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REVIEW 3 major objections 4 minor 104 references

Onset of Bjorken Flow in Quantum Evolution of the Massive Schwinger Model

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A strongly coupled quantum gauge theory can develop Bjorken-like flow on its own, the paper claims.

desk verdict A serious tensor-network study that plausibly sees Bjorken-like hydrodynamic onset in the massive Schwinger model from a local excitation, but missing convergence checks in bond dimension and lattice spacing leave the central claim not yet fully established. read the letter →

arxiv 2509.10855 v1 pith:L6UIT3MG submitted 2025-09-13 hep-ph hep-latnucl-th

Haiyang Shao , Shile Chen , Shuzhe Shi This is my paper · ORCID
classification hep-phhep-latnucl-th
keywords massiveSchwingermodelBjorkenflowhydrodynamizationtensornetworksmatrixproductstatesreal-timelatticegaugetheorythermalizationquark-gluonplasma
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Using a real-time tensor-network simulation of the massive Schwinger model, this paper asks whether a pure gauge-theory state can turn into a fluid by itself. It prepares the vacuum with one localized energy bump, mimicking energy deposited by colliding nuclei, and follows the full unitary evolution. For strong coupling (fermion mass $m=g/10$), the system shows exactly the signatures of generalized Bjorken flow: the flow velocity approaches $v_z=z/t$, the energy density forms a widening rapidity plateau and decays with proper time as $\varepsilon\propto\tau^{-1-c_s^2}$, and the bulk pressure falls inside the Navier-Stokes bound $\Pi\propto\tau^{-2}$. For weak coupling and heavy mass ($m=2g$), none of this appears. The point is that local thermal equilibrium and hydrodynamization would then emerge from first-principle quantum dynamics, not from external sources or classical approximations.

What carries the argument

The load-bearing object is the full energy-momentum tensor $T^{\mu\nu}(t,z)$, computed as the expectation value of lattice stress-tensor operators minus vacuum contributions and decomposed in the Landau-Lifshitz frame into energy density $\varepsilon$, fluid velocity $u^\mu$, and bulk pressure $\Pi$. The test is whether these components satisfy the generalized Bjorken-flow solution: $u^z=z/t$, $\varepsilon$ depending only on proper time $\tau$, and $\Pi$ bounded by the Navier-Stokes formula $\zeta s\theta\propto\tau^{-2}$. Numerically, this is carried by staggered (Kogut-Susskind) fermions on a 100-site lattice and by evolving a matrix product state with bond dimension $D=500$ using the time-evolving block decimation algorithm; the equation of state $P(\varepsilon)$ and the speed of sound $c_s^2$ are computed separately from finite-temperature traces.

What would settle it

Repeat the same evolution with bond dimension $D=1000$ or $D=2000$ and with lattice spacing $a=0.25/g$; if the energy-density scaling $\varepsilon\propto\tau^{-1-c_s^2}$ or the bulk-pressure decay $\Pi\propto\tau^{-2}$ changes materially, the reported hydrodynamic onset is a truncation artifact. Averaging over many localized-excitation initial states would also test whether the large-rapidity quantum fluctuations are event-by-event noise that disappears in the ensemble.

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Extended reading notes

Core claim

The central claim is that in the strongly coupled massive Schwinger model at $m=g/10$, a pure state prepared as a localized excitation on the vacuum evolves, without external sources, into a state whose energy-momentum tensor matches the generalized Bjorken solution with a finite collision time. The simulation finds $v_z=z/t$ inside the light cone, a mid-rapidity plateau in energy density that widens with time, mid-rapidity $\varepsilon$ scaling as $\tau^{-1-c_s^2}$ with $c_s^2\approx 0.5$ taken from the equation of state, and a bulk pressure that starts large, then oscillates and decays within the Navier-Stokes band $\Pi\propto\tau^{-2}$ after $\tau-\tau_{\mathrm{ini}}\gtrsim 1/g$. Local thermal equilibrium, judged by the effective pressure matching the thermal equation of state, is reached on a timescale $\tau\sim 10/g$, which matches the thermalization time of the quantum momentum distribution. The same protocol at $m=2g$ shows neither Bjorken energy scaling nor decaying bulk-pressure oscillations, supporting the conclusion that strong coupling, equivalently small fermion mass, drives the onset of hydrodynamic behavior.

Load-bearing premise

The result rests on the assumption that the simulation's truncation limit, bond dimension $D=500$, and lattice spacing $a=0.5/g$ faithfully represent the true real-time quantum evolution; no convergence checks with $D$ or $a$ are shown, so if the truncation error is substantial, the apparent hydrodynamization could be a numerical artifact rather than a physical property of the continuum model.

Editorial extensions

If this is right

  • If the central claim holds, a pure, isolated, strongly coupled gauge theory can hydrodynamize without any external source, semi-classical input, or assumption of local equilibrium, meaning the fluid description is emergent.
  • Hydrodynamization, judged by small viscous corrections and a Bjorken velocity profile, sets in before full global thermalization, consistent with the picture of early fluid behavior in far-from-equilibrium heavy-ion collisions.
  • The contrast between $m=g/10$ and $m=2g$ identifies strong coupling and light fermions as the microscopic ingredient responsible: enhanced scattering and pair production accelerate thermalization, while heavy fermions do not thermalize on the accessible timescale.
  • The onset time $\tau\sim 10/g$ coincides with the thermalization time of the quantum momentum distribution, suggesting that a single dynamical mechanism controls both hydrodynamization and thermalization.
  • The calculation provides a benchmark set of observables, $\varepsilon$, $u^\mu$, and $\Pi$, for future real-time non-perturbative simulations of hydrodynamization in higher-dimensional gauge theories.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that averaging over many localized-excitation initial states, analogous to event averaging in heavy-ion collisions, should suppress the $O(0.1)$ quantum fluctuations seen in the large-rapidity tails and sharpen the Bjorken comparison; this is directly testable with the same simulation setup.
  • Because 1+1 dimensions have no shear tensor, this test probes only bulk viscosity; whether the same rapid onset survives in 2+1 or 3+1 non-Abelian theories, where shear modes dominate, remains an open question the paper does not address.
  • A natural falsifying check is to repeat the simulation at larger bond dimension and smaller lattice spacing; the absence of published convergence checks means the present claim is only as strong as the truncation control.
  • One could probe local thermalization more stringently by extracting local entropy density from entanglement or particle-number fluctuations, going beyond the stress-tensor criteria used here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper uses matrix product state (MPS) time-evolving block decimation to simulate the real-time quantum evolution of the massive Schwinger model on a lattice with N=100 sites, lattice spacing a=0.5/g, and bond dimension D=500. Starting from a local excitation placed on the vacuum, it computes the full stress-energy tensor T^{\mu\nu}, defines the fluid velocity in the Landau-Lifshitz frame, and compares the resulting energy density, velocity, and bulk pressure with the generalized Bjorken flow with finite collision time. For the strong-coupling case m=g/10, the authors report that v_z approaches z/t, the rapidity profile develops a plateau, the mid-rapidity energy density follows \varepsilon \propto \tau^{-1-c_s^2} with c_s^2=0.5 from the independently computed equation of state, and the bulk pressure decays within a \tau^{-2} envelope; they interpret the timescale \tau~10 g^{-1} as the onset of hydrodynamics and find that this behavior is absent for m=2g.

Significance. If the numerical results are converged, this would be a notable first-principles demonstration that an isolated, strongly coupled gauge theory can hydrodynamize without external sources or semiclassical assumptions, and the m=g/10 versus m=2g contrast provides a falsifiable, qualitative prediction about when Bjorken-like behavior emerges. The paper's strengths include computing the equation of state and c_s^2 independently on the same lattice rather than fitting the exponent, checking energy-momentum conservation at the operator level, and directly using the stress tensor rather than proxy observables. The main weakness is numerical provenance: the central claim rests on a single bond dimension D=500 and a single lattice spacing a=0.5/g with no reported convergence checks, and the comparison to the Bjorken power law uses an unspecified amplitude normalization. These issues are fixable, so the manuscript is suitable for major revision rather than rejection.

major comments (3)
  1. [Model and Numerical Setup] The central claim rests on real-time evolution with D=500 and a=0.5/g, but the manuscript reports no convergence checks in either the bond dimension or the lattice spacing. Fixed-bond-dimension TEBD is a dissipative approximation: each truncation discards Schmidt weights and can induce apparent relaxation and entropy growth, which could mimic hydrodynamization even if the true unitary evolution is far from local equilibrium. The concern is sharpened because the initial excitation in Eq. (5) has width a, exactly one lattice spacing, so the initial state contains UV structure whose entanglement must be resolved. Please add convergence tests (for example D=250, 500, and 1000 at fixed volume, and a=0.25/g, 0.5/g, and 0.75/g at fixed physical volume) and report the truncation error or discarded weight over the full simulated time; without these, the late-time profiles in Figs. 3-5 cannot be taken as properties of the continuum model.
  2. [Hydrodynamic Evolution, Fig. 5(b)] The comparison between the simulated energy density and the Bjorken power law \varepsilon \propto \tau^{-1-c_s^2} requires an amplitude normalization that is not specified in the text. If the normalization is simply matched to the simulation at one time, the agreement is partly a fit and the power-law claim is weakened. Please state the normalization procedure explicitly, or show the unnormalized product \varepsilon(t) \tau^{1+c_s^2} so that the reader can judge whether the exponent is actually reproduced without a free amplitude.
  3. [Summary and Discussion] All results are obtained from a single pure initial state, and the text itself notes O(0.1) quantum fluctuations. Because the stated motivation is to mimic event-by-event heavy-ion collisions, a single state cannot distinguish state-specific fluctuations from generic hydrodynamic behavior. Please provide either an average over several initial excitations or a quantitative sensitivity study varying the excitation width and profile, so that the claimed onset time \tau~10 g^{-1} is not tied to one specially chosen initial condition.
minor comments (4)
  1. [Model, Eq. (5)] The initial state is defined by a phase profile \phi(z) of width a, but the text does not state the resulting deposited energy or its relation to the scales in the problem; adding this would make the analogy to heavy-ion energy deposition more concrete.
  2. [References] Reference [98] is listed as arXiv:2509.xxxxx; please update it to the actual companion paper identifier before publication so that the numerical details can be verified.
  3. [Hydrodynamic Evolution, Fig. 3] The caption of Fig. 3 does not define all curves or panels clearly; please label the three rows as (a), (b), and (c) and specify the plotted quantity in each panel.
  4. [Hydrodynamic Evolution, Fig. 5(a)] The rapidity-distribution panel would benefit from a legend or color bar identifying the values of \tau-\tau_{\rm ini}, since the caption lists several times but does not indicate which curve corresponds to which time.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Bjorken comparison is anchored to an independently computed EoS and an analytic solution, with self-citations used only for ancillary timescale comparisons.

full rationale

The paper's derivation chain is: define the Schwinger-model Hamiltonian and T^{mu nu} (Eqs. 2-3), evolve a pure initial state by TEBD (Eqs. 4-5), extract T^{mu nu}(t,z) (Eq. 6), compute the equilibrium EoS P(epsilon) and c_s^2 from thermal ensembles (Fig. 2), and compare the dynamical profiles to the analytic Bjorken solution epsilon ~ tau^{-1-c_s^2}. The speed of sound is obtained from the finite-temperature partition function of the same lattice Hamiltonian, not fitted to the dynamical epsilon(tau) curve, so the Bjorken exponent is an independent input to the comparison. The normalization of the dashed Bjorken curves in Fig. 5(b) is not specified and may be matched to the simulation; if so, only the power-law shape is tested. That weakens the quantitative claim but is not circular, since the exponent is still fixed by the EoS rather than by the data. The only self-citations are (i) Ref. [87], by two of the present authors, for the thermalization time of the Wigner function, used solely to compare with the hydrodynamization time; (ii) Ref. [101], by a co-author, for the generalized Bjorken analytic solution used as a comparison; and (iii) the placeholder companion paper [98] for numerical details ('Details of the numerical method... can be referred to our companion paper [98]'). None of these carries the load of the central hydrodynamization claim. Absent convergence checks in bond dimension D and lattice spacing a, and the unavailable companion paper, are reproducibility/correctness concerns but do not constitute circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the lattice discretization and the tensor network truncation. The main computed inputs are the EoS and c_s^2, which are produced by the same numerical setup and are not fitted to the dynamical data, apart from the amplitude normalization in the Bjorken comparison.

free parameters (2)
  • Bjorken amplitude normalization = not stated
    In Fig. 5(b), the dashed Bjorken curve epsilon proportional to tau^{-1-c_s^2} must be normalized to compare with the simulation; the normalization is not specified, effectively matching the amplitude to simulation data.
  • Initial excitation width = a = 0.5 g^{-1}
    The initial localized excitation has width equal to one lattice spacing; the dependence of the result on this width is not explored.
assumptions (3)
  • domain assumption Kogut-Susskind staggered fermion discretization with N=100 sites and open boundaries approximates the continuum massive Schwinger model.
    The lattice regularization is standard but the continuum limit is not taken; a=0.5/g is fixed and no a-dependence study is presented.
  • standard math The Landau-Lifshitz frame defines a unique fluid velocity in non-equilibrium systems.
    Used to extract u^mu from T^{mu nu}; this is conventional in relativistic hydrodynamics.
  • domain assumption The finite-temperature EoS computed on the same lattice parameters provides the equilibrium pressure-energy relation.
    Lattice artifacts could bias P(epsilon) and hence the c_s^2 used in the Bjorken comparison.

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Cite this review

Pith. "Pith review of Onset of Bjorken Flow in Quantum Evolution of the Massive Schwinger Model." pith.science (2026). https://pith.science/paper/L6UIT3MG

@misc{pith2026250910855,
  author       = {Pith},
  title        = {Pith review of: Onset of Bjorken Flow in Quantum Evolution of the Massive Schwinger Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L6UIT3MG}},
  note         = {Machine review of arXiv:2509.10855}
}
read the original abstract

The onset of hydrodynamics in the hot medium created in relativistic heavy-ion collisions is a crucial theoretical question. A first-principle simulation requires a real-time, non-perturbative calculation of the quantum system. In this Letter, we perform such simulations using the tensor network method, which enables large-scale quantum many-body simulations by retaining only the most essential quantum states for collective behaviors. We focus on the massive Schwinger model, a low-dimensional analog of quantum chromodynamics (QCD), as they share important properties such as confinement and chiral symmetry breaking. Starting from an initial state that puts a localized excitation atop the vacuum and mimics the energy deposition from colliding nuclei, we observe hydrodynamic behavior consistent with Bjorken flow in all relevant degrees of freedom: energy density, fluid velocity, and bulk pressure. The time scale for hydrodynamic onset aligns with the thermalization time of the quantum distribution function.

Figures

Figures reproduced from arXiv: 2509.10855 by the authors.

Figure 1
Figure 1. FIG. 1. Staggered fermion (circles) and gauge link (lines) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Pressure (top) and speed of sound (bottom) versus [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Energy density (top), fluid velocity (middle), and [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Energy density rapidity distributions. (b-d) [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

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Reference graph

Works this paper leans on

104 extracted references · 1 canonical work pages

  1. [86]

    Efficient and precise quantum simulation of ultrarela- tivistic quark-nucleus scattering,

    Sihao Wu, Weijie Du, Xingbo Zhao, and James P. Vary, “Efficient and precise quantum simulation of ultrarela- tivistic quark-nucleus scattering,” Phys. Rev. D110, 056044 (2024), arXiv:2404.00819 [quant-ph]

    arXiv 2024

  2. [1]

    Local thermalization throughP(ε) convergence,

  3. [2]

    Flow patterns viau µ profiles,

  4. [3]

    Studying hydrodynam- ics requires as many space grids as possible, which in- duces computational complexity because of the exponen- tial scaling of Hilbert space

    Dissipative dynamics from Π evolution, allowing us to establish hydrodynamic onset without phenomenological assumptions. Studying hydrodynam- ics requires as many space grids as possible, which in- duces computational complexity because of the exponen- tial scaling of Hilbert space. Thus, we use tensor net- arXiv:2509.10855v1 [hep-ph] 13 Sep 2025 2 ε ⋯⋯ F...

    arXiv 2025

  5. [4]

    Strongly coupled quark-gluon plasma in heavy ion collisions,

    Edward Shuryak, “Strongly coupled quark-gluon plasma in heavy ion collisions,” Rev. Mod. Phys.89, 035001 (2017), arXiv:1412.8393 [hep-ph]

    arXiv 2017

  6. [5]

    The iEBE- VISHNU code package for relativistic heavy-ion colli- sions,

    Chun Shen, Zhi Qiu, Huichao Song, Jonah Bern- hard, Steffen Bass, and Ulrich Heinz, “The iEBE- VISHNU code package for relativistic heavy-ion colli- sions,” Comput. Phys. Commun.199, 61–85 (2016), arXiv:1409.8164 [nucl-th]

    arXiv 2016

  7. [6]

    El- liptic and triangular flow in event-by-event (3+1)D vis- cous hydrodynamics,

    Bjorn Schenke, Sangyong Jeon, and Charles Gale, “El- liptic and triangular flow in event-by-event (3+1)D vis- cous hydrodynamics,” Phys. Rev. Lett.106, 042301 (2011), arXiv:1009.3244 [hep-ph]

    arXiv 2011

  8. [7]

    A 3+1 dimensional viscous hydrodynamic code for relativistic heavy ion collisions,

    Iu. Karpenko, P. Huovinen, and M. Bleicher, “A 3+1 dimensional viscous hydrodynamic code for relativistic heavy ion collisions,” Comput. Phys. Commun.185, 3016–3027 (2014), arXiv:1312.4160 [nucl-th]

    arXiv 2014

Show all 104 references
  1. [8]

    Fully Dynamical Simulation of Central Nu- clear Collisions,

    Wilke van der Schee, Paul Romatschke, and Scott Pratt, “Fully Dynamical Simulation of Central Nu- clear Collisions,” Phys. Rev. Lett.111, 222302 (2013), arXiv:1307.2539 [nucl-th]

    2013 arXiv

  2. [9]

    Pseudorapidity distribution and decorrelation of anisotropic flow within the open-computing-language implementation CLVisc hydrodynamics,

    Long-Gang Pang, Hannah Petersen, and Xin-Nian Wang, “Pseudorapidity distribution and decorrelation of anisotropic flow within the open-computing-language implementation CLVisc hydrodynamics,” Phys. Rev. C 97, 064918 (2018), arXiv:1802.04449 [nucl-th]

    2018 arXiv

  3. [10]

    (3+1)-dimensional dissipa- tive relativistic fluid dynamics at non-zero net baryon density,

    Lipei Du and Ulrich Heinz, “(3+1)-dimensional dissipa- tive relativistic fluid dynamics at non-zero net baryon density,” Comput. Phys. Commun.251, 107090 (2020), arXiv:1906.11181 [nucl-th]

    2020 arXiv

  4. [11]

    Lev Davidovich Landau and E. M. Lifshitz,Fluid me- chanics(1959)

    1959

  5. [12]

    Hydrodynam- ics Beyond the Gradient Expansion: Resurgence and Resummation,

    Michal P. Heller and Michal Spalinski, “Hydrodynam- ics Beyond the Gradient Expansion: Resurgence and Resummation,” Phys. Rev. Lett.115, 072501 (2015), arXiv:1503.07514 [hep-th]

    2015 arXiv

  6. [13]

    Hydrodynamic Gradient Expansion in Gauge Theory Plasmas,

    Michal P. Heller, Romuald A. Janik, and Przemys- law Witaszczyk, “Hydrodynamic Gradient Expansion in Gauge Theory Plasmas,” Phys. Rev. Lett.110, 211602 (2013), arXiv:1302.0697 [hep-th]

    2013 arXiv

  7. [14]

    Relativistic Fluid Dynamics Far From Local Equilibrium,

    Paul Romatschke, “Relativistic Fluid Dynamics Far From Local Equilibrium,” Phys. Rev. Lett.120, 012301 (2018), arXiv:1704.08699 [hep-th]

    2018 arXiv

  8. [15]

    Relativistic Hydrodynamic Attrac- tors with Broken Symmetries: Non-Conformal and Non- Homogeneous,

    Paul Romatschke, “Relativistic Hydrodynamic Attrac- tors with Broken Symmetries: Non-Conformal and Non- Homogeneous,” JHEP12, 079 (2017), arXiv:1710.03234 [hep-th]

    2017 arXiv

  9. [16]

    Fluid dynamics of out of equilibrium boost invariant plasmas,

    Jean-Paul Blaizot and Li Yan, “Fluid dynamics of out of equilibrium boost invariant plasmas,” Phys. Lett. B 780, 283–286 (2018), arXiv:1712.03856 [nucl-th]

    2018 arXiv

  10. [17]

    On the hydrodynamic attractor of Yang–Mills plasma,

    Micha l Spali´ nski, “On the hydrodynamic attractor of Yang–Mills plasma,” Phys. Lett. B776, 468–472 (2018), arXiv:1708.01921 [hep-th]

    2018 arXiv

  11. [18]

    Bjorken flow at- tractors with transverse dynamics,

    Victor E. Ambrus, Sergiu Busuioc, Jan A. Fotakis, Kai Gallmeister, and Carsten Greiner, “Bjorken flow at- tractors with transverse dynamics,” Phys. Rev. D104, 094022 (2021), arXiv:2102.11785 [nucl-th]

    2021 arXiv

  12. [19]

    Early- and Late-Time Behavior of Attractors in Heavy-Ion Collisions,

    Aleksi Kurkela, Wilke van der Schee, Urs Achim Wiede- mann, and Bin Wu, “Early- and Late-Time Behavior of Attractors in Heavy-Ion Collisions,” Phys. Rev. Lett. 124, 102301 (2020), arXiv:1907.08101 [hep-ph]

    2020 arXiv

  13. [20]

    Nonequilibrium Attractor in High-Temperature QCD Plasmas,

    Dekrayat Almaalol, Aleksi Kurkela, and Michael Strick- land, “Nonequilibrium Attractor in High-Temperature QCD Plasmas,” Phys. Rev. Lett.125, 122302 (2020), arXiv:2004.05195 [hep-ph]

    2020 arXiv

  14. [21]

    Emergence of hydro- dynamical behavior in expanding ultra-relativistic plasmas,

    Jean-Paul Blaizot and Li Yan, “Emergence of hydro- dynamical behavior in expanding ultra-relativistic plasmas,” Annals Phys.412, 167993 (2020), arXiv:1904.08677 [nucl-th]

    2020 arXiv

  15. [22]

    Analytical attractor for Bjorken flows,

    Jean-Paul Blaizot and Li Yan, “Analytical attractor for Bjorken flows,” Phys. Lett. B820, 136478 (2021), arXiv:2006.08815 [nucl-th]

    2021 arXiv

  16. [23]

    Attractor and fixed points in Bjorken flows,

    Jean-Paul Blaizot and Li Yan, “Attractor and fixed points in Bjorken flows,” Phys. Rev. C104, 055201 (2021), arXiv:2106.10508 [nucl-th]

    2021 arXiv

  17. [24]

    Attractor for (1+1)D vis- cous hydrodynamics with general rapidity distribution,

    Shile Chen and Shuzhe Shi, “Attractor for (1+1)D vis- cous hydrodynamics with general rapidity distribution,” Phys. Rev. C111, L021902 (2025), arXiv:2407.15209 [hep-ph]

    2025 arXiv

  18. [25]

    ’Bottom up’ thermalization in heavy ion colli- sions,

    R. Baier, Alfred H. Mueller, D. Schiff, and D. T. Son, “’Bottom up’ thermalization in heavy ion colli- sions,” Phys. Lett. B502, 51–58 (2001), arXiv:hep- ph/0009237

    2001

  19. [26]

    Isotropization and hydro- dynamization in weakly coupled heavy-ion collisions,

    Aleksi Kurkela and Yan Zhu, “Isotropization and hydro- dynamization in weakly coupled heavy-ion collisions,” Phys. Rev. Lett.115, 182301 (2015), arXiv:1506.06647 [hep-ph]

    2015 arXiv

  20. [27]

    Bose– Einstein Condensation and Thermalization of the Quark Gluon Plasma,

    Jean-Paul Blaizot, Francois Gelis, Jin-Feng Liao, Larry McLerran, and Raju Venugopalan, “Bose– Einstein Condensation and Thermalization of the Quark Gluon Plasma,” Nucl. Phys. A873, 68–80 (2012), arXiv:1107.5296 [hep-ph]

    2012 arXiv

  21. [28]

    Gluon Transport Equation in the Small Angle Approx- imation and the Onset of Bose-Einstein Condensation,

    Jean-Paul Blaizot, Jinfeng Liao, and Larry McLerran, “Gluon Transport Equation in the Small Angle Approx- imation and the Onset of Bose-Einstein Condensation,” Nucl. Phys. A920, 58–77 (2013), arXiv:1305.2119 [hep- ph]

    2013 arXiv

  22. [29]

    Bose condensation far from equilibrium,

    J. Berges and D. Sexty, “Bose condensation far from equilibrium,” Phys. Rev. Lett.108, 161601 (2012), arXiv:1201.0687 [hep-ph]

    2012 arXiv

  23. [30]

    Universality far from equilibrium: From su- perfluid Bose gases to heavy-ion collisions,

    J. Berges, K. Boguslavski, S. Schlichting, and R. Venu- gopalan, “Universality far from equilibrium: From su- perfluid Bose gases to heavy-ion collisions,” Phys. Rev. Lett.114, 061601 (2015), arXiv:1408.1670 [hep-ph]

    2015 arXiv

  24. [31]

    Quark pro- duction, Bose–Einstein condensates and thermalization of the quark–gluon plasma,

    Jean-Paul Blaizot, Bin Wu, and Li Yan, “Quark pro- duction, Bose–Einstein condensates and thermalization of the quark–gluon plasma,” Nucl. Phys. A930, 139– 162 (2014), arXiv:1402.5049 [hep-ph]

    2014 arXiv

  25. [32]

    Thermalization of gluons with Bose-Einstein 6 condensation,

    Zhe Xu, Kai Zhou, Pengfei Zhuang, and Carsten Greiner, “Thermalization of gluons with Bose-Einstein 6 condensation,” Phys. Rev. Lett.114, 182301 (2015), arXiv:1410.5616 [hep-ph]

    2015 arXiv

  26. [33]

    Dissipative Dynamics of Highly Anisotropic Systems,

    Mauricio Martinez and Michael Strickland, “Dissipative Dynamics of Highly Anisotropic Systems,” Nucl. Phys. A848, 183–197 (2010), arXiv:1007.0889 [nucl-th]

    2010 arXiv

  27. [34]

    Boost-Invariant (2+1)-dimensional Anisotropic Hydrodynamics,

    Mauricio Martinez, Radoslaw Ryblewski, and Michael Strickland, “Boost-Invariant (2+1)-dimensional Anisotropic Hydrodynamics,” Phys. Rev. C85, 064913 (2012), arXiv:1204.1473 [nucl-th]

    2012 arXiv

  28. [35]

    Highly- anisotropic hydrodynamics in 3+1 space-time dimen- sions,

    Radoslaw Ryblewski and Wojciech Florkowski, “Highly- anisotropic hydrodynamics in 3+1 space-time dimen- sions,” Phys. Rev. C85, 064901 (2012), arXiv:1204.2624 [nucl-th]

    2012 arXiv

  29. [36]

    Anisotropic nonequilibrium hydrodynamic attrac- tor,

    Michael Strickland, Jorge Noronha, and Gabriel Deni- col, “Anisotropic nonequilibrium hydrodynamic attrac- tor,” Phys. Rev. D97, 036020 (2018), arXiv:1709.06644 [nucl-th]

    2018 arXiv

  30. [37]

    The non-equilibrium attractor for ki- netic theory in relaxation time approximation,

    M. Strickland, “The non-equilibrium attractor for ki- netic theory in relaxation time approximation,” JHEP 12, 128 (2018), arXiv:1809.01200 [nucl-th]

    2018 arXiv

  31. [38]

    Paul Romatschke and Ulrike Romatschke,Relativistic Fluid Dynamics In and Out of Equilibrium, Cambridge Monographs on Mathematical Physics (Cambridge Uni- versity Press, 2019) arXiv:1712.05815 [nucl-th]

    2019 arXiv

  32. [39]

    QCD thermalization: Ab ini- tio approaches and interdisciplinary connections,

    J¨ urgen Berges, Michal P. Heller, Aleksas Mazeliauskas, and Raju Venugopalan, “QCD thermalization: Ab ini- tio approaches and interdisciplinary connections,” Rev. Mod. Phys.93, 035003 (2021), arXiv:2005.12299 [hep- th]

    2021 arXiv

  33. [40]

    Spectral BBGKY: a scal- able scheme for nonlinear Boltzmann and correlation kinetics,

    Xingjian Lu and Shuzhe Shi, “Spectral BBGKY: a scal- able scheme for nonlinear Boltzmann and correlation kinetics,” (2025), arXiv:2507.14243 [nucl-th]

    2025

  34. [41]

    Relativistic anisotropic hydrody- namics,

    Mubarak Alqahtani, Mohammad Nopoush, and Michael Strickland, “Relativistic anisotropic hydrody- namics,” Prog. Part. Nucl. Phys.101, 204–248 (2018), arXiv:1712.03282 [nucl-th]

    2018 arXiv

  35. [42]

    New theories of relativistic hydrodynamics in the LHC era,

    Wojciech Florkowski, Michal P. Heller, and Michal Spalinski, “New theories of relativistic hydrodynamics in the LHC era,” Rept. Prog. Phys.81, 046001 (2018), arXiv:1707.02282 [hep-ph]

    2018 arXiv

  36. [43]

    Recent development of hydro- dynamic modeling in heavy-ion collisions,

    Chun Shen and Li Yan, “Recent development of hydro- dynamic modeling in heavy-ion collisions,” Nucl. Sci. Tech.31, 122 (2020), arXiv:2010.12377 [nucl-th]

    2020 arXiv

  37. [44]

    Hydrodynamic attractors in heavy ion collisions: a review,

    Alexander Soloviev, “Hydrodynamic attractors in heavy ion collisions: a review,” Eur. Phys. J. C82, 319 (2022), arXiv:2109.15081 [hep-th]

    2022 arXiv

  38. [45]

    Hydrodynamic attractors in ultrarelativistic nuclear collisions,

    Jakub Jankowski and Micha l Spali´ nski, “Hydrodynamic attractors in ultrarelativistic nuclear collisions,” Prog. Part. Nucl. Phys.132, 104048 (2023), arXiv:2303.09414 [nucl-th]

    2023 arXiv

  39. [46]

    Hydrodynamization and re- summed viscous hydrodynamics,

    Michael Strickland, “Hydrodynamization and re- summed viscous hydrodynamics,” Int. J. Mod. Phys. E33, 2430004 (2024), arXiv:2402.09571 [nucl-th]

    2024 arXiv

  40. [47]

    Gauge Invariance and Mass. 2

    Julian S. Schwinger, “Gauge Invariance and Mass. 2.” Phys. Rev.128, 2425–2429 (1962)

    1962

  41. [48]

    Quantum electro- dynamics in two-dimensions,

    J. H. Lowenstein and J. A. Swieca, “Quantum electro- dynamics in two-dimensions,” Annals Phys.68, 172–195 (1971)

    1971

  42. [49]

    SCHWINGER MODEL ON S(2),

    C. Jayewardena, “SCHWINGER MODEL ON S(2),” Helv. Phys. Acta61, 636–711 (1988)

    1988

  43. [50]

    On the fermion condensate in Schwinger model,

    Andrei V. Smilga, “On the fermion condensate in Schwinger model,” Phys. Lett. B278, 371–376 (1992)

    1992

  44. [51]

    Instantons and vacuum expectation values in the Schwinger model,

    C. Adam, “Instantons and vacuum expectation values in the Schwinger model,” Z. Phys. C63, 169–180 (1994)

    1994

  45. [52]

    Massive Schwinger model within mass perturbation theory,

    Christoph Adam, “Massive Schwinger model within mass perturbation theory,” Annals Phys.259, 1–63 (1997), arXiv:hep-th/9704064

    1997 arXiv

  46. [53]

    More About the Massive Schwinger Model,

    Sidney R. Coleman, “More About the Massive Schwinger Model,” Annals Phys.101, 239 (1976)

    1976

  47. [54]

    The Schwinger mass in the massive Schwinger model,

    Christoph Adam, “The Schwinger mass in the massive Schwinger model,” Phys. Lett. B382, 383–388 (1996), arXiv:hep-ph/9507331

    1996 arXiv

  48. [55]

    General bound state structure of the massive Schwinger model,

    C. Adam, “General bound state structure of the massive Schwinger model,” Phys. Lett. B382, 111–116 (1996), arXiv:hep-th/9602175

    1996 arXiv

  49. [56]

    The boson-boson bound state in the massive Schwinger model,

    Christoph Adam, “The boson-boson bound state in the massive Schwinger model,” Z. Phys. C74, 727–730 (1997), arXiv:hep-ph/9601227

    1997 arXiv

  50. [57]

    Quantum-classical computation of Schwinger model dynamics using quantum comput- ers,

    N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Morris, R. C. Pooser, M. Sanz, E. Solano, P. Lougov- ski, and M. J. Savage, “Quantum-classical computation of Schwinger model dynamics using quantum comput- ers,” Phys. Rev. A98, 032331 (2018), arXiv:1803.03326 [quant-ph]

    2018 arXiv

  51. [58]

    Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits,

    Roland C. Farrell, Marc Illa, Anthony N. Ciavarella, and Martin J. Savage, “Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits,” PRX Quan- tum5, 020315 (2024), arXiv:2308.04481 [quant-ph]

    2024 arXiv

  52. [59]

    Quantum simulations of hadron dynamics in the Schwinger model using 112 qubits,

    Roland C. Farrell, Marc Illa, Anthony N. Ciavarella, and Martin J. Savage, “Quantum simulations of hadron dynamics in the Schwinger model using 112 qubits,” Phys. Rev. D109, 114510 (2024), arXiv:2401.08044 [quant-ph]

    2024 arXiv

  53. [60]

    Dynamical Topo- logical Transitions in the Massive Schwinger Model with aθTerm,

    T. V. Zache, N. Mueller, J. T. Schneider, F. Jendrze- jewski, J. Berges, and P. Hauke, “Dynamical Topo- logical Transitions in the Massive Schwinger Model with aθTerm,” Phys. Rev. Lett.122, 050403 (2019), arXiv:1808.07885 [quant-ph]

    2019 arXiv

  54. [61]

    Real-time dynamics of Chern-Simons fluctuations near a critical point,

    Kazuki Ikeda, Dmitri E. Kharzeev, and Yuta Kikuchi, “Real-time dynamics of Chern-Simons fluctuations near a critical point,” Phys. Rev. D103, L071502 (2021), arXiv:2012.02926 [hep-ph]

    2021 arXiv

  55. [62]

    Real-time chi- ral dynamics from a digital quantum simulation,

    Dmitri E. Kharzeev and Yuta Kikuchi, “Real-time chi- ral dynamics from a digital quantum simulation,” Phys. Rev. Res.2, 023342 (2020), arXiv:2001.00698 [hep-ph]

    2020 arXiv

  56. [63]

    Quantum simulation of nonequilibrium dynamics and thermaliza- tion in the Schwinger model,

    Wibe A. de Jong, Kyle Lee, James Mulligan, Mateusz P losko´ n, Felix Ringer, and Xiaojun Yao, “Quantum simulation of nonequilibrium dynamics and thermaliza- tion in the Schwinger model,” Phys. Rev. D106, 054508 (2022), arXiv:2106.08394 [quant-ph]

    2022 arXiv

  57. [64]

    Variational thermal quantum simulation of the lattice Schwinger model,

    Xu-Dan Xie, Xingyu Guo, Hongxi Xing, Zheng-Yuan Xue, Dan-Bo Zhang, and Shi-Liang Zhu (QuNu), “Variational thermal quantum simulation of the lattice Schwinger model,” Phys. Rev. D106, 054509 (2022), arXiv:2205.12767 [quant-ph]

    2022 arXiv

  58. [65]

    High-Energy Collision of Quarks and Mesons in the Schwinger Model: From Ten- sor Networks to Circuit QED,

    Ron Belyansky, Seth Whitsitt, Niklas Mueller, Ali Fahimniya, Elizabeth R. Bennewitz, Zohreh Davoudi, and Alexey V. Gorshkov, “High-Energy Collision of Quarks and Mesons in the Schwinger Model: From Ten- sor Networks to Circuit QED,” Phys. Rev. Lett.132, 091903 (2024), arXiv:23...

    2024 arXiv

  59. [66]

    Real-Time Nonperturbative Dynamics of Jet Production in Schwinger Model: Quantum Entangle- ment and Vacuum Modification,

    Adrien Florio, David Frenklakh, Kazuki Ikeda, Dmitri Kharzeev, Vladimir Korepin, Shuzhe Shi, and Kwang- min Yu, “Real-Time Nonperturbative Dynamics of Jet Production in Schwinger Model: Quantum Entangle- ment and Vacuum Modification,” Phys. Rev. Lett.131, 021902 (2023), arXiv:...

    2023 arXiv

  60. [67]

    Quantum real-time evolution of entan- glement and hadronization in jet production: Lessons from the massive Schwinger model,

    Adrien Florio, David Frenklakh, Kazuki Ikeda, Dmitri E. Kharzeev, Vladimir Korepin, Shuzhe Shi, and Kwangmin Yu, “Quantum real-time evolution of entan- glement and hadronization in jet production: Lessons from the massive Schwinger model,” Phys. Rev. D110, 094029 (2024), arXiv...

    2024 arXiv

  61. [68]

    Thermalization from quantum entanglement: jet simulations in the massive Schwinger model,

    Adrien Florio, David Frenklakh, Sebastian Grieninger, Dmitri E. Kharzeev, Andrea Palermo, and Shuzhe Shi, “Thermalization from quantum entanglement: jet simulations in the massive Schwinger model,” (2025), arXiv:2506.14983 [hep-ph]

    2025 arXiv

  62. [69]

    Two-fermion negativity and confine- ment in the Schwinger model,

    Adrien Florio, “Two-fermion negativity and confine- ment in the Schwinger model,” Phys. Rev. D109, L071501 (2024), arXiv:2312.05298 [hep-th]

    2024 arXiv

  63. [70]

    Realtime dynamics of hyperon spin correlations from string fragmentation in a deformed four-flavor Schwinger model,

    Jo˜ ao Barata, Wenjie Gong, and Raju Venugopalan, “Realtime dynamics of hyperon spin correlations from string fragmentation in a deformed four-flavor Schwinger model,” Phys. Rev. D109, 116003 (2024), arXiv:2308.13596 [hep-ph]

    2024 arXiv

  64. [71]

    Detecting the critical point through entan- glement in the Schwinger model,

    Kazuki Ikeda, Dmitri E. Kharzeev, Ren´ e Meyer, and Shuzhe Shi, “Detecting the critical point through entan- glement in the Schwinger model,” Phys. Rev. D108, L091501 (2023), arXiv:2305.00996 [hep-ph]

    2023 arXiv

  65. [72]

    Nonlinear chiral magnetic waves,

    Kazuki Ikeda, Dmitri E. Kharzeev, and Shuzhe Shi, “Nonlinear chiral magnetic waves,” Phys. Rev. D108, 074001 (2023), arXiv:2305.05685 [hep-ph]

    2023 arXiv

  66. [73]

    Liouvillian dynamics of the open Schwinger model: String breaking and kinetic dissipation in a thermal medium,

    Kyle Lee, James Mulligan, Felix Ringer, and Xiao- jun Yao, “Liouvillian dynamics of the open Schwinger model: String breaking and kinetic dissipation in a thermal medium,” Phys. Rev. D108, 094518 (2023), arXiv:2308.03878 [quant-ph]

    2023 arXiv

  67. [74]

    Digital Quan- tum Simulation for Spectroscopy of Schwinger Model,

    Dongwook Ghim and Masazumi Honda, “Digital Quan- tum Simulation for Spectroscopy of Schwinger Model,” PoSLA TTICE2023, 213 (2024), arXiv:2404.14788 [hep-lat]

    2024 arXiv

  68. [75]

    Partonic collinear structure by quantum computing,

    Tianyin Li, Xingyu Guo, Wai Kin Lai, Xiaohui Liu, Enke Wang, Hongxi Xing, Dan-Bo Zhang, and Shi- Liang Zhu (QuNu), “Partonic collinear structure by quantum computing,” Phys. Rev. D105, L111502 (2022), arXiv:2106.03865 [hep-ph]

    2022 arXiv

  69. [76]

    Exploring light-cone distribution amplitudes from quantum computing,

    Tianyin Li, Xingyu Guo, Wai Kin Lai, Xiaohui Liu, Enke Wang, Hongxi Xing, Dan-Bo Zhang, and Shi- Liang Zhu (QuNu), “Exploring light-cone distribution amplitudes from quantum computing,” Sci. China Phys. Mech. Astron.66, 281011 (2023), arXiv:2207.13258 [hep-ph]

    2023 arXiv

  70. [77]

    Scattering amplitude from quantum comput- ing with reduction formula,

    Tianyin Li, Wai Kin Lai, Enke Wang, and Hongxi Xing (QuNu), “Scattering amplitude from quantum comput- ing with reduction formula,” Phys. Rev. D109, 036025 (2024), arXiv:2301.04179 [hep-ph]

    2024 arXiv

  71. [78]

    Quantum simulation of chiral phase tran- sitions,

    Alexander M. Czajka, Zhong-Bo Kang, Henry Ma, and Fanyi Zhao, “Quantum simulation of chiral phase tran- sitions,” JHEP08, 209 (2022), arXiv:2112.03944 [hep- ph]

    2022 arXiv

  72. [79]

    Improved Hamiltonians for Quantum Simulations of Gauge Theories,

    Marcela Carena, Henry Lamm, Ying-Ying Li, and Wanqiang Liu, “Improved Hamiltonians for Quantum Simulations of Gauge Theories,” Phys. Rev. Lett.129, 051601 (2022), arXiv:2203.02823 [hep-lat]

    2022 arXiv

  73. [80]

    SU(2) gauge theory in 2+1 dimensions on a plaquette chain obeys the eigenstate thermaliza- tion hypothesis,

    Xiaojun Yao, “SU(2) gauge theory in 2+1 dimensions on a plaquette chain obeys the eigenstate thermaliza- tion hypothesis,” Phys. Rev. D108, L031504 (2023), arXiv:2303.14264 [hep-lat]

    2023 arXiv

  74. [81]

    Eigenstate thermalization in (2+1)-dimensional SU(2) lattice gauge theory,

    Lukas Ebner, Berndt M¨ uller, Andreas Sch¨ afer, Clemens Seidl, and Xiaojun Yao, “Eigenstate thermalization in (2+1)-dimensional SU(2) lattice gauge theory,” Phys. Rev. D109, 014504 (2024), arXiv:2308.16202 [hep-lat]

    2024 arXiv

  75. [82]

    Real-time chiral dy- namics at finite temperature from quantum simulation,

    Kazuki Ikeda, Zhong-Bo Kang, Dmitri E. Kharzeev, Wenyang Qian, and Fanyi Zhao, “Real-time chiral dy- namics at finite temperature from quantum simulation,” JHEP10, 031 (2024), arXiv:2407.21496 [hep-ph]

    2024 arXiv

  76. [83]

    String-net for- mulation of Hamiltonian lattice Yang-Mills theories and quantum many-body scars in a nonabelian gauge the- ory,

    Tomoya Hayata and Yoshimasa Hidaka, “String-net for- mulation of Hamiltonian lattice Yang-Mills theories and quantum many-body scars in a nonabelian gauge the- ory,” JHEP09, 126 (2023), arXiv:2305.05950 [hep-lat]

    2023 arXiv

  77. [84]

    Dense QCD 2 with matrix product states,

    Tomoya Hayata, Yoshimasa Hidaka, and Kentaro Nishimura, “Dense QCD 2 with matrix product states,” JHEP07, 106 (2024), arXiv:2311.11643 [hep-lat]

    2024 arXiv

  78. [85]

    Quan- tum circuit for Z3 lattice gauge theory at nonzero baryon density,

    Yoshimasa Hidaka and Arata Yamamoto, “Quan- tum circuit for Z3 lattice gauge theory at nonzero baryon density,” Phys. Rev. D111, 014510 (2025), arXiv:2409.17349 [hep-lat]

    2025 arXiv

  79. [87]

    Quantum error thresholds for gauge- redundant digitizations of lattice field theories,

    Marcela Carena, Henry Lamm, Ying-Ying Li, and Wanqiang Liu, “Quantum error thresholds for gauge- redundant digitizations of lattice field theories,” Phys. Rev. D110, 054516 (2024), arXiv:2402.16780 [hep-lat]

    2024 arXiv

  80. [88]

    Efficient quantum simulation of QCD jets on the light front,

    Wenyang Qian, Meijian Li, Carlos A. Salgado, and Michael Kreshchuk, “Efficient quantum simulation of QCD jets on the light front,” Phys. Rev. D111, 096001 (2025), arXiv:2411.09762 [hep-ph]

    2025 arXiv

  81. [89]

    Universality and emergent effec- tive fluid from jets and string breaking in the mas- sive Schwinger model using tensor networks,

    Romuald A. Janik, Maciej A. Nowak, Marek M. Rams, and Ismail Zahed, “Universality and emergent effec- tive fluid from jets and string breaking in the mas- sive Schwinger model using tensor networks,” (2025), arXiv:2502.12901 [hep-ph]

    2025 arXiv

  82. [90]

    Quan- tum thermalization of Quark-Gluon Plasma,

    Shile Chen, Li Yan, and Shuzhe Shi, “Quan- tum thermalization of Quark-Gluon Plasma,” (2024), arXiv:2412.00662 [hep-ph]

    2024 arXiv

  83. [91]

    Emergent hydro- dynamic mode on SU(2) plaquette chains and quan- tum simulation,

    Francesco Turro and Xiaojun Yao, “Emergent hydro- dynamic mode on SU(2) plaquette chains and quan- tum simulation,” Phys. Rev. D111, 094502 (2025), arXiv:2502.17551 [hep-ph]

    2025 arXiv

  84. [92]

    Quantum Simulation for High-Energy Physics,

    Christian W. Baueret al., “Quantum Simulation for High-Energy Physics,” PRX Quantum4, 027001 (2023), arXiv:2204.03381 [quant-ph]

    2023 arXiv

  85. [93]

    Quantum simulation of fundamen- tal particles and forces,

    Christian W. Bauer, Zohreh Davoudi, Natalie Klco, and Martin J. Savage, “Quantum simulation of fundamen- tal particles and forces,” Nature Rev. Phys.5, 420–432 (2023), arXiv:2404.06298 [hep-ph]

    2023 arXiv

  86. [94]

    Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group,

    Stefan Rommer and Stellan Ostlund, “Class of ansatz wave functions for one-dimensional spin systems and their relation to the density matrix renormalization group,” Phys. Rev. B55, 2164–2181 (1997), arXiv:cond- mat/9606213

    1997

  87. [95]

    Matrix prod- uct states for gauge field theories,

    Boye Buyens, Jutho Haegeman, Karel Van Acoleyen, Henri Verschelde, and Frank Verstraete, “Matrix prod- uct states for gauge field theories,” Phys. Rev. Lett. 113, 091601 (2014), arXiv:1312.6654 [hep-lat]

    2014 arXiv

  88. [96]

    Hamiltonian simulation of the Schwinger model at finite temperature,

    Boye Buyens, Frank Verstraete, and Karel Van Acoleyen, “Hamiltonian simulation of the Schwinger model at finite temperature,” Phys. Rev. D 94, 085018 (2016), arXiv:1606.03385 [hep-lat]

    2016 arXiv

  89. [97]

    Hamiltonian Formulation of Wilson’s Lattice Gauge Theories,

    John B. Kogut and Leonard Susskind, “Hamiltonian Formulation of Wilson’s Lattice Gauge Theories,” Phys. Rev. D11, 395–408 (1975). 8

    1975

  90. [98]

    Lattice Fermions,

    Leonard Susskind, “Lattice Fermions,” Phys. Rev. D 16, 3031–3039 (1977)

    1977

  91. [99]

    Many-body localization dynam- ics from gauge invariance,

    Marlon Brenes, Marcello Dalmonte, Markus Heyl, and Antonello Scardicchio, “Many-body localization dynam- ics from gauge invariance,” Phys. Rev. Lett.120, 030601 (2018), arXiv:1706.05878 [cond-mat.str-el]

    2018 arXiv

  92. [100]

    Efficient simulation of one-dimensional quan- tum many-body systems,

    G. Vidal, “Efficient simulation of one-dimensional quan- tum many-body systems,” Phys. Rev. Lett.93, 040502 (2004), arXiv:quant-ph/0310089

    2004 arXiv

  93. [101]

    Collec- tive motion in the massive Schwinger model via Tensor Network,

    Haiyang Shao, Shile Chen, and Shuzhe Shi, “Collec- tive motion in the massive Schwinger model via Tensor Network,” (2025), arXiv:2509.xxxxx [hep-ph]

    2025

  94. [102]

    Statistical Description of Hadron Constituents as a Basis for the Fluid Model of High- Energy Collisions,

    Rudolph C. Hwa, “Statistical Description of Hadron Constituents as a Basis for the Fluid Model of High- Energy Collisions,” Phys. Rev. D10, 2260 (1974)

    1974

  95. [103]

    Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region,

    J. D. Bjorken, “Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region,” Phys. Rev. D27, 140–151 (1983)

    1983

  96. [104]

    Fam- ily of new exact solutions for longitudinally expand- ing ideal fluids,

    Shuzhe Shi, Sangyong Jeon, and Charles Gale, “Fam- ily of new exact solutions for longitudinally expand- ing ideal fluids,” Phys. Rev. C105, L021902 (2022), arXiv:2201.06670 [hep-ph]

    2022 arXiv

Pith tools

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