Source capture
Authors Yuan Yang, Xin Lu, Yuan Wan, Wei-Qiang Chen, Shou-Shu Gong
arXiv / Publisher https://arxiv.org/abs/2507.07545
PDF https://arxiv.org/pdf/2507.07545
Relevance score 4.833
Primary category cond-mat.str-el
Published 2026-11-06
Research paradigm Theoretical
Sample form Unknown

Summary

This study uses the bilayer t-J-J⊥ ladder as a theoretical model (inspired by bilayer cuprate superconductors and La3Ni2O7), where t is the in-plane electron hopping, J is the in-plane spin exchange, and J⊥ is the interlayer spin exchange; using the density matrix renormalization group method, it tunes J⊥ over a wide doping range δ=1/8–1/2 and, with spin-singlet pairing correlations as the main criterion, constructs the quantum phase diagram of the bilayer two-leg ladder. The results show that sufficiently large J⊥ can drive interlayer superconductivity in two parent states, the Luther-Emery liquid and the Luttinger liquid: when two Luther-Emery liquids are coupled, in-plane superconductivity can transform into interlayer superconductivity either through an intermediate charge-density-wave (CDW) phase over a finite doping interval or directly, and this CDW phase likely develops from the CDW state of the two-leg ladder at δ=1/4; when two Luttinger liquids are coupled, large J⊥ drives the system into interlayer superconductivity, whereas at intermediate J⊥ the in-plane Luttinger-liquid features remain stable and interlayer superconductivity exhibits enhanced quasi-long-range order with power-law exponent KzzSC~1. Results for the three-leg ladder further indicate that strong J⊥ can stabilize interlayer superconductivity over a wider doping range, providing a numerical basis for understanding interlayer pairing in bilayer cuprates and nickelates.

Materials

  • bilayer t-J-J⊥ two-leg ladders
  • bilayer t-J-J⊥ three-leg ladders
  • bilayer t-J-J⊥ four-leg ladders
  • La3Ni2O7
  • bilayer cuprate superconductors

Methods

  • DMRG
  • SU(2) spin-rotational symmetry
  • U(1) charge symmetry
  • correlation function calculations
  • entanglement entropy central charge fitting

Keywords

  • interlayer superconductivity
  • intralayer superconductivity
  • luther emery liquid
  • luttinger liquid
  • charge density wave
  • spin singlet pairing
  • quasi long range order
  • interlayer spin exchange
  • quantum phase diagram
  • c2s0 state
  • c2s2 state
  • c0s1 state

Highlights

  • A rich quantum phase diagram of bilayer two-leg t-J-J⊥ ladders is mapped out by tuning J⊥ over a large doping range δ=1/8–1/2.
  • An intermediate CDW phase emerges between in-plane and inter-plane superconductivity at smaller doping, originating from the δ=1/4 CDW state of the two-leg ladder.
  • At intermediate J⊥, interlayer pairing correlation becomes stronger than the single-particle correlation square, indicating emergent inter-plane SC while in-plane Luttinger-liquid features persist.
  • The bilayer four-leg ladder at δ=1/4 and strong J⊥ shows the same ZZ-SC behavior, demonstrating robust inter-plane superconductivity in wider systems.
  • No evidence of phase separation is found in the studied parameter region despite the relatively large J⊥.
  • The results suggest a potential connection to the strong interlayer coupling of La3Ni2O7 under pressure, consistent with a ZZ-SC phase near δ=1/4 with dominant interlayer orbital pairing.

Conclusions

  • A large interlayer spin exchange J⊥ can always drive 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 phase or directly, depending on doping ratio.
  • The emergent CDW phase exists over a finite doping range and appears to develop from the CDW state of the two-leg ladder at δ=1/4.
  • By coupling two Luttinger liquids, the in-plane Luttinger liquids show a transition to the inter-plane superconducting phase at large J⊥.
  • In the intermediate J⊥ regime, the in-plane Luttinger-liquid features remain stable while the inter-plane superconductivity develops an enhanced quasi-long-range order with power exponent KzzSC ≈ 1.
  • The interlayer superconductivity is also stable by coupling bilayer three-leg t-J ladders with a strong J⊥ interaction, from both the Luther-Emery liquid and Luttinger-liquid states.
  • In the direct evolution from in-plane to inter-plane SC, the DMRG data find no evidence of a coexistence regime of in-plane and inter-plane superconductivity.
  • The absence of superconductivity at δ=1/4 in the bilayer two-leg system is a special consequence of coupling two C0S1 states, rather than a generic feature of wider bilayer ladders.

Main claims

  • A sufficiently large interlayer spin exchange Jperp always drives interlayer superconductivity when coupling two Luther-Emery liquid or two Luttinger liquid layers.
    • Evidence: DMRG results show inter-plane ZZ-SC at large Jperp for both LEL and TLL parent states,interlayer pairing correlation decays algebraically with small power exponent at large Jperp
  • Coupling two Luther-Emery liquids, in-plane superconductivity evolves to interlayer superconductivity either through an intermediate CDW phase or directly, depending on doping.
    • Evidence: at delta=1/8 a CDW state with central charge near zero appears at small Jperp before ZZ-SC,at delta=1/6 the LEL persists up to intermediate Jperp then transitions directly to ZZ-SC,fitted central charge approaches 2 for the C2S0 state at delta=1/6
  • The emergent CDW phase develops from the CDW state of the two-leg ladder at delta=1/4 and extends over a finite doping range.
    • Evidence: CDW pattern shows one hole per CDW period persisting at studied dopings,central charge near zero at delta=1/8, 1/6, 1/5, 1/4
  • At intermediate Jperp coupling two Luttinger liquids, in-plane Luttinger-liquid features remain stable while interlayer superconductivity develops enhanced quasi-long-range order with KzzSC ≈ 1.
    • Evidence: interlayer pairing correlation becomes stronger than the square of single-particle correlation at intermediate Jperp,power exponent of interlayer pairing correlation decreases to small values,central charge approaches 2 supporting C2S2 state
  • Interlayer superconductivity is also stable in bilayer three-leg and four-leg ladders under strong Jperp over a wide doping range.
    • Evidence: bilayer three-leg ladders at delta=1/6 and 1/4 show ZZ-SC at strong Jperp,bilayer four-leg ladder at delta=1/4 shows power-law interlayer pairing and exponential in-plane pairing
  • The absence of superconductivity at delta=1/4 in the bilayer two-leg system is a special consequence of coupling two C0S1 states rather than a generic feature of wider ladders.
    • Evidence: at delta=1/4 spin correlation becomes exponential while pairing correlations remain exponential at large Jperp,central charge changes from 1 to 0 upon coupling

Workflow

  • model_definition — The bilayer t-J-Jperp ladder captures the competition between intralayer and interlayer pairing relevant to bilayer cuprates and nickelates.
    • Materials: bilayer t-J-Jperp ladder (two-leg and three-leg, finite width)
    • Methods: Hamiltonian construction with in-plane hopping t, in-plane spin exchange J, interlayer spin exchange Jperp; no-double-occupancy constraint
    • Observations: model motivated by bilayer cuprates and La3Ni2O7
  • ground_state_calculation — DMRG with SU(2) and U(1) symmetries yields converged ground states of the bilayer ladders.
    • Materials: bilayer two-leg and three-leg t-J-Jperp ladders
    • Methods: density matrix renormalization group (DMRG); SU(2) spin-rotational and U(1) charge symmetries
    • Observations: accurate ground-state wavefunctions obtained across doping and Jperp
  • correlation_measurement — Decay behavior of pairing, density, spin, and single-particle correlations plus fitted central charge identify the quantum phases.
    • Materials: ground-state wavefunctions
    • Methods: charge density profile (column-averaged); spin-spin correlation; single-particle Green's function; density-density correlation; spin-singlet pairing correlation (in-plane and interlayer); bipartite entanglement entropy
    • Observations: power-law vs exponential decay of correlations distinguishes LEL, TLL, CDW, and ZZ-SC phases; central charge fitted from entanglement entropy
  • phase_diagram_construction — A rich quantum phase diagram emerges with doping and Jperp, including CDW, coupled LEL, and interlayer ZZ-SC phases.
    • Materials: bilayer two-leg ladders over doping delta = 1/8 - 1/2
    • Methods: tuning Jperp at fixed doping; fitting central charge at various dopings
    • Observations: CDW states at small Jperp for delta = 1/8, 1/6, 1/5, 1/4 (central charge near zero); coupled LEL (C2S0) at larger doping; ZZ-SC at large Jperp
  • interpretation — Strong Jperp robustly stabilizes interlayer superconductivity in wider bilayer ladders, connecting to La3Ni2O7 and bilayer cuprates.
    • Materials: DMRG phase diagram and correlation exponents
    • Methods: comparison with single-layer two-leg ladder phases; comparison with bilayer three-leg and four-leg results
    • Observations: interlayer SC stable in three-leg and four-leg ladders at strong Jperp; no evidence of phase separation