Predicting critical temperature in quantum simulators for high-$T_c$ superconductivity: the matrix product state plus mean field approach

Adrian Kantian; Thomas K\"ohler

REVIEW

Predicting critical temperature in quantum simulators for high-$T_c$ superconductivity: the matrix product state plus mean field approach

Thomas Köhler, Adrian Kantian

Superconductivity · Submitted 19 Aug 2026 · 58 references

Pith's verdict: A careful MPS+MF study that makes the first concrete Tc predictions for 2D anisotropic mixD bilayers, but the paper's strongest sentence goes beyond what its own caveats allow.

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CitationAdrian Kantian, Thomas K\"ohler. "Predicting critical temperature in quantum simulators for high-$T_c$ superconductivity: the matrix product state plus mean field approach." arXiv:2608.18861 (2026). https://pith.science/paper/ARRYSU5R

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Editor's note

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The genuinely new thing here is the application of MPS+MF to the 2D mixD bilayer with anisotropic tunneling, at system sizes an experiment could actually build. The paper gives concrete Tc values across densities, shows a clean Ep versus Es trade-off, and identifies that Tc drops as density increases. That last point is physically interesting because it goes against the naive intuition that strong pairing should help. The finite-size heuristic for when the speed of sound calculation picks up the wrong excited state is also a useful practical contribution, and validating it against ED is the right thing to do. The DMRG work looks careful: bond dimensions up to 2048, extrapolation to zero discarded weight, and data deposited on Zenodo.

The soft spots are mostly about how far the central claim can stretch. The sentence in Sec. 4.2 that these systems 'absolutely would realize' high-Tc superconducting states is stronger than the evidence supports. The paper itself lists three caveats immediately after, and the first one—competing many-body instabilities not ruled out—is load-bearing. If an insulating phase wins, the predicted superconducting state simply does not form. This is not a minor technicality; it is the difference between a prediction and a conjecture. The authors know this and defer the calculation to future work, but then the abstract and the Sec. 4.2 wording should be softened accordingly. The second caveat, that the framework overestimates Tc by a factor of order four, is less damaging to the qualitative conclusions. It does mean the absolute Tc numbers should be read as upper bounds, not estimates. The missing entropy calculation is also important for the cold-atom audience, but the authors acknowledge it and treat it as future work.

On self-citation: the reliance on the MPS+MF framework from the authors' own prior work is legitimate. The framework has been published in peer-reviewed venues, and the paper is an application, not a validation of the method. The ratio function in Eq. (14) comes from [19], and applying it at finite Lx is the kind of extension that should be scrutinized but is not obviously wrong.

This paper deserves serious peer review. It is a solid application of a known method to a timely problem, and the qualitative results about the Ep/Es trade-off and the density dependence of Tc are likely to be useful for experimental design. I would ask for revisions that tone down the strong claims and explicitly frame the superconducting phase as a candidate rather than a certainty, unless the competing-instability check is added.

Referee report

Significance

If the quantitative predictions survive scrutiny, the paper provides a concrete, experimentally motivated route toward quantum simulation of repulsively mediated high-T_c superconductivity, and it substantially extends the characterization of 1D mixD ladders beyond existing work. The numerical work is careful in several respects: DMRG energies are extrapolated in bond dimension, the level-crossing heuristic is checked against exact diagonalization, and the data are promised to be available via Zenodo. The T_c computation is not fitted to a target value and is therefore not circular in the usual sense. However, the significance is weakened by the gap between the unconditional wording of the central result and the paper's own caveats: the stable phase of the 2D model is not established, the entropy budget is unknown, and the quoted T_c values are raw outputs of a framework stated to overestimate T_c by a factor of roughly 4.

Major objections (3)

[Sec. 4.2, first key finding; Sec. 5] The sentence "These systems, once implemented experimentally, thus absolutely would realize the analogues of repulsively mediated high-Tc superconducting states" goes beyond what the paper's own caveat (1) permits. Competing many-body instabilities are not ruled out, and Sec. 5 explicitly defers this check to future work. Because the superconducting saddle point found in the MPS+MF calculation may not be the global ground state, the unconditional prediction is unsupported; the same framework has been used elsewhere to resolve CDW versus SC order [21], so the check is feasible. At minimum, the sentence should be rephrased as a conditional prediction, and ideally the competing-order calculation should be performed or its absence explicitly justified.
[Sec. 4.2, caveat (3) and Fig. 6] The paper states that the MPS+MF framework overestimates T_c by a quasi-constant factor that may be of order 4, yet the headline numbers in Fig. 6 are the raw, uncorrected values. The claim that "Tc reaches significant fractions of Jy across all densities" is therefore not a statement about the physical T_c unless the correction factor is applied or the values are explicitly labeled as upper bounds. Please report the corrected range and state whether the qualitative conclusions, including the monotonic density dependence and the linear scaling in t_z^2/E_p, survive the factor-4 uncertainty.
[Sec. 4.2, caveat (2); Sec. 5] For ultracold-gas quantum simulators, entropy per particle rather than temperature is the experimentally relevant figure of merit, and the paper does not compute S/N at the predicted T_c. The claim that these systems "absolutely would realize" high-Tc states once implemented is thus conditional on an unverified entropy budget. This is acknowledged in caveat (2), but the manuscript should either provide an estimate or clearly frame the T_c prediction as a necessary, not sufficient, condition for experimental realization.

Minor objections (6)

[Abstract and Sec. 1] "Lays the groundwork to purpose" should read "lays the groundwork to propose", and "of which the 2D system is comprized" should be "of which the 2D system is comprised"; both typos appear in the abstract.
[Sec. 4.1] "The sweep spot of tx and density" should be "the sweet spot of tx and density".
[Sec. 4.2] The phrase "non-¸trivial" contains a stray symbol and should read "non-trivial".
[Sec. 4.3] "The introgression of a state" is nonstandard; consider "the intrusion" or "the ingression" of a state from the antisymmetric density sector.
[Appendix A] The statement "All shown data, including their fits, are accessible via zenodo on reasonable request" is internally contradictory: data are either publicly accessible via Zenodo or available on reasonable request, not both.
[Sec. 3, Eq. (16)] The finite-size applicability of the ratio function R(K_c) is not directly validated in this work; the heuristic of Sec. 4.3 validates the selection of the correct first excited state for v_c but not the accuracy of Eq. (14) at L_x=8-32. A sentence citing the earlier benchmark [19] and stating the expected finite-size error would make the T_c estimate easier to assess.

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The paper

The paper claims

This paper claims that a stack of ultracold-atom ladders can realize high-temperature superconductivity, with predicted critical temperatures of a tenth to half the spin-exchange energy.

The reading

This paper tries to establish that a two-dimensional bilayer made of many coupled one-dimensional 'mixD' ladders—models in which the only inter-layer coupling is spin exchange—can serve as a cold-atom quantum simulator of high-temperature superconductivity. Working at parameters close to those already realized in an experiment on a single ladder (tx = 0.7Jy, tz = 0.2Jy), it computes the critical temperature Tc with a matrix-product-state plus mean-field method and finds values around 0.1 to 0.5 times the spin-exchange energy Jy at densities below half-filling. The authors are explicit that this framework overestimates Tc by a roughly constant factor, possibly close to four, and that competing insulating phases have not been ruled out; even so, the paper's claim is that, once implemented, such systems would realize the analogues of repulsively mediated high-Tc superconducting states. The wider point is to redirect quantum-simulation efforts away from the 2D Hubbard model toward these experimentally accessible ladder stacks.

What carries the argument

The load-bearing object is the array of two-leg t-J 'mixD' ladders, solved one at a time as matrix-product states, with the rest of the stack encoded in self-consistent mean-field amplitudes α and β (pairing and hopping fields proportional to $tz^{2}$/Ep). From the isolated ladder one computes the Tomonaga-Luttinger parameter Kc via its compressibility and speed of sound, and from the self-consistent effective Hamiltonian one extracts the gap Δ between ground and first excited state. The critical temperature is the ratio Tc = Δ/R(Kc), where R(Kc) is the ratio function derived in Ref. [19]; Δ supplies the energy scale for breaking coherence in the direction transverse to the array, while R(Kc) converts that scale into a thermal temperature by accounting for phase fluctuations in the coupled system. The heuristic for finite-size reliability is a kink in the first-excited-state energy that signals contamination from the antisymmetric density sector, marking where the TLL input Kc stops being trustworthy.

Extended reading notes

Core claim

Starting from the t-J Hamiltonian of a single mixD ladder (intra-leg tunnelling tx, strong spin exchange Jy between legs, no inter-leg tunnelling), the paper constructs a 2D bilayer by stacking ladders and coupling them with a small tunnelling tz. The MPS+MF approximation replaces the stack by one ladder subject to self-consistent pairing and hopping mean fields, and from the converged effective Hamiltonian yields the gap Δ between the ground and first excited state. The critical temperature is Tc = Δ/R(Kc), with Kc the Tomonaga-Luttinger parameter of the isolated ladder and R(Kc) the ratio function of Ref. [19]. For tx=0.7Jy and tz=0.2Jy, the paper obtains Tc between roughly 0.1 and 0.5Jy across densities, and states that, once implemented, these systems would realize analogues of repulsively mediated high-Tc superconducting states, subject to three explicit caveats: competing many-body instabilities are not ruled out; the entropy per particle (the practical figure of merit in ultracold gases) is not computed; and the mean-field step makes the Tc values overestimates by a factor earlier work places near four.

Load-bearing premise

The prediction assumes that the superconducting solution is the lowest-energy state of the 2D bilayer mixD model, i.e., that no competing charge- or spin-ordered insulating phase has lower free energy; the paper leaves this unverified, and if such a phase wins, the predicted high-Tc state would not form.

Editorial extensions

If this is right

A 2D bilayer mixD system at tx≈0.7Jy and tz≈0.2Jy is predicted to realize a high-Tc superconducting state at attainable temperatures in a cold-atom quantum simulator, with Tc up to roughly half the spin-exchange energy.
At fixed tz, Tc is highest at lowest density, where Ep is lowest, so lowering density rather than raising tx is the practical lever for raising Tc.
The identified sweet spot in tx (near 3.5Jy) and density (0.5–0.75) balances large Ep against the 2Es constraint, and should guide the design of future mixD experiments.
The finite-size heuristic (watch for the level-crossing kink in E1) defines the parameter range where the TLL-based Tc calculation is valid, and where larger ladders are required.

Reading between the lines

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

Because the overestimation factor is claimed to be roughly constant across parameters, the predicted relative trends (density dependence, tz/Ep scaling) are more trustworthy than the absolute Tc values; the absolute numbers should be read as upper bounds.
The same Tc = Δ/R(Kc) construction should transfer to other quasi-1D pairing building blocks, so the paper's approach is a template for predicting critical temperatures in anisotropic 2D and 3D arrays of other ladder systems.
If charge or spin ordering is later found to win in part of the parameter space, the density dependence of Tc would probably shift the optimal operating point rather than destroy the superconducting region entirely.

Original abstract

Quantum simulation based on ultra cold atomic lattice gases is one of the most promising platforms to investigate high-$T_c$ superconductivity beyond the limited capabilities of quantum many body numerics on classical computers. Yet, despite enormous progress since the field's inception, realizing a high-$T_c$ superconducting state still remains out of reach. The present work lays the groundwork to purpose the recently proposed, and already partly realized, mixed-dimensional (mixD) models, towards this end. These systems offer the proven capability to realize very high pairing energies while retaining appreciable mobility of pairs. We specifically investigate the potential of 2D mixD-models with anisotropic tunneling, using the matrix product state plus mean field theory (MPS+MF) for fermions, and show that these models may enter a high-$T_c$ superconducting phase. These simulations in turn are based on a comprehensive characterization of the 1D mixD-systems, which are the sub-units of which the 2D system is comprized. In this, we cover the range of currently experimentally relevant system sizes, and establish practical heuristics to determine when finite\hyp size effects preclude the use of a 1D mixD-system to build the 2D ones.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,796 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.81 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.12 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Quantum simulation based on ultra cold atomic lattice gases is one of the most promising platforms to investigate high-$T_c$ superconductivity beyond the limited capabilities of quantum many body numerics on classical computers. Yet, despite enormous progress since the field's inception, realizing a high-$T_c$ superconducting state still remains out of reach. The present work lays the groundwork to purpose the recently proposed, and already partly realized, mixed-dimensional (mixD) models, towards this end. These systems offer the proven capability to realize very high pairing energies while retaining appreciable mobility of pairs. We specifically investigate the potential of 2D mixD-models with anisotropic tunneling, using the matrix product state plus mean field theory (MPS+MF) for fermions, and show that these models may enter a high-$T_c$ superconducting phase. These simulations in turn are based on a comprehensive characterization of the 1D mixD-systems, which are the sub-units of which the 2D system is comprized. In this, we cover the range of currently experimentally relevant system sizes, and establish practical heuristics to determine when finite\hyp size effects preclude the use of a 1D mixD-system to build the 2D ones.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Quantum simulation based on ultra cold atomic lattice gases is one of the most promising platforms to investigate high-$T_c$ superconductivity beyond the limited capabilities of quantum many body numerics on classical computers.	background 1.00	0.04	0.02	asserted 0.89
s1	Yet, despite enormous progress since the field's inception, realizing a high-$T_c$ superconducting state still remains out of reach.	background 0.55	0.05	0.02	asserted 1.00
s2	The present work lays the groundwork to purpose the recently proposed, and already partly realized, mixed-dimensional (mixD) models, towards this end.	contribution 0.95	0.45	0.06	asserted 0.80
s3	These systems offer the proven capability to realize very high pairing energies while retaining appreciable mobility of pairs.	background 0.93	0.09	0.06	asserted 0.99
s4	We specifically investigate the potential of 2D mixD-models with anisotropic tunneling, using the matrix product state plus mean field theory (MPS+MF) for fermions, and show that these models may enter a high-$T_c$ superconducting phase.	contribution 0.86	0.94	0.83	hedged possibility 0.82
s5	These simulations in turn are based on a comprehensive characterization of the 1D mixD-systems, which are the sub-units of which the 2D system is comprized.	method 0.96	0.79	0.05	asserted 1.00
s6	In this, we cover the range of currently experimentally relevant system sizes, and establish practical heuristics to determine when finite\hyp size effects preclude the use of a 1D mixD-system to build the 2D ones.	contribution 0.58	0.93	0.06	asserted 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 3.** As also illustrated in Fig. 7, the introgression of a state from the antisymmetric

Showing 1 of the paper's 3 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2608.18861 by the authors.

References

Works cited (58)

[19]

G. Bollmark, T. K¨ ohler, L. Pizzino, Y. Yang, J. S. Hofmann, H. Shi, S. Zhang, T. Giamarchi and A. Kantian,Solving 2D and 3D Lattice Models of Correlated Fermions—Combining Matrix Product States with Mean-Field Theory, Phys. Rev. X 13, 011039 (2023), doi:https://doi.org/10.1103/PhysRevX.13.011039

View on Pith doi:10.1103/physrevx.13.011039 2023
[21]

G. Bollmark, T. K¨ ohler and A. Kantian,Resolving competition of charge density wave and superconducting phases using the matrix product state plus mean field algorithm, Physical Review B111(12), 125141 (2025), doi:10.1103/PhysRevB.111.125141

View on Pith doi:10.1103/physrevb.111.125141 2025
[1]

H. J. Schulz,Superconductivity and Antiferromagnetism in the Two-Dimensional Hubbard Model: Scaling Theory, Europhys. Lett.4(5), 609 (1987), doi:10.1209/0295- 5075/4/5/016

View on Pith doi:10.1209/0295- 1987
[2]

P. Lederer, G. Montambaux and D. Poilblanc,Antiferromagnetism and supercon- ductivity in a quasi two-dimensional electron gas. Scaling theory of a generic Hubbard model, J. Phys. France48(10), 1613 (1987), doi:10.1051/jphys:0198700480100161300

View on Pith doi:10.1051/jphys:0198700480100161300 1987
[3]

G. Kotliar and J. Liu,Superexchange mechanism and d-wave superconductivity, Phys- ical Review B38(7), 5142 (1988), doi:10.1103/PhysRevB.38.5142

View on Pith doi:10.1103/physrevb.38.5142 1988
[4]

M. Inui, S. Doniach, P. Hirschfeld and A. Ruckenstein,Coexistence of antiferromag- netism and superconductivity in a mean-field theory of high-T {c}superconductors, Phys. Rev. B37(4), 2320 (1988), doi:10.1103/PhysRevB.37.2320

View on Pith doi:10.1103/physrevb.37.2320 1988
[5]

D. J. Scalapino,The case for pairing in the cuprate superconductors &2 ,,2, Phys. Reps.250, 329 (1995)

View on Pith 1995
[6]

N. Furukawa, T. M. Rice and M. Salmhofer,Truncation of a Two-Dimensional Fermi Surface due to Quasiparticle Gap Formation at the Saddle Points, Phys. Rev. Lett. 81(15), 3195 (1998), doi:10.1103/PhysRevLett.81.3195

View on Pith doi:10.1103/physrevlett.81.3195 1998
[7]

J. Orenstein,Advances in the Physics of High-Temperature Superconductivity, Science 288(5465), 468 (2000), doi:10.1126/science.288.5465.468

View on Pith doi:10.1126/science.288.5465.468 2000
[8]

W. Metzner, J. Reiss and D. Rohe,Magnetic and superconducting correlations in the two-dimensional Hubbard model, physica status solidi (b)243(1), 46 (2006), doi:10.1002/pssb.200562445

View on Pith doi:10.1002/pssb.200562445 2006
[9]

T. A. Maier, M. Jarrell and D. J. Scalapino,Pairing interaction in the two- dimensional Hubbard model studied with a dynamic cluster quantum Monte Carlo approximation, Phys. Rev. B74(9), 094513 (2006), doi:10.1103/PhysRevB.74.094513

View on Pith doi:10.1103/physrevb.74.094513 2006
[10]

S. Raghu, S. A. Kivelson and D. J. Scalapino,Superconductivity in the repulsive Hubbard model: An asymptotically exact weak-coupling solution, Phys. Rev. B81(22), 224505 (2010), doi:10.1103/PhysRevB.81.224505. 17 SciPost Physics Submission

View on Pith doi:10.1103/physrevb.81.224505 2010
[11]

P. W. Anderson,Twenty-five years of high-temperature superconductivity - A personal review, J. Phys.: Conf. Ser449(1) (2013), doi:10.1088/1742-6596/449/1/012001

View on Pith doi:10.1088/1742-6596/449/1/012001 2013
[12]

T. A. Maier, M. Jarrell, T. C. Schulthess, P. R. C. Kent and J. B. White,Systematic Study of d -Wave Superconductivity in the 2D Repulsive Hubbard Model, Phys. Rev. Lett.95(23), 237001 (2005), doi:10.1103/PhysRevLett.95.237001

View on Pith doi:10.1103/physrevlett.95.237001 2005
[13]

C. C. Chang and S. Zhang,Spin and charge order in the doped hubbard model: Long-wavelength collective modes, Phys. Rev. Lett.104(11), 1 (2010), doi:10.1103/PhysRevLett.104.116402

View on Pith doi:10.1103/physrevlett.104.116402 2010
[14]

J. P. F. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M. Henderson, C. A. Jim´ enez-Hoyos, E. Kozik, X.-W. Liu et al.,Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms, Phys. Rev. X5(4), 041041 (2015), doi:10.1103/PhysRevX.5.041041, Publish...

View on Pith doi:10.1103/physrevx.5.041041 2015
[15]

B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang and G. K.-L. Chan,Stripe order in the underdoped re- gion of the two-dimensional Hubbard model., Science358(6367), 1155 (2017), doi:10.1126/science.aam7127

View on Pith doi:10.1126/science.aam7127 2017
[16]

M. Qin, C. M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollw¨ ock, S. R. White and S. Zhang,Absence of Superconductivity in the Pure Two-Dimensional Hubbard Model, Phys. Rev. X10(3), 31016 (2020), doi:10.1103/PhysRevX.10.031016, Publisher: American Physical Society

View on Pith doi:10.1103/physrevx.10.031016 2020
[17]

H. Xu, H. Shi, E. Vitali, M. Qin and S. Zhang,Stripes and spin-density waves in the doped two-dimensional Hubbard model: Ground state phase diagram, Physical Review Research4(1), 013239 (2022), doi:10.1103/PhysRevResearch.4.013239

View on Pith doi:10.1103/physrevresearch.4.013239 2022
[18]

H. Xu, C.-M. Chung, M. Qin, U. Schollw¨ ock, S. R. White and S. Zhang,Coexistence of superconductivity with partially filled stripes in the Hubbard model, Science384(6696), eadh7691 (2024), doi:10.1126/science.adh7691

View on Pith doi:10.1126/science.adh7691 2024
[20]

S. Marten, G. Bollmark, T. K¨ ohler, S. R. Manmana and A. Kantian,Transient su- perconductivity in three-dimensional Hubbard systems by combining matrix-product states and self-consistent mean-field theory, SciPost Physics15(6), 236 (2023), doi:10.21468/SciPostPhys.15.6.236

View on Pith doi:10.21468/scipostphys.15.6.236 2023
[22]

T. Giamarchi,Quantum Physics in One Dimension, Oxford University Press, 1st edn., ISBN 978-0-19-852500-4 (2003)

View on Pith 2003
[23]

G. Karakonstantakis, E. Berg, S. R. White and S. A. Kivelson,Enhanced pairing in the checkerboard Hubbard ladder, Physical Review B - Condensed Matter and Materials Physics83(5), 054508 (2011), doi:10.1103/PhysRevB.83.054508. 18 SciPost Physics Submission

View on Pith doi:10.1103/physrevb.83.054508 2011
[24]

S. Trebst, U. Schollw¨ ock, M. Troyer and P. Zoller,d-Wave Resonating Valence Bond States of Fermionic Atoms in Optical Lattices, Phys. Rev. Lett.96(25), 250402 (2006), doi:10.1103/PhysRevLett.96.250402

View on Pith doi:10.1103/physrevlett.96.250402 2006
[25]

M. Dolfi, B. Bauer, S. Keller and M. Troyer,Pair correlations in doped Hubbard ladders, Phys. Rev. B92(19), 195139 (2015), doi:10.1103/PhysRevB.92.195139, Publisher: American Physical Society

View on Pith doi:10.1103/physrevb.92.195139 2015
[26]

A. Bohrdt, L. Homeier, I. Bloch, E. Demler and F. Grusdt,Strong pairing in mixed dimensional bilayer antiferromagnetic Mott insulators, Nat. Phys. (2022), doi:10.1038/s41567-022-01561-8

View on Pith doi:10.1038/s41567-022-01561-8 2022
[27]

S. Hirthe, T. Chalopin, D. Bourgund, P. Bojovi´ c, A. Bohrdt, E. Demler, F. Grusdt, I. Bloch and T. A. Hilker,Magnetically mediated hole pairing in fermionic ladders of ultracold atoms, Nature613(7944), 463 (2023), doi:10.1038/s41586-022-05437-y

View on Pith doi:10.1038/s41586-022-05437-y 2023
[28]

H. Lange, A. Chen, A. Georges, F. Grusdt, A. Bohrdt and C. Roth,Simulating super- conductivity in mixed-dimensional$t \parallel$-${J}\parallel$-${J}\perp$bilayers with neural quantum states, doi:10.48550/ARXIV.2602.10091, Version Number: 1 (2026)

View on Pith doi:10.48550/arxiv.2602.10091 2026
[29]

T. Kaneko, H. Sakakibara, M. Ochi and K. Kuroki,Pair correlations in the two-orbital Hubbard ladder: Implications on superconductivity in the bi- layer nickelate La$ 3$Ni$ 2$O$ 7$, Physical Review B109(4), 045154 (2024), doi:10.1103/PhysRevB.109.045154, ArXiv:2310.01952 [cond-mat.supr-con]

View on Pith Pith review arXiv 2024
[30]

X.-Z. Qu, D.-W. Qu, J. Chen, C. Wu, F. Yang, W. Li and G. Su,Bilayer $t$-$J$-$J\perp$Model and Magnetically Mediated Pairing in the Pressurized Nickelate La$ 3$Ni$ 2$O$ 7$, Physical Review Letters132(3), 036502 (2024), doi:10.1103/PhysRevLett.132.036502, ArXiv:2307.16873 [cond-mat.str-el]

View on Pith Pith review arXiv 2024
[31]

I. Bloch, J. Dalibard and W. Zwerger,Many-body physics with ultracold gases, Rev. Mod. Phys.80(3), 885 (2008), doi:10.1103/RevModPhys.80.885, Publisher: Ameri- can Physical Society

View on Pith doi:10.1103/revmodphys.80.885 2008
[32]

M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif and M. Greiner,Site- resolved measurement of the spin-correlation function in the Fermi-Hubbard model, Science353(6305) (2016)

View on Pith 2016
[33]

L. W. Cheuk, M. A. Nichols, K. R. Lawrence, M. Okan, H. Zhang, E. Khatami, N. Trivedi, T. Paiva, M. Rigol and M. W. Zwierlein,Observation of spatial charge and spin correlations in the 2D Fermi-Hubbard model, Science353(6305) (2016)

View on Pith 2016
[34]

D. Greif, M. F. Parsons, A. Mazurenko, C. S. Chiu, S. Blatt, F. Huber, G. Ji and M. Greiner,Site-resolved imaging of a fermionic Mott insulator, Science351(6276), 953 (2016), doi:10.1126/science.aad9041

View on Pith doi:10.1126/science.aad9041 2016
[35]

A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kan´ asz-nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif and M. Greiner,A cold-atom Fermi–Hubbard an- tiferromagnet, Nature545(7655), 462 (2017), doi:10.1038/nature22362

View on Pith doi:10.1038/nature22362 2017
[36]

J. Koepsell, J. Vijayan, P. Sompet, F. Grusdt, T. A. Hilker, E. Demler, G. Salomon, I. Bloch and C. Gross,Imaging magnetic polarons in the doped Fermi–Hubbard model, Nature572(7769), 358 (2019), doi:10.1038/s41586-019-1463-1

View on Pith doi:10.1038/s41586-019-1463-1 2019
[37]

A. Bohrdt, C. S. Chiu, G. Ji, M. Xu, D. Greif, M. Greiner, E. Demler, F. Grusdt and M. Knap,Classifying snapshots of the doped Hubbard model with machine learning, Nat. Phys.15(9), 921 (2019), doi:10.1038/s41567-019-0565-x. 19 SciPost Physics Submission

View on Pith doi:10.1038/s41567-019-0565-x 2019
[38]

A. Bohrdt, L. Homeier, C. Reinmoser, E. Demler and F. Grusdt,Exploration of doped quantum magnets with ultracold atoms, Ann. Phys.435, 168651 (2021), doi:10.1016/j.aop.2021.168651, ArXiv: 2107.08043

View on Pith arXiv 2021
[39]

M. Xu, L. H. Kendrick, A. Kale, Y. Gang, C. Feng, S. Zhang, A. W. Young, M. Lebrat and M. Greiner,A neutral-atom Hubbard quantum simulator in the cryogenic regime, Nature642(8069), 909 (2025), doi:10.1038/s41586-025-09112-w

View on Pith doi:10.1038/s41586-025-09112-w 2025
[40]

L. H. Kendrick, A. Kale, Y. Gang, A. D. Deters, M. Lebrat, A. W. Young and M. Greiner,Pseudogap in a Fermi-Hubbard quantum simulator, doi:10.48550/arXiv.2509.18075, ArXiv:2509.18075 [cond-mat] (2025)

View on Pith Pith review doi:10.48550/arxiv.2509.18075 2025
[41]

W. S. Bakr, J. I. Gillen, A. Peng, S. F¨ olling and M. Greiner,A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice., Nature 462(7269), 74 (2009), doi:10.1038/nature08482

View on Pith doi:10.1038/nature08482 2009
[42]

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch and S. Kuhr,Single- atom-resolved fluorescence imaging of an atomic Mott insulator., Nature467(7311), 68 (2010), doi:10.1038/nature09378

View on Pith doi:10.1038/nature09378 2010
[43]

L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe and M. W. Zwierlein,Quantum-Gas Microscope for Fermionic Atoms, Phys. Rev. Lett.114(19), 193001 (2015), doi:10.1103/PhysRevLett.114.193001

View on Pith doi:10.1103/physrevlett.114.193001 2015
[44]

P. Rabl, A. J. Daley, P. O. Fedichev, J. I. Cirac and P. Zoller,Defect-Suppressed Atomic Crystals in an Optical Lattice, Phys. Rev. Lett.91(11), 110403 (2003), doi:10.1103/PhysRevLett.91.110403

View on Pith doi:10.1103/physrevlett.91.110403 2003
[45]

A. Kantian, A. J. Daley, P. T¨ orm¨ a and P. Zoller,Atomic lattice excitons: from conden- sates to crystals, New J. Phys.9(11), 407 (2007), doi:10.1088/1367-2630/9/11/407, Publisher: IOP Publishing

View on Pith doi:10.1088/1367-2630/9/11/407 2007
[46]

A. Kantian, A. J. Daley and P. Zoller,ηCondensate of Fermionic Atom Pairs via Adiabatic State Preparation, Phys. Rev. Lett.104(24), 240406 (2010), doi:10.1103/PhysRevLett.104.240406

View on Pith doi:10.1103/physrevlett.104.240406 2010
[47]

U. Schollw¨ ock,Schollw¨ ock, U. (2011). The density-matrix renormaliza- tion group in the age of matrix product states. Annals of Physics, 326(1), 96–192. https://doi.org/10.1016/j.aop.2010.09.012The density-matrix renormaliza- tion group in the age of matrix product states, Ann. Phys.326(1), 96 (2011), doi:10.1016/j.aop.2010.09.012, ISBN: 0003-4916

View on Pith doi:10.1016/j.aop.2010.09.012the 2011
[48]

H. Fehske, G. Hager and E. Jeckelmann,Metallicity in the half-filled Holstein-Hubbard model, Epl84(5) (2008), doi:10.1209/0295-5075/84/57001

View on Pith doi:10.1209/0295-5075/84/57001 2008
[49]

G. Bollmark, N. Laflorencie and A. Kantian,Dimensional crossover and phase tran- sitions in coupled chains : Density matrix renormalization group results, Phys. Rev. B102(195145), 1 (2020), doi:10.1103/PhysRevB.102.195145

View on Pith doi:10.1103/physrevb.102.195145 2020
[50]

J. Tempere and J. P. A. Devreese,Path-Integral Description of Cooper Pairing, In Superconductors - Materials, Properties and Applications. InTech, doi:10.5772/48458 (2012)

View on Pith doi:10.5772/48458 2012
[51]

A. Sewer, X. Zotos and H. Beck,Quantum Monte Carlo study of the three- dimensional attractive Hubbard model, Phys. Rev. B66(14), 140504 (2002), doi:10.1103/PhysRevB.66.140504. 20 SciPost Physics Submission

View on Pith doi:10.1103/physrevb.66.140504 2002
[52]

V. J. Emery and S. A. Kivelson,Importance of phase fluctuations in superconductors with small superfluid density, Nature374(6521), 434 (1995), doi:10.1038/374434a0, Publisher: Nature Publishing Group

View on Pith doi:10.1038/374434a0 1995
[53]

H. Nguyen, H. Shi, J. Xu and S. Zhang,CPMC-Lab: A Matlab package for Con- strained Path Monte Carlo calculations, Comput. Phys. Commun.185(12), 3344 (2014), doi:10.1016/j.cpc.2014.08.003, ISBN: 9783893368846 Publisher: Elsevier B.V

View on Pith doi:10.1016/j.cpc.2014.08.003 2014
[54]

M. Qin, H. Shi and S. Zhang,Coupling quantum Monte Carlo and independent- particle calculations: Self-consistent constraint for the sign problem based on the density or the density matrix, Phys. Rev B235119(23), 1 (2016), doi:10.1103/PhysRevB.94.235119

View on Pith doi:10.1103/physrevb.94.235119 2016
[55]

Y. Y. He, M. Qin, H. Shi, Z. Y. Lu and S. Zhang,Finite-temperature auxiliary- field quantum Monte Carlo: Self-consistent constraint and systematic approach to low temperatures, Phys. Rev. B99(4) (2019), doi:10.1103/PhysRevB.99.045108, ArXiv: 1811.07290

View on Pith Pith review arXiv 2019
[56]

S. R. White,Density matrix formulation for quantum renormalization groups, Phys- ical Review Letters69(19), 2863 (1992), doi:10.1103/PhysRevLett.69.2863

View on Pith doi:10.1103/physrevlett.69.2863 1992
[57]

S. R. White,Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B48(14), 10345 (1993), doi:10.1103/PhysRevB.48.10345

View on Pith doi:10.1103/physrevb.48.10345 1993
[58]

T. K¨ ohler and A. Kantian,Predicting critical temperature in quantum simulators for high-T c superconductivity: the matrix product state plus mean field approach, doi:10.5281/zenodo.20737857 (2026). 21

View on Pith doi:10.5281/zenodo.20737857 2026

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites as background 2 papers

The second strand are the recently proposed socalled mixD systems, bilayer systems in 1D and 2D in which singlefermion tunneling is only present within each layer, while interlayer coupling consists only of spin exchanges [ref] .

Strong pairing in mixed dimensional bilayer antiferromagnetic Mott insulators · Introduction · 0.90
The second strand are the recently proposed socalled mixD systems, bilayer systems in 1D and 2D in which singlefermion tunneling is only present within each layer, while interlayer coupling consists only of spin exchanges [ref] .

2602.10091 · Introduction · 0.90

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [19]

G. Bollmark, T. K¨ ohler, L. Pizzino, Y. Yang, J. S. Hofmann, H. Shi, S. Zhang, T. Giamarchi and A. Kantian,Solving 2D and 3D Lattice Models of Correlated Fermions—Combining Matrix Product States with Mean-Field Theory, Phys. Rev. X 13, 011039 (2023), doi:https://doi.org/10.1103/PhysRevX.13.011039

View on Pith doi:10.1103/physrevx.13.011039 2023

[2] [21]

G. Bollmark, T. K¨ ohler and A. Kantian,Resolving competition of charge density wave and superconducting phases using the matrix product state plus mean field algorithm, Physical Review B111(12), 125141 (2025), doi:10.1103/PhysRevB.111.125141

View on Pith doi:10.1103/physrevb.111.125141 2025

[3] [1]

H. J. Schulz,Superconductivity and Antiferromagnetism in the Two-Dimensional Hubbard Model: Scaling Theory, Europhys. Lett.4(5), 609 (1987), doi:10.1209/0295- 5075/4/5/016

View on Pith doi:10.1209/0295- 1987

[4] [2]

P. Lederer, G. Montambaux and D. Poilblanc,Antiferromagnetism and supercon- ductivity in a quasi two-dimensional electron gas. Scaling theory of a generic Hubbard model, J. Phys. France48(10), 1613 (1987), doi:10.1051/jphys:0198700480100161300

View on Pith doi:10.1051/jphys:0198700480100161300 1987

[5] [3]

G. Kotliar and J. Liu,Superexchange mechanism and d-wave superconductivity, Phys- ical Review B38(7), 5142 (1988), doi:10.1103/PhysRevB.38.5142

View on Pith doi:10.1103/physrevb.38.5142 1988

[6] [4]

M. Inui, S. Doniach, P. Hirschfeld and A. Ruckenstein,Coexistence of antiferromag- netism and superconductivity in a mean-field theory of high-T {c}superconductors, Phys. Rev. B37(4), 2320 (1988), doi:10.1103/PhysRevB.37.2320

View on Pith doi:10.1103/physrevb.37.2320 1988

[7] [5]

D. J. Scalapino,The case for pairing in the cuprate superconductors &2 ,,2, Phys. Reps.250, 329 (1995)

View on Pith 1995

[8] [6]

N. Furukawa, T. M. Rice and M. Salmhofer,Truncation of a Two-Dimensional Fermi Surface due to Quasiparticle Gap Formation at the Saddle Points, Phys. Rev. Lett. 81(15), 3195 (1998), doi:10.1103/PhysRevLett.81.3195

View on Pith doi:10.1103/physrevlett.81.3195 1998

[9] [7]

J. Orenstein,Advances in the Physics of High-Temperature Superconductivity, Science 288(5465), 468 (2000), doi:10.1126/science.288.5465.468

View on Pith doi:10.1126/science.288.5465.468 2000

[10] [8]

W. Metzner, J. Reiss and D. Rohe,Magnetic and superconducting correlations in the two-dimensional Hubbard model, physica status solidi (b)243(1), 46 (2006), doi:10.1002/pssb.200562445

View on Pith doi:10.1002/pssb.200562445 2006

[11] [9]

T. A. Maier, M. Jarrell and D. J. Scalapino,Pairing interaction in the two- dimensional Hubbard model studied with a dynamic cluster quantum Monte Carlo approximation, Phys. Rev. B74(9), 094513 (2006), doi:10.1103/PhysRevB.74.094513

View on Pith doi:10.1103/physrevb.74.094513 2006

[12] [10]

S. Raghu, S. A. Kivelson and D. J. Scalapino,Superconductivity in the repulsive Hubbard model: An asymptotically exact weak-coupling solution, Phys. Rev. B81(22), 224505 (2010), doi:10.1103/PhysRevB.81.224505. 17 SciPost Physics Submission

View on Pith doi:10.1103/physrevb.81.224505 2010

[13] [11]

P. W. Anderson,Twenty-five years of high-temperature superconductivity - A personal review, J. Phys.: Conf. Ser449(1) (2013), doi:10.1088/1742-6596/449/1/012001

View on Pith doi:10.1088/1742-6596/449/1/012001 2013

[14] [12]

T. A. Maier, M. Jarrell, T. C. Schulthess, P. R. C. Kent and J. B. White,Systematic Study of d -Wave Superconductivity in the 2D Repulsive Hubbard Model, Phys. Rev. Lett.95(23), 237001 (2005), doi:10.1103/PhysRevLett.95.237001

View on Pith doi:10.1103/physrevlett.95.237001 2005

[15] [13]

C. C. Chang and S. Zhang,Spin and charge order in the doped hubbard model: Long-wavelength collective modes, Phys. Rev. Lett.104(11), 1 (2010), doi:10.1103/PhysRevLett.104.116402

View on Pith doi:10.1103/physrevlett.104.116402 2010

[16] [14]

J. P. F. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M. Henderson, C. A. Jim´ enez-Hoyos, E. Kozik, X.-W. Liu et al.,Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms, Phys. Rev. X5(4), 041041 (2015), doi:10.1103/PhysRevX.5.041041, Publish...

View on Pith doi:10.1103/physrevx.5.041041 2015

[17] [15]

B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang and G. K.-L. Chan,Stripe order in the underdoped re- gion of the two-dimensional Hubbard model., Science358(6367), 1155 (2017), doi:10.1126/science.aam7127

View on Pith doi:10.1126/science.aam7127 2017

[18] [16]

M. Qin, C. M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollw¨ ock, S. R. White and S. Zhang,Absence of Superconductivity in the Pure Two-Dimensional Hubbard Model, Phys. Rev. X10(3), 31016 (2020), doi:10.1103/PhysRevX.10.031016, Publisher: American Physical Society

View on Pith doi:10.1103/physrevx.10.031016 2020

[19] [17]

H. Xu, H. Shi, E. Vitali, M. Qin and S. Zhang,Stripes and spin-density waves in the doped two-dimensional Hubbard model: Ground state phase diagram, Physical Review Research4(1), 013239 (2022), doi:10.1103/PhysRevResearch.4.013239

View on Pith doi:10.1103/physrevresearch.4.013239 2022

[20] [18]

H. Xu, C.-M. Chung, M. Qin, U. Schollw¨ ock, S. R. White and S. Zhang,Coexistence of superconductivity with partially filled stripes in the Hubbard model, Science384(6696), eadh7691 (2024), doi:10.1126/science.adh7691

View on Pith doi:10.1126/science.adh7691 2024

[21] [20]

S. Marten, G. Bollmark, T. K¨ ohler, S. R. Manmana and A. Kantian,Transient su- perconductivity in three-dimensional Hubbard systems by combining matrix-product states and self-consistent mean-field theory, SciPost Physics15(6), 236 (2023), doi:10.21468/SciPostPhys.15.6.236

View on Pith doi:10.21468/scipostphys.15.6.236 2023

[22] [22]

T. Giamarchi,Quantum Physics in One Dimension, Oxford University Press, 1st edn., ISBN 978-0-19-852500-4 (2003)

View on Pith 2003

[23] [23]

G. Karakonstantakis, E. Berg, S. R. White and S. A. Kivelson,Enhanced pairing in the checkerboard Hubbard ladder, Physical Review B - Condensed Matter and Materials Physics83(5), 054508 (2011), doi:10.1103/PhysRevB.83.054508. 18 SciPost Physics Submission

View on Pith doi:10.1103/physrevb.83.054508 2011

[24] [24]

S. Trebst, U. Schollw¨ ock, M. Troyer and P. Zoller,d-Wave Resonating Valence Bond States of Fermionic Atoms in Optical Lattices, Phys. Rev. Lett.96(25), 250402 (2006), doi:10.1103/PhysRevLett.96.250402

View on Pith doi:10.1103/physrevlett.96.250402 2006

[25] [25]

M. Dolfi, B. Bauer, S. Keller and M. Troyer,Pair correlations in doped Hubbard ladders, Phys. Rev. B92(19), 195139 (2015), doi:10.1103/PhysRevB.92.195139, Publisher: American Physical Society

View on Pith doi:10.1103/physrevb.92.195139 2015

[26] [26]

A. Bohrdt, L. Homeier, I. Bloch, E. Demler and F. Grusdt,Strong pairing in mixed dimensional bilayer antiferromagnetic Mott insulators, Nat. Phys. (2022), doi:10.1038/s41567-022-01561-8

View on Pith doi:10.1038/s41567-022-01561-8 2022

[27] [27]

S. Hirthe, T. Chalopin, D. Bourgund, P. Bojovi´ c, A. Bohrdt, E. Demler, F. Grusdt, I. Bloch and T. A. Hilker,Magnetically mediated hole pairing in fermionic ladders of ultracold atoms, Nature613(7944), 463 (2023), doi:10.1038/s41586-022-05437-y

View on Pith doi:10.1038/s41586-022-05437-y 2023

[28] [28]

H. Lange, A. Chen, A. Georges, F. Grusdt, A. Bohrdt and C. Roth,Simulating super- conductivity in mixed-dimensional$t \parallel$-${J}\parallel$-${J}\perp$bilayers with neural quantum states, doi:10.48550/ARXIV.2602.10091, Version Number: 1 (2026)

View on Pith doi:10.48550/arxiv.2602.10091 2026

[29] [29]

T. Kaneko, H. Sakakibara, M. Ochi and K. Kuroki,Pair correlations in the two-orbital Hubbard ladder: Implications on superconductivity in the bi- layer nickelate La$ 3$Ni$ 2$O$ 7$, Physical Review B109(4), 045154 (2024), doi:10.1103/PhysRevB.109.045154, ArXiv:2310.01952 [cond-mat.supr-con]

View on Pith Pith review arXiv 2024

[30] [30]

X.-Z. Qu, D.-W. Qu, J. Chen, C. Wu, F. Yang, W. Li and G. Su,Bilayer $t$-$J$-$J\perp$Model and Magnetically Mediated Pairing in the Pressurized Nickelate La$ 3$Ni$ 2$O$ 7$, Physical Review Letters132(3), 036502 (2024), doi:10.1103/PhysRevLett.132.036502, ArXiv:2307.16873 [cond-mat.str-el]

View on Pith Pith review arXiv 2024

[31] [31]

I. Bloch, J. Dalibard and W. Zwerger,Many-body physics with ultracold gases, Rev. Mod. Phys.80(3), 885 (2008), doi:10.1103/RevModPhys.80.885, Publisher: Ameri- can Physical Society

View on Pith doi:10.1103/revmodphys.80.885 2008

[32] [32]

M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif and M. Greiner,Site- resolved measurement of the spin-correlation function in the Fermi-Hubbard model, Science353(6305) (2016)

View on Pith 2016

[33] [33]

L. W. Cheuk, M. A. Nichols, K. R. Lawrence, M. Okan, H. Zhang, E. Khatami, N. Trivedi, T. Paiva, M. Rigol and M. W. Zwierlein,Observation of spatial charge and spin correlations in the 2D Fermi-Hubbard model, Science353(6305) (2016)

View on Pith 2016

[34] [34]

D. Greif, M. F. Parsons, A. Mazurenko, C. S. Chiu, S. Blatt, F. Huber, G. Ji and M. Greiner,Site-resolved imaging of a fermionic Mott insulator, Science351(6276), 953 (2016), doi:10.1126/science.aad9041

View on Pith doi:10.1126/science.aad9041 2016

[35] [35]

A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kan´ asz-nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif and M. Greiner,A cold-atom Fermi–Hubbard an- tiferromagnet, Nature545(7655), 462 (2017), doi:10.1038/nature22362

View on Pith doi:10.1038/nature22362 2017

[36] [36]

J. Koepsell, J. Vijayan, P. Sompet, F. Grusdt, T. A. Hilker, E. Demler, G. Salomon, I. Bloch and C. Gross,Imaging magnetic polarons in the doped Fermi–Hubbard model, Nature572(7769), 358 (2019), doi:10.1038/s41586-019-1463-1

View on Pith doi:10.1038/s41586-019-1463-1 2019

[37] [37]

A. Bohrdt, C. S. Chiu, G. Ji, M. Xu, D. Greif, M. Greiner, E. Demler, F. Grusdt and M. Knap,Classifying snapshots of the doped Hubbard model with machine learning, Nat. Phys.15(9), 921 (2019), doi:10.1038/s41567-019-0565-x. 19 SciPost Physics Submission

View on Pith doi:10.1038/s41567-019-0565-x 2019

[38] [38]

A. Bohrdt, L. Homeier, C. Reinmoser, E. Demler and F. Grusdt,Exploration of doped quantum magnets with ultracold atoms, Ann. Phys.435, 168651 (2021), doi:10.1016/j.aop.2021.168651, ArXiv: 2107.08043

View on Pith arXiv 2021

[39] [39]

M. Xu, L. H. Kendrick, A. Kale, Y. Gang, C. Feng, S. Zhang, A. W. Young, M. Lebrat and M. Greiner,A neutral-atom Hubbard quantum simulator in the cryogenic regime, Nature642(8069), 909 (2025), doi:10.1038/s41586-025-09112-w

View on Pith doi:10.1038/s41586-025-09112-w 2025

[40] [40]

L. H. Kendrick, A. Kale, Y. Gang, A. D. Deters, M. Lebrat, A. W. Young and M. Greiner,Pseudogap in a Fermi-Hubbard quantum simulator, doi:10.48550/arXiv.2509.18075, ArXiv:2509.18075 [cond-mat] (2025)

View on Pith Pith review doi:10.48550/arxiv.2509.18075 2025

[41] [41]

W. S. Bakr, J. I. Gillen, A. Peng, S. F¨ olling and M. Greiner,A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice., Nature 462(7269), 74 (2009), doi:10.1038/nature08482

View on Pith doi:10.1038/nature08482 2009

[42] [42]

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch and S. Kuhr,Single- atom-resolved fluorescence imaging of an atomic Mott insulator., Nature467(7311), 68 (2010), doi:10.1038/nature09378

View on Pith doi:10.1038/nature09378 2010

[43] [43]

L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe and M. W. Zwierlein,Quantum-Gas Microscope for Fermionic Atoms, Phys. Rev. Lett.114(19), 193001 (2015), doi:10.1103/PhysRevLett.114.193001

View on Pith doi:10.1103/physrevlett.114.193001 2015

[44] [44]

P. Rabl, A. J. Daley, P. O. Fedichev, J. I. Cirac and P. Zoller,Defect-Suppressed Atomic Crystals in an Optical Lattice, Phys. Rev. Lett.91(11), 110403 (2003), doi:10.1103/PhysRevLett.91.110403

View on Pith doi:10.1103/physrevlett.91.110403 2003

[45] [45]

A. Kantian, A. J. Daley, P. T¨ orm¨ a and P. Zoller,Atomic lattice excitons: from conden- sates to crystals, New J. Phys.9(11), 407 (2007), doi:10.1088/1367-2630/9/11/407, Publisher: IOP Publishing

View on Pith doi:10.1088/1367-2630/9/11/407 2007

[46] [46]

A. Kantian, A. J. Daley and P. Zoller,ηCondensate of Fermionic Atom Pairs via Adiabatic State Preparation, Phys. Rev. Lett.104(24), 240406 (2010), doi:10.1103/PhysRevLett.104.240406

View on Pith doi:10.1103/physrevlett.104.240406 2010

[47] [47]

U. Schollw¨ ock,Schollw¨ ock, U. (2011). The density-matrix renormaliza- tion group in the age of matrix product states. Annals of Physics, 326(1), 96–192. https://doi.org/10.1016/j.aop.2010.09.012The density-matrix renormaliza- tion group in the age of matrix product states, Ann. Phys.326(1), 96 (2011), doi:10.1016/j.aop.2010.09.012, ISBN: 0003-4916

View on Pith doi:10.1016/j.aop.2010.09.012the 2011

[48] [48]

H. Fehske, G. Hager and E. Jeckelmann,Metallicity in the half-filled Holstein-Hubbard model, Epl84(5) (2008), doi:10.1209/0295-5075/84/57001

View on Pith doi:10.1209/0295-5075/84/57001 2008

[49] [49]

G. Bollmark, N. Laflorencie and A. Kantian,Dimensional crossover and phase tran- sitions in coupled chains : Density matrix renormalization group results, Phys. Rev. B102(195145), 1 (2020), doi:10.1103/PhysRevB.102.195145

View on Pith doi:10.1103/physrevb.102.195145 2020

[50] [50]

J. Tempere and J. P. A. Devreese,Path-Integral Description of Cooper Pairing, In Superconductors - Materials, Properties and Applications. InTech, doi:10.5772/48458 (2012)

View on Pith doi:10.5772/48458 2012

[51] [51]

A. Sewer, X. Zotos and H. Beck,Quantum Monte Carlo study of the three- dimensional attractive Hubbard model, Phys. Rev. B66(14), 140504 (2002), doi:10.1103/PhysRevB.66.140504. 20 SciPost Physics Submission

View on Pith doi:10.1103/physrevb.66.140504 2002

[52] [52]

V. J. Emery and S. A. Kivelson,Importance of phase fluctuations in superconductors with small superfluid density, Nature374(6521), 434 (1995), doi:10.1038/374434a0, Publisher: Nature Publishing Group

View on Pith doi:10.1038/374434a0 1995

[53] [53]

H. Nguyen, H. Shi, J. Xu and S. Zhang,CPMC-Lab: A Matlab package for Con- strained Path Monte Carlo calculations, Comput. Phys. Commun.185(12), 3344 (2014), doi:10.1016/j.cpc.2014.08.003, ISBN: 9783893368846 Publisher: Elsevier B.V

View on Pith doi:10.1016/j.cpc.2014.08.003 2014

[54] [54]

M. Qin, H. Shi and S. Zhang,Coupling quantum Monte Carlo and independent- particle calculations: Self-consistent constraint for the sign problem based on the density or the density matrix, Phys. Rev B235119(23), 1 (2016), doi:10.1103/PhysRevB.94.235119

View on Pith doi:10.1103/physrevb.94.235119 2016

[55] [55]

Y. Y. He, M. Qin, H. Shi, Z. Y. Lu and S. Zhang,Finite-temperature auxiliary- field quantum Monte Carlo: Self-consistent constraint and systematic approach to low temperatures, Phys. Rev. B99(4) (2019), doi:10.1103/PhysRevB.99.045108, ArXiv: 1811.07290

View on Pith Pith review arXiv 2019

[56] [56]

S. R. White,Density matrix formulation for quantum renormalization groups, Phys- ical Review Letters69(19), 2863 (1992), doi:10.1103/PhysRevLett.69.2863

View on Pith doi:10.1103/physrevlett.69.2863 1992

[57] [57]

S. R. White,Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B48(14), 10345 (1993), doi:10.1103/PhysRevB.48.10345

View on Pith doi:10.1103/physrevb.48.10345 1993

[58] [58]

T. K¨ ohler and A. Kantian,Predicting critical temperature in quantum simulators for high-T c superconductivity: the matrix product state plus mean field approach, doi:10.5281/zenodo.20737857 (2026). 21

View on Pith doi:10.5281/zenodo.20737857 2026