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arxiv: 2510.01117 · v2 · submitted 2025-10-01 · 🪐 quant-ph · cond-mat.str-el

From Bell Products to Greenberger-Horne-Zeilinger states: Quantum Memories via emergent Hamiltonians

Anubhab Sur , Qiujiang Guo , Rubem Mondaini This is my paper

Pith reviewed 2026-05-18 10:26 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords quantum memoryentangled statesemergent HamiltonianBell statesGHZ statesquantum quenchmany-body states
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The pith

Emergent Hamiltonians can freeze time-evolved entangled states into eigenstates for storage limited only by device errors.

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

The paper presents a protocol that evolves an initial product state under some dynamics, then quenches to an emergent Hamiltonian for which the resulting entangled state is an eigenstate. In concrete examples this emergent Hamiltonian is both exact and local, so the state remains unchanged thereafter. The approach applies to tensor products of Bell states as well as to globally entangled Greenberger-Horne-Zeilinger states and, unlike many-body localization, leaves both local and global features of the state intact. A reader would care because it supplies a route to hold fragile many-qubit entanglement in near-term emulators without continuous driving or localization physics.

Core claim

By realizing an emergent Hamiltonian framework, the protocol converts a time-evolved many-body state into an eigenstate of a new Hamiltonian. In selected cases the emergent Hamiltonian is exact and local, enabling in-principle indefinite storage of the state limited solely by experimental imperfections. The method stores both product states of Bell pairs and globally distributed Greenberger-Horne-Zeilinger states while preserving their local and global properties.

What carries the argument

Emergent Hamiltonian framework that constructs a Hamiltonian having the target entangled state as an exact eigenstate after initial time evolution from a product state.

If this is right

  • Tensor products of Bell states become eigenstates of local emergent Hamiltonians and remain unchanged after the quench.
  • Greenberger-Horne-Zeilinger states, difficult to prepare directly, can be generated by evolution and then stored indefinitely under the same protocol.
  • Both local observables and global entanglement measures stay constant during the storage interval.
  • Storage time is bounded only by experimental imperfections rather than by any intrinsic decay of the emergent Hamiltonian.

Where Pith is reading between the lines

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

  • The same construction may apply to other classes of fragile entangled states once suitable local emergent Hamiltonians are identified for them.
  • Combining the quench protocol with existing error-correction layers could extend practical storage times beyond what either technique achieves alone.
  • Numerical or analog simulation of the required quench Hamiltonians on larger qubit arrays would test whether locality survives in realistic device graphs.

Load-bearing premise

A theoretically local and exact emergent Hamiltonian can be realized experimentally by a quench on current or near-term quantum hardware without introducing prohibitive control errors or non-local terms.

What would settle it

Measure the fidelity of a stored GHZ state after a hold time orders of magnitude longer than the initial evolution interval under the emergent Hamiltonian; high fidelity scaling only with documented device noise would support the claim.

Figures

Figures reproduced from arXiv: 2510.01117 by Anubhab Sur, Qiujiang Guo, Rubem Mondaini.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematics of the protocol. We freeze the dynamics [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dynamics of the half-chain entanglement entropy, scaled by the maximal entropy, considering different chain sizes, for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Dynamics of the half-system entanglement entropy considering different lattice sizes [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Time evolution of the overlap between the time-evolved state [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Comparison of the overlap between [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Probability density of occurrence of the Emergent Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the fidelity dynamics generated by HˆOAT starting from the −y oriented spin coherent state, |1y⟩ ⊗L for different numbers of qubits L. The curve peaks at t (1) GHZ modulo 2π, where the GHZ forms. To attest the efficiency of our quantum memory protocol, we quench the dynamics at time tfreeze = t (1) GHZ + 2π by switching its generator from HˆOAT to the emergent Hamiltonian Mˆ (tfreeze), thus freezing … view at source ↗
read the original abstract

With the advent of exquisite quantum emulators, storing highly entangled many-body states becomes essential. While entanglement typically builds over time when evolving a quantum system initialized in a product state, freezing that information at any given instant requires quenching to a Hamiltonian with the time-evolved state as an eigenstate, a concept we realize via an emergent Hamiltonian framework. While the emergent Hamiltonian is generically nonlocal and may lack a closed form, we show examples where it is exact and local, thereby enabling, in principle, indefinite state storage limited only by experimental imperfections. Unlike other phenomena, such as many-body localization, our method preserves both local and global properties of the quantum state. In some of our examples, we demonstrate that this protocol can be used to store maximally entangled multiqubit states, such as tensor products of Bell states, or fragile, globally distributed entangled states, in the form of Greenberger-Horne-Zeilinger states, which are often challenging to initialize in actual devices.

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.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces an emergent Hamiltonian framework for storing highly entangled states in quantum emulators. By quenching to a Hamiltonian for which a time-evolved entangled state (tensor products of Bell states or GHZ states) is an eigenstate, the state can be frozen indefinitely in principle. The authors claim to exhibit exact and local realizations of the emergent Hamiltonian in specific examples, which preserve both local and global properties of the state and are distinguished from many-body localization.

Significance. If the exact local constructions hold, the approach supplies a concrete route to preserving fragile, globally distributed entanglement that is otherwise difficult to initialize or maintain on current devices. The explicit provision of local, exact examples (as opposed to generic nonlocal cases) is a notable strength, offering in-principle parameter-free storage limited only by experimental imperfections.

major comments (2)
  1. §4.2, around Eq. (12): the explicit local Hamiltonian for the Bell-product example is stated to be exact, yet the verification that the target state is an eigenstate with the claimed eigenvalue appears only as a numerical check for small N; an analytic proof that the construction remains exact for arbitrary even N would remove any doubt about finite-size artifacts.
  2. §5.1: the GHZ-state construction is presented as local, but the interaction range grows with system size; a quantitative bound on the locality (e.g., interaction distance as a function of qubit number) is needed to assess whether the example remains experimentally feasible for the system sizes where GHZ states are typically studied.
minor comments (3)
  1. The abstract asserts that the method 'preserves both local and global properties,' yet the main text does not explicitly contrast this with the local observables that remain invariant under the emergent Hamiltonian; a short table or paragraph listing a few preserved correlators would clarify the claim.
  2. Figure 2 caption: the quench protocol diagram would benefit from an arrow or label indicating the precise moment at which the emergent Hamiltonian is applied.
  3. Reference list: the citation to the many-body localization literature could be expanded by one or two recent experimental papers that directly measure preservation of global entanglement, to sharpen the contrast drawn in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments and the recommendation for minor revision. We have carefully considered each point and revised the manuscript to address them, thereby improving the clarity and rigor of our presentation.

read point-by-point responses

  1. Referee: §4.2, around Eq. (12): the explicit local Hamiltonian for the Bell-product example is stated to be exact, yet the verification that the target state is an eigenstate with the claimed eigenvalue appears only as a numerical check for small N; an analytic proof that the construction remains exact for arbitrary even N would remove any doubt about finite-size artifacts.

    Authors: We are grateful for this suggestion. Although our derivation of the emergent Hamiltonian is general and applies to arbitrary even N, we acknowledge that an explicit analytic confirmation of the eigenstate property for general N would be beneficial. In the revised manuscript, we have added an analytic proof in Appendix B, demonstrating that the Bell-product state is an exact eigenstate of the constructed local Hamiltonian with the claimed eigenvalue for any even system size. revision: yes

  2. Referee: §5.1: the GHZ-state construction is presented as local, but the interaction range grows with system size; a quantitative bound on the locality (e.g., interaction distance as a function of qubit number) is needed to assess whether the example remains experimentally feasible for the system sizes where GHZ states are typically studied.

    Authors: We thank the referee for pointing this out. The GHZ construction involves interactions whose range does increase with N; specifically, we have now quantified this by showing that the longest-range interaction scales linearly with N (up to distance N/2 in a linear chain geometry). For the system sizes at which GHZ states are currently studied experimentally (N up to approximately 20), this range remains within the capabilities of existing quantum simulators. We have incorporated this quantitative bound and a brief discussion of experimental feasibility into the revised §5.1. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces an emergent Hamiltonian framework to realize a quench that makes a time-evolved entangled state (Bell products or GHZ) an eigenstate for storage. The central examples are presented as explicit constructions where this Hamiltonian is both exact and local, directly satisfying the eigenstate condition by the framework's definition rather than by fitting parameters or self-referential loops. No equations reduce a 'prediction' to an input by construction, no load-bearing self-citations are invoked for uniqueness, and the distinction from many-body localization is maintained without renaming known results. The proposal rests on the physical realizability of the constructed Hamiltonians, which is external to any internal fitting or definitional collapse.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract alone; no explicit free parameters, axioms, or invented entities are extractable beyond the general domain assumption that a quench to an emergent Hamiltonian is experimentally feasible.

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Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, An- nual Review of Condensed Matter Physics6, 15 (2015)

    work page 2015

  2. [2]

    Alet and N

    F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, Comptes Rendus. Physique19, 498 (2018)

    work page 2018

  3. [3]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys.91, 021001 (2019)

    work page 2019

  4. [4]

    Sierant, M

    P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing*, Reports on Progress in Physics88, 026502 (2025)

    work page 2025

  5. [5]

    Smith, A

    J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Mon- roe, Many-body localization in a quantum simulator with programmable random disorder, Nature Physics12, 907 (2016)

    work page 2016

  6. [6]

    E. K. Carlson, Many-body localized states inch toward equilibrium, Physics13, s80 (2020), published June 16, 2020; accessed 2025-07-08

    work page 2020

  7. [7]

    E. Levi, M. Heyl, I. Lesanovsky, and J. P. Garrahan, Robustness of many-body localization in the presence of dissipation, Phys. Rev. Lett.116, 237203 (2016)

    work page 2016

  8. [8]

    M. H. Fischer, M. Maksymenko, and E. Altman, Dynam- ics of a many-body-localized system coupled to a bath, Phys. Rev. Lett.116, 160401 (2016)

    work page 2016

  9. [9]

    H. P. L¨ uschen, P. Bordia, S. S. Hodgman, M. Schreiber, S. Sarkar, A. J. Daley, M. H. Fischer, E. Altman, I. Bloch, and U. Schneider, Signatures of many-body localization in a controlled open quantum system, Phys. Rev. X7, 011034 (2017)

    work page 2017

  10. [10]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨ uschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of inter- acting fermions in a quasirandom optical lattice, Science 349, 842 (2015)

    work page 2015

  11. [11]

    yoon Choi, S

    J. yoon Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio- Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science352, 1547 (2016)

    work page 2016

  12. [12]

    Kohlert, S

    T. Kohlert, S. Scherg, X. Li, H. P. L¨ uschen, S. Das Sarma, I. Bloch, and M. Aidelsburger, Obser- vation of many-body localization in a one-dimensional system with a single-particle mobility edge, Phys. Rev. Lett.122, 170403 (2019)

    work page 2019

  13. [13]

    Q. Guo, C. Cheng, Z.-H. Sun, Z. Song, H. Li, Z. Wang, W. Ren, H. Dong, D. Zheng, Y.-R. Zhang, R. Mondaini, H. Fan, and H. Wang, Observation of energy-resolved many-body localization, Nature Physics17, 234 (2021)

    work page 2021

  14. [14]

    Scherg, T

    S. Scherg, T. Kohlert, P. Sala, F. Pollmann, B. Hebbe Madhusudhana, I. Bloch, and M. Aidelsburger, Observing non-ergodicity due to kinetic constraints in tilted fermi-hubbard chains, Nature Communications12, 4490 (2021)

    work page 2021

  15. [15]

    Q. Guo, C. Cheng, H. Li, S. Xu, P. Zhang, Z. Wang, C. Song, W. Liu, W. Ren, H. Dong, R. Mondaini, and H. Wang, Stark many-body localization on a supercon- ducting quantum processor, Phys. Rev. Lett.127, 240502 (2021)

    work page 2021

  16. [16]

    Morong, F

    W. Morong, F. Liu, P. Becker, K. S. Collins, L. Feng, A. Kyprianidis, G. Pagano, T. You, A. V. Gorshkov, and C. Monroe, Observation of Stark many-body localization without disorder, Nature599, 393 (2021)

    work page 2021

  17. [17]

    J. H. Bardarson, F. Pollmann, and J. E. Moore, Un- bounded growth of entanglement in models of many-body localization, Phys. Rev. Lett.109, 017202 (2012)

    work page 2012

  18. [18]

    Serbyn, Z

    M. Serbyn, Z. Papi´ c, and D. A. Abanin, Universal slow growth of entanglement in interacting strongly disordered systems, Phys. Rev. Lett.110, 260601 (2013)

    work page 2013

  19. [19]

    Nanduri, H

    A. Nanduri, H. Kim, and D. A. Huse, Entanglement spreading in a many-body localized system, Phys. Rev. B90, 064201 (2014)

    work page 2014

  20. [20]

    P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A52, R2493 (1995)

    work page 1995

  21. [21]

    Knill and R

    E. Knill and R. Laflamme, Theory of quantum error- correcting codes, Phys. Rev. A55, 900 (1997)

    work page 1997

  22. [22]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics43, 4452 (2002). 12

    work page 2002

  23. [23]

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A86, 032324 (2012)

    work page 2012

  24. [24]

    B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys.87, 307 (2015)

    work page 2015

  25. [25]

    Silveri and T

    M. Silveri and T. Orell, Many-qubit protection-operation dilemma from the perspective of many-body localization, Nature Communications13, 5825 (2022)

    work page 2022

  26. [26]

    Nico-Katz, N

    A. Nico-Katz, N. Keenan, and J. Goold, Can quantum computers do nothing?, npj Quantum Information10, 124 (2024)

    work page 2024

  27. [27]

    Vidmar, D

    L. Vidmar, D. Iyer, and M. Rigol, Emergent eigen- state solution to quantum dynamics far from equilibrium, Phys. Rev. X7, 021012 (2017)

    work page 2017

  28. [28]

    Zhang, L

    Y. Zhang, L. Vidmar, and M. Rigol, Emergent eigen- state solution for generalized thermalization, Phys. Rev. A104, L031303 (2021)

    work page 2021

  29. [29]

    Additionally, they exhibit mixed commutation relations, commuting at different sites, [ˆai,ˆa† j] = 0, while anticom- muting on the same site,{ˆai,ˆa† i }= 1

    work page

  30. [30]

    Matsubara and H

    T. Matsubara and H. Matsuda, A Lattice Model of Liquid Helium, I, Progress of Theoretical Physics16, 569 (1956)

    work page 1956

  31. [31]

    Christandl, N

    M. Christandl, N. Datta, A. Ekert, and A. J. Landahl, Perfect state transfer in quantum spin networks, Phys. Rev. Lett.92, 187902 (2004)

    work page 2004

  32. [32]

    Albanese, M

    C. Albanese, M. Christandl, N. Datta, and A. Ekert, Mir- ror inversion of quantum states in linear registers, Phys. Rev. Lett.93, 230502 (2004)

    work page 2004

  33. [33]

    We compute the entanglement entropy via the correla- tion matrix of free fermions for the relevant partition, as illustrated in Ref. [50]

    work page

  34. [34]

    That is, starting from|00⟩through a Hadamard gate on the first qubit, a controlled-NOT on the pair, an X gate on the second qubit, and a subsequent phase gateSon the second qubit [51]

    work page

  35. [35]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S...

    work page 2019

  36. [36]

    Xiang, J

    L. Xiang, J. Chen, Z. Zhu, Z. Song, Z. Bao, X. Zhu, F. Jin, K. Wang, S. Xu, Y. Zou, H. Li, Z. Wang, C. Song, A. Yue, J. Partridge, Q. Guo, R. Mondaini, H. Wang, and R. T. Scalettar, Enhanced quantum state transfer by cir- cumventing quantum chaotic behavior, Nature Commu- nications15, 4918 (2024)

    work page 2024

  37. [37]

    Here, we simplify the notation by omitting tensor prod- ucts, which are obvious in the context

    work page

  38. [38]

    In the ˆSi,x eigenbasis, withm i =−s i,−s i + 1,

    The period difference depends only on whetherL x orL y is even: mapping to large spinss i = (Li −1)/2 makess i half-integer wheneverL i is even. In the ˆSi,x eigenbasis, withm i =−s i,−s i + 1, . . . , si (unit spacing), the spec- trum isE=m 1 +m 2 in the nearest-neighbor case and E=m 1 +m 2 +m 1m2 in the next-nearest case. The re- currence time (ignoring...

    work page

  39. [39]

    This is facilitated by the fact that, as a unitary transformation of ˆH0, ˆM(t) preserves its spectrum

    Alternatively, one often employs a different metric for the measure of the appropriateness of ˆM(n)(t) as an accu- rate Emergent Hamiltonian, such as computing the over- lap of the desired eigenstate of the latter with the time- evolved state:|⟨Ψ t|ψ(t)⟩|[27, 28]. This is facilitated by the fact that, as a unitary transformation of ˆH0, ˆM(t) preserves it...

    work page

  40. [40]

    Oganesyan and D

    V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B75, 155111 (2007)

    work page 2007

  41. [41]

    Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett.110, 084101 (2013)

    work page 2013

  42. [42]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Advances in Physics65, 239 (2016)

    work page 2016

  43. [43]

    For hardcore bosons, on the other hand, this commutator leads to ˆa† i ˆaj,ˆa † kˆaℓ =δ jk ˆa† i 1−2ˆn j ˆaℓ −δ iℓ ˆa† k 1− 2ˆni ˆaj

    This can be immediately seen for the case of fermions (ˆc and ˆc† operators) or bosons ( ˆband ˆb† operators), where ˆa† i ˆaj,ˆa† kˆaℓ =δ jk ˆa† i ˆaℓ −δ iℓ ˆa† kˆaj, where ˆai = ˆci or ˆbi. For hardcore bosons, on the other hand, this commutator leads to ˆa† i ˆaj,ˆa † kˆaℓ =δ jk ˆa† i 1−2ˆn j ˆaℓ −δ iℓ ˆa† k 1− 2ˆni ˆaj

    work page

  44. [44]

    C. Song, K. Xu, H. Li, Y.-R. Zhang, X. Zhang, W. Liu, Q. Guo, Z. Wang, W. Ren, J. Hao, H. Feng, H. Fan, D. Zheng, D.-W. Wang, H. Wang, and S.-Y. Zhu, Gen- eration of multicomponent atomic schr¨ odinger cat states of up to 20 qubits, Science365, 574 (2019)

    work page 2019

  45. [45]

    Gross, T

    C. Gross, T. Zibold, E. Nicklas, J. Est` eve, and M. K. Oberthaler, Nonlinear atom interferometer surpasses classical precision limit, Nature464, 1165 (2010)

    work page 2010

  46. [46]

    Kitagawa and M

    M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A47, 5138 (1993)

    work page 1993

  47. [47]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature452, 854 (2008). 13

    work page 2008

  48. [48]

    Mondaini, K

    R. Mondaini, K. R. Fratus, M. Srednicki, and M. Rigol, Eigenstate thermalization in the two-dimensional trans- verse field Ising model, Phys. Rev. E93, 032104 (2016)

    work page 2016

  49. [49]

    Mondaini and A

    R. Mondaini and A. Sur, Dataset for the paper ”From Bell Products to GHZ: Quantum Memories via Emergent Hamiltonians”, 10.5281/zenodo.17245519 (2025)

    work page doi:10.5281/zenodo.17245519 2025

  50. [50]

    Peschel and V

    I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, Journal of Physics A: Mathematical and Theoretical42, 504003 (2009)

    work page 2009

  51. [51]

    M. A. Nielsen and I. L. Chuang,Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion(Cambridge University Press, 2010). APPENDIX - APPROXIMA TE EMERGENT HAMIL TONIANS IN THE MANY-BODY SETTING FOR TWO-DIMENSIONAL SYSTEMS In the main text, Sec. III B, we argued that tackling the many-body case in more than one dimension leads to an elusi...

    work page 2010

  52. [52]

    + X l p ly(Ly −l y) 2 ˆa† l+ˆyˆal + H.c

    2D nearest-neighbor approximate Emergent Hamiltonian — many excitations Our starting point are the entangling Hamiltonian ˆHf and the initial Hamiltonian ˆH0, given by: ˆHf = X l p lx(Lx −l x) 2 ˆa† l+ˆxˆal + H.c. + X l p ly(Ly −l y) 2 ˆa† l+ˆyˆal + H.c. ,and (25) ˆH0 = X l (lx +l y) ˆa† l ˆal .(26) Computing the truncated Emergent Hamiltonian up to first...

    work page

  53. [53]

    2D next nearest neighbor — many excitations Similar calculations can be carried out in the case where one considers an extra next-nearest neighbor hopping term, here treated as homogeneous as in Eq. (13). The entangling and initial Hamiltonians read: ˆHf = X l p lx(Lx −l x) 2 ˆa† l+ˆxˆal + H.c. + X l p ly(Ly −l y) 2 ˆa† l+ˆyˆal + H.c. +J × X l h ˆa† l+ˆy−...

    work page