Evolution from intralayer to interlayer superconductivity in a bilayer t-J-J_{\bot} ladder

摘要

该研究以双层t-J-J⊥梯子为理论模型（受双层铜氧化物超导体和La₃Ni₂O₇启发），其中t为面内电子跃迁、J为面内自旋交换、J⊥为层间自旋交换；作者采用密度矩阵重整化群方法，在δ=1/8–1/2的大掺杂范围内调节J⊥，并以自旋单态配对关联为主要判据，构建了双层两腿梯子的量子相图。结果表明，足够大的J⊥可在Luther-Emery液体和Luttinger液体两类母态中驱动层间超导：耦合两个Luther-Emery液体时，面内超导可经有限掺杂区间的中间电荷密度波相或直接转变为层间超导，该CDW相似由δ=1/4时两腿梯子的CDW态发展而来；耦合两个Luttinger液体时，大J⊥使体系进入层间超导，而在中间J⊥下，面内Luttinger液体特征保持稳定，层间超导则呈现幂指数Kzz_SC~1的增强准长程有序。三层梯子结果进一步表明，强J⊥可在更宽掺杂范围稳定层间超导，为理解双层铜氧化物和镍酸盐中的层间配对提供了数值依据。

材料

• 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

方法

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

关键词

• 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

亮点

• 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.

结论

• 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 Kzz_SC ≈ 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.

主要论断

• A sufficiently large interlayer spin exchange J_perp always drives interlayer superconductivity when coupling two Luther-Emery liquid or two Luttinger liquid layers.
  • 证据: DMRG results show inter-plane ZZ-SC at large J_perp for both LEL and TLL parent states,interlayer pairing correlation decays algebraically with small power exponent at large J_perp
• Coupling two Luther-Emery liquids, in-plane superconductivity evolves to interlayer superconductivity either through an intermediate CDW phase or directly, depending on doping.
  • 证据: at delta=1/8 a CDW state with central charge near zero appears at small J_perp before ZZ-SC,at delta=1/6 the LEL persists up to intermediate J_perp 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.
  • 证据: 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 J_perp coupling two Luttinger liquids, in-plane Luttinger-liquid features remain stable while interlayer superconductivity develops enhanced quasi-long-range order with K^zz_SC ≈ 1.
  • 证据: interlayer pairing correlation becomes stronger than the square of single-particle correlation at intermediate J_perp,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 J_perp over a wide doping range.
  • 证据: bilayer three-leg ladders at delta=1/6 and 1/4 show ZZ-SC at strong J_perp,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.
  • 证据: at delta=1/4 spin correlation becomes exponential while pairing correlations remain exponential at large J_perp,central charge changes from 1 to 0 upon coupling

研究流程

• model_definition — The bilayer t-J-J_perp ladder captures the competition between intralayer and interlayer pairing relevant to bilayer cuprates and nickelates.
  • 材料: bilayer t-J-J_perp ladder (two-leg and three-leg, finite width)
  • 方法: Hamiltonian construction with in-plane hopping t, in-plane spin exchange J, interlayer spin exchange J_perp; no-double-occupancy constraint
  • 观察: 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.
  • 材料: bilayer two-leg and three-leg t-J-J_perp ladders
  • 方法: density matrix renormalization group (DMRG); SU(2) spin-rotational and U(1) charge symmetries
  • 观察: accurate ground-state wavefunctions obtained across doping and J_perp
• correlation_measurement — Decay behavior of pairing, density, spin, and single-particle correlations plus fitted central charge identify the quantum phases.
  • 材料: ground-state wavefunctions
  • 方法: 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
  • 观察: 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 J_perp, including CDW, coupled LEL, and interlayer ZZ-SC phases.
  • 材料: bilayer two-leg ladders over doping delta = 1/8 - 1/2
  • 方法: tuning J_perp at fixed doping; fitting central charge at various dopings
  • 观察: CDW states at small J_perp for delta = 1/8, 1/6, 1/5, 1/4 (central charge near zero); coupled LEL (C2S0) at larger doping; ZZ-SC at large J_perp
• interpretation — Strong J_perp robustly stabilizes interlayer superconductivity in wider bilayer ladders, connecting to La₃Ni₂O₇ and bilayer cuprates.
  • 材料: DMRG phase diagram and correlation exponents
  • 方法: comparison with single-layer two-leg ladder phases; comparison with bilayer three-leg and four-leg results
  • 观察: interlayer SC stable in three-leg and four-leg ladders at strong J_perp; no evidence of phase separation