Dear Masaki
Indeed in 1d you have to multiply with L
Matthias
On Jun 11, 2009, at 11:26 AM, Masaki Oshikawa <oshikawa@issp.u-tokyo.ac.jp
wrote:
Hello,
Many thanks to the ALPS developers for providing such a great software.
I have a question on the stiffness (I am dealing with spin system now.)
To my knowledge, the stiffness is usually defined by the gradient expansion of the free energy density ¥[ f ¥sim ¥mbox{const.} + ¥frac{¥rho_s}{2} (¥nabla ¥theta)^2, ¥] where $¥theta$ is the phase angle (rotation angle in the xy plane, in the case of a planar magnet).
For the groundstate ($T=0$) of a 1D spin chain, the above implies ¥[ ¥rho_s = L ¥frac{¥partial^2 E_0(¥varphi)}{¥partial ¥varphi^2}, ¥] where $E_0(¥varphi)$ is the groundstate energy of a ring of circumference $L$ for the twisted boundary condition with the twist angle $¥varphi$. ($E_0$ is $L$ times the groundstate energy density.) (I am following, e.g. N. Laflorencie et al., Eur. Phys. J. B 24, 77 (2001) and references therein. )
I was bit confused by the results from ALPS so I tried to check S=1/2 XXZ chain, for which exact values are known.
Just for example I chose Jxy=1, Jz=0.5 (easy-plane antiferromagnetic chain). I haven't done a very systematic study, but for a given length (I did $L=30, 60, 120$) the stiffness seems to converge to a constant as the temperature is lowered. I took the "stiffness" output of ALPS at low enough temperature as the groundstate value. (I mostly used ALPS/looper code here, but I checked that dirloop_sse also produces similar results.)
L=30 : 0.0104 L=60 : 0.00517 L=120: 0.00258
This decreases for larger size $L$.
On the other hand, the exact value of the stiffness is known to be $9 ¥sqrt{3}/(16 ¥pi) ¥sim 0.310$ (in the infinite size limit, at T=0). This is in apparent contradition to the ALPS result if they refer to the same quantity.
Actually if I multiply the output of ALPS by the length $L$, it agrees well with the exact value.
So it seems that the "stiffness" output of ALPS is $1/L$ times the standard(??) definition. (Am I missing something trivial?)
For the $S=1/2$ chain I think I can convert the output to what I want. But more generally what is the definition of the "stiffness" in ALPS? I looked at ALPS documentation but could not find it. I would appreciate if you could give me a pointer.
I am now interested in S>1/2 case, and would like to make sure I will not make wrong observation due to mismatch in definition/convention.
Thanks.
Regards,
Masaki
-- Masaki Oshikawa oshikawa@issp.u-tokyo.ac.jp
Institute for Solid State Physics, University of Tokyo 5-1-5 Kashiwanoha, Kashiwa, 277-8581 Japan URL http://oshikawa.issp.u-tokyo.ac.jp/