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$d=2$维量子伊辛模型的块重正化：应用于纯铁磁体和随机铁磁体以及自旋玻璃

Block Renormalization for quantum Ising models in dimension $d=2$ : applications to the pure and random ferromagnet, and to the spin-glass

Cecile Monthus

备注：
v2=final version (21 pages, 6 figures)
发表信息：
J. Stat. Mech. (2015) P01023
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摘要：

对于量子伊辛链，Fernando-Pacheco [Phys.Rev.D19，3173（1979）]的自对偶块重整化过程可以精确地再现零温度临界点的位置和相关长度指数$ν=1$。最近，Miyazaki和Nishimori [Phys.Rev.E87，032154（2013）]已经提出通过在每个方向上连续应用Fernando-Pacheco过程来研究$d>1$维中的无序量子伊辛模型。为了避免他们的方法的方向的不等价性，我们在这里提出了一个替代的程序，$d$方向被视为在同一个立足点。对于纯模型，这导致$d=2$中的相关长度指数$ν\simeq 0.625$（与3D经典伊辛模型指数$ν\simeq 0.63$比较）和$ν\simeq 0.5018$（与4D经典伊辛模型平均场指数$ν=1/2$比较）。对于无序模型的尺寸$d=2$，无论是铁磁或自旋玻璃，数值应用重整化规则的线性尺寸$L=4096$的样品产生的过渡是由一个无限无序不动点，与激活指数$ψ\simeq 0.65$，典型的相关指数$ν_{typ} \simeq 0.44$和有限尺寸的相关指数$ν_{FS} \simeq 1.25$。我们讨论了与强无序重整化结果的异同。

英文摘要：

For the quantum Ising chain, the self-dual block renormalization procedure of Fernandez-Pacheco [Phys. Rev. D 19, 3173 (1979)] is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent $ν=1$. Recently, Miyazaki and Nishimori [Phys. Rev. E 87, 032154 (2013)] have proposed to study the disordered quantum Ising model in dimensions $d>1$ by applying the Fernandez-Pacheco procedure successively in each direction. To avoid the inequivalence of directions of their approach, we propose here an alternative procedure where the $d$ directions are treated on the same footing. For the pure model, this leads to the correlation length exponents $ν\simeq 0.625$ in $d=2$ (to be compared with the 3D classical Ising model exponent $ν\simeq 0.63$) and $ν\simeq 0.5018$ (to be compared with the 4D classical Ising model mean-field exponent $ν=1/2$). For the disordered model in dimension $d=2$, either ferromagnetic or spin-glass, the numerical application of the renormalization rules to samples of linear size $L=4096$ yields that the transition is governed by an Infinite Disorder Fixed Point, with the activated exponent $ψ\simeq 0.65$, the typical correlation exponent $ν_{typ} \simeq 0.44$ and the finite-size correlation exponent $ν_{FS} \simeq 1.25$. We discuss the similarities and differences with the Strong Disorder Renormalization results.

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