狄拉克台球求解器-Dirac billiard solver-matlab

function [Phim,Xim,X,Y,E,area]=DB(L1s,L1l,... L2s,L2l,Nx,Ny,G,Vm,Ve,Ax,Ay,enumber,ecenter,norm_func) % The function computes the eigenstates and eigenvalues of a Dirac billiard (DB) % with abitrary shape and of arbitrary potentials. % % For details about the % method, please consult [email protected]. A acedminc paper is to be % potentially published later with reference here. % We appreciate your reference to this article if it is used for % for academic purpose! %%%% Input: % % Ls1,L1l,L2s,L2l - define a rectangular space [Ls1,L1l]*[L2s,L2l] that % encompasses the billiard. % Nx and Ny - the number of discretization in x and y direction, such % that the finite difference unit is h1=[L1l-L2s]/Nx, h2=[L2l-L2s]/Ny; % G - cell-list fucntion handle. G=[G1,G2,G3,...], where G1 is the % outer-boundary function handle such that G1(x,y)==1 contains all the % points inside. This also applies to G2,G3 ..., which are function handles % for the inner-boundaries. % Vm - scalar function handle @(x,y), the mass potential % Ve - scalar fcuntion hanlde @(x,y), the electric potential % Ax - scalar function handle @(x,y), the x direction vector potential % Ay - scalar function handle @(x,y), the y direction vector potential % enumber, ecenter - number of eigenvalues (enumber) around the center (ecenter) % norm_func - user defined linear and homogenous boundary condition % cell-list function handle @(x,y) % norm_func=[norm_func_1,norm_func_2,....] with a length same as G with the corresponding meaning % for example, norm_func_1(x,y) gives the ratio between chi and phi % component at the outer-boundary %%%% Output: % % Phim,Chim - matrices of phi and chi component of the eigenvector % X, Y - matrices of the discrete points % E - eigenvalues % area - actual area that the discretized billiard takes place %%%% Example: % An AB billiard % clear % R1=0.3;Rd1=0.65;Rd2=0.85;R2=1;V=5;alpha=0.5; % L1=2.1;L2=2.1;N=200; % L1s=-L1/2;L1l=L1/2;L2s=-L2/2;L2l=L2/2; % Nx=N;Ny=N; % number=10;center=0;L1=2.1;L2=2.1; % PhiB=alpha*2*pi;NL=@(x,y) x.*0; % Ax=@(x,y) PhiB/(2*pi)*(-(y)./((y).^2+(x).^2)); % Ay=@(x,y) PhiB/(2*pi)*((x)./((y).^2+(x).^2)); % Ve=@(x,y) V*(x.^2+y.^2>=Rd1^2).*(x.^2+y.^2<=Rd2^2);Vm=@(x,y) x.*0; % G1=@(x,y) x.^2+y.^2<=R2^2; % G2=@(x,y) x.^2+y.^2<=R1^2; % G={G1,G2}; % circlefun=@(x,y) 1i*(x+1i*y)./sqrt((x).^2+(y).^2); % norm_func={@(x,y) circlefun(x,y),@(x,y) -1*circlefun(x,y)}; % tic; % [Phim,Xim,X,Y,E,area]=DB(-L1/2,L1/2,-L2/2,L2/2,Nx,Ny,G,Vm,Ve,... % Ax,Ay,number,center,norm_func); % toc; % pcolor(X,Y,(real(Phim(:,:,1)))) % shading interp % title(['Re(Phi), E=',num2str(real(E(1)))]) h1=(L1l-L1s)/Nx;h2=(L2l-L2s)/Ny; sigma_x=[0,1;1,0];sigma_y=[0,-1i;1i,0];sigma_z=[1,0;0,-1]; x = linspace(L1s, L1l, Nx+1);y = linspace(L2s,L2l, Ny+1);[X, Y] = meshgrid(x, y); area=zeros(numel(G),1); n_Bt=zeros(numel(G),1); % Find the boundary points given by G for t=1:numel(G) if t==1 binary_image = G{t}(X, Y) ==1; [yt,xt]=find(binary_image==1); B = bwboundaries(binary_image,4); boundary_points = [B{1}(1:end-1,:)]; boundary_x = boundary_points(:,2);boundary_y = boundary_points(:,1); area(t)=polyarea(boundary_points(:,2),boundary_points(:,1))*h1*h1; n_Bt(t)=numel(boundary_x); else binary_image = G{t}(X, Y) ==1; [ytn,xtn]=find(binary_image==1); B = bwboundaries(binary_image,4); boundary_pointsn = [B{1}(1:end-1,:)]; boundary_xn = boundary_pointsn(:,2);boundary_yn = boundary_pointsn(:,1); XYtot=[setdiff([xt,yt],[xtn,ytn],'rows');[boundary_xn,boundary_yn]]; xt=XYtot(:,1);yt=XYtot(:,2); boundary_x=[boundary_x;boundary_xn]; boundary_y=[boundary_y;boundary_yn];n_Bt(t)=numel(boundary_xn); area(t)=polyarea(boundary_pointsn(:,2),boundary_pointsn(:,1))*h1*h1; end end % Construct the discretized Hamiltonians n_tot=numel(xt); n_B=sum(n_Bt); tab=zeros(n_tot,2); tab(1:n_B,:)=[boundary_x,boundary_y]; tab(n_B+1:end,:)=setdiff([xt,yt],[boundary_x,boundary_y],"rows"); tabinv = sparse(tab(:,1),tab(:,2),1:n_tot); xright_up=xt+0.5;yright_up=yt+0.5; xleft_up=xt-0.5;yleft_up=yt+0.5; xright_down=xt+0.5;yright_down=yt-0.5; xleft_down=xt-0.5;yleft_down=yt-0.5; XYnew=intersect(intersect(intersect([xright_up,yright_up],[xright_down,yright_down],"rows")... ,[xleft_up,yleft_up],'rows'),[xleft_down,yleft_down],'rows'); n_Dirac=size(XYnew,1); xm=XYnew(1:end,1)-0.5;xp=XYnew(1:end,1)+0.5;ym=XYnew(1:end,2)-0.5;yp=XYnew(1:end,2)+0.5; C=sparse([1:n_Dirac,1:n_Dirac,1:n_Dirac,1:n_Dirac],[tabinv(sub2ind(size(tabinv), xm,ym ));... tabinv(sub2ind(size(tabinv), xm,yp ));... tabinv(sub2ind(size(tabinv), xp,yp ));... tabinv(sub2ind(size(tabinv), xp,ym ))],1,n_Dirac,n_tot); Dx=sparse([1:n_Dirac,1:n_Dirac,1:n_Dirac,1:n_Dirac],[tabinv(sub2ind(size(tabinv), xm,ym ));... tabinv(sub2ind(size(tabinv), xm,yp ));... tabinv(sub2ind(size(tabinv), xp,yp ));... tabinv(sub2ind(size(tabinv), xp,ym ))],kron(ones(n_Dirac,1),... [-1,-1,1,1]),n_Dirac,n_tot); Dy=sparse([1:n_Dirac,1:n_Dirac,1:n_Dirac,1:n_Dirac],[tabinv(sub2ind(size(tabinv), xm,ym ));... tabinv(sub2ind(size(tabinv), xm,yp ));... tabinv(sub2ind(size(tabinv), xp,yp ));... tabinv(sub2ind(size(tabinv), xp,ym ))],kron(ones(n_Dirac,1),... [-1,1,1,-1]),n_Dirac,n_tot); V=C.*transpose(Vm(x(xm)+h1/2,y(ym)+h2/2)); Vel=C.*transpose(Ve(x(xm)+h1/2,y(ym)+h2/2)); Axx=C.*transpose(Ax(x(xm)+h1/2,y(ym)+h2/2)); Ayy=C.*transpose(Ay(x(xm)+h1/2,y(ym)+h2/2)); loss1=setdiff(1:2*n_tot,(1:n_B)*2-1); loss2=setdiff(loss1,(1:n_B)*2); % This permutation is just to make the boudary condition could be written % easier, not nessasary. pert=[loss2,(1:n_B)*2,(1:n_B)*2-1]; P=sparse(1:2*n_tot,pert(1:2*n_tot),1,2*n_tot,2*n_tot); Hprime=((-2i/h1)*kron(Dx,sigma_x)+(-2i/h2)*kron(Dy,sigma_y)+kron(V,sigma_z)... +kron(Vel,eye(2))-kron(Axx,sigma_x)-kron(Ayy,sigma_y))*transpose(P); Cprime=kron(C,eye(2))*transpose(P); Pi=2*n_tot-(2*n_Dirac)-n_B; if Pi==2 Bprime=spalloc(n_B+2,2*n_tot,n_B+2); zeroprime=spalloc(n_B+2,2*n_tot,0); else Bprime=spalloc(n_B,2*n_tot,n_B); zeroprime=spalloc(n_B,2*n_tot,0); end % Apply boundary conditions for i=1:numel(G) if i==1 for m=1:n_Bt(1) %[m+2*(n_tot-n_B),2*(n_tot-n_B)+m+n_B] xx=x(tab(m,1)); yy=y(tab(m,2)); buff=norm_func{i}(xx,yy); Bprime(m,m+2*(n_tot-n_B))=1; Bprime(m,2*(n_tot-n_B)+m+n_B)=-buff; end else for m=sum(n_Bt(1:i-1))+1:sum(n_Bt(1:i)) xx=x(tab(m,1)); yy=y(tab(m,2)); buff=norm_func{i}(xx,yy); Bprime(m,m+2*(n_tot-n_B))=1; Bprime(m,2*(n_tot-n_B)+m+n_B)=-buff; end end end % Extra constraint for Pi=2 if Pi==2 % psedomode1=zeros(1,2*n_tot); psedomode2=zeros(1,2*n_tot); for k=1:n_tot psedomode1(2*k)=(-1)^(tab(k,1)+tab(k,2)); psedomode2(2*k-1)=(-1)^(tab(k,1)+tab(k,2)); end buff1=Bprime(1,:); buff2=Bprime(2,:); Bprime(1,:)=psedomode1*transpose(P); Bprime(2,:)=psedomode2*transpose(P); Bprime(n_B+1,:)=buff1; Bprime(n_B+2,:)=buff2; end hh1=[Hprime;Bprime]; hh2=[Cprime;zeroprime]; % Different hamiltonian dependent Pi if numel(G)<3 [V,E]=eigs(hh1,hh2,enumber,ecenter); else [V,E]=eigs(hh2'*hh1,hh2'*hh2,enumber,ecenter); end E=diag(E);V=transpose(P)*V; Phi=V(1:2:end,:);Xi=V(2:2:end,:); [~,sq2]=sort(real(E),'ascend'); Phi=Phi(:,sq2);Xi=Xi(:,sq2);E=E(sq2); % Convert one-dimensional vector back to two dimensional [Phim,Xim]=tomatrix(Phi,Xi,tabinv,Nx,Ny); function [P,X]=tomatrix(Phi,Xi,tabinv,Nx,Ny) P=zeros(Nx+1,Ny+1,size(Phi,2)); X=zeros(Nx+1,Ny+1,size(Phi,2)); for tt=1:size(Phi,2) for ll=1:size(tabinv,1) for j=1:size(tabinv,2) if tabinv(ll,j)~=0 P(j,ll,tt)=Phi(tabinv(ll,j),tt); X(j,ll,tt)=Xi(tabinv(ll,j),tt); end end end end end end