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H_Eff.m
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H_Eff.m
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function [op] = H_Eff(mps, V, target, op, para)
%% [op] = H_Eff(mps, Vtarget, op, para)
% creates the effective operators for 'target' to allow fast contractions
% in matrix exponentials and eigs()
%
% mps : single-site A matrix, mps{sitej}
% Vmat : single-site V matrix, Vmat{sitej}
% e.g.
% [op] = H_Eff(Amat, [] , 'V' , op, para);
% [op] = H_Eff([] , Vmat, 'A' , op, para); % transforms op.h1j, h2j to OBB
% [op] = H_Eff(Amat, [] , 'CA', op, para); % relies on previous OBB trafo
%
% Only MC-TDVP code (HOSVD): 'MC-OBB', 'MC-CV', 'MC-V', 'MC-VS', 'MC-A'
%
% [op] = H_Eff([] ,Vtens, 'MC-OBB', op, para); % transforms all chain h1j/h2j into MCOBB, calls MC-CV
% [op] = H_Eff([] ,Vtens, 'MC-CV' , op, para); % transforms single chain into OBB; defined by para.currentChain
% [op] = H_Eff([] ,Vtens, 'MC-V' , op, para); % create HnonInt and Hleft/rightAS for single-chain V evolution
% [op] = H_Eff([] ,[] , 'MC-VS' , op, para); % create h12jAV terms for VS evolution
% [op] = H_Eff([] ,Vtens, 'MC-A' , op, para); % create h1/2jOBB terms, depends on MC-OBB
%
% Only in Star-MPS code:
%
% [op] = H_Eff(mps{1}, [], 'ST-CA', op, para); % transforms System + all other chains into Hleft, Opleft defined by para.currentChain
%
% Only in Tree-MPS code:
%
% [op] = H_Eff(treeMPS, [], 'TR-A' , [], para); % transforms node h1h/h2j into OBB; for leaf/chain use standard 'A'
% [op] = H_Eff(treeMPS, [], 'TR-CA', [], para); % transforms Node + all other chains into H/Opleft defined by para.currentChain
% [op] = H_Eff(treeMPS, [], 'TR-CP', [], para); % transforms Node + all chains into H/Opright for parent node
%
% Created 04/08/2015 by FS
M = para.M;
switch target
case 'V'
%% multiply MPS into op.Hright, Hleft, Opright, Opleft
% HleftA_(n~',n~) = A*_(l',r,n~') [Hl_(l',l) * A_(l,r,n~)]_(l',r,n~)
op.HleftA = contracttensors(op.Hleft,2,2,mps,3,1);
op.HleftA = contracttensors(conj(mps),3,[1 2], op.HleftA,3,[1 2]);
% Hright_(n~',n~) = A*_(l,r',n~') [Hr_(r',r) * A_(l,r,n~)]_(r',l,n~)
op.HrightA = contracttensors(op.Hright,2,2,mps,3,2);
op.HrightA = contracttensors(conj(mps),3,[1 2], op.HrightA,3,[2 1]);
op.OpleftA = cell(M,1);
op.OprightA = cell(M,1);
for k=1:M
% same contractions as above for Hleft/Hright
if isempty(op.Opleft{k})
op.OpleftA{k} = [];
else
op.OpleftA{k}= contracttensors(op.Opleft{k}, 2,2, mps,3,1);
op.OpleftA{k}= contracttensors(conj(mps),3,[1 2], op.OpleftA{k},3,[1 2]);
end
if isempty(op.Opleft{k})
op.OpleftA{k} = [];
else
op.OprightA{k} = contracttensors(op.Opright{k},2,2, mps,3,2);
op.OprightA{k} = contracttensors(conj(mps),3,[1 2], op.OprightA{k},3,[2 1]);
end
end
case 'A'
%% multiply Vmat into op.h1j, h2j
% transform all bare H terms of current sitej into OBB
% works with multi-chain at OBB level
if ~isfield(op,'h1j') || isempty(op.h1j)
op = gen_sitej_h1h2(op,para,para.sitej);
end
if para.nChains == 1 || (~iscell(V) && ~iscell(op.h1j))
op.h1jOBB = V' * (op.h1j * V); % faster and more accurate
op.h2jOBB = cell(M,2);
for i=1:M
op.h2jOBB{i,1} = V' * (op.h2j{i,1} * V); % faster and more accurate
op.h2jOBB{i,2} = V' * (op.h2j{i,2} * V); % faster and more accurate
end
else % nChains > 1 && iscell(V)
h1jnew = 0;
for i = find(~cellfun('isempty',op.h1j'))
H1 = cell(para.nChains,1);
H1(i) = op.h1j(i);
h1jnew = h1jnew + contractMultiChainOBB(V, H1, para);
end
op.h1jOBB = h1jnew;
op.h2jOBB = cell(M,2);
for i = 1:M
op.h2jOBB{i,1} = contractMultiChainOBB(V, op.h2j(i,1,:), para);
op.h2jOBB{i,2} = contractMultiChainOBB(V, op.h2j(i,2,:), para);
end
end
case 'CA'
%% contract op.Hleft,Hright,Opleft,Opright with op.h1jOBB,h2jOBB and MPS matrix
% save into op.HleftAV,HrightAV,OprightAV,h2jAV
switch para.sweepto
case 'r' % MPS and V contracted into left operators only
% HleftAV_(r~',r~) = A*_(l',r~',n~) [Hl_(l',l) * A_(l,r~,n~)]_(l',r~,n~)
op.HleftAV = updateCleft(op.Hleft, mps, [], [], mps, []);
% h1jAV_(r~',r~) = A*_(l,r~',n~') [A_(l,r~,n~), h1j_(n~',n~)]_(l,r~,n~')
op.HleftAV = op.HleftAV + updateCleft([],mps,[],op.h1jOBB,mps,[]);
% op.Opleft will be summed over and added to HleftAV, since
% it does not interact across CA
op.h2jAV = cell(M,1);
for k = 1:M
% OpleftAV_(r~',r~) = A*_(l',r~',n~')[[OPl_(l',l) A_(l,r~,n~)]_(l',r~,n~) h2j_(n~',n~)]_(l',r~,n~')
op.HleftAV = op.HleftAV + updateCleft(op.Opleft{k}, mps, [], op.h2jOBB{k,2}, mps, []);
% h2jAV_(r~',r~) = A*_(l,r~',n~') [A_(l,r~,n~) h2j_(n~',n~)]_(l,r~,n~')
op.h2jAV{k} = updateCleft([], mps, [], op.h2jOBB{k,1}, mps, []);
end
case 'l' % MPS and V contracted into right operators only
% HHrightAV_(l~',l~) = A*_(l~',r',n~) [Hr_(r',r) * A_(l~,r,n~)]_(r',l~,n~)
op.HrightAV = updateCright(op.Hright, mps, [], [], mps, []);
% h1jAV_(l~',l~) = A*_(l~',r,n~') [A_(l~,r,n~), h1j_(n~',n~)]_(l~,r,n~')
op.HrightAV = op.HrightAV + updateCright([],mps,[],op.h1jOBB,mps,[]);
% op.OprightAV = cell(M,1); % add to op.HrightAV directly
op.h2jAV = cell(M,1);
for k = 1:M
% OprightAV_(l~',l~) = A*_(l~',r',n~')[[A_(l~,r,n~) OPr_(r',r)]_(l~,n~,r') h2j_(n~',n~)]_(l~,r',n~')
op.HrightAV = op.HrightAV + updateCright(op.Opright{k},mps,[],op.h2jOBB{k,1},mps,[]);
% h2jAV_(l~',l~) = A*_(l~',r,n~') [A_(l~,r,n~) h2j_(n~',n~)]_(l~,r,n~')
op.h2jAV{k} = updateCright([],mps,[],op.h2jOBB{k,2},mps,[]);
end
end
case 'MC-V'
%% contract Tensors of Multi-Chain HOSVD-TDVP for evolution of V{k}
% builds upon 'V' preparation. -> [op] = H_Eff(Amat, [] , 'V' , op, para);
% V has to be cell with V{end} = VS the tensor-center, unpermuted: n1 x n2 x ... x nNC x n~
% Saves in [H/op][left/right]AS
% leave out ~ for clarity
NC = para.nChains;
nc = para.currentChain; % # chain I am working on
nTerms = para.M/NC; % # interacting terms per chain
% 1. contract VS into 4th-order tensor
IndContract = 1:NC; IndContract(nc) = [];
VSfullContract = contracttensors(conj(V{end}),NC+1, IndContract, V{end}, NC+1, IndContract); % VS_(nk', n', nk, n)
op.HleftAS = contracttensors(op.HleftA, 2,[1,2], VSfullContract,4,[2,4]); % H_(nk',nk)
op.HrightAS = contracttensors(op.HrightA,2,[1,2], VSfullContract,4,[2,4]);
% this could collect all terms non-interacting with current chain!
op.HnonInt = op.HleftAS + op.HrightAS;
% h1j terms of other chains
for mc = 1:NC
if mc == nc || isempty(op.h1jMCOBB{mc}), continue; end;
IndContract = 1:NC+1; IndContract([mc,nc]) = [];
VS = contracttensors(conj(V{end}),NC+1, IndContract, V{end}, NC+1, IndContract);
if mc < nc % -> VS_(nmc',nk',nmc,nk)
order = [1,3];
else % -> VS_(nk',nmc',nk,nmc)
order = [2,4];
end
op.HnonInt = op.HnonInt + contracttensors(op.h1jMCOBB{mc},2,[1,2], VS,4,order);
end
for m = 1:para.M
mc = ceil(m/nTerms); % the chain number for current m
if nc == mc
op.OpleftAS{m} = contracttensors(op.OpleftA{m}, 2,[1,2], VSfullContract,4,[2,4]);
op.OprightAS{m} = contracttensors(op.OprightA{m}, 2,[1,2], VSfullContract,4,[2,4]);
else % sum into op.HnonInt
if isempty(op.h2jMCOBB{m,2,mc}) && isempty(op.h2jMCOBB{m,1,mc}), continue; end;
if NC > 2
% better contraction scheme?:
[idxA, idxB] = getIdxTensChain(NC+1,mc,nc);
% idxA = 1:NC+1; idxB = idxA;
% idxA([mc,nc]) = []; idxA = [idxA,mc]; % Do not contract nc
% if mc < nc
% idxB(nc-1) = []; % since nc shifted
% else
% idxB(nc) = [];
% end
OpTemp = contracttensors(V{end}, NC+1, NC+1, op.OpleftA{m}.',2,1); %_(ni..,nk)
OpTemp = contracttensors(OpTemp, NC+1, mc, op.h2jMCOBB{m,2,mc}.',2,1); %_(ni,...,nk,mc)
op.HnonInt = op.HnonInt + contracttensors(conj(V{end}),NC+1, idxA, OpTemp, NC+1, idxB);
OpTemp = contracttensors(V{end}, NC+1, NC+1, op.OprightA{m}.',2,1); % _(ni..,nk)
OpTemp = contracttensors(OpTemp, NC+1, mc, op.h2jMCOBB{m,1,mc}.',2,1); % _(nc,nk,mc)
op.HnonInt = op.HnonInt + contracttensors(conj(V{end}),NC+1, idxA, OpTemp, NC+1, idxB);
else % need more efficient code! NC+1 = 3
assert(NC+1 == 3, 'this code only works under this assumption!');
OpTemp = contracttensors(V{end}, NC+1, NC+1, op.OpleftA{m}.',2,1); % _(n1,n2,nk)
OpTemp = contracttensors(OpTemp, NC+1, mc, op.h2jMCOBB{m,2,mc}.',2,1); % _(nc,nk,mc)
op.HnonInt = op.HnonInt + contracttensors(conj(V{end}),NC+1, [mc,3], OpTemp, NC+1, [3,2]);
OpTemp = contracttensors(V{end}, NC+1, NC+1, op.OprightA{m}.',2,1); % _(n1,n2,nk)
OpTemp = contracttensors(OpTemp, NC+1, mc, op.h2jMCOBB{m,1,mc}.',2,1); % _(nc,nk,mc)
op.HnonInt = op.HnonInt + contracttensors(conj(V{end}),NC+1, [mc,3], OpTemp, NC+1, [3,2]);
end
end
end
% finish with terms: op.HnonInt, op.OpleftAS{m}, op.OprightAS{m}
case 'MC-CV'
%% contract to get each single chain h1j/h2j into its OBB.
% equivalent to creating the H_Eff for VS
if isempty(op.h1j)
op = gen_sitej_h1h2(op,para,para.sitej);
end
nTerms = para.M/para.nChains;
nc = para.currentChain;
M = 1:para.M;
M(ceil(M/nTerms) ~=nc) = []; % find which m to operate on
if ~isempty(op.h1j{nc})
op.h1jMCOBB{nc} = V{nc}' * (op.h1j{nc} * V{nc});
else
op.h1jMCOBB{nc} = [];
end
for m = M
if ~isempty(op.h2j{m,1,nc})
op.h2jMCOBB{m,1,nc} = V{nc}' * (op.h2j{m,1,nc} * V{nc});
op.h2jMCOBB{m,2,nc} = V{nc}' * (op.h2j{m,2,nc} * V{nc});
else
op.h2jMCOBB{m,1,nc} = [];
op.h2jMCOBB{m,2,nc} = [];
end
end
case 'MC-OBB'
%% Create all OBB terms for each chain
for i = 1:para.nChains
para.currentChain = i;
[op] = H_Eff([] , V, 'MC-CV', op, para);
end
case 'MC-VS'
%% Create the kronecker product terms for faster VS exponential
d = para.d_opt(:,para.sitej);
nTerms = para.M/para.nChains;
for k = 1:para.nChains
M = 1:para.M;
M(ceil(M/nTerms) ~=k) = []; % find which m to operate on
op.h12jAV{k} = kron(eye(d(end)), op.h1jMCOBB{k});
for m = M
op.h12jAV{k} = op.h12jAV{k} + kron(op.OpleftA{m} , op.h2jMCOBB{m,2,k});
op.h12jAV{k} = op.h12jAV{k} + kron(op.OprightA{m}, op.h2jMCOBB{m,1,k});
end
end
case 'MC-A'
%% Create the OBB terms for A, gathering all chains
% needs all h1jMCOBB to be up-to-date (MC-OBB)
% use contractMultiChainOBB with objects from here! Copied from 'A'
h1jnew = 0;
for i = find(~cellfun('isempty',op.h1jMCOBB))
H1 = cell(para.nChains,1);
H1(i) = op.h1jMCOBB(i);
h1jnew = h1jnew + contractMultiChainOBB(V{end}, H1, para);
end
op.h1jOBB = h1jnew;
op.h2jOBB = cell(M,2);
for i = 1:M
op.h2jOBB{i,1} = contractMultiChainOBB(V{end}, op.h2jMCOBB(i,1,:), para);
op.h2jOBB{i,2} = contractMultiChainOBB(V{end}, op.h2jMCOBB(i,2,:), para);
end
case 'ST-CA'
%% Create the effective Chain-System terms for STAR-MPS
% Store in op.chain(para.currentChain).H/Opleft
% Only used once for entering a chain
% similar to MC-V in structure
nc = para.currentChain;
NC = para.nChains;
M = para.M / NC; % what is M for each single chain?
d = size(mps);
% MPS is only mps{1}: 1 x D1 x D2 x D3 x ... X D(NC) x dk
% chain index in mps is shifted by 1 due to first singleton
% dimension!! -> Compatibility with 1-chain models and future Boundary Conditions?
op.chain(nc).Hleft = 0; % contains effective H of all other chains
op.chain(nc).Opleft = cell(para.M/NC,1); % contains only terms interacting with Chain #nc
for mc = 1:NC
if mc == nc, continue; end;
% always make sure that Hlrstorage{1} and Opstorage{:,2,1} are up-to-date!
if para.tdvp.HEffSplitIsometry == 0
% 1. Contract to non-interacting parts -> Hleft
[idxA, idxB] = getIdxTensChain(NC+2,mc+1,nc+1);
OpTemp = contracttensors(mps, NC+2, mc+1, op.chain(mc).Hlrstorage{1}.',2,1); %_(ni..,nk)
op.chain(nc).Hleft = op.chain(nc).Hleft + contracttensors(conj(mps),NC+2, idxA, OpTemp, NC+2, idxB);
% 2. Contract the other chain-system interacting parts -> Hleft
for m = 1:M
% this chain's m
systemM = M*(mc-1) + m;
OpTemp = contracttensors(mps , NC+2, NC+2, op.h2jOBB{systemM,1}.' ,2,1);
OpTemp = contracttensors(OpTemp, NC+2, mc+1, op.chain(mc).Opstorage{m,2,1}.',2,1); % (m,2,1) should be the operator of site 2 in the effective left basis for system site 1
op.chain(nc).Hleft = op.chain(nc).Hleft + contracttensors(conj(mps),NC+2, idxA, OpTemp, NC+2, idxB);
end
else
% 1. Split-off isometry
% but need D(mc), D(nc) and dk in Atens!
% para.currentChain = mc;
[Atens,dOut] = tensShape(mps, 'unfoldiso', [mc+1,nc+1,NC+2], d); % +1 for leading singleton
[Iso, Atens] = qr(Atens,0); % Iso is isometry with all unused chains
% TODO: comment if not testing!!
% [Iso2,A2,err]= rrQR(Atens, floor(0.1*prod(dOut(end-2:end))),0); % low rank QR approximation
% fprintf('H_eff: rank(A) = %d, dim(A,2) = %d, rrQR error = %g; %g\n', rank(Atens), prod(dOut(end-2:end)),norm(Atens - Iso2*A2), err);
Atens = reshape(Atens,[size(Atens,1),dOut(end-2:end)]); % D(mc)*D(nc)*dk x D(mc) x D(nc) x dk
idxA = [1, 2, 4]; idxB = [1, 4, 3];
% 2. Contract to non-interacting parts -> Hleft
OpTemp = contracttensors(Atens, 4, 2, op.chain(mc).Hlrstorage{1}.',2,1); %_(ni..,nk)
op.chain(nc).Hleft = op.chain(nc).Hleft + contracttensors(conj(Atens),4, idxA, OpTemp, 4, idxB);
% 3. Contract the other chain-system interacting parts -> Hleft
for m = 1:M
% this chain's m
systemM = M*(mc-1) + m;
OpTemp = contracttensors(Atens , 4, 4, op.h2jOBB{systemM,1}.' ,2,1);
OpTemp = contracttensors(OpTemp, 4, 2, op.chain(mc).Opstorage{m,2,1}.',2,1); % (m,2,1) should be the operator of site 2 in the effective left basis for system site 1
op.chain(nc).Hleft = op.chain(nc).Hleft + contracttensors(conj(Atens),4, idxA, OpTemp, 4, idxB);
end
end
end
% 3. Contract h1jOBB -> Hleft
idx = 1:NC+2; idx(nc+1) = [];
OpTemp = contracttensors(mps, NC+2, NC+2, op.h1jOBB.', 2, 1);
op.chain(nc).Hleft = op.chain(nc).Hleft + contracttensors(conj(mps), NC+2, idx, OpTemp,NC+2, idx);
% 4. Contract h2jOBB which will interact with chain nc
for m = 1:M
systemM = M*(nc-1) + m;
OpTemp = contracttensors(mps, NC+2, NC+2, op.h2jOBB{systemM,1}.', 2, 1);
op.chain(nc).Opleft{m} = contracttensors(conj(mps), NC+2, idx, OpTemp,NC+2, idx);
end
% overwrite main vars for evolve Kn
% op.Hleft = op.chain(nc).Hleft;
% op.Opleft = op.chain(nc).Opleft;
op.Hright = op.chain(nc).Hlrstorage{1};
op.Opright = op.chain(nc).Opstorage(:,2,1);
case 'TR-A'
%% multiply Vmat into h1j/h2j for node
% transform all bare H terms of current node into OBB
% only for treeMPS
op = H_Eff_TR_A(mps);
case 'TR-CA'
%% Create the effective Hleft terms for the children of TREE-MPS
% Store in treeMPS.child(para.currentChain).op.H/Opleft
% since Hlrstorage{1} still contains Hright for CA
% Only used once for entering a child
% similar to MC-V and ST-CA in structure
%
% mps: treeMPS - contains node-specific para
% V: []
% op: []
% para: para - contains calculation-specific para
% do call to subfunction to translate variable names -> more consistent
op = H_Eff_TR_CA(mps,para);
case 'TR-CP'
%% Create the effective Hright terms for the parent of TREE-MPS
% TRee - CenterParent
% Store in treeMPS.op.H/Opright
% since Hlrstorage{1} still has Hleft for CA
% similar to MC-V and ST-CA in structure
%
% mps: treeMPS - contains node-specific para
% V: []
% op: []
% para: para - contains calculation-specific para
if mps.hasSite
op = H_Eff_TR_CP_Site(mps,para);
else
op = H_Eff_TR_CP_NoSite(mps,para); % special version to treat chain-combination nodes
end
end
end
function op = H_Eff_TR_A(treeMPS)
%% function op = H_Eff_TR_A(treeMPS)
%
% fetches h1j/h2j if empty and transforms into OBB
% Only for internal nodes of tree
op = treeMPS.op;
if ~isfield(op,'h1j') || isempty(op.h1j)
op = gen_sitej_h1h2(op,treeMPS);
end
V = treeMPS.Vmat{1};
op.h1jOBB = V' * (op.h1j * V);
op.h2jOBB = cell(size(op.h2j));
for i = 1:size(op.h2j,1)
for j = 1:size(op.h2j,3)
op.h2jOBB{i,1,j} = V' * (op.h2j{i,1,j} * V); % faster than cellfun and more accurate
op.h2jOBB{i,2,j} = V' * (op.h2j{i,2,j} * V);
end
end
end
function op = H_Eff_TR_CA(treeMPS,para)
%% function op = H_Eff_TR_CA(treeMPS,para)
%
% Creates Hleft for entering a child/chain = walking down the tree
% Child has no site and is an entanglement renormalisation tensor!
%
% created by FS 23/02/2016
nc = treeMPS.currentChild; % = child to prepare operators for
NC = treeMPS.degree; % = nChild for node
M = treeMPS.M;
d = size(treeMPS.mps{1}); % numel(d) = NC+1 or NC+2
Nd = numel(d);
mps = treeMPS.mps{1}; % get handle, should not copy object!
% MPS is only mps{1}: Dl x D1 x D2 x D3 x ... X D(NC) x dk
% chain index in mps is shifted by 1 due to first left Bond
for i = 1:length(treeMPS.child)
NCoup(i) = length(treeMPS.child(i).chainIdx); % much faster than arrayfun()
end
op = treeMPS.child(nc).op; % only return op -> needs overwriting in function call
op.Hleft = 0; % contains effective H of all other chains
op.Opleft = cell(M,NCoup(nc)); % contains only terms interacting with Chain #nc
% 1. construct non-interacting Hleft for child
for mc = 0:NC
if treeMPS.hasSite
dProd = prod(d([mc+1,nc+1,NC+2])); % mc interacts with local site
else
dProd = prod(d([mc+1,nc+1,1])); % mc interacts with parent
end
% mc: other chain to contract with MPS to get into left basis of nc
if mc == nc, continue, end
% always make sure that Hlrstorage{1} and Opstorage{:,2,1} are up-to-date!
if para.tdvp.HEffSplitIsometry == 0 || (prod(d)/dProd^2) < 1.3
if mc == 0 && treeMPS.isRoot
continue
elseif mc == 0 && ~treeMPS.isRoot % Node-parent exists -> contract
idx = 1:Nd; idx(nc+1) = [];
% 1. Contract to non-interacting parts
w = contracttensors(treeMPS.op.Hlrstorage{1},2,2, mps, Nd, 1); % order unchanged, Focused on Node -> Hlrstorage{1} is Hleft of Node
if treeMPS.hasSite
% 2. Contract the parent-local site interacting parts
for m = 1:M
OpTemp = contracttensors(mps, Nd, Nd, treeMPS.op.h2jOBB{m,2,1}.',2,1); % parent is 1st, node is 2nd position in H_int !!
w = w + contracttensors(treeMPS.op.Opstorage{m,1,1},2,2, OpTemp, Nd, 1); % (m,1,1) should be the operator of parent in the effective left basis for node site 1
end
end
op.Hleft = op.Hleft + contracttensors(conj(mps),Nd, idx, w, Nd, idx);
continue
end
% 1. Contract to non-interacting parts -> Hleft
w = contracttensors(mps, Nd, mc+1, treeMPS.child(mc).op.Hlrstorage{1}.',2,1); %_(ni..,nk)
% 2. Contract the other chain-system interacting parts -> Hleft
% NCoup(mc) indicates how many terms are needed: 1 for leaf/chain; more for ER-tensor
NCoupOffset = sum(NCoup(1:mc-1)); % how many terms belong to other children -> Offset
if treeMPS.hasSite
for nn = 1:NCoup(mc)
for m = 1:M
OpTemp = contracttensors(mps , Nd, Nd, treeMPS.op.h2jOBB{m,1,nn+NCoupOffset}.' ,2,1);
w = w + contracttensors(OpTemp, Nd, mc+1, treeMPS.child(mc).op.Opstorage{m,2,1,nn}.',2,1); % (m,2,1) should be the operator of site 2 in the effective left basis for system site 1
end
end
else % ~hasSite
for nn = 1:NCoup(mc)
for m = 1:M
OpTemp = contracttensors(treeMPS.op.Opstorage{m,1,1, nn+NCoupOffset},2,2, mps , Nd, 1); % Take interaction from parent
w = w + contracttensors(OpTemp, Nd, mc+1, treeMPS.child(mc).op.Opstorage{m,2,1,nn}.',2,1); % (m,2,1) should be the operator of site 2 in the effective left basis for system site 1
end
end
end
[idxA, idxB] = getIdxTensChain(Nd,mc+1,nc+1);
% contraction of mps with chain ops of mc moved its index to the end
% -> generate idx pair for <mps|H|MPS>, where everything except nc shall be contracted
op.Hleft = op.Hleft + contracttensors(conj(mps),Nd, idxA, w, Nd, idxB);
else
if mc == 0 && treeMPS.isRoot
continue
end
% 1. Split-off isometry
% but need D(mc), D(nc) and dk in Atens!
[Atens,dOut] = tensShape(mps, 'unfoldiso', [mc+1,nc+1,NC+2], d); % +1 for leading singleton
[~, Atens] = qr(Atens,0); % Iso is isometry with all unused chains
% TODO: comment if not testing!!
% rEst = rank(Atens,10^-4.5);
% fprintf('H_eff: rank(A) = %d, dim(A,2) = %d\n',rEst,prod(dOut(end-2:end)));
% [Q, ~] = qr(Atens,0); % Iso is isometry with all unused chains
% A2 = Q'*Atens;
% [Iso2,A3,err]= rrQR(Atens, rEst,0); % low rank QR approximation
% fprintf('H_eff: rank(A) = %d, dim(A,2) = %d, rrQR error = %g; %g\n', rank(Atens), prod(dOut(end-2:end)),norm(Atens - Iso2*A2), err);
% Atens = A1;
if mc == 0 % TODO: if these assignments take too long, then remove this shortcut!
Hstor = treeMPS.op.Hlrstorage{1}; % Hleft of node
h2jOBB = treeMPS.op.h2jOBB(:,2,1);
Opstor = treeMPS.op.Opstorage(:,1,1); % Opleft of node
else
Hstor = treeMPS.child(mc).op.Hlrstorage{1}; % Hright of child
h2jOBB = treeMPS.op.h2jOBB(:,1,mc);
Opstor = treeMPS.child(mc).op.Opstorage(:,2,1); % Opleft of child
end
Atens = reshape(Atens,[size(Atens,1),dOut(end-2:end)]); % D(mc)*D(nc)*dk x D(mc) x D(nc) x dk
idxA = [1, 2, 4]; idxB = [1, 4, 3];
% 2. Contract to non-interacting parts -> Hleft
OpTemp = contracttensors(Atens, 4, 2, Hstor.',2,1); %_(ni..,nk)
op.Hleft = op.Hleft + contracttensors(conj(Atens),4, idxA, OpTemp, 4, idxB);
% 3. Contract the other chain-system interacting parts -> Hleft
for m = 1:M
OpTemp = contracttensors(Atens , 4, 4, h2jOBB{m}.', 2, 1);
OpTemp = contracttensors(OpTemp, 4, 2, Opstor{m}.', 2, 1); % (m,2,1) should be the operator of site 2 in the effective left basis for system site 1
op.Hleft = op.Hleft + contracttensors(conj(Atens),4, idxA, OpTemp, 4, idxB);
end
end
end
idx = 1:Nd; idx(nc+1) = [];
NCoupOffset = sum(NCoup(1:nc-1)); % how many terms belong to other children -> Offset
if treeMPS.hasSite
% 3. Contract h1jOBB -> Hleft
OpTemp = contracttensors(mps, Nd, Nd, treeMPS.op.h1jOBB.', 2, 1);
op.Hleft = op.Hleft + contracttensors(conj(mps), Nd, idx, OpTemp,Nd, idx);
% 4. Contract h2jOBB which will interact with chain nc
for nn = 1:NCoup(nc)
for m = 1:M
OpTemp = contracttensors(mps, Nd, Nd, treeMPS.op.h2jOBB{m,1,nn+NCoupOffset}.', 2, 1);
op.Opleft{m,nn} = contracttensors(conj(mps), Nd, idx, OpTemp,Nd, idx);
end
end
else % ~hasSite: carry over parent interactions
for nn = 1:NCoup(nc)
for m = 1:M
OpTemp = contracttensors(treeMPS.op.Opstorage{m,1,1,nn+NCoupOffset}, 2, 2, mps, Nd, 1);
op.Opleft{m,nn} = contracttensors(conj(mps), Nd, idx, OpTemp,Nd, idx);
end
end
end
op.Hright = treeMPS.child(nc).op.Hlrstorage{1}; % copy into temporary H/Opright to allow overwriting in updateop.
op.Opright = squeeze(treeMPS.child(nc).op.Opstorage(:,2,1,:)); % (M,NCoup)
end
function op = H_Eff_TR_CP_Site(treeMPS,para)
%% function op = H_Eff_TR_CP_Site(treeMPS,para)
%
% Creates Hright for entering the parent = walking up the tree
% Applies if current node carries a physical site which couples to the parent and which all children are coupled to
%
% created by FS 23/02/2016
NChild = treeMPS.degree; % = nChildren for node
NChain = treeMPS.nChains; % = nChains of node
M = treeMPS.M;
d = size(treeMPS.mps{1}); % numel(d) = NC+2 if hasSite; elseif ~hasSite: NC+1
mps = treeMPS.mps{1}; % get handle, should not copy object!
% MPS is only mps{1}: Dl x D1 x D2 x D3 x ... X D(NC) [x dk ]
% chain index in mps is shifted by 1 due to first left Bond
op = treeMPS.op; % only return op -> needs overwriting in function call
op.Hright = 0; % contains effective H of node and all chains
op.Opright = cell(M,1); % contains only terms interacting with parent
if treeMPS.isRoot
error('VMPS:H_Eff:NotSupported','This function shall only be used for non-root nodes');
end
% 1. construct non-interacting Hright for parent
for mc = 1:NChild
% mc: chain to contract with MPS to get into right basis of parent
% always make sure that Hlrstorage{1} and Opstorage{:,2,1} are up-to-date!
if para.tdvp.HEffSplitIsometry == 0
[idxA, idxB] = getIdxTensChain(NChild+2,mc+1,1);
% contraction of mps with chain ops of mc will move its index to the end
% -> generate idx pair for <mps|H|MPS>, where everything except 1 shall be contracted
% 1. Contract to non-interacting parts -> Hright
OpTemp = contracttensors(mps, NChild+2, mc+1, treeMPS.child(mc).op.Hlrstorage{1}.',2,1);
op.Hright = op.Hright + contracttensors(conj(mps),NChild+2, idxA, OpTemp, NChild+2, idxB);
% 2. Contract the other chain-system interacting parts -> Hright
for m = 1:M
OpTemp = contracttensors(mps , NChild+2, NChild+2, treeMPS.op.h2jOBB{m,1,mc}.' ,2,1);
OpTemp = contracttensors(OpTemp, NChild+2, mc+1, treeMPS.child(mc).op.Opstorage{m,2,1}.',2,1); % (m,2,1) should be the operator of site 2 in the effective left basis for system site 1
op.Hright = op.Hright + contracttensors(conj(mps),NChild+2, idxA, OpTemp, NChild+2, idxB);
end
else
% 1. Split-off isometry
% but need D(mc), D(nc) and dk in Atens!
[Atens,dOut] = tensShape(mps, 'unfoldiso', [mc+1,1,NChild+2], d); % +1 for leading singleton
[~, Atens] = qr(Atens,0); % Iso is isometry with all unused chains
% TODO: comment if not testing!!
% [Iso2,A2,err]= rrQR(Atens, floor(0.1*prod(dOut(end-2:end))),0); % low rank QR approximation
% fprintf('H_eff: rank(A) = %d, dim(A,2) = %d, rrQR error = %g; %g\n', rank(Atens), prod(dOut(end-2:end)),norm(Atens - Iso2*A2), err);
Hstor = treeMPS.child(mc).op.Hlrstorage{1}; % Hright of child
h2jOBB = treeMPS.op.h2jOBB(:,1,mc);
Opstor = treeMPS.child(mc).op.Opstorage(:,2,1); % Opleft of child
Atens = reshape(Atens,[size(Atens,1),dOut(end-2:end)]); % D(mc)*D(nc)*dk x D(mc) x D(nc) x dk
idxA = [1, 2, 4]; idxB = [1, 4, 3];
% 2. Contract to non-interacting parts -> Hright
OpTemp = contracttensors(Atens, 4, 2, Hstor.',2,1);
op.Hright = op.Hright + contracttensors(conj(Atens),4, idxA, OpTemp, 4, idxB);
% 3. Contract the other chain-system interacting parts -> Hright
for m = 1:M
OpTemp = contracttensors(Atens , 4, 4, h2jOBB{m}.', 2, 1);
OpTemp = contracttensors(OpTemp, 4, 2, Opstor{m}.', 2, 1); % (m,2,1) should be the operator of site 2 in the effective left basis for system site 1
op.Hright = op.Hright + contracttensors(conj(Atens),4, idxA, OpTemp, 4, idxB);
end
end
end
% 3. Contract h1jOBB -> Hright
idx = 2:NChild+2;
OpTemp = contracttensors(mps, NChild+2, NChild+2, treeMPS.op.h1jOBB.', 2, 1);
op.Hright = op.Hright + contracttensors(conj(mps), NChild+2, idx, OpTemp,NChild+2, idx);
% 4. Contract h2jOBB which will interact with parent
for m = 1:M
OpTemp = contracttensors(mps, NChild+2, NChild+2, op.h2jOBB{m,2,1}.', 2, 1);
op.Opright{m} = contracttensors(conj(mps), NChild+2, idx, OpTemp,NChild+2, idx);
end
op.Hleft = treeMPS.op.Hlrstorage{1}; % copy into temporary H/Opleft to allow overwriting in updateop.
op.Opleft = treeMPS.op.Opstorage(:,1,1);
end
function op = H_Eff_TR_CP_NoSite(treeMPS,para)
%% function op = H_Eff_TR_CP_NoSite(treeMPS,para)
%
% Creates Hright for entering the parent = walking up the tree
% Applies if current node carries NO physical site, but serves to combine several nodes/leaves for entanglement renormalisation
% All children of this node couple to the parent, thus need to channel through treeMPS.child(:).op.Opstorage{:,2,1,:}
%
% created by FS 30/08/2016
NChild = treeMPS.degree; % = nChildren for node
NCoup = length(treeMPS.chainIdx); % = number of coupling channels from children to parent node
% if child node hasSite -> chainIdx = min(child.chainIdx)
% elseif child ~hasSite -> chainIdx = [child.chainIdx]
% thus correctly counts the number of couplings through the ER-Node
M = treeMPS.M;
d = size(treeMPS.mps{1}); % numel(d) = NC+2 if hasSite; elseif ~hasSite: NC+1
mps = treeMPS.mps{1}; % get handle, should not copy object!
% MPS is only mps{1}: Dl x D1 x D2 x D3 x ... X D(NC) [x dk ]
% chain index in mps is shifted by 1 due to first left Bond
op = treeMPS.op; % only return op -> needs overwriting in function call
op.Hright = 0; % contains effective H of node and all chains
op.Opright = cell(M,NCoup); % if no Site, then all chains from right must be channelled through to the left, thus need all coupling terms
NCoupOffset = 0; % necessary to correctly & easily fill op.Opright
if treeMPS.isRoot
error('VMPS:H_Eff:NotSupported','This function shall only be used for non-root nodes');
end
for mc = 1:NChild
% mc: chain to contract with MPS to get into right basis of parent
HStor = treeMPS.child(mc).op.Hlrstorage{1}; % Hright of child
OpStor = treeMPS.child(mc).op.Opstorage(:,2,1,:); % Opright of child: (M,NCoup)
[MM,NN] = size(OpStor);
ratio = (d(1)*d(mc+1))^2/prod(d); % d(needed legs)/d(rest); Benefit from Isometry only if << 1
% always make sure that Hlrstorage{1} and Opstorage{:,2,1} are up-to-date!
if para.tdvp.HEffSplitIsometry == 0 || ratio >= 1 % TODO: need to find optimal threshold
[idxA, idxB] = getIdxTensChain(NChild+1,mc+1,1);
% contraction of mps with chain ops of mc will move its index to the end
% -> generate idx pair for <mps|H|MPS>, where everything except 1 shall be contracted
% 1. Contract to non-interacting parts -> Hright
OpTemp = contracttensors(mps, NChild+1, mc+1, HStor.',2,1);
op.Hright = op.Hright + contracttensors(conj(mps),NChild+1, idxA, OpTemp, NChild+1, idxB);
% 2. Contract the other parts for the chain-parent interacting parts -> Opright
for n = 1:NN
for m = 1:MM
OpTemp = contracttensors(mps, NChild+1, mc+1, OpStor{m,n}.',2,1);
op.Opright{m,n+NCoupOffset} = contracttensors(conj(mps),NChild+1, idxA, OpTemp, NChild+1, idxB);
end
end
NCoupOffset = NCoupOffset + NN;
else
% 1. Split-off isometry
% but need D(mc), D(nc=1) in Atens! (no dk, since no Site)
[Atens,dOut] = tensShape(mps, 'unfoldiso', [1, mc+1], d); % +1 for leading singleton
[~, Atens] = qr(Atens,0); % [Iso,R] = qr(A); Iso is isometry with all unused chains -> contracts to Iso'*Iso = eye
% TODO: comment if not testing!!
% [Iso2,A2,err]= rrQR(Atens, floor(0.1*prod(dOut(end-2:end))),0); % low rank QR approximation
% fprintf('H_eff: rank(A) = %d, dim(A,2) = %d, rrQR error = %g; %g\n', rank(Atens), prod(dOut(end-2:end)),norm(Atens - Iso2*A2), err);
Atens = reshape(Atens,[size(Atens,1),dOut(end-1:end)]); % D(1)*D(mc) x D(1) x D(mc)
idxA = [1, 3]; idxB = [1, 3];
% 2. Contract to non-interacting parts -> Hright
OpTemp = contracttensors(Atens, 3, 3, HStor.',2,1);
op.Hright = op.Hright + contracttensors(conj(Atens),3, idxA, OpTemp, 3, idxB);
% 3. Contract the other parts for the chain-parent interacting parts -> Opright
for n = 1:NN
for m = 1:MM
OpTemp = contracttensors(Atens, 3, 3, OpStor{m,n}.',2,1);
op.Opright{m,n+NCoupOffset} = contracttensors(conj(Atens),3, idxA, OpTemp, 3, idxB);
end
end
NCoupOffset = NCoupOffset + NN;
end
end
assert(size(op.Opright,2) == NCoup, 'VMPS:H_Eff:H_EFF_TR_CP_NoSite:WrongNCOUP','The number of couplings was wrong, Opright grew! Please check chainIdx for each node for correctness!');
op.Hleft = treeMPS.op.Hlrstorage{1}; % copy into temporary H/Opleft to allow overwriting in updateop.
op.Opleft = treeMPS.op.Opstorage(:,1,1,:);
end
function [idxA, idxB] = getIdxTensChain(N,m,n)
% generate two arrays with indices corresponding to the desired contractions in the case:
% all bonds ~= n will be contracted over
% m : 1 position which got moved to the end in Tensor 1 (e.g. by contraction)
% n : 1 position which should be the output bonds
%
idxA = 1:N; idxB = idxA;
idxA([m,n]) = []; idxA = [idxA,m];
if m < n
idxB(n-1) = [];
elseif m > n
idxB(n) = [];
else
error('This is only meant to be used for m ~= n');
end
end