AI 中文总结
该研究为O(n)普适族开发双轴插值框架，通过连接空间维度D和自旋分量数n的精确极限解进行插值。在空间轴和自旋轴上分别得到相关结果，确立其为临界现象的统一预测方法，还阐明了插值成功的条件。
AI 中文摘要
我们为O(n)普适族开发了一个双轴插值框架，将空间维度D和自旋分量数n视为连接精确极限解的独立连续参数。在空间轴上，在D = 2的昂萨格解和D→∞的平均场理论之间进行锚定，得到了3D伊辛临界耦合的封闭形式预测，与蒙特卡罗基准值K_c = 0.2204（基准值：0.22165）吻合良好，且无可调参数。威尔逊 - 费希尔约束多项式插值在D = 3时给出ν = 2/3，β = 31/96，η = 35/864，并在3≤D < 4范围内重现了共形引导结果。在自旋轴上，建立了必要的兼容性标准：双锚点插值仅对在锚点值之间单调变化的可观测量成功。临界耦合K_c(n)违反此标准，而关联长度指数ν(n)满足该标准。微扰1/n²展开得到ν(3) = 0.7493（基准值：0.7112），通过精确标度关系得到β(3) = 0.3797（基准值：0.3689）和γ(3) = 1.489（基准值：1.396），且无需引入额外参数。该框架自然扩展到非整数自旋，给出了O(2.5)普适类的预测ν(2.5) = 0.7143。这些结果将维度和自旋插值确立为一种统一且具有预测性的临界现象方法，同时阐明了插值成功的结构条件。
英文摘要
We develop a two-axis interpolation framework for the O$(n)$ universality family, treating the spatial dimension $D$ and the spin-component number $n$ as independent continuous parameters connecting exact limiting solutions. On the spatial axis, anchoring between the Onsager solution at $D=2$ and mean-field theory at $D\to\infty$ yields a closed-form prediction for the 3D Ising critical coupling that agrees well with Monte Carlo benchmarks $K_c = 0.2204$ (benchmark: $0.22165$) with no adjustable parameters. Wilson--Fisher-constrained polynomial interpolation gives $ν=2/3$, $β=31/96$, and $η=35/864$ at $D=3$ (benchmarks: $0.6299$, $0.3265$, $0.0362$), and reproduces conformal-bootstrap results across $3 \le D < 4$. On the spin axis, we establish a necessary compatibility criterion: two-anchor interpolation succeeds only for observables that vary monotonically between the anchor values. The critical coupling $K_c(n)$ violates this criterion because the Heisenberg value falls below the spherical limit, whereas the correlation-length exponent $ν(n)$ satisfies it. A perturbative $1/n^2$ expansion yields $ν(3) = 0.7493$ (benchmark: $0.7112$), and propagation through exact scaling relations gives $β(3) = 0.3797$ (benchmark: $0.3689$) and $γ(3) = 1.489$ (benchmark: $1.396$), without introducing additional parameters. The framework naturally extends to non-integer spin, producing the prediction $ν(2.5) = 0.7143$ for the O$(2.5)$ universality class. These results establish dimensional and spin interpolation as a unified and predictive approach to critical phenomena, while clarifying the structural conditions under which interpolation succeeds.
Comments19 pages, 4 figures
