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getObservable.m
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getObservable.m
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function out = getObservable(type,mps,Vmat,para)
%% Calculates the following Observables:
% SBM: 'spin', 'occupation', 'current', 'shift', 'rdm', 'staroccupation', 'starpolaron',
% 'energy', 'displacement', 'bath2correlators'
% MLSBM: 'participation', 'tunnelenergy', 'staroccupation', 'starpolaron', 'energy'
%
% Use as: first argument is cell, with descriptor as type{1}
% all other type{n} are options defined in each function
% getObservable({'spin'} ,mps,Vmat,para)
% getObservable({'occupation'} ,mps,Vmat,para)
% getObservable({'current',AnAm} ,mps,Vmat,para)
% getObservable({'shift'} ,mps,Vmat,para)
% getObservable({'rdm',2} ,mps,Vmat,para)
% getObservable({'tunnelenergy',op} ,mps,Vmat,para)
% getObservable({'displacement'} ,mps,Vmat,para)
% getObservable({'bath2correlators'} ,mps,Vmat,para)
% getObservable({'starpolaron'} ,mps,Vmat,para)
% getObservable({'staroccupation',AnAm} ,mps,Vmat,para)
% getObservable({'energy',op} ,mps,Vmat,para)
% getObservable({'hshi',op} ,mps,Vmat,para) % op is optional
% getObservable({'stateproject',i,j} ,mps,Vmat,para) % i,j optional
% getObservable({'sysheff',i} ,mps,Vmat,para) % i optional; possibly only TreeMPS operational
%
% In 'current' and 'staroccupation': AnAm optional!
%
% The output 'out' can be of different form:
% 'spin': struct, fields: sx, sy, sz
% 'occupation': array (L x NC)
% 'shift': array
% 'rdm': matrix
% 'participation': number
% 'tunnelenergy': number
% 'displacement': vector (L x 1)
% 'bath2correlators': matrix (L x L x NC)
% 'staroccupation': array (3 x X x NC)
% 'current': array (NC x L-1)
% 'starpolaron': array (3 x X x NC)
% 'energy' scalar
% 'hshi' scalar
% 'coherence' not implemented yet
% 'stateproject' scalar
% 'state' matrix (dk x dk)
%
% Created 03/06/2014 by Florian Schroeder @ University of Cambridge
% TODO: - Implement Boson Site Shift to export the routine from optimizesite.m. Only get a single shift value.
%
% Modified:
% - 21/12/14 FS: replaced OBB contractions with faster matrix products.
% - 21/08/15 FS: added Multi-Chain Vmat / Vtens capability
% - 28/02/16 FS: added TreeMPS support
switch type{1}
case 'spin'
% applicable for spin-boson model and for folded SBM2
out = calSpin(mps,Vmat,para);
case 'occupation'
% applicable to all single stranded chains. Extensible to 2-chains
if para.useTreeMPS == 1
out = calBosonOcc_Tree(mps,para); % (L x NC)
return;
elseif para.foldedChain == 0
if para.nChains == 1 && ~para.useStarMPS
out = calBosonOcc(mps,Vmat,para);
else % Multi-chain with Vmat / V-tensor-network
out = calBosonOcc_MC(mps,Vmat,para); % (NC x L)
end
elseif para.foldedChain == 1
out(1,:) = calBosonOcc(mps,Vmat,para,1);
out(2,:) = calBosonOcc(mps,Vmat,para,2);
end
if size(out,1) > 1
out = out.'; % (L x NC) for compatibility with tresults
end
case 'shift'
% applicable to all single stranded chains. Extendable to 2-chains
out = calBosonShift(mps,Vmat,para);
case '1siteshift'
% intended for use in optimizesite. Only for focused single-site
% mps and Vmat are single-site matrices
out = cal1siteShift(mps,Vmat,para);
case 'rdm'
% applicable to all systems.
if type{2} <= para.L
if para.useTreeMPS
out = calRDM(mps.mps,mps.Vmat,para,1); % only for site 1 for now.
else
out = calRDM(mps,Vmat,para,type{2});
end
else
out = [];
end
case 'rdm_adiabatic'
% applicable to site 1 for now only!
if type{2} ~= 1
error('rdm_adiabatic only available for site 1');
end
if para.nChains > 1
error('rdm_adiabatic only for single chains!');
end
assert( type{3} <= para.dk(1,1), 'Need valid number for state selection');
k = type{3}; % the (adiabatic) bond state to construct the (adiabatic) RDM from
% SVD of mps{1} and delete unwanted bond states!
A = mps{1};
[D1, D2, d] = size(A); % d is site dimension
A = permute(A, [3, 1, 2]);
A = reshape(A, [d * D1, D2]); % reshapes to (a1 d),a2
[B, S, U] = svd2(A); % Could also use QR decomposition if nargin !=5
DB = size(S,1);
B = B* sparse(k,k,1,DB,DB) *U;
B = reshape(B, [d, D1, D2]);
mps{1} = permute(B, [2, 3, 1]); % now only contains the desired bond state
out = calRDM(mps,Vmat,para,type{2});
case 'rdm_adiabatic2'
% applicable to site 1 for now only!
% A different way of selecting the bond-state, without SVD.
% Seems to be equivalent to 'rdm_adiabatic'
if type{2} ~= 1
error('rdm_adiabatic only available for site 1');
end
if para.nChains > 1
error('rdm_adiabatic only for single chains!');
end
assert( type{3} <= para.dk(1,1), 'Need valid number for state selection');
k = type{3}; % the (adiabatic) bond state to construct the (adiabatic) RDM from
[~, D2, ~] = size(mps{1}); % d is site dimension
for ii = 1:D2
if ii ~= k
mps{1}(1,ii,:) = 0; % delete 2 for k=1 or 1 for k=2
end
end
mps{1} = mps{1}./norm(reshape(mps{1},[],1)); % normalise the state!
out = calRDM(mps,Vmat,para,type{2});
case 'state'
% applicable to site 1 for now only!
% saves the MPS of the site with diabatic & adiabatic information!
% closely related to 'rdm_adiabatic'
% {'state',sitej}
if type{2} ~= 1
error('VMPS:getObservable:state','state only available for site 1');
end
if para.nChains > 1 || para.useTreeMPS
if para.useStarMPS
d = size(mps{1});
A = reshape(mps{1},[],d(end)); % ... x dk
[U,S,~] = svd2(A.'); % transpose to benefit from speedup in svd2
elseif para.useTreeMPS
d = size(mps.mps{1});
[~,V] = prepare_onesiteAmat(reshape(mps.mps{1},d(1),[],d(end)),para,1); % A_(1,D(1)*..*D(NC),dOBB') * V_(dOBB',dk)
if diff(size(V)) ~= 0 % if dOBB < dk then expand
V(d(end),d(end)) = 0;
end
out = V';
return;
else
error('VMPS:getObservable:NotImplemented','Needs to be implemented!')
end
else
d = size(mps{1});
A = reshape(mps{1},[],d(end)); % ... x dk
[U,S,~] = svd2(A.'); % transpose to benefit from speedup in svd2
if size(S,1) < d(end)
% add zeros
U(d(end),d(end)) = 0;
S(d(end),d(end)) = 0;
end
end
out = U*S; % should be dk x D(1:dk) for D > dk
case 'sys-env-state'
% saves the MPS of site 1 & 2 containing all diabatic & adiabatic information!
% decompose also 1st site to have adiabatic information available
% returns out: struct
% {'sys-env-state'}
if para.nChains > 1
if para.useStarMPS
d = size(mps{1});
A = reshape(mps{1},[],d(end)); % ... x dk
[U,S,~] = svd2(A.'); % transpose to benefit from speedup in svd2
else
error('VMPS:getObservable:NotImplemented','Needs to be implemented!')
end
else
d = size(mps{1});
A = reshape(mps{1},[],d(end)); % ... x dk
[U,S,~] = svd2(A.'); % transpose to benefit from speedup in svd2
if size(S,1) < d(end)
% add zeros
U(d(end),d(end)) = 0;
S(d(end),d(end)) = 0;
end
end
Vmat{1} = U*S; % should be dk x D(1:dk) for D > dk; sets focus on Vmat!! This is simpler for StarMPS
% save the system and reaction coord state
out.mps = mps([1,2]);
out.Vmat = Vmat([1,2]);
case 'participation'
% applicable to all systems, only for first site.
out = calParticipation(calRDM(mps,Vmat,para,1));
case 'tunnelenergy'
% useful for MLSBM, applicable to all
% needs type{2} = op
if length(type) == 2
out = calTunnelingEnergy(mps,Vmat,para,type{2});
else
out = calTunnelingEnergy(mps,Vmat,para);
end
case 'bath1correlators'
% = [state projected] chain observable
% returns a L [x dk(1,1)] x nChains Vector
%
% obs: ['bp'|'x'|'bp^2'|'x^2'|'n'] selects observable to calculate
%
% by default:
% {'bath1correlators',obs} : no projection - TODO!
% {'bath1correlators',obs,'diabatic'} : projects onto diabatic states
% {'bath1correlators',obs,'adiabatic'}: projects onto adiabatic states
NC = para.nChains;
if para.useTreeMPS && isstruct(mps)
if length(type) == 2
out = calBath1SiteCorrelators_Tree(mps,para,type{2}); % L x NC
else
out = zeros(para.L, mps.dk(1), NC);
% One SVD to dk to separate diabatic or adiabatic basis!
% Assume, Vmat of node = eye; mps == treeMPS
assert(all(all(mps.Vmat{1} == eye(mps.dk(1)))),'Please implement for Vmat ~= eye, if this happens!');
d = size(mps.mps{1});
[A,V] = prepare_onesiteAmat(reshape(mps.mps{1},d(1),[],d(end)),para,1); % A_(1,D(1)*..*D(NC),dOBB') * V_(dOBB',dk)
A = reshape(A,[],size(A,3));
% Now dOBB is adiabatic basis, dk is diabatic. V carries probabilities for each state, which will be removed later!
for kk = 1:mps.dk(1)
newV = zeros(size(V));
if strcmpi(type{3},'diabatic')
% Project onto root nodes' diabatic states
newV(:,kk) = V(:,kk);
elseif strcmpi(type{3},'adiabatic')
% Project onto root nodes' adiabatic states
newV(kk,:) = V(kk,:);
elseif strcmpi(type{3},'lettcoherence') % get |TT><LE+| mode-mediated coherence terms
newV = V;
projOp = zeros(size(V,2));
projOp(1,2) = 1;% projOp(2,1) = 1;
end
if norm(newV) ~= 0
newV = newV./norm(newV); % remove weight by /norm()
end
mps.mps{1} = reshape(A*newV,d);
if strcmpi(type{3},'lettcoherence')
out(:,kk,:) = calBath1SiteCorrelators_Tree(mps,para,type{2},projOp); % projection needed for cross terms
break; % only do once!
else
out(:,kk,:) = calBath1SiteCorrelators_Tree(mps,para,type{2}); % no projection needed anymore inside calBath1SiteCorrelators
end
end
end
return;
end
if length(type) == 3
out = zeros(para.L, para.dk(1,1), NC);
elseif length(type) == 2
out = zeros(para.L,NC);
else
error('VMPS:getObservable:bath1correlators','Please call with correct input');
end
if para.useStarMPS
for mc = 1:NC
% extract MPS for each chain and calculate <anam> separately
cL = para.chain{mc}.L;
mpsC = [mps{1}, cellfun(@(x) x{mc},mps(2:cL),'UniformOutput',false)];
VmatC = [Vmat{1}, cellfun(@(x) x{mc},Vmat(2:cL),'UniformOutput',false)];
mpsC{1} = permute(mpsC{1}, [1:mc,mc+2:NC+1, mc+1, NC+2]);
mpsC{1} = reshape(mpsC{1}, [],para.D(mc,1),para.dk(1,1));
paraC = para;
paraC.nChains = 1; paraC.L = para.chain{mc}.L;
paraC.dk = para.dk(mc,:); paraC.dk(1) = para.dk(1,1); % need to preserve system dk
paraC.shift = para.shift(mc,:);
% calculate <anam>
if length(type) == 2
out(1:paraC.L,mc) = calBath1SiteCorrelatorsProject(mpsC,VmatC,paraC,type{2},[],[]);
elseif strcmpi(type{3},'diabatic')
% diabatic projections (H0 eigenstates)
for kk = 1:paraC.dk(1,1)
stateProj = zeros(paraC.dk(1,1)); stateProj(kk,kk) = 1;
bondProj = [];
out(1:paraC.L,kk,mc) = calBath1SiteCorrelatorsProject(mpsC,VmatC,paraC,type{2},stateProj,bondProj);
end
elseif strcmpi(type{3},'adiabatic')
% adiabatic states
for kk = 1:para.D(mc,1)
stateProj = [];
bondProj = zeros(para.D(mc,1)); bondProj(kk,kk) = 1;
out(1:paraC.L,kk,mc) = calBath1SiteCorrelatorsProject(mpsC,VmatC,paraC,type{2},stateProj,bondProj);
end
end
end
else
if length(type) == 2
out(1:para.L,:) = calBath1SiteCorrelatorsProject(mps,Vmat,para,type{2},[],[]);
elseif strcmpi(type{3},'diabatic')
% diabatic projections (H0 eigenstates)
for kk = 1:para.dk(1,1)
stateProj = zeros(para.dk(1,1)); stateProj(kk,kk) = 1;
bondProj = [];
out(1:para.L,kk,:) = calBath1SiteCorrelatorsProject(mps,Vmat,para,type{2},stateProj,bondProj);
end
elseif strcmpi(type{3},'adiabatic')
% adiabatic states
for kk = 1:para.dk(1,1)
stateProj = [];
bondProj = zeros(para.dk(1,1)); bondProj(kk,kk) = 1;
out(1:para.L,kk,:) = calBath1SiteCorrelatorsProject(mps,Vmat,para,type{2},stateProj,bondProj);
end
end
% out(:,1,:) = calBath1SiteCorrelators_MC(mps,Vmat,para,[1,0;0,0]); % spin up
% out(:,2,:) = calBath1SiteCorrelators_MC(mps,Vmat,para,[0,0;0,1]); % spin down
end
case 'bath2correlators'
% needed for mapping from chain to star
% returns triangular L x L Matrix
if para.useStarMPS
NC = para.nChains;
out = zeros(para.L, para.L, NC);
for mc = 1:NC
% extract MPS for each chain and calculate <anam> separately
cL = para.chain{mc}.L;
mpsC = [mps{1}, cellfun(@(x) x{mc},mps(2:cL),'UniformOutput',false)];
VmatC = [Vmat{1}, cellfun(@(x) x{mc},Vmat(2:cL),'UniformOutput',false)];
mpsC{1} = permute(mpsC{1}, [1:mc,mc+2:NC+1, mc+1, NC+2]);
mpsC{1} = reshape(mpsC{1}, [],para.D(mc,1),para.dk(1,1));
paraC = para;
paraC.nChains = 1; paraC.L = para.chain{mc}.L;
paraC.dk = para.dk(mc,:); paraC.dk(1) = para.dk(1,1); % need to preserve system dk
paraC.shift = para.shift(mc,:);
% calculate <anam>
out(1:paraC.L,1:paraC.L,mc) = calBath2SiteCorrelators_MC(mpsC,VmatC,paraC);
end
else
out = calBath2SiteCorrelators_MC(mps,Vmat,para);
end
case 'staroccupation'
%% does mapping chain -> star
% uses bath2correlators
if para.foldedChain == 1, error('Not yet implemented'); end
% Get the correlator
if length(type) == 2
AmAn = real(type{2});
else
AmAn = real(getObservable({'bath2correlators'},mps,Vmat,para)); % L x L x nChains
end
% preallocate output array
[x,~,~] = getStarMapping(para,1);
out = zeros(2,length(x),para.nChains);
for mc = 1:para.nChains
[x,hsquared,pxn] = getStarMapping(para,mc);
if strcmp(para.chain{mc}.mapping, 'LanczosTriDiag')
result = diag(pxn * (AmAn(:,:,mc) + triu(AmAn(:,:,mc),1).') * pxn.'); % pxn == U map -> no need for hsquared
out(:,1:length(x),mc) = [x,result(2:end)].';
else
% out(:,1:length(x),mc) = [x, hsquared.*((pxn.^2)*diag(AmAn(:,:,mc)) + 2.* diag(pxn*(AmAn(:,:,mc)-diag(diag(AmAn(:,:,mc))))*pxn.') )]';
out(:,1:length(x),mc) = [x, hsquared.*diag(pxn*(AmAn(:,:,mc)+triu(AmAn(:,:,mc),1).')*pxn.')].';
end
end
case 'current'
%% gets current through each bond.
% works with Single and Multi-Chain Vmat / Vtens
L = para.L; NC = para.nChains;
if length(type) == 2
AmAn = imag(type{2}); % (L x L x NC)
else
AmAn = imag(getObservable({'bath2correlators'},mps,Vmat,para)); % tridiagonal
end
out = zeros(L-1,NC);
for mc = 1:NC
out(:,mc) = (para.chain{mc}.t.*diag(AmAn(1:end-1,2:end,mc))); % t(n)*a(n)*a(n+1)^+
end
case 'displacement'
% = state projected chain polaron
% needed for mapping from chain to star, starpolaron
% returns a L x dk(1,1) x nChains Vector
%
% by default:
% {'displacement'} : no projection - TODO!
% {'displacement','diabatic'} : projects onto diabatic states
% {'displacement','adiabatic'}: projects onto adiabatic states
NC = para.nChains;
out = zeros(para.L, para.dk(1,1), NC);
if para.useStarMPS
for mc = 1:NC
% extract MPS for each chain and calculate <anam> separately
cL = para.chain{mc}.L;
mpsC = [mps{1}, cellfun(@(x) x{mc},mps(2:cL),'UniformOutput',false)];
VmatC = [Vmat{1}, cellfun(@(x) x{mc},Vmat(2:cL),'UniformOutput',false)];
mpsC{1} = permute(mpsC{1}, [1:mc,mc+2:NC+1, mc+1, NC+2]);
mpsC{1} = reshape(mpsC{1}, [],para.D(mc,1),para.dk(1,1));
paraC = para;
paraC.nChains = 1; paraC.L = para.chain{mc}.L;
paraC.dk = para.dk(mc,:); paraC.dk(1) = para.dk(1,1); % need to preserve system dk
paraC.shift = para.shift(mc,:);
% calculate <anam>
if length(type) == 1 || strcmpi(type{2},'diabatic')
% diabatic projections (H0 eigenstates)
for kk = 1:paraC.dk(1,1)
stateProj = zeros(paraC.dk(1,1)); stateProj(kk,kk) = 1;
bondProj = [];
out(1:paraC.L,kk,mc) = calBath1SiteCorrelatorsProject(mpsC,VmatC,paraC,'x',stateProj,bondProj);
end
elseif strcmpi(type{2},'adiabatic')
% adiabatic states
for kk = 1:para.D(mc,1)
stateProj = [];
bondProj = zeros(para.D(mc,1)); bondProj(kk,kk) = 1;
out(1:paraC.L,kk,mc) = calBath1SiteCorrelatorsProject(mpsC,VmatC,paraC,'x',stateProj,bondProj);
end
end
end
else
if length(type) == 1 || strcmpi(type{2},'diabatic')
% diabatic projections (H0 eigenstates)
for kk = 1:para.dk(1,1)
stateProj = zeros(para.dk(1,1)); stateProj(kk,kk) = 1;
bondProj = [];
out(1:para.L,kk,:) = calBath1SiteCorrelatorsProject(mps,Vmat,para,'x',stateProj,bondProj);
end
elseif strcmpi(type{2},'adiabatic')
% adiabatic states
for kk = 1:para.dk(1,1)
stateProj = [];
bondProj = zeros(para.dk(1,1)); bondProj(kk,kk) = 1;
out(1:para.L,kk,:) = calBath1SiteCorrelatorsProject(mps,Vmat,para,'x',stateProj,bondProj);
end
end
% out(:,1,:) = calBath1SiteCorrelators_MC(mps,Vmat,para,[1,0;0,0]); % spin up
% out(:,2,:) = calBath1SiteCorrelators_MC(mps,Vmat,para,[0,0;0,1]); % spin down
end
case 'starpolaron'
%% does mapping chain -> star
% uses displacement
% x stands for continuous variable (momentum k)
% Does state projection!
% Works with Single and Multi Chain Vmat / Vtens
%
%
if para.foldedChain == 1, error('FoldedChain not yet implemented for polaron'); end
NC = para.nChains;
% Get correlators
if length(type) == 1 || strcmpi(type{2},'diabatic')
An = real(getObservable({'displacement','diabatic'},mps,Vmat,para)); % L x states x nChains
elseif strcmpi(type{2},'adiabatic')
An = real(getObservable({'displacement','adiabatic'},mps,Vmat,para)); % L x states x nChains
end
% AnUp = real(calBath1SiteCorrelators_MC(mps,Vmat,para,1)); % L x nChains
% AnDown = real(calBath1SiteCorrelators_MC(mps,Vmat,para,-1)); % L x nChains
[x,~,~] = getStarMapping(para,1);
out = zeros(size(An,2)+1, length(x), NC);
for mc = 1:NC % for each chain:
[x,h,pxn] = getStarMapping(para,mc);
if strcmp(para.chain{mc}.mapping, 'LanczosTriDiag')
result = 2.* pxn * An(:,:,mc); % pxn == U map -> no need for hsquared
out(:,1:length(x),mc) = [x,result(2:end,:)].';
else
% out(:,:,mc) = [x, h.*(pxn*An(:,1,mc)),h.*(pxn*An(:,2,mc))]';
out(:,:,mc) = [x, (h*ones(1,size(An,2))).*(pxn*An(:,:,mc))]';
end
end
case 'energy'
% Calculates the entire energy of the chain
% only usable for focus on spin site!
% needs type{2} = op
if length(type) == 2
out = calEnergy(mps,Vmat,para,type{2});
else
error('Need op as 2nd argument');
end
case 'coherence'
% only for spin in SBM
case 'hshi'
% Calculates the entire energy of the system hamiltonian and first interaction terms
% needs type{2} = op
if length(type) == 2
out = calHsHi(mps,Vmat,para,type{2});
else
op.h1term = cell(1,2); % One body terms of the Hamiltonian
op.h2term = cell(para.M, 2,2); % default: nearest neighbor interaction
for s=1:2
op=genh1h2term_onesite(para,op,s);
end
out = calHsHi(mps,Vmat,para,op);
end
case 'stateproject'
% Calculates the amplitude of the projection <P|Psi> of the MPS
% Here |P> is a simple state like |1>|0000000> (System-Bath)
%
% by default:
% {'stateproject'} : projects onto |2>|0000000>
% {'stateproject',i} : projects onto |i>|0000000>
% {'stateproject',i,j} : projects onto |i>|jjjjjjj>
% default settings
systemState = 2; % take by default state 2
envState = 1; % the state on each boson to project on
if length(type) >= 2
systemState = type{2};
end
if length(type) == 3
envState = type{3};
end
out = calStateProject(mps,Vmat,para,systemState,envState);
case 'sysheff'
% Calculates the effective Hamiltonian representing the adiabatic total energy surfaces (TES)
% on which the system evolves.
% For now: optimised for treeMPS
%
% {'sysheff'} : returns (dk x dk)^2 operator and C
% {'sysheff',i} : returns (dk x i)^2, operator with the i strongest adiabats
% out = {Heff,sysState} returns the effective hamiltonian together with |psi> mapping from diabatic to the adiabatic basis
out = cell(1,3);
if para.useTreeMPS
[out{1},out{2},out{3}] = calSysHeff(mps,para);
else
error('Needs to be implemented');
end
case 'syspes'
% Calculates the effective potenital representing the adiabatic potential energy surfaces (PES)
% on which the system evolves.
out = cell(1,3);
if para.useTreeMPS
[out{1},out{2},out{3}] = calSysPES(mps,para,type{2});
else
error('Needs to be implemented');
end
end
end
function spin = calSpin(mps,Vmat,para)
% Calculate the spin expectation value
% Has to be modified if site 1 changes dimension!!
% Modified:
% FS 24/05/2014: excluded MLSpinBoson. Use with try-catch block
% FS 19/08/2015: - Can be used with Multi-Chain Vmat and Vtens code
%
% TODO: Needs extension to use information about dimension of spin site.
%
if para.useVtens || para.useStarMPS % only Quick Fix for Vtens code, Could be used as Standard code!
McOp = cell(1,1,3); % L x NC x N
[McOp{1}, McOp{2}, McOp{3}] = spinop(para.spinbase);
if para.useVtens
spinVal = real(expectation_allsites_MC(McOp,mps,Vmat,para));
else
spinVal = real(expectation_allsites_StarMPS(McOp,mps,Vmat,para));
end
spin.sx = spinVal(1);
spin.sy = spinVal(2);
spin.sz = spinVal(3);
return;
end
if para.useTreeMPS
McOp = cell(1,1,3); % L x NC x N
[McOp{1}, McOp{2}, McOp{3}] = spinop(para.spinbase);
error('VMPS:getObservable:NotImplementedYet','Please implement this feature!')
end
N=para.L;
assert(N==length(mps) && N==length(Vmat));
assert(~strcmp(para.model,'MLSpinBoson'),'not possible for MLSBM');
ndset=cell(1,N);
for j=1:N
ndset{1,j}=eye(size(Vmat{j},1));
end
sx=ndset;sy=sx;sz=sy;
%debug:
% sx{1,1}
[sigmaX,sigmaY,sigmaZ]=spinop(para.spinbase);
sx{para.spinposition(end)}=sigmaX;
sy{para.spinposition(end)}=sigmaY;
sz{para.spinposition(end)}=sigmaZ;
if strcmp(para.model,'2SpinPhononModel')
sx{para.spinposition}=kron(sigmaZ,eye(2)); %measures excitation of site1
sz{para.spinposition}=kron(eye(2),sigmaZ); %measures excitation of site2
%try to find a good way to measure this!
sy{para.spinposition}=kron(sigmaY,eye(2)); %measures only sy of site1
end
spin.sx=expectationvalue(sx,mps,Vmat,mps,Vmat);
spin.sy=expectationvalue(sy,mps,Vmat,mps,Vmat);
spin.sz=expectationvalue(sz,mps,Vmat,mps,Vmat);
spin.sx=real(spin.sx);
spin.sy=real(spin.sy);
spin.sz=real(spin.sz);
end
function nx = calBosonOcc(mps,Vmat,para,varargin)
% Calculate the boson occupation on x chain
% The operator on the spin site is set to zero
% left or right chain occupation via varargin{1} = 1 or 2
%
% Modified:
% FS 23/01/2014: - Introduced '~' to ignore unused returned values
% - support for folded Chain models
% FS 10/03/2014: - Using correlator_allsites which is more general
% FS 04/06/2014: - updated for spinposition array.
% FS 17/07/2015: - varargin = {1,2} to calculate nx/nz for folded chain
n_op = cell(1,para.L);
for j=1:para.L
if prod(j~=para.spinposition)
if para.foldedChain == 0
% 1-chain SBM
[~,~,n_op{j}] = bosonop(para.dk(j),para.shift(j),para.parity);
%Modification for 2chain model!! Not perfect or right yet!
elseif para.foldedChain == 1
% only kron(n,1) chain occupation calculated.
if para.parity == 'n'
[~,~,n] = bosonop(sqrt(para.dk(j)),para.shift(j),para.parity);
idm = eye(size(n));
if varargin{1} == 1
nr = kron(n,idm);
elseif varargin{1} == 2
nr = kron(idm,n);
end
else
[bp,~,~] = bosonop(para.dk(j),para.shift(j),para.parity); % Why without sqrt??
[~,~,nr,~,~,~]=paritykron(bp,para.bosonparity);
end
n_op{j} = nr;
else
disp('Not Implemented: getObservable, calBosonOcc, foldedchain >1');
end
else
n_op{j}=zeros(para.dk(j)); % don't measure spin.
end
end
%
nx = expectation_allsites(n_op,mps,Vmat);
nx = real(nx); % imag(nx) = eps -> neglect
end
function n = calBosonOcc_MC(mps,Vtens,para)
%% calculate the boson occupation for Single / Multi-Chain + Vmat / V-tensor-network
% Zero operator for spin site
% calculate for all chains simultaneously
%
% Created 18/08/2015 by FS
assert(para.foldedChain == 0, 'not for folded chains');
L = para.L; NC = para.nChains;
% create Operator for expectationvalue: need nc^2 each (i,:,j) is one operator [] x [] x n x [] x []
McOp = cell(L, NC, NC);
for mc = 1:NC
for j = 1:para.chain{mc}.L
if j ~= para.spinposition % works with array
[~,~,McOp{j,mc,mc}] = bosonop(para.dk(mc,j),para.shift(mc,j),para.parity);
else
McOp{j,mc,mc} = zeros(para.dk(1,j)); % don't measure spin.
end
end
end
% if para.useStarMPS
% n = real(expectation_allsites_StarMPS(McOp,mps,Vtens,para));
% else
n = real(expectation_allsites_MC(McOp,mps,Vtens,para)); % (NC x L)
% end
end
function n = calBosonOcc_Tree(treeMPS,para)
%% calculate the boson occupation for treeMPS
% Zero operator for spin site
% calculate for all chains simultaneously
%
% Created by FS 28/02/2016
%
% L = para.L; NC = para.nChains; % total number of site and chains
% Implement recursively for now!
if treeMPS.height == 0
% this is leaf / chain
% create Operator for expectationvalue: need nc^2 each (i,:,j) is one operator [] x [] x n x [] x []
Lc = treeMPS.L; % length of chain
Op = cell(1,Lc);
for j = 1:Lc
if isempty(treeMPS.spinposition) || j ~= treeMPS.spinposition % works with array
[~,~,Op{1,j}] = bosonop(treeMPS.dk(1,j),treeMPS.shift(1,j),para.parity);
else
Op{1,j} = zeros(treeMPS.dk(1,j)); % don't measure spin.
end
end
n = real(expectation_allsites(Op,treeMPS.mps,treeMPS.Vmat,treeMPS.BondCenter)); % (NC x L)
n = reshape(n,[],1); % L x 1
else
% this is node -> more recursive calls
% assume, Focused on MPS!
% save n into temp cell array, since total number of chains at this node is unknown
nc = treeMPS.degree; % number of subchains at node
nTemp = cell(1,nc);
Atemp = contracttensors(treeMPS.BondCenter,2,2,treeMPS.mps{1},nc+2,1);
for ii = 1:nc
treeMPS.child(ii).BondCenter = contracttensors(conj(treeMPS.mps{1}),nc+2,[1:ii,ii+2:nc+2],Atemp,nc+2,[1:ii,ii+2:nc+2]); % contract all except D_(ii+1)
nTemp{ii} = calBosonOcc_Tree(treeMPS.child(ii),para); % L x nc(child)
end
lengths = cellfun(@(x) size(x,1),nTemp);
nChains = cellfun(@(x) size(x,2),nTemp);
n = zeros(max(lengths)+1,sum(nChains)); % L x nc_max
mc = 0;
for ii = 1:nc
n(1+(1:lengths(ii)),mc+(1:nChains(ii))) = nTemp{ii}; % copy all together
mc = mc + nChains(ii);
end
end
end
function bosonshift = calBosonShift(mps,Vmat,para)
% Calculate boson shift x, x^2, var(x)
% The operator on the spin site is set to zero
% Modified:
% FS 22/01/2014: - changed to using para.foldedChain.
% FS 04/06/2014: - updated for spinposition array
%
L = para.L;
x_opx = cell(1,L);
x2_opx = cell(1,L);
for j = 1:L
if j ~= para.spinposition
if para.foldedChain == 1
% Constructs Supersite Operators
% Only measures kron(x,1) chain part
[bp,~,n] = bosonop(sqrt(para.dk(j)),para.shift(j),para.parity);
idm = eye(size(n));
bpx = kron(bp,idm); bmx = bpx'; %nx = kron(n,idm); unused
x_opx{j} = sqrt(2)/2.*(bpx+bmx);
x2_opx{j} = x_opx{j}*x_opx{j};
elseif para.foldedChain == 0
[bp,bm,~] = bosonop(para.dk(1,j),para.shift(1,j),para.parity);
x_opx{j} = sqrt(2)/2.*(bp+bm);
x2_opx{j} = x_opx{j}*x_opx{j};
end
else
x_opx{j} = zeros(para.dk(j));
x2_opx{j} = zeros(para.dk(j));
end
end
bosonshift.x = expectation_allsites(x_opx,mps,Vmat);
bosonshift.xsquare = expectation_allsites(x2_opx,mps,Vmat);
bosonshift.xvariant = sqrt(bosonshift.xsquare-bosonshift.x.^2);
bosonshift.xerror = mean(abs(para.shift-bosonshift.x));
end
function shift = cal1siteShift(mps,Vmat,para)
%% Calculates shift for a single site, para.sitej!
% for use of shifting procedure in optimizesite.m
% mps and Vmat are single-site matrices
% mps has to be focused
NC = para.nChains; j = para.sitej;
xOp_OBB = cell(1,NC); % one x-OBB operator per chain
shift = zeros(1,NC); % one value per chain
% Generate operators and contract with OBB
if para.foldedChain == 1
switch para.model
case 'SpinDoubleBoson' % calculates shift only for x chain!
bp = bosonop(sqrt(para.dk(j)),para.shift(j),para.parity);
if para.parity == 'n'
idm = eye(size(bp));
bpx = kron(bp,idm); bmx = bpx';
else
[bpx,bmx,~,~,~,~]=paritykron(bp,para.bosonparity);
end
x = sqrt(2)/2*(bpx+bmx);
case '2SpinPhononModel'
bp = bosonop(sqrt(para.dk(j)),para.shift(j),para.parity);
if para.parity=='n'
idm=eye(size(bp));
bpr=kron(bp,idm); bmr=bpr'; % right chain
% bpl=kron(idm,bp); bml=bpl'; % left chain
else
[bpr,bmr,~,~,~,~]=paritykron(bp,para.bosonparity);
end
x = sqrt(2)/2*(bpr+bmr); % why only evaluate it for right part?
end
if para.useVmat
xOp_OBB{1} = Vmat' * x * Vmat;
else
xOp_OBB{1} = x;
end
else
for mc = 1:NC
x = cell(1,NC);
[bp,bm,~] = bosonop(para.dk(mc,j),para.shift(mc,j),para.parity);
x{mc} = sqrt(2)/2*(bp+bm);
xOp_OBB{mc} = contractMultiChainOBB(Vmat, x, para); % _(n~',n~)
end
end
temp = contracttensors(conj(mps),3,[1,2],mps,3,[1,2]); % _(n~',n~)
for mc = 1:NC
shift(mc) = real(sum(sum(temp.*xOp_OBB{mc}))); % fast tr(A * B.')
end
end
function reducedDensity = calRDM(mps,Vmat,para,k)
% calculates the reduced density matrix of any single site, mps{k} for size(mps{k})=a_{k-1} x a_k x n_k
%
% created 24/05/2014 by Florian Schroeder @ University of Cambridge
%
%
% copied from prepare.m:
% does l -> r sweep to create state in local picture of k
para.sweepto = 'r';
if length(k) == 1 % single-site RDM
for i = 1:k-1
if para.useVmat==1
[Vmat{i},V] = prepare_onesiteVmat(Vmat{i},para); % Vmat = U * S * V' ; Vmat := U; V:= S * V'
mps{i} = contracttensors(mps{i},3,3,V,2,2); % = Ai_{l,r,n} * V'_{p,n}; This contraction is defined differently to the paper.
end
[mps{i}, U] = prepare_onesite(mps{i}, para,i); % SVD(Ai_(l,r,n)) = Ai_(l,m,n) * U_(m,r)
mps{i+1} = contracttensors(U,2,2,mps{i+1},3,1); % U_(m,l) * A(i+1)_(l,r,n)
para=gennonzeroindex(mps,Vmat,para,i); % only if parity not 'n'
para=gennonzeroindex(mps,Vmat,para,i+1); % only if parity not 'n'
end
% now in form: Al{1}...Al{k-1} M{k} Ar{k+1}...Ar{L}
% with Al = left-normalized, Ar: right-normalized.
n = ndims(mps{k});
reducedDensity = contracttensors(mps{k},n,1:n-1,conj(mps{k}),n,1:n-1); % contract rD_nm = Mk_abn Mk*_abm
% reducedDensity = Vmat{k} * (reducedDensity * Vmat{k}');
reducedDensity = contracttensors(reducedDensity,2,2,conj(Vmat{k}),2,2); % contract rD_nj = rD_nm Vmat*_jm
reducedDensity = contracttensors(Vmat{k},2,2,reducedDensity,2,1); % contract rD_ij = Vmat_in rd_nj
elseif length(k) == 2 && k(2) == k(1) + 1
% 2-site RDM of nearest neighbours!
% move to k(1)
for i = 1:k(1)-1
if para.useVmat==1
[Vmat{i},V] = prepare_onesiteVmat(Vmat{i},para); % Vmat = U * S * V' ; Vmat := U; V:= S * V'
mps{i} = contracttensors(mps{i},3,3,V,2,2); % = Ai_{l,r,n} * V'_{p,n}; This contraction is defined differently to the paper.
end
[mps{i}, U] = prepare_onesite(mps{i}, para,i); % SVD(Ai_(l,r,n)) = Ai_(l,m,n) * U_(m,r)
mps{i+1} = contracttensors(U,2,2,mps{i+1},3,1); % U_(m,l) * A(i+1)_(l,r,n)
para=gennonzeroindex(mps,Vmat,para,i); % only if parity not 'n'
para=gennonzeroindex(mps,Vmat,para,i+1); % only if parity not 'n'
end
% use ncon, since easy and not performance-critical!
reducedDensity = ncon({mps{k(1)}, mps{k(2)}, conj(mps{k(1)}), conj(mps{k(2)}), Vmat{k(1)}, Vmat{k(2)}, conj(Vmat{k(1)}), conj(Vmat{k(2)})},...
{[1,2,3], [2,4,5], [1,6,7], [6,4,8], [-2,3], [-1,5], [-4,7], [-3,8]});
end
end
function participation = calParticipation(rdm)
% rdm: a reduced density matrix of a single site
% e.g. rdm = calRDM(mps,Vmat,para,k)
%
% takes a rdm and calculates the participation ratio
participation = 1/sum(diag(rdm).^2);
end
function tunnelE = calTunnelingEnergy(mps,Vmat,para,op)
%% calculates the tunneling energy in MLSBM of the PPC system
% <Psi|H0-diag(H0)|Psi>
%
% could be applied to any system.
% Interacting System Hamiltonian: HI
HI = cell(1);
if ~exist('op','var') || ~isfield(op,'h1term')
disp('guessing H0 from para for MLSBM');
op.h1term{1,1} = Hamiltonian_PPC(para);
end
HI{1} = op.h1term{1,1}-diag(diag(op.h1term{1,1}));
tunnelE = expectationvalue(HI,mps,Vmat,mps,Vmat);
end
function An = calBath1SiteCorrelatorsProject(mps,Vmat,para,opSelect,stateProj,bondProj)
% calculates the (projected) single-site expectation value
% output is matrix containing all values for all chains (L x nChains)
% custom made solution for biggest speedup, exact solution!
% opSelect: ['bp'|'x'|'bp^2'|'x^2'|'n']
% stateProj: equals state-projection operator, normalised!
% []: no selection, take all
% bondProj: equals bond-projection operator, normalised!
% []: no selection, take entire bond
%
% supports Multi-Chain with Vmat and Vtens, Works with single chain!
% does NOT support StarMPS or TreeMPS input! This needs pre-processing
assert(para.foldedChain == 0, 'Please use single-chain code for folded Chains');
An = zeros(para.L, para.nChains); % initialize results array
op_OBB = cell( para.L, para.nChains);
%% generate all operators & contract with OBB Vmat / Vtens:
for j = 1:para.L
para.sitej = j;
for mc = 1:para.nChains
op = cell(1, para.nChains); % containing single-site, single-chain operator
if j ~= para.spinposition
switch opSelect
case 'bp'
op{mc} = bosonop(para.dk(mc,j),para.shift(mc,j),para.parity);
case 'x'
bp = bosonop(para.dk(mc,j),para.shift(mc,j),para.parity);
op{mc} = (bp+bp')/2;
case 'bp^2'
bp = bosonop(para.dk(mc,j),para.shift(mc,j),para.parity);
op{mc} = bp^2;
case 'x^2'
bp = bosonop(para.dk(mc,j),para.shift(mc,j),para.parity);
op{mc} = (bp+bp')^2/4;
case 'n'
[~,~,op{mc}] = bosonop(para.dk(mc,j),para.shift(mc,j),para.parity);
end
op_OBB{j, mc} = contractMultiChainOBB(Vmat{j}, op, para);
else % Spin & Multi-Level System
if ~isempty(stateProj)
op_OBB{j,mc} = Vmat{j}' * stateProj * Vmat{j}; % Spin-site Vmat, project onto system state!
else
op_OBB{j,mc} = Vmat{j}' * Vmat{j};
end
end
end
end
if isempty(stateProj) && ~isempty(bondProj)
% do SVD from site 1 to 2 in order to get projections onto the dominant system states!
% otherwise have projection onto dominant Bath states (from previous sweep to left)
A = mps{1};
[D1, D2, d] = size(A); % d is site dimension
A = permute(A, [3, 1, 2]);
A = reshape(A, [d * D1, D2]); % reshapes to (a1 d),a2
[B, S, U] = svd2(A); % Could also use QR decomposition if nargin !=5
DB = size(S, 1); % new a2 dimension of B should == d
B = reshape(B, [d, D1, DB]);
mps{1} = permute(B, [2, 3, 1]); % not used anymore!
% do not contract S into mps{2}, to avoid the displacement to be contaminated with the state amplitude!
mps{2} = contracttensors(U,2,2, mps{2},3,1);
end
%% compute all partial contractions
% such that (1+sz/4) gets updated to the right in row 1;
% Trace(contraction for result) = Am(j)
opContract = cell(para.L, 1); % same for all chains! since only 1 common MPS backbone
for j = 1:para.L
for mc = 1:para.nChains
if j ~= 1
An(j,mc) = trace(updateCleft(opContract{j-1,1},mps{j},[],op_OBB{j,mc},mps{j},[])); % this is the result!
end
end
if j ~=1