Dear Zhidong Zhang,

First, I want to stress that we disagree with your statement, that gamma

matrices can have anything to do with the low or high temperature series. If

you work with the local spin variables it is easy to get the series. If you use

the nonlocal Gamma matrices it will be extremely difficult, but the result must

be the same.

I will only answer a few of your questions, as some of your questions are too

vague, but may not be a question if the following answers are given:

The equipartition theorem states that the temperature is related to the

kinetic energy of the particles.  Particles move faster as temperature

increases. Thus as the temperature increases, the particles move faster and

faster. In the spin languages, the spins flip more and more quickly as the

temperature increases.

The ensemble theory states that the probability of a system being in a state 

is given by W=e^{-E/kT}/Z, where E is the energy of the system and Z the

partition function.  In the Ising model the energy is

E=J sum sigma sigma',

and the summation is over all the nearest-neighbor pair of spins sigma and

sigma', and T is the temperature, k is the Boltzmann constant, while J is

another arbitrary constant (interaction constant).  As an example, for an Ising

model with L spins sigma_1,...,sigma_L=+1 or -1, there are 2^L states for L spins.

At very high temperature $1/kT$ is very small, and the probabilities of the

different states are almost the same, even though they have different

energies. In the Ising model the spins become independent random variables

 (in probability theory), or in another words that means they are not

correlated. The free energy U-TS is then almost entirely given by the entropy contribution

and the free energy per site -kT log(Z)/L is then almost -kT log(2). Therefore,

theorists consider the reduced free energy per spin f/kT as the fundamental

quantity at high temperature as it is almost -S/kL, up to a constant factor the

entropy per spin.

While as T decreases, the lower energies E have higher probabilities, and the

states with high energies have low probability. Thus at zero temperature only

the lowest state is occupied. The entropy contribution -TS to the free energy

goes to zero as T goes to zero and the free energy U-TS becomes U=E_{min}, the

ground state energy. In the Ising model it is the state with all spins up or the

state with all spins down. Some people say that the spins are frozen. Or, in the

solid state language, the atoms are frozen with no vibrations (except quantum

zero-point motion, which does not apply to the classical Ising model. The 3d

Ising model is classical, not quantum, and there is therefore no Heisenberg

uncertainty relation, as long as one discusses the Ising model by itself.).

As the probabilities are given, we can define the expectation  values, such

as spontaneous magnetization, internal energy, etc.

 As there are two (so-called extremal) states at zero temperature, one with all

spins up and one with all spins down, there is a phase transition as one changes

the sign of the magnetic field. However, in your paper you discuss zero magnetic

field only. Below the critical point you must choose one of the two states when

you calculate the spontaneous magnetization. Then you find that this

magnetization is analytic in the temperature except at the critical point. The

free energy, internal energy and specific heat are even functions of the

magnetization (same value for both signs of H). Therefore, their values do not

depend on the choice of the equilibrium state and they are also analytic in T

for all T except T_c. (For T=infinity, you must use f/T as function of 1/T).

Let us consider the isotropic case for simplicity). All your results in your

paper are analytic in exp(-2JT) below T_c and have the correct value at T=0. So

they have unique series expansions and the series coefficients must agree with

the known ones. Similarly, your f/kT is analytic in 1/T, or equivalently

beta=1/kT and it goes to the correct entropic value at T infinite.

Again, f/kT has a unique series expansion and you must agree with the rest of

the world. You do not agree, so that it is rigorously known that your conjecture

fails. Any counter argument just confuses the real matter.

 Do you agree with these statements?

If you disagree, then why do you study the 3D Ising model?

Helen