper cites For eachC∈ {A, B}, letx C,l(τ) =±1 be its directed-link worldline variable and letn C,p(τ)∈ {0,1} record a plaquette-flip event between adjacent slices.

Quantized topological invariant of symmetry-projected Gibbs states For eachC∈ {A, B}, letx C,l(τ) =±1 be its directed-link worldline variable and letn C,p(τ)∈ {0,1} record a plaquette-flip event between adjacent slices

Reference 59

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:29.869739Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation b5e2e2bc-a559-4469-b073-0ce25910780d · outbound

This paper cites ThusN τ and ∆τ eff are fixed before a chain is run.

Quantized topological invariant of symmetry-projected Gibbs states ThusN τ and ∆τ eff are fixed before a chain is run

Reference 60

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:29.707411Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 8d849e2c-8675-4c79-a40e-0beb8dd23e99 · outbound

This paper cites In round 0 the offsetsb j learned dur- ing adaptation are frozen, and the round passes only if every bin is visited.

Quantized topological invariant of symmetry-projected Gibbs states In round 0 the offsetsb j learned dur- ing adaptation are frozen, and the round passes only if every bin is visited

Reference 61

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:29.595674Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 93e0bb11-bde7-46ee-85ac-37aeb6dc262b · outbound

This paper cites At (s, T) = (0.30,0.30), the three refinement factors give I (f=1) 4 =−0.9999995712, I (f=2) 4 =−0.9999994446, I (f=4) 4 =−0.9999994161.

Quantized topological invariant of symmetry-projected Gibbs states At (s, T) = (0.30,0.30), the three refinement factors give I (f=1) 4 =−0.9999995712, I (f=2) 4 =−0.9999994446, I (f=4) 4 =−0.9999994161

Reference 62

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T19:23:29.452629Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

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Reference resolution

Outbound references

Pith citing papers