Comment on ’ Symmetry properties of magnetization in the Hubbard model at finite temperature ’ ” Gang

In replying to the comment (cond-mat/9807402), we point out that the statements by Avella, Mancini and Villani on our papers are simply wrong. Actually, they made simple mistakes and performed false calculations. We take cautions for the assertion that the exact relations for finite temperatures we found before can be simply extended down to zero temperature. In their recent comment [1], Avella, Mancini and Villani stated that they can extend the exact relations we found before for finite temperatures [2,3] down to zero temperature, and based on the numerical calculations they asserted that our solutions for both spin and pseudospin correlation functions are incorrect. In this reply, we point out that the statements by Avella, Mancini and Villani on our papers are simply wrong. Actually, they made simple mistakes and performed false calculations. We take cautions for the assertion that the exact relations for finite temperatures we found before can be simply extended down to zero temperature. 1. The extrapolation of thermal averages (e.g. spin correlation functions) down to zero temperature is in principle not so simple as these authors did. Actually, when one gets the thermal average from the spectral theorem of Green’s function, one implicitly multiplies the exponential factor like exp±βh to the two sides of the equation, which makes that the extrapolating to the zero temperature should be cautions, as the ground state property is probably more complicated than one expects. For instance, these authors obtained Eqs. (11) and the zero-temperature version of Eqs. (28) by simply extrapolating the thermal averages down to zero temperature while they did not confirm if such extrapolations are reliable. If the limit of the left-hand side of Eqs. (11) or (28) does not exist, their results would be incorrect. 2. In Ref. [2] we obtained a somewhat general formula for the magnetization as a function of the applied field at finite temperature in the Hubbard model. We also presented a special solution only for two limiting cases, namely, the noninteracting case and the result at halffilling obtained by Sutö [4] several years ago. While these authors confirmed Sutö’s result in the large U limit, they claimed that this special solution for the noninteracting case is incorrect. This is really a surprising assertion, because from any textbooks one can find that the magnetization as a function of the magnetic field for noninteracting systems takes the form of our Eq. (18) in Ref. [2]. It is a simple exercise to get this formula.