w********i
发帖数: 795 | 1 自从我验算了三维海森堡模型后，我的导师又让我验证二维海森堡模型，
理论上说二维的经典海森堡模型（反铁齿） Tc应当收敛到零。但是怎么才能证明我的
收敛就是正确的收
敛呢？这真是一个伤脑筋的问题。请问哪位同学有相关的计算经验，帮我看看我的计算
结果对不对，或
者有什么文献有过类似记录呢？
我计算的30X30晶格，Jx=Jy=Jz=1.0
计算的magnetization 的susceptibilities 发现这个Tc是0.7左右。
我还算了6X6的晶格，Tc是1.2 左右。
请问大家，我这个结果正确吗？一般要到多大晶格，才能明显看出Tc衰减到零附近？ |
s*s
发帖数: 100 | 2 I don't think it is easy to see T_c go to zero in 2D. In finite systems, T_c
is defined as the temperature at which the correlation length l_c reaches
the size of the whole lattice. In 2D Heisenberg model, the correlation
length l_c scales like c_1*(J/T)exp(c_2*J/T) (c1 and c2 are dimensionless
constants), which increases extremely fast as T->0. This means your system
size need to be enlarged very much to just see a little decrease in T_c.
What method do you use, exact diagonalization? In that case it is impossible
to go to large systems especially in 3D. |
w********i
发帖数: 795 | 3
systems, T_c
reaches
dimensionless
system
T_c.
impossible
非常感谢您的讲解，我用的只是最简单的经典海森堡模型，现在我算到了80X80的晶格
了，Tc的下降
速度成加快趋势，达到0.5 左右。
我听了您的解释，应该说晶格还不够大吧。
我找了一位同学，对比了10X10晶格的结果，证明我算的结果也是正确的。
我查了很多文献，发现有个叫做二维T_SK（ Stanley kaplan 1966 PRL)的相变温度，
对于二
维系统，该理论是正确的吗？只是一个近似结果？我用这篇文章去问一位博士后，被骂
了一顿，说我
在搞伪科学。
【在 s*s 的大作中提到】
: I don't think it is easy to see T_c go to zero in 2D. In finite systems, T_c
: is defined as the temperature at which the correlation length l_c reaches
: the size of the whole lattice. In 2D Heisenberg model, the correlation
: length l_c scales like c_1*(J/T)exp(c_2*J/T) (c1 and c2 are dimensionless
: constants), which increases extremely fast as T->0. This means your system
: size need to be enlarged very much to just see a little decrease in T_c.
: What method do you use, exact diagonalization? In that case it is impossible
: to go to large systems especially in 3D.
|
e**********n
发帖数: 359 | 4 It is hard to draw any conclusion from estimating Tc for a finite 2D system.
First of all, how do you pin down Tc from the finite and smooth observables
? Whatever your choice is, it is rather arbitrary. I bet your boss is not
familiar with literature on this subject. The 'correct' approach is finite
size scaling analysis. In your model, the scaling formulas are predicted by
2D non-linear sigma model. If you look at the total magnetization, it will
asymptotically take the form
M(T,L) = f(T/Le^{c_2/L}).
The functional form of f() is not important (yet it can be derived). The
good thing is you can rescale T so that the magnetization curves of
different sizes shrink to a single curve.
This approach is correct given that the renormalization and scaling theory
are correct, as people generally accept these theory now. However there is
no rigorous proof that they are correct for any experimental system or
computational model at arbitrarily low temperature scale. In particular,
arguing for something being exactly zero in an experimental system, without
reference to any energy scale, is sort of psuedo-science indeed.
【在 w********i 的大作中提到】
:
: systems, T_c
: reaches
: dimensionless
: system
: T_c.
: impossible
: 非常感谢您的讲解，我用的只是最简单的经典海森堡模型，现在我算到了80X80的晶格
: 了，Tc的下降
: 速度成加快趋势，达到0.5 左右。
|
s*s
发帖数: 100 | 5 I didn't know you used classical Heisenberg model. I thought it was quantum.
But what I said about correlation length should not change anyway.
And I think evenhorizon's suggestion is good. Finite scaling is by far most
convincing.
【在 w********i 的大作中提到】
:
: systems, T_c
: reaches
: dimensionless
: system
: T_c.
: impossible
: 非常感谢您的讲解，我用的只是最简单的经典海森堡模型，现在我算到了80X80的晶格
: 了，Tc的下降
: 速度成加快趋势，达到0.5 左右。
|
w********i
发帖数: 795 | 6 谢谢牛人的指点。我一定遵循您的教导，把该问题梳理清楚。
我也不知道我导师知不知道，但估计他知道，但他从来不管我，知道也故意不说，隔几
个月问我一两
个问题启发启发我。您的提到的2D non-linear sigma model让我耳目一新。我正在就
该方法展
开调研工作。
system.
observables
not
finite
predicted by
will
The
【在 e**********n 的大作中提到】
: It is hard to draw any conclusion from estimating Tc for a finite 2D system.
: First of all, how do you pin down Tc from the finite and smooth observables
: ? Whatever your choice is, it is rather arbitrary. I bet your boss is not
: familiar with literature on this subject. The 'correct' approach is finite
: size scaling analysis. In your model, the scaling formulas are predicted by
: 2D non-linear sigma model. If you look at the total magnetization, it will
: asymptotically take the form
: M(T,L) = f(T/Le^{c_2/L}).
: The functional form of f() is not important (yet it can be derived). The
: good thing is you can rescale T so that the magnetization curves of
|
w********i
发帖数: 795 | 7
quantum.
most
非常感谢牛人的分析，您和上上一位牛人共同强调的Finite scaling 作为最令人信服
的论据的观点
使我受到很大启发。
虽然大家都这么说，但是我以前一直对该Finite scaling没有产生足够思想上的重视。
现在通过实
践，意识到确实是问题的关键。
【在 s*s 的大作中提到】
: I didn't know you used classical Heisenberg model. I thought it was quantum.
: But what I said about correlation length should not change anyway.
: And I think evenhorizon's suggestion is good. Finite scaling is by far most
: convincing.
|
N***m
发帖数: 4460 | 8 最后一句话真是比较搞笑
【在 w********i 的大作中提到】
: 谢谢牛人的指点。我一定遵循您的教导，把该问题梳理清楚。
: 我也不知道我导师知不知道，但估计他知道，但他从来不管我，知道也故意不说，隔几
: 个月问我一两
: 个问题启发启发我。您的提到的2D non-linear sigma model让我耳目一新。我正在就
: 该方法展
: 开调研工作。
:
: system.
: observables
: not
|
c****e
发帖数: 2097 | 9
you can read Alexander Polyakov's paper "Interaction of goldstone particles
in two dimensions. Applications to ferromagnets and massive Yang-Mills
fields "
【在 w********i 的大作中提到】
: 谢谢牛人的指点。我一定遵循您的教导，把该问题梳理清楚。
: 我也不知道我导师知不知道，但估计他知道，但他从来不管我，知道也故意不说，隔几
: 个月问我一两
: 个问题启发启发我。您的提到的2D non-linear sigma model让我耳目一新。我正在就
: 该方法展
: 开调研工作。
:
: system.
: observables
: not
|
w********i
发帖数: 795 | 10
particles
非常感谢您提供的宝贵线索，我看到该文献后，非常感慨，没想到看似简单的海森堡模
型，居然蕴含了
这么纷繁复杂博大精深的物理理论。穷尽了人类的智慧。
【在 c****e 的大作中提到】
:
: you can read Alexander Polyakov's paper "Interaction of goldstone particles
: in two dimensions. Applications to ferromagnets and massive Yang-Mills
: fields "
|
c****e
发帖数: 2097 | 11 yeah, for sure.
【在 w********i 的大作中提到】
:
: particles
: 非常感谢您提供的宝贵线索，我看到该文献后，非常感慨，没想到看似简单的海森堡模
: 型，居然蕴含了
: 这么纷繁复杂博大精深的物理理论。穷尽了人类的智慧。
|
