Pith. sign in

REVIEW 2 major objections 4 minor 82 references

A topologically trivial monitored s-wave superconductor can host super-logarithmic entanglement S(L) ~ ln² L from an SO(R) weak-anti-localization flow.

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 · grok-4.5

2026-07-10 17:02 UTC pith:O7ZR7ITP

load-bearing objection Solid one-loop derivation that super-log entanglement can arise in a topologically trivial s-wave chain via complementary mass constraints; the only real soft spot is the usual R o1 continuation of the SO(R) beta function. the 2 major comments →

arxiv 2607.07835 v1 pith:O7ZR7ITP submitted 2026-07-08 quant-ph cond-mat.stat-mechcond-mat.supr-con

Super-Logarithmic Entanglement Scaling in a Monitored Superconducting Chain

Rui-Jing Guo , Zhi-Yuan Wei This is my paper
classification quant-ph cond-mat.stat-mechcond-mat.supr-con
keywords measurement-induced phase transitionnonlinear sigma modelweak anti-localizations-wave BCS chainentanglement entropyKeldysh replicasymmetry class Csuper-logarithmic scaling
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 builds a Keldysh-replica field theory for a one-dimensional spinful s-wave BCS chain that is continuously monitored by rare spin-resolved density measurements. Although the clean Hamiltonian sits in symmetry class CI, the measurements and projection onto a conserved sector reduce the effective problem to class C. Measurement backaction and the pairing gap together mass out different fluctuation channels, leaving only soft modes that live on an SO(R) manifold in replica space. A one-loop renormalization-group analysis of that nonlinear sigma model yields a negative beta function when the replica number is continued to R → 1. The resulting weak-anti-localization flow produces a steady-state entanglement entropy that grows as the square of a logarithm of subsystem size. The result shows that super-logarithmic criticality does not require Majorana topology or a Wess–Zumino–Witten term; it can arise dynamically from the interplay of pairing and measurement in an ordinary superconductor.

Core claim

In the rare-measurement regime, complementary mass constraints from measurement vertices and the pairing amplitude dynamically project the parent symplectic soft modes onto an SO(R) nonlinear sigma model. Its one-loop beta function becomes negative in the replica limit R → 1, driving weak anti-localization and the steady-state entanglement scaling S_s(L) ∼ ln² L.

What carries the argument

The SO(R) nonlinear sigma model obtained by projecting the parent USp(4R)/[USp(2R)×USp(2R)] coset with measurement and pairing mass terms; its one-loop beta function β(g) = (R−2)g²/32π continues to β(g) = −g²/32π at R → 1 and supplies the running stiffness that integrates to ln² L entanglement.

Load-bearing premise

The algebraic continuation of the dual Coxeter number that appears in the one-loop beta function remains valid down to the formal replica limit R = 1, even though the geometric dimension of SO(R) vanishes there.

What would settle it

A controlled numerical extraction of the entanglement scaling for a long monitored s-wave chain in the rare-measurement window that fails to show a clear ln² L growth (or that shows only ordinary logarithmic growth with a vanishing quadratic coefficient) would falsify the claimed weak-anti-localization flow.

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

If this is right

  • Super-logarithmic entanglement criticality is not restricted to topological Majorana chains; ordinary s-wave pairing plus measurements can produce the same SO(R) flow.
  • The entire rare-measurement regime is a critical phase rather than a single critical point, with a crossover to area-law behavior only when the measurement rate becomes comparable to hopping or pairing.
  • Because the model has no Wess–Zumino–Witten term, any eventual localization transition at strong measurement is not protected by topological interference and can proceed by ordinary vortex proliferation.
  • The same complementary-mass projection mechanism may generate SO(R) soft modes in other monitored superconductors that lack static topological protection.

Where Pith is reading between the lines

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

  • If the replica continuation is accepted, similar super-logarithmic phases should appear in any monitored system whose residual soft modes land on an orthogonal group after pairing and measurement projections.
  • The vanishing of the first-order time derivative on the SO(R) manifold (forcing z = 1 rather than diffusion) is a geometric selection rule that may reappear in other projected Keldysh sigma models.
  • Finite-size numerics will typically sit in a crossover regime containing both ln L and ln² L pieces; clean asymptotic ln² L requires exponentially large systems set by the bare stiffness.

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 / 4 minor

Summary. The manuscript constructs a Keldysh-replica nonlinear sigma model for the entanglement dynamics of a one-dimensional monitored spinful s-wave BCS chain in the rare-measurement regime γ ≪ J, Δ. Although the clean Hamiltonian is in class CI, spin-resolved measurements and projection onto a conserved f-sector reduce the effective problem to class C. Complementary mass constraints from measurement backaction and the pairing amplitude project the parent USp(4R) soft modes onto an SO(R) target manifold. A one-loop RG analysis yields a negative beta function β(g) = −g²/32π in the replica limit R o 1, producing a weak-anti-localization flow and the super-logarithmic steady-state entanglement scaling S_s(L) ∼ ln² L. The result explains companion numerics and shows that a topologically trivial s-wave superconductor can realize this critical phase without a WZW term.

Significance. If correct, the work establishes that measurement-induced super-logarithmic criticality is not restricted to topological Majorana chains: the same SO(R) weak-anti-localizing phase can emerge dynamically from the interplay of s-wave pairing and measurement backaction on a class-C saddle. The derivation is complete at one-loop order (SCBA saddle, exhaustive mass-gap analysis of Keldysh–Nambu channels, gradient expansion of stiffnesses, vanishing of the first-order temporal term on the projected manifold, and the standard SO(R) beta function), with detailed appendices. It supplies a controlled field-theoretic explanation of the companion Letter’s numerics without fitting parameters into the flow, and cleanly separates the rare-measurement critical regime from the expected Zeno crossover at larger γ. These are genuine strengths of the manuscript.

major comments (2)
  1. Sec. V A (paragraph after Eq. 53) and Sec. VI B, Eqs. (63)–(64): The sign of the beta function and the ln² L scaling rest on analytic continuation of the dual Coxeter number (R−2) of the principal chiral model on SO(R) down to the replica limit R o 1. The geometric dimension of SO(R) vanishes at R = 1, so the manifold is only a formal target. The manuscript follows the standard Keldysh-replica procedure used for monitored free fermions and p-wave chains, and the algebraic coefficient is smooth in R, but the continuation is not re-derived from first principles for this projected class-C saddle. A short, explicit discussion of the domain of validity of this step (or a precise reference establishing it for the projected manifold) would make the central claim more robust.
  2. Sec. VI C and App. E: The geometric prefactor C_geo is defined via a derivative of the permutation-generator trace and left as a universal O(1) constant. While the asymptotic form S_s ∼ ln² L is insensitive to its precise value, the manuscript never evaluates or bounds it from the eigenvalues of the SO(N) monodromy generator. A one-line evaluation (or an explicit statement that only the scaling form is claimed) would remove ambiguity when comparing coefficients with the companion numerics.
minor comments (4)
  1. Throughout (e.g., abstract and Sec. I): the notation for the entanglement entropy alternates between S(L), S_s(L) and Ss; a single consistent symbol would improve readability.
  2. Sec. III A and Eq. (26): the origin of the prefactor 2/4^{2R} is explained, but a brief parenthetical reminder that the two Nambu sectors each contribute an identical SO(R) copy would help readers tracking the overall multiplicity that later enters C_geo.
  3. App. C 2: the all-order vanishing of the first-order temporal term is carefully shown; a one-sentence cross-reference in the main text (Sec. V B) to the fact that the cancellation holds only after projection onto SO(R) would prevent misreading of the parent-manifold dynamics.
  4. References: the companion Letter is cited as arXiv:2604.04375; once published, the journal reference should be updated for archival permanence.

Circularity Check

0 steps flagged

No significant circularity: the SO(R) projection, one-loop beta function, and ln^{2}L scaling are derived from the microscopic BCS+measurement action, not forced by fit or self-citation.

full rationale

The load-bearing chain is self-contained. The microscopic Hamiltonian (Eq. 1) and Poissonian projectors generate the Keldysh-replica action; SCBA and complementary mass constraints from measurement vertices M̂_β and pairing Δ project the parent USp(4R) coset onto SO(R) (Sec. V A, Eqs. 49–53). The one-loop beta function β(g)=(R−2)g²/32π is the standard dual-Coxeter coefficient of the principal chiral model on SO(R) (App. D, Eq. 63); the replica limit R→1 is the usual analytic continuation of that algebraic coefficient, not a fit to data. Entanglement scaling follows by inserting the running stiffness into the free energy of replica twist defects (Eqs. 69–71). The companion Letter [1] is cited only as numerical evidence that the theory explains; it is not an input to the RG flow or the manifold projection. No parameter is fitted and then re-predicted; no uniqueness theorem is imported from the authors to forbid alternatives; no ansatz is smuggled in via self-citation. The analytic continuation of (R−2) to R=1 is a standard (and potentially delicate) assumption of the replica method, but that is a correctness/validity risk, not circularity by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The derivation rests on the standard Keldysh-replica and NLSM toolkit plus a small set of domain assumptions (rare-measurement continuum limit, SCBA, infinite-temperature Keldysh saddle, sector decoupling under S_z). No free parameters are fitted to data; J, Δ, γ are microscopic inputs. No new particles or forces are postulated—the SO(R) manifold is an emergent soft-mode target.

axioms (6)
  • domain assumption Replica trick with sequential limits R → 1 then N → 1 recovers the ensemble-averaged von Neumann entropy from integer moments (Sec. II B).
    Standard in monitored-fermion literature but unproved in full generality; used throughout the entanglement calculation.
  • domain assumption In the rare-measurement limit γ ≪ J, Δ the discrete Poissonian projectors may be coarse-grained into a continuous spacetime disorder and treated by SCBA (Secs. III–IV, App. A).
    Controls the gradient expansion and the existence of a finite subgap density of states that sets the NLSM stiffness.
  • domain assumption Unconditioned ensemble average under continuous monitoring drives the system to an infinite-temperature (F = 0) Keldysh saddle with vanishing off-diagonal Keldysh component (Sec. IV B).
    Fixes the saddle Q_0 = τ_z ⊗ μ_0 ⊗ I_R and the causality structure used for all subsequent expansions.
  • standard math One-loop beta function of the principal chiral model on SO(R) is β(g) = (R − 2)g² / 32π, and the dual Coxeter number R − 2 may be continued to R = 1 (Sec. VI B, App. D).
    Standard result for SO(R) PCM; the analytic continuation is the load-bearing step that flips the sign of the flow.
  • domain assumption WZW level vanishes (k = 0) because the s-wave BdG vector has winding number W = 0 (Sec. V B).
    Argued from topology rather than computed microscopically from the Keldysh action; used to assert that the critical phase needs no WZW term.
  • domain assumption Spin-resolved measurements plus S_z conservation decouple the chain into two independent class-C f-sectors, each contributing an identical SO(R) NLSM (Sec. III A).
    Reduces the parent class-CI problem to class C and doubles the geometric prefactor C_geo.

pith-pipeline@v1.1.0-grok45 · 44676 in / 3240 out tokens · 39945 ms · 2026-07-10T17:02:40.349909+00:00 · methodology

0 comments
read the original abstract

We develop a Keldysh-replica non-linear sigma model (NLSM) for the entanglement dynamics of a monitored one-dimensional spinful $s$-wave BCS chain in the rare-measurement regime, $\gamma \ll J,\Delta$. Although the clean spinful $s$-wave BCS Hamiltonian belongs to symmetry class CI, spin-resolved measurements and projection to a conserved $f$-sector reduce the effective problem to class C. Starting from the corresponding parent symplectic saddle, we show that measurement backaction and the pairing amplitude impose complementary mass constraints that gap out different fluctuation channels. Their interplay dynamically projects the surviving massless modes onto an $\textrm{SO(R)}$ target manifold in replica space. A one-loop renormalization group analysis of this $\textrm{SO(R)}$ NLSM shows that, in the replica limit $R\to1$, the beta function becomes negative, producing a weak-anti-localization flow. This flow yields a super-logarithmic steady-state entanglement scaling $S(L)\sim \ln^2 L$ in the rare-measurement regime. Our field-theoretic result explains the numerical evidence reported in the companion Letter [arXiv:2604.04375] and shows that a topologically trivial monitored $s$-wave superconductor can realize an $\textrm{SO(R)}$ weak-anti-localizing critical phase without relying on a Wess-Zumino-Witten term.

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

82 extracted references · 82 canonical work pages · 8 internal anchors

  1. [1]

    Measurement-enhanced entanglement in a monitored superconducting chain

    R.-J. Guo, J.-Y. Chen, and Z.-Y. Wei, Measurement- enhanced entanglement in a monitored superconducting chain (2026), arXiv:2604.04375 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    Y. Li, X. Chen, and M. P. A. Fisher, Quantum zeno effect and the many-body entanglement transition, Phys. Rev. B98, 205136 (2018)

    work page 2018

  3. [3]

    M. A. Nielsen and I. L. Chuang,Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion(Cambridge University Press, 2010)

    work page 2010

  4. [4]

    C.-M. Jian, B. Bauer, A. Keselman, and A. W. W. Lud- wig, Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions, Phys. Rev. B106, 134206 (2022)

    work page 2022

  5. [5]

    C.-M. Jian, H. Shapourian, B. Bauer, and A. W. W. Lud- wig, Measurement-induced entanglement transitions in quantum circuits of non-interacting fermions: Born-rule versus forced measurements (2023), arXiv:2302.09094 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  6. [6]

    S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum er- ror correction in scrambling dynamics and measurement- induced phase transition, Phys. Rev. Lett.125, 030505 (2020)

    work page 2020

  7. [7]

    Agrawal, A

    U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur, Entanglement and charge-sharpening transitions in u(1) symmetric monitored quantum circuits, Phys. Rev. X12, 041002 (2022)

    work page 2022

  8. [8]

    Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B101, 104301 (2020)

    work page 2020

  9. [9]

    Jian, Y.-Z

    C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Measurement-induced criticality in random quantum cir- cuits, Phys. Rev. B101, 104302 (2020)

    work page 2020

  10. [10]

    Y. Li, X. Chen, and M. P. A. Fisher, Measurement- driven entanglement transition in hybrid quantum cir- cuits, Phys. Rev. B100, 134306 (2019)

    work page 2019

  11. [11]

    M. J. Gullans and D. A. Huse, Dynamical purifica- tion phase transition induced by quantum measurements, Phys. Rev. X10, 041020 (2020)

    work page 2020

  12. [12]

    M. J. Gullans and D. A. Huse, Scalable probes of measurement-induced criticality, Phys. Rev. Lett.125, 070606 (2020)

    work page 2020

  13. [13]

    Iaconis, A

    J. Iaconis, A. Lucas, and X. Chen, Measurement-induced phase transitions in quantum automaton circuits, Phys. Rev. B102, 224311 (2020)

    work page 2020

  14. [14]

    Lavasani, Y

    A. Lavasani, Y. Alavirad, and M. Barkeshli, Measurement-induced topological entanglement transi- tions in symmetric random quantum circuits, Nature Physics17, 342 (2021)

    work page 2021

  15. [15]

    Sang and T

    S. Sang and T. H. Hsieh, Measurement-protected quan- tum phases, Phys. Rev. Res.3, 023200 (2021)

    work page 2021

  16. [16]

    Nahum, S

    A. Nahum, S. Roy, B. Skinner, and J. Ruhman, Mea- surement and entanglement phase transitions in all-to- all quantum circuits, on quantum trees, and in landau- ginsburg theory, PRX Quantum2, 010352 (2021)

    work page 2021

  17. [17]

    Block, Y

    M. Block, Y. Bao, S. Choi, E. Altman, and N. Y. Yao, Measurement-induced transition in long-range in- teracting quantum circuits, Phys. Rev. Lett.128, 010604 (2022)

    work page 2022

  18. [18]

    Z.-Y. Wei, J. Nelson, J. Rajakumar, E. Cruz, A. V. Gor- shkov, M. J. Gullans, and D. Malz, Measurement-induced entanglement in noisy 2d random clifford circuits (2026), arXiv:2510.12743 [quant-ph]

    work page arXiv 2026

  19. [19]

    M. Fava, L. Piroli, D. Bernard, and A. Nahum, Monitored 16 fermions with conservedu(1) charge, Phys. Rev. Res.6, 043246 (2024)

    work page 2024

  20. [20]

    Coppola, E

    M. Coppola, E. Tirrito, D. Karevski, and M. Collura, Growth of entanglement entropy under local projective measurements, Phys. Rev. B105, 094303 (2022)

    work page 2022

  21. [21]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- induced phase transitions in the dynamics of entangle- ment, Phys. Rev. X9, 031009 (2019)

    work page 2019

  22. [22]

    Szyniszewski, A

    M. Szyniszewski, A. Romito, and H. Schomerus, Entan- glement transition from variable-strength weak measure- ments, Phys. Rev. B100, 064204 (2019)

    work page 2019

  23. [23]

    Travaglino, C

    R. Travaglino, C. Rylands, and P. Calabrese, Quench dy- namics of entanglement entropy under projective charge measurements: the free fermion case, Journal of Statis- tical Mechanics: Theory and Experiment2025, 123101 (2025)

    work page 2025

  24. [24]

    Swann, D

    T. Swann, D. Bernard, and A. Nahum, Spacetime pic- ture for entanglement generation in noisy fermion chains (2023), arXiv:2302.12212 [cond-mat.stat-mech]

    work page arXiv 2023

  25. [25]

    A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99, 224307 (2019)

    work page 2019

  26. [26]

    X. Cao, A. Tilloy, and A. D. Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Phys.7, 024 (2019)

    work page 2019

  27. [27]

    Ladewig, S

    B. Ladewig, S. Diehl, and M. Buchhold, Monitored open fermion dynamics: Exploring the interplay of measure- ment, decoherence, and free hamiltonian evolution, Phys. Rev. Res.4, 033001 (2022)

    work page 2022

  28. [28]

    Carollo and V

    F. Carollo and V. Alba, Entangled multiplets and spread- ing of quantum correlations in a continuously monitored tight-binding chain, Phys. Rev. B106, L220304 (2022)

    work page 2022

  29. [29]

    L´ oio, A

    H. L´ oio, A. De Luca, J. De Nardis, and X. Turkeshi, Purification timescales in monitored fermions, Phys. Rev. B108, L020306 (2023)

    work page 2023

  30. [30]

    Turkeshi, L

    X. Turkeshi, L. Piroli, and M. Schir´ o, Enhanced entangle- ment negativity in boundary-driven monitored fermionic chains, Phys. Rev. B106, 024304 (2022)

    work page 2022

  31. [31]

    Turkeshi, M

    X. Turkeshi, M. Dalmonte, R. Fazio, and M. Schir´ o, En- tanglement transitions from stochastic resetting of non- hermitian quasiparticles, Phys. Rev. B105, L241114 (2022)

    work page 2022

  32. [32]

    Van Regemortel, Z.-P

    M. Van Regemortel, Z.-P. Cian, A. Seif, H. Dehghani, and M. Hafezi, Entanglement entropy scaling transition under competing monitoring protocols, Phys. Rev. Lett. 126, 123604 (2021)

    work page 2021

  33. [33]

    Kells, D

    G. Kells, D. Meidan, and A. Romito, Topological tran- sitions in weakly monitored free fermions, SciPost Phys. 14, 031 (2023)

    work page 2023

  34. [34]

    Poboiko, I

    I. Poboiko, I. V. Gornyi, and A. D. Mirlin, Measurement- induced phase transition for free fermions above one di- mension, Phys. Rev. Lett.132, 110403 (2024)

    work page 2024

  35. [35]

    Poboiko, P

    I. Poboiko, P. P¨ opperl, I. V. Gornyi, and A. D. Mirlin, Theory of free fermions under random projective mea- surements, Phys. Rev. X13, 041046 (2023)

    work page 2023

  36. [36]

    M. Fava, L. Piroli, T. Swann, D. Bernard, and A. Nahum, Nonlinear sigma models for monitored dynamics of free fermions, Phys. Rev. X13, 041045 (2023)

    work page 2023

  37. [37]

    G.-Y. Zhu, N. Tantivasadakarn, and S. Trebst, Struc- tured volume-law entanglement in an interacting, moni- tored majorana spin liquid, Phys. Rev. Res.6, L042063 (2024)

    work page 2024

  38. [38]

    Oshima, K

    H. Oshima, K. Mochizuki, R. Hamazaki, and Y. Fuji, Topology and spectrum in measurement-induced phase transitions, Phys. Rev. Lett.134, 240401 (2025)

    work page 2025

  39. [39]

    Nahum and B

    A. Nahum and B. Skinner, Entanglement and dynamics of diffusion-annihilation processes with majorana defects, Phys. Rev. Res.2, 023288 (2020)

    work page 2020

  40. [40]

    C. Yin, F. Bo, and A. M. Garc´ia-Garc´ia, No mea- surement induced phase transition in the entanglement dynamics of monitored non-interacting one-dimensional fermions in a disordered or quasiperiodic potential (2026), arXiv:2605.10758 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  41. [41]

    Niederegger, T

    C. Niederegger, T. Vovk, E. Starchl, and L. M. Sieberer, Absence of measurement- and unraveling-induced en- tanglement transitions in continuously monitored one- dimensional free fermions, Phys. Rev. B113, 144317 (2026)

    work page 2026

  42. [42]

    Poboiko and A

    I. Poboiko and A. D. Mirlin, Quantum dynamics of mon- itored free fermions: Evolution of quantum correlations and scaling at measurement-induced phase transitions, Phys. Rev. B113, 144311 (2026)

    work page 2026

  43. [43]

    Poboiko, P

    I. Poboiko, P. P¨ opperl, I. V. Gornyi, and A. D. Mirlin, Measurement-induced transitions for interacting fermions, Phys. Rev. B111, 024204 (2025)

    work page 2025

  44. [44]

    Goto and I

    S. Goto and I. Danshita, Measurement-induced transi- tions of the entanglement scaling law in ultracold gases with controllable dissipation, Phys. Rev. A102, 033316 (2020)

    work page 2020

  45. [45]

    Fuji and Y

    Y. Fuji and Y. Ashida, Measurement-induced quantum criticality under continuous monitoring, Phys. Rev. B 102, 054302 (2020)

    work page 2020

  46. [46]

    E. V. H. Doggen, Y. Gefen, I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Generalized quantum measurements with matrix product states: Entanglement phase transi- tion and clusterization, Phys. Rev. Res.4, 023146 (2022)

    work page 2022

  47. [47]

    E. V. H. Doggen, Y. Gefen, I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Evolution of many-body systems under ancilla quantum measurements, Phys. Rev. B107, 214203 (2023)

    work page 2023

  48. [48]

    M. S. Foster, H. Guo, C.-M. Jian, and A. W. W. Ludwig, Free-fermion measurement-induced volume- to area-law entanglement transition in the presence of fermion inter- actions (2025), arXiv:2510.23706 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  49. [49]

    Tang and W

    Q. Tang and W. Zhu, Measurement-induced phase transi- tion: A case study in the nonintegrable model by density- matrix renormalization group calculations, Phys. Rev. Res.2, 013022 (2020)

    work page 2020

  50. [50]

    Pokharel, H

    B. Pokharel, H. Pan, K. Aziz, L. C. G. Govia, S. Gane- shan, T. Iadecola, J. H. Wilson, B. A. Jones, A. Desh- pande, J. H. Pixley, and M. Takita, Order from chaos with adaptive circuits on quantum hardware (2025), arXiv:2509.18259 [quant-ph]

    work page arXiv 2025

  51. [51]

    Z. Yang, A. W. W. Ludwig, and C.-M. Jian, Decoding co- herent errors in toric codes on honeycomb and square lat- tices: duality to majorana monitored dynamics and sym- metry classes (2026), arXiv:2604.08650 [cond-mat.stat- mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  52. [52]

    Szyniszewski, O

    M. Szyniszewski, O. Lunt, and A. Pal, Disordered moni- tored free fermions, Phys. Rev. B108, 165126 (2023)

    work page 2023

  53. [53]

    S.-K. Jian, C. Liu, X. Chen, B. Swingle, and P. Zhang, Measurement-induced phase transition in the monitored sachdev-ye-kitaev model, Phys. Rev. Lett.127, 140601 (2021)

    work page 2021

  54. [54]

    Altland, M

    A. Altland, M. Buchhold, S. Diehl, and T. Micklitz, Dy- namics of measured many-body quantum chaotic sys- tems, Phys. Rev. Res.4, L022066 (2022)

    work page 2022

  55. [55]

    C. Noel, P. Niroula, D. Zhu, A. Risinger, L. Egan, 17 D. Biswas, M. Cetina, A. V. Gorshkov, M. J. Gullans, D. A. Huse, and C. Monroe, Measurement-induced quan- tum phases realized in a trapped-ion quantum computer, Nature Physics18, 760 (2022)

    work page 2022

  56. [56]

    J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout, Nature Physics19, 1314–1319 (2023)

    work page 2023

  57. [57]

    Z. Wu, X. Sun, S. Wang, J. Zhang, X. Yang, J. Chu, J. Niu, Y. Zhong, X. Chen, Z.-C. Yang, and D. Yu, Measurement-and feedback-driven non- equilibrium phase transitions on a quantum processor (2025), arXiv:2512.07966 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  58. [58]

    J. C. Hoke, M. Ippoliti, E. Rosenberg, D. Abanin, R. Acharya, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, J. C. Bardin, A. Bengts- son, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, Z. Chen, B. Chiaro, D. Chik, J. Cogan, R. Collins, P. Conner, W. Courtn...

    work page 2023

  59. [59]

    Mart´in-V´ azquez, T

    G. Mart´in-V´ azquez, T. Tolppanen, and M. Silveri, Phase transitions induced by standard and feedback mea- surements in transmon arrays, Phys. Rev. B109, 214308 (2024)

    work page 2024

  60. [60]

    Altland and M

    A. Altland and M. R. Zirnbauer, Nonstandard symme- try classes in mesoscopic normal-superconducting hybrid structures, Physical Review B55, 1142–1161 (1997)

    work page 1997

  61. [61]

    S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Lud- wig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New Journal of Physics 12, 065010 (2010)

    work page 2010

  62. [62]

    Replica Keldysh field theory of quantum-jump processes: General formalism and application to imbalanced and inefficient fermion counting

    F. Kloiber-Tollinger and L. M. Sieberer, Replica keldysh field theory of quantum-jump processes: General formal- ism and application to imbalanced and inefficient fermion counting (2025), arXiv:2512.16520 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  63. [63]

    Bertini, M

    B. Bertini, M. Fagotti, L. Piroli, and P. Calabrese, Entan- glement evolution and generalised hydrodynamics: non- interacting systems, Journal of Physics A: Mathematical and Theoretical51, 39LT01 (2018)

    work page 2018

  64. [64]

    S. F. Edwards and P. W. Anderson, Theory of spin glasses, Journal of Physics F: Metal Physics5, 965 (1975)

    work page 1975

  65. [65]

    Exploring critical systems under measurements and decoherence via Keldysh field theory

    R. Ma, Exploring critical systems under measure- ments and decoherence via keldysh field theory (2023), arXiv:2304.08277 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  66. [66]

    Wegner, The mobility edge problem: Continuous sym- metry and a conjecture, Zeitschrift f¨ ur Physik B Con- densed Matter35, 207 (1979)

    F. Wegner, The mobility edge problem: Continuous sym- metry and a conjecture, Zeitschrift f¨ ur Physik B Con- densed Matter35, 207 (1979)

    work page 1979

  67. [67]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of Statistical Mechanics: Theory and Experiment2004, P06002 (2004)

    work page 2004

  68. [68]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, Journal of Physics A: Mathemat- ical and Theoretical42, 504005 (2009)

    work page 2009

  69. [69]

    Chen, Intrinsic Axion Statistical Topological In- sulator, Physical Review Letters134, 10.1103/Phys- RevLett.134.226601 (2025)

    F. Barratt, U. Agrawal, S. Gopalakrishnan, D. A. Huse, R. Vasseur, and A. C. Potter, Field theory of charge sharpening in symmetric monitored quantum circuits, Physical Review Letters129, 10.1103/phys- revlett.129.120604 (2022)

    work page doi:10.1103/phys- 2022

  70. [70]

    P¨ opperl, I

    P. P¨ opperl, I. V. Gornyi, and Y. Gefen, Measurements on an anderson chain, Phys. Rev. B107, 174203 (2023)

    work page 2023

  71. [71]

    T. Jin, J. a. S. Ferreira, M. Filippone, and T. Giamarchi, Exact description of quantum stochastic models as quan- tum resistors, Phys. Rev. Res.4, 013109 (2022)

    work page 2022

  72. [72]

    Peng and S.-X

    J.-P. Peng and S.-X. Zhou, Theory of quantum magneto- transport in two-dimensional electron systems, Commu- nications in Theoretical Physics17, 275 (1992)

    work page 1992

  73. [73]

    Altland and B

    A. Altland and B. D. Simons,Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, 2010)

    work page 2010

  74. [74]

    Witten, Non-abelian bosonization in two dimen- sions, Communications in Mathematical Physics92, 455 (1984)

    E. Witten, Non-abelian bosonization in two dimen- sions, Communications in Mathematical Physics92, 455 (1984)

    work page 1984

  75. [75]

    K. G. Wilson and J. Kogut, The renormalization group and theϵexpansion, Physics Reports12, 75 (1974). Appendix A: Saddle Point Equation and SCBA The saddle-point approximation of the total action S[ ˆG, ˆΣ] =S 0[ ˆG, ˆΣ]+γ R d2xL M[ ˆG] determines the mean- field configuration of the system. Variation with respect to the conjugate self-energy field ˆΣ giv...

    work page 1974

  76. [76]

    Here Tr denotes the functional trace over both spatial coordinates and dis- crete indices

    Evaluating this derivative relies on the identity δdet(M) = det(M) Tr(M −1δM). Here Tr denotes the functional trace over both spatial coordinates and dis- crete indices. Under variation, the non-commuting ma- trices inside the trace must respect the cyclic property, so thatδln det(M) = Tr(M −1δM) with the operator or- der preserved. Applying this identity...

    work page

  77. [77]

    We restrict the search for station- arity points to the subspace exhibiting a diagonal matrix form, ˆG0 ∝τ K z

    This constraint eliminates the remaining off-diagonal Keldysh component. We restrict the search for station- arity points to the subspace exhibiting a diagonal matrix form, ˆG0 ∝τ K z . Within this configuration, the first-order tadpole contribution vanishes. This is enforced by the causality structure of the theory: the saddle-point ma- trix (∝τ K z ) an...

    work page

  78. [78]

    The exponential parameterization ˆQ=e ˆW /2 ˆQ0e− ˆW /2 encapsulates the deviations from the saddle point within ˆW

    Geometry of Fluctuations on the Parent Manifold We derive the explicit matrix structure of the fluctua- tion generator ˆWby enforcing the geometric constraints associated with the parent manifold in this subsection. The exponential parameterization ˆQ=e ˆW /2 ˆQ0e− ˆW /2 encapsulates the deviations from the saddle point within ˆW. Ensuring that ˆQremains ...

    work page

  79. [79]

    We start from the exact determinant representation of the measurement Lagrangian density derived in the main text Eq

    Perturbative Expansion of the Measurement Term We derive the measurement action expanded to the second order in the matrix field ˆQ. We start from the exact determinant representation of the measurement Lagrangian density derived in the main text Eq. (38). We evaluate the fluctuations around the saddle point ˆQ0. Instead of a Taylor expansion of the local...

    work page

  80. [80]

    Spatial Stiffness and Frequency Renormalization To extract the stiffness coefficients of the field theory, we perform a gradient expansion of the bare actionS 0[ ˆQ] derived in Eq. (45). This requires evaluating the trace- logarithm functional in the presence of the saddle-point self-energy. The slow fluctuations of the matrix field are parameterized thro...

    work page

Showing first 80 references.