% ising_ground   Ground state energy of the quantum Ising model %   ising_ground(B) gives the ground state energy of %   Ising model in transverse field in the%   thermodynamic limit, if the %   field strength is B and the coefficient %   of the nearest neighbor coupling is 1.%   ising_ground(B,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_ground(BField,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;   k=1;          % Boltzmann constant   % Using the paper   % Pfeuty, Ann. Phys. 57, 79-90 (1970)   % Eq. (3.2)   J=4;   Gamma=2*BField;   lambda=J/2/Gamma;   % Elliptic integral   dphi=pi/1000000;   phi=0:dphi:pi/2;   theta=sqrt(4*lambda/(1+lambda)^2);   Eint=sum(sqrt(1-theta^2*sin(phi).^2))*dphi;   % Ground state energy/N   E0=-Gamma*2/pi*(1+lambda)*Eint/2;   % /2 is added compared to the paper ...else   % Numerical solution for N qubits   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);   E0=min(real(eig(H)))/N;end %if