% ising_thermal - Internal energy of the Ising model in thermal equilibrium%   ising_thermal(B,T) gives the internal energy of %   Ising model in transverse field in the%   thermodynamic limit, if the temperature is T, the  %   field strength is B and the coefficient %   of the nearest neighbor coupling is 1.%   ising_thermal(B,T,N) does the same for N qubits.% Copyright (C) 2005  Geza Toth    E.mail: toth@alumni.nd.edu%% This program is free software; you can redistribute it and/or% modify it under the terms of the GNU General Public License% as published by the Free Software Foundation; see gpl.txt% of this subroutine package.%% This program is distributed in the hope that it will be useful,% but WITHOUT ANY WARRANTY; without even the implied warranty of% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the% GNU General Public License for more details.% % You should have received a copy of the GNU General Public License% along with this program; if not, write to the Free Software% Foundation, Inc., 51 Franklin Street, Fifth Floor, % Boston, MA  02110-1301, USA.function E0=ising_thermal(BField,T,varargin)if isempty(varargin),    N=Inf;else    if length(varargin)~=1,        error('Wrong number of input arguments.')    end %if    N=varargin{1};end %if% Thermodynamical limitif N==Inf;   % Using Pfeuty, Ann. Phys. 57, 79-90 (1970)   J=4;   Gamma=2*BField;   lambda=J/2/Gamma;   k=1;   N=100;   m=-N/2:1:N/2-1;   q=2*pi*m/N;   dq=1;   wq=sqrt(1+2*lambda*cos(q)+lambda^2);   % Normalization   E0=sum(Gamma.*wq.*1./(exp(Gamma.*wq./k./T)+1))/N+ising_ground(BField);else   % Numerics for finite N   x=[0 1;1 0];   y=[0 -i;i 0];    z=[1 0; 0 -1];   e=[1 0; 0 1];   % Using routines from the library to make the Ising Hamiltonian   H=-nnchainp(x,x,N)-BField*nnchainp(z,e,N);   rho=expm(-H/T);   rho=rho/trace(rho);   E0=trace(H*rho)/N;end %if