Evolution from intralayer to interlayer superconductivity in a bilayer $t$-$J$ model

Shou-Shu Gong; Wei-Qiang Chen; Xin Lu; Yuan Wan; Yuan Yang

REVIEW 4 major objections 5 minor 1 cited by

Evolution from intralayer to interlayer superconductivity in a bilayer $t$-$J$ model

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A bilayer $t$-$J$ ladder model shows that a large interlayer spin exchange $J_\perp$ drives an interlayer superconducting state across nearly the entire doping range studied, with the intralayer phase giving way either through a…

desk verdict Solid DMRG phase diagram for the bilayer t-J-J⊥ ladder; the load-bearing caveat is finite-size convergence of the CDW and SC boundaries, but the paper is worth refereeing. read the letter →

arxiv 2507.07545 v1 pith:6SWY5BL6 submitted 2025-07-10 cond-mat.str-el

Yuan Yang , Xin Lu , Yuan Wan , Wei-Qiang Chen , Shou-Shu Gong This is my paper · ORCID

classification cond-mat.str-el

keywords bilayert-JmodelinterlayersuperconductivityLuther-EmeryliquidLuttingerchargedensitywavematrixrenormalizationgrouptwo-legladdernickelatesuperconductors

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

This paper asks how superconductivity changes when two layers of a strongly correlated electron system are coupled by an interlayer magnetic exchange interaction $J_\perp$. Studying a bilayer two-leg $t$-$J$-$J_\perp$ ladder with density matrix renormalization group calculations, it claims that a sufficiently large $J_\perp$ always drives the system into an interlayer (ZZ) superconducting phase over the doping range $\delta = 1/8$ to $1/2$, with $\delta = 0.5$ as the only exception. The path depends on doping: coupling two Luther-Emery liquids at low doping passes through an intermediate charge-density-wave phase, while at higher doping the transition from intralayer to interlayer pairing is direct. In the Luttinger-liquid regime at intermediate $J_\perp$, the interlayer pairing correlation develops an enhanced power-law exponent $K^{zz}_{\rm SC} \sim 1$, indicating quasi-long-range interlayer superconducting order that coexists with surviving intralayer liquid features. If correct, the interlayer spin exchange alone is enough to stabilize interlayer superconductivity over a broad doping window, which is directly relevant to bilayer cuprates and nickelate superconductors.

What carries the argument

The central diagnostic objects are the spin-singlet pairing correlation functions $P^{yy}(r)$ and $P^{zz}(r)$ for in-plane and interlayer bonds, together with the spin, single-particle, density, and central-charge measurements that identify which modes are gapless. The argument classifies each phase by whether these correlations decay algebraically or exponentially: the interlayer ZZ superconducting phase is defined by algebraic $P^{zz}(r)$ decay with a small power exponent, a gapped single-particle Green's function $G(r)$, and a gapped spin correlation $F(r)$. The phase diagram is organized by starting from the known phases of the single-layer two-leg $t$-$J$ ladder at $t/J=3$ — Luther-Emery liquid, a CDW state at $\delta=1/4$, and Luttinger liquid — and tracking how $J_\perp$ couples two copies of those phases, using the central charge extracted from the entanglement entropy $S(x) = (c/6)\log[(L_x/\pi)\sin(\pi x/L_x)] + g$ to distinguish gapless from fully gapped states.

What would settle it

Perform DMRG at Lx = 64 and 96 with bond dimensions beyond 14000 and re-fit the power exponent $K^{zz}_{\rm SC}$ and the central charge $c$ in the CDW and intermediate-$J_\perp$ regions; if $K^{zz}_{\rm SC}$ drifts toward the free-fermion value 2 with growing length, or the CDW central charge rises away from 0, the claimed phase boundaries and the enhanced-pairing regime are not converged.

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Extended reading notes

Core claim

The central claim is that strong interlayer spin exchange $J_\perp$ acts as a universal driver of interlayer pairing in the bilayer two-leg $t$-$J$-$J_\perp$ model: at large $J_\perp$, the ground state becomes an interlayer (ZZ) spin-singlet superconductor for essentially every doping ratio from $\delta = 1/8$ to $\delta = 1/2$, except the special filling $\delta = 0.5$. Coupling two Luther-Emery liquid states produces either an intermediate fully gapped charge-density-wave phase at low doping, or a direct transition to the interlayer superconducting phase at higher doping such as $\delta = 0.375$, with no numerical evidence of intralayer and interlayer superconductivity coexisting. Coupling two Luttinger liquids at $\delta \approx 0.4583$ requires a larger $J_\perp$ to reach the interlayer superconducting phase, but at intermediate coupling $J_\perp/J = 1.0$ the interlayer pairing correlation $P^{zz}(r)$ decays algebraically with exponent $K^{zz}_{\rm SC} \simeq 1.14$, much stronger than the square of the single-particle Green's function, showing enhanced interlayer pairing while in-plane Luttinger-liquid features persist. The same interlayer superconducting phase appears when strongly coupling two three-leg ladders, indicating the mechanism is not special to the two-leg geometry.

Load-bearing premise

The whole phase diagram is inferred from numerical simulations of a 48-site open ladder, with algebraic versus exponential decay judged over distances up to about 30 sites; if those fitted exponents and central charges shift with system size or numerical bond dimension, the phase boundaries and the enhanced-pairing regime would move.

Editorial extensions

If this is right

At large $J_\perp$, the interlayer ZZ superconducting phase occupies essentially the whole doping range $1/8 \le \delta \le 1/2$ except $\delta = 0.5$, regardless of whether each layer starts as a Luther-Emery liquid or a Luttinger liquid.
When two Luther-Emery liquids are coupled at low doping, the evolution to the ZZ phase passes through a fully gapped CDW phase with near-zero central charge; at higher doping, such as $\delta = 0.375$, the transition is direct with no evidence of coexisting intralayer and interlayer superconductivity.
In the Luttinger-liquid regime at $\delta \approx 0.4583$, an intermediate $J_\perp/J \approx 1.0$ produces interlayer pairing with $K^{zz}_{\rm SC} \simeq 1.14$, an order stronger than the square of the single-particle Green's function while the in-plane liquid features remain stable.
Coupling two three-leg ladders with a strong $J_\perp$ also yields an interlayer superconducting phase at both $\delta = 0.375$ and $\delta = 0.5$, so the mechanism is not particular to the two-leg geometry.
At $\delta = 0.5$, where each two-leg layer is a C0S1 state, a large $J_\perp$ gaps the spin mode but the interlayer pairing correlation stays exponentially decaying, so no interlayer superconductivity forms at that special filling.

Reading between the lines

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

The paper leaves open whether the enhanced regime with $K^{zz}_{\rm SC} \sim 1$ survives in wider, more two-dimensional bilayer systems; if it does, the interlayer pairing susceptibility would diverge strongly at low temperature, a mechanism that could be tested in broader cylinders.
The claim that the bilayer CDW phase develops from the $\delta=1/4$ ladder CDW suggests a testable prediction: the CDW wavevector should lock to one hole per period and the phase should persist in wider ladders, but establishing that would require going beyond the presented two-leg data.
For the nickelate and bilayer-cuprate motivation, the result implies that pressure-induced enhancement of $J_\perp$ could switch the dominant pairing from intralayer to interlayer without any change in the intralayer Hamiltonian; that should be checked in a model that also includes interlayer hopping $t_\perp$, which this paper does not.
Because the paper does not resolve the weak-coupling limit $J_\perp \to 0$, the Kosterlitz-Thouless transition expected between the weakly coupled LEL and the CDW phase remains a natural target for future work with larger system sizes.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper maps out the J⊥–δ phase diagram of the bilayer two-leg t-J-J⊥ model and finds that a strong interlayer spin exchange drives interlayer (ZZ) pairing over a wide doping range, with an intermediate CDW phase at low doping and an enhanced interlayer pairing regime at intermediate J⊥ inside the TLL phase. That is genuinely new: prior ladder work focused near δ ≈ 0.5, and the CDW intermediate phase and the K_zz^SC ≈ 1 enhancement are not in the earlier literature. The DMRG is state-of-the-art for this problem — bond dimensions up to 14000, truncation errors ~1e-6, multiple correlation functions, central charge fits, and careful comparison with the single-layer baseline. The comparison of P_zz(r) against G^2(r)/4 is a good, honest diagnostic that distinguishes a true interlayer pairing enhancement from a trivial single-particle contribution. The three-leg check in Appendix B adds useful robustness.

The soft spots are real but not fatal. The phase diagram is built almost entirely from Lx=48 runs with open boundaries, and phase boundaries are drawn by distinguishing algebraic from exponential decay over distances up to about 30 sites. That is standard for DMRG ladders, but it does mean the CDW phase edges and the SC boundary at J⊥/J ≈ 0.8 carry an uncertainty that is not quantified by systematic finite-size scaling. The stress-test worry about the CDW phase at δ = 0.1875 is legitimate: the entanglement entropy shows strong oscillations, and the c ≈ 0 fit is not as clean as one would like. Still, the authors support the CDW assignment with density profiles, the K_c exponent from a Friedel-oscillation fit, and the exponential decay of P_yy(r); the evidence is consistent. The "always drives interlayer SC" claim slightly overreaches from discrete dopings, but the trend across δ = 1/8 to 1/2 is clear. The approximation t⊥ = 0 is a model choice that could be flagged more prominently, but it does not invalidate the phase diagram for the model they actually solved.

Who gets value from this? Anyone working on bilayer cuprates or nickelates who wants a systematic DMRG reference for how intralayer pairing crosses over to interlayer pairing in a minimal one-band model. It is not the last word — a follow-up with finite-size scaling at a few representative points and explicit t⊥ would make the boundaries firmer — but it is a serious numerical study with a clear central claim. It deserves peer review, and a good referee would ask for the convergence checks rather than rejection.

Recommendation: send it to review, with the finite-size caveat as the main requested revision.

Referee Report

4 major / 5 minor

Summary. This paper reports DMRG ground-state calculations for a bilayer two-leg t-J-J⊥ model with t/J=3, doping δ=1/8–1/2, and interlayer exchange J⊥/J≥0.1. The authors map a J⊥–δ phase diagram in which a sufficiently large J⊥ drives an interlayer (ZZ) superconducting phase, either through an intermediate CDW phase (at lower doping) or directly (at higher doping), while at intermediate J⊥ in the Luttinger-liquid regime the interlayer pairing correlation is enhanced with exponent K^zz_SC≈1. They also extend the strong-J⊥ ZZ-SC result to coupled three-leg ladders. The central physical claim is that strong interlayer spin exchange generically stabilizes quasi-long-range interlayer pairing in this model family over a broad doping range.

Significance. If the phase diagram is correct, the paper provides a concrete numerical scenario for how intralayer pairing evolves into interlayer pairing as J⊥ grows, which is directly relevant to bilayer cuprates and to bilayer nickelate models such as La3Ni2O7. The numerical work is careful in several respects: SU(2) and U(1) symmetries are implemented, bond dimensions up to 14000 SU(2) multiplets are kept, truncation errors are about 10^-6, multiple correlation functions are cross-checked, central charges are extracted from entanglement entropy, and the single-layer ladder results are explicitly compared with the weakly coupled bilayer. The three-leg extension in Appendix B is a useful robustness check. The main weakness is that the phase classification rests overwhelmingly on Lx=48 open-boundary runs without systematic finite-size scaling, and several phase assignments depend on distinguishing algebraic from exponential decay over distances of roughly 30 sites.

major comments (4)

[§III A 1, Figs. 2 and 3] The CDW phase at δ=0.1875 for J⊥/J≲0.6 is load-bearing for the claimed intermediate phase, but the supporting evidence is incomplete. The charge density profile n(x) shows strong oscillations, the in-plane pairing P_yy(r) is fit as exponential, and the central charge is extracted from an entanglement entropy S(x) that itself exhibits strong oscillations in Fig. 2(f); fitting Eq. (7) to an oscillating S(x) can bias c toward a small or zero value. No density-density correlation D(r), defined in Eq. (5), is shown for this doping, although D(r) would be the most direct diagnostic of CDW order. The authors should present D(r) for δ=0.1875, fit its decay, and show at least two system sizes (e.g., Lx=48 and 64) so that the CDW assignment is not a finite-size or boundary artifact.
[§III A 3 and Fig. 1(c)] The phase boundaries in Fig. 1(c), especially the CDW/LEL and LEL/ZZ-SC boundaries, are drawn from DMRG data almost entirely at Lx=48, and the algebraic-versus-exponential distinction is made over distances up to about 30 sites. At δ=0.3125, for example, the c≈2 versus c≈0 classification in Fig. 6 is made at isolated J⊥ values, and no finite-size or bond-dimension extrapolation is shown. Because the central 'always drives interlayer SC' claim requires that these boundaries not shift significantly with system size, the authors should provide representative finite-size scaling for at least one point in each phase (e.g., δ=0.1875 at J⊥/J=0.4 and 0.9, δ=0.3125 at J⊥/J=0.4, and δ=0.375 at J⊥/J=0.7 and 0.9).
[Abstract and §IV, Fig. 1(c)] The statement that a large J⊥ 'always' drives interlayer superconductivity over δ=1/8–1/2 extrapolates from a discrete set of doping values and J⊥ values (J⊥/J≥0.1 and typically up to J⊥/J≈2, with only a few larger values). The special case δ=0.5 is explicitly excepted, and the data at δ=0.1875, 0.25, 0.375, and 0.4583 do not cover the full interval. This wording should be tempered to reflect the computed doping points, or additional points (e.g., δ=0.3, δ=0.45) should be added to support the 'always' phrasing.
[§III B, Figs. 7 and 9] The enhanced interlayer pairing at intermediate J⊥ in the TLL regime, with K^zz_SC≈1.14 at J⊥/J=1.0, is a highlighted result, but its robustness is not established. The power-law fit of P_zz(r) is performed over r≲30 at Lx=48, and the comparison with G^2(r)/4 in Fig. 9 is suggestive rather than conclusive evidence of a distinct pairing phase. The authors should show that K^zz_SC is stable when Lx is increased to 64 (or more) and when the fit window is varied, and they should report the statistical or fit uncertainty. Without such checks, the claimed exponent K^zz_SC∼1 could be an artifact of the limited fitting range.

minor comments (5)

[Throughout] There are numerous typographical errors, for example 'bilayter' in the model section, 'dnesity' in the Fig. 5 caption, 'obtaind' in the Introduction, and inconsistent spacing in 't-J-J⊥'. These should be corrected.
[Fig. 1(c)] The caption states that circles mark the CDW phase and rhombuses denote the LEL phase, but the legend in the figure is not fully legible in the current rendering; the symbols and their phase labels should be clarified.
[Eq. (7) and Fig. 2(f)] The paper does not specify the fitting window used for the central charge extraction in Eq. (7), nor how the oscillations in S(x) are handled. A brief description of the fitting procedure and the uncertainty in c would improve reproducibility.
[§III A 1] The sentence 'The two coupled LEL states are gapped at either infinitesimal J⊥ or a very small J⊥/J<0.1' is ambiguous, since the manuscript does not study J⊥/J<0.1; this should be rephrased to indicate that the CDW phase appears already at the smallest studied J⊥/J=0.1.
[Appendix A] The discussion of δ=0.5 is a useful exception, but the claim that P_zz(r) remains exponential at large J⊥ would be more convincing if the corresponding correlation length were shown to remain finite as Lx grows; the current figure shows only one system size.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the bilayer phase diagram is built from new DMRG data, and the only self-citation (ref [89]) supplies a non-load-bearing single-layer baseline.

full rationale

The central claims, that large J_perp drives interlayer ZZ-SC, that a CDW intermediate phase appears at low doping, and that K^zz_SC about 1 signals enhanced interlayer pairing in the TLL regime, are supported by DMRG correlation functions, central-charge fits, and bond-dimension checks presented in Figs. 2-9. These are newly computed bilayer quantities, not derived by construction from any fitted parameter or from an input result. The single-layer phase diagram of ref [89], which shares co-authors with the present paper, is used as a reference baseline, but the paper also reproduces the single-layer LEL behavior directly in Figs. 3 and 5, so the bilayer conclusions do not reduce to that citation. No equation in the paper defines the target phase in terms of the input, and no fitted quantity is renamed as a prediction. The load-bearing premise is numerical rather than definitional: phases are assigned at Lx=48 by distinguishing algebraic from exponential decay and by extracting c from Eq. (7), which is a finite-size-convergence risk rather than a circularity. Under the stated rules this is a normal honest non-finding, with one minor self-citation that is not load-bearing.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No model parameters are fitted in this work; t/J = 3 and the doping range are fixed inputs, and the fitted correlation exponents are diagnostics rather than free parameters. The main axioms are the validity of the t-J-J⊥ model as a description of the target materials, the single-layer ladder phase diagram taken from prior work by the same group, and the convergence of finite-size DMRG data. No invented entities are introduced.

assumptions (5)

domain assumption The bilayer t-J-J⊥ model with no double occupancy captures the essential low-energy physics of bilayer cuprates and La3Ni2O7.
The introduction motivates the model via these materials, but no derivation is given in this paper; it relies on earlier strong-coupling derivations (refs [25,68,69,74,81]).
domain assumption The single-layer two-leg t-J ladder at t/J = 3 has the phase diagram used here (LEL, CDW at δ=1/4, TLL, C0S1 at δ=1/2).
Taken from ref [89] by the same group; used as the baseline to interpret the bilayer results.
domain assumption DMRG results at Lx = 48-64 with bond dimensions 6000-14000 SU(2) multiplets are converged enough to classify phases.
The authors show truncation error ~1e-6 and bond-dimension checks, but no systematic finite-size scaling to Lx -> infinity; phase boundaries are read off from these finite systems.
domain assumption Power-law vs exponential decay of correlation functions over distances up to ~30 sites can be reliably distinguished from the data.
Used throughout Sec. III to identify SC, CDW, LEL, TLL phases; no error bars on fitted exponents.
standard math Conformal field theory formula for entanglement entropy (Eq. 7) applies to the gapless phases studied.
Used to extract central charge c from the dependence of entanglement entropy on subsystem size.

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Cite this review

Pith. "Pith review of Evolution from intralayer to interlayer superconductivity in a bilayer $t$-$J$ model." pith.science (2026). https://pith.science/paper/6SWY5BL6

@misc{pith2026250707545,
  author       = {Pith},
  title        = {Pith review of: Evolution from intralayer to interlayer superconductivity in a bilayer $t$-$J$ model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6SWY5BL6}},
  note         = {Machine review of arXiv:2507.07545}
}

abstract

Motivated by the bilayer cuprate superconductors and nickelate superconductor La$_3$Ni$_2$O$_7$, we investigate the evolution from intralayer to interlayer superconductivity based on a bilayer two-leg $t$-$J$-$J_{\bot}$ model, where $t$ is the in-plane electron hopping, $J$ is the in-plane spin interaction, and $J_{\bot}$ is the inter-plane spin interaction. By means of the density matrix renormalization group calculations, we obtain the quantum phase diagram of the system by tuning $J_{\bot}$ in a large doping range $\delta = 1/8 - 1/2$. We find that a large $J_{\bot}$ can always drive an interlayer superconductivity by coupling the two layers in both the Luther-Emery liquid and Luttinger liquid states. By coupling two Luther-Emery liquid states, the in-plane superconductivity evolves to inter-plane superconductivity either through an intermediate charge density wave (CDW) phase or directly, depending on doping ratio. This emergent CDW phase, which exists over a finite doping range, appears to develop from the CDW state of the two-leg ladder at $\delta = 1/4$. By coupling two Luttinger liquids, the in-plane Luttinger liquids show a transition to the inter-plane superconducting phase at large $J_{\bot}$, as reported in previous literature. Interestingly, in the intermediate $J_{\bot}$ regime we find that while the in-plane Luttinger-liquid features remain stable, the inter-plane superconductivity can develop an enhanced quasi-long-range order with the power exponent $K^{zz}_{\rm SC} \sim 1$. At last, we show that the interlayer superconductivity is also stable by coupling the bilayer three-leg $t$-$J$ ladders by a strong $J_{\bot}$ interaction, from both the Luther-Emery liquid and Luttinger-liquid states.

Figures

Figures reproduced from arXiv: 2507.07545 by the authors.

**Figure 2.** FIG. 2. Measurements of the bilayer two-leg [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Comparison of (a) local charge density [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4. Measurements of the bilayer two-leg [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Comparison of (a) local charge dnesity [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Central charge extracted from the entanglement entr [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Measurements of the bilayer two-leg [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9. Comparison of the inter-plane pairing correlation [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10. Measurements of the bilayer two-leg [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11. Correlation functions of the three-leg [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Filling and Interlayer Superexchange Control Superconductivity in La$_3$Ni$_2$O$_7$
cond-mat.supr-con 2026-03 unverdicted novelty 5.0 of 10

The superconducting T_c of La3Ni2O7 is controlled by the d_x2-y2 orbital filling and the interlayer magnetic exchange J_perp, so clean electron doping should raise T_c.

Reference graph

Works this paper leans on

108 extracted references · 71 canonical work pages · cited by 1 Pith paper

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Evolution to the interlayer SC with an intermediate CDW ph ase We begin by studying the bilayer t-J-J⊥ model at δ =
[2]

The two-leg t-J ladder at this doping level is the LEL [89]

1875. The two-leg t-J ladder at this doping level is the LEL [89]. In Fig. 2, we present the DMRG results of the t-J- J⊥ model with growing J⊥. The charge density proﬁle n(x) exhibits a strong CDW oscillation at 0. 1 ≤ J⊥/J ≲ 0. 6, but is strongly suppressed at larger J⊥ [Fig. 2(a)], which suggests a possible CDW order at small J⊥. In Figs. 2(b) and 2(c),...
[3]

375, the phase diagram with growing J⊥ shows a notable differ- ence from that at δ = 0

Direct transition from in-plane to inter-plane SC phase When coupling two single-layer LEL states at δ = 0 . 375, the phase diagram with growing J⊥ shows a notable differ- ence from that at δ = 0 . 1875, as illustrated in Fig. 1(c). The system remains in the LEL phase up to J⊥/J ≈ 0. 6. For the stronger J⊥, the system has a transition to the ZZ-SC phase. ...
[4]

1875 and 0

Mapping out the phase diagram Since the bilayer model at δ = 0 . 1875 and 0. 375 hosts dif- ferent ground states at small J⊥, we further investigate the doping range of δ = 0 . 125 − 0. 375 with growing J⊥. As we have shown in Fig. 3(c) and Fig. 5(c), here the central charge could be an effective diagnosis of the different stat es, thus we ﬁt the central ...
[5]

2, 0. 4, 0. 6 are presented in Fig
[6]

For δ = 0 . 125, 0. 1875, and 0. 25, the ﬁtted central charge is close to zero, indicating a fully gapped state. For δ = 0 . 3125, the obtained central charge c ∼ 2 is similar to δ = 0 . 375. Thus, we can identify the CDW phase and the coupled LEL phase (C2S0) as shown in Fig. 1(c). Notice that the two-leg t-J ladder at δ = 0 . 25 is a fully gapped CDW st...
[7]

Below J⊥/J ∼ 1

5. Below J⊥/J ∼ 1. 5, both G(r) and F (r) exhibit power- law decay, with the power exponents KG ∼ 1 and KF ∼ 2, respectively [Figs. 7(b) and 7(c)]. Above this J⊥ value, G(r) and F (r) clearly become exponential decay. Meanwhile, den- 7 5 10 15 20 25 x 0.56 0.58 0.60 0.62 0.64 0.66 0.68 n(x) (a) δ = 0.375 /uni00000036/uni0000004c/uni00000051/uni0000004a/un...
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14 at J⊥/J = 1 . 0. In Fig. 9, we further compare P zz (r) and the single-particle correlation square G2(r)/ 4. For a weak interlayer coupling such as J⊥/J = 0 . 1 and 0. 25, we ﬁnd P zz (r) ≈ G2(r)/ 4 indicating that the power-law decay of P zz (r) is solely from the contribution of G(r). With in- creasing J⊥, P zz (r) is enhanced and becomes stronger th...

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Evolution to the interlayer SC with an intermediate CDW ph ase We begin by studying the bilayer t-J-J⊥ model at δ =

[2] [2]

The two-leg t-J ladder at this doping level is the LEL [89]

1875. The two-leg t-J ladder at this doping level is the LEL [89]. In Fig. 2, we present the DMRG results of the t-J- J⊥ model with growing J⊥. The charge density proﬁle n(x) exhibits a strong CDW oscillation at 0. 1 ≤ J⊥/J ≲ 0. 6, but is strongly suppressed at larger J⊥ [Fig. 2(a)], which suggests a possible CDW order at small J⊥. In Figs. 2(b) and 2(c),...

[3] [3]

375, the phase diagram with growing J⊥ shows a notable differ- ence from that at δ = 0

Direct transition from in-plane to inter-plane SC phase When coupling two single-layer LEL states at δ = 0 . 375, the phase diagram with growing J⊥ shows a notable differ- ence from that at δ = 0 . 1875, as illustrated in Fig. 1(c). The system remains in the LEL phase up to J⊥/J ≈ 0. 6. For the stronger J⊥, the system has a transition to the ZZ-SC phase. ...

[4] [4]

1875 and 0

Mapping out the phase diagram Since the bilayer model at δ = 0 . 1875 and 0. 375 hosts dif- ferent ground states at small J⊥, we further investigate the doping range of δ = 0 . 125 − 0. 375 with growing J⊥. As we have shown in Fig. 3(c) and Fig. 5(c), here the central charge could be an effective diagnosis of the different stat es, thus we ﬁt the central ...

[5] [5]

2, 0. 4, 0. 6 are presented in Fig

[6] [6]

For δ = 0 . 125, 0. 1875, and 0. 25, the ﬁtted central charge is close to zero, indicating a fully gapped state. For δ = 0 . 3125, the obtained central charge c ∼ 2 is similar to δ = 0 . 375. Thus, we can identify the CDW phase and the coupled LEL phase (C2S0) as shown in Fig. 1(c). Notice that the two-leg t-J ladder at δ = 0 . 25 is a fully gapped CDW st...

[7] [7]

Below J⊥/J ∼ 1

5. Below J⊥/J ∼ 1. 5, both G(r) and F (r) exhibit power- law decay, with the power exponents KG ∼ 1 and KF ∼ 2, respectively [Figs. 7(b) and 7(c)]. Above this J⊥ value, G(r) and F (r) clearly become exponential decay. Meanwhile, den- 7 5 10 15 20 25 x 0.56 0.58 0.60 0.62 0.64 0.66 0.68 n(x) (a) δ = 0.375 /uni00000036/uni0000004c/uni00000051/uni0000004a/un...

[8] [8]

14 at J⊥/J = 1 . 0. In Fig. 9, we further compare P zz (r) and the single-particle correlation square G2(r)/ 4. For a weak interlayer coupling such as J⊥/J = 0 . 1 and 0. 25, we ﬁnd P zz (r) ≈ G2(r)/ 4 indicating that the power-law decay of P zz (r) is solely from the contribution of G(r). With in- creasing J⊥, P zz (r) is enhanced and becomes stronger th...