tensor product factorization and entanglement properties ... · florian gebhard, reinhard m. noack,...

Tensor product factorization and entanglement propertiesin models with long range interactions

Ors Legeza

Wigner Research Centre for Physics, Budapest, Hungary

in collaboration with

◮ Jeno Solyom, Edina Szirmai, Gergely Barcza, Szilard Szalay, Tibor Szilvasi

◮ Florian Gebhard, Reinhard M. Noack, Georg Ehlers (Marburg)

◮ Valentin Murg, Frank Verstraete (Vienna, Ghent)

◮ Markus Reiher, Konrad H. Marti, Stefan Knecht (ETH-Zurich)

◮ Katharina Boguslawski, Pawel Tecmer, Paul Ayers (McMaster)

◮ Max Pfeffer and Reinhold Schneider (Tu-Berlin)

◮ Christian Krumnow, Jens Eisert, Edoardo Fertitta, Beate Paulus(Fu-Berlin)

◮ L. Veis, J. Pittner (U-Prague)

Natal, 24.06.2015

Topics to be covered

1. Motivations2. Tensor product factorization:

• Density matrix renormalization group (DMRG)• Matrix Product State (MPS)• Tensor Network States (TNS)

3. Two-site mutual information→Classical and quantum correlations

4. Examples for models with long range interactions:• Quantum chemical systems• ab-initio treatment of solids• Nuclear shell problems• Momentum space representation

5. One- and Two-orbital mutual information→Entanglement• Optimizing the algorithms• Efficient construction of active spaces• Description of bond formation and breaking procedures

6. Multi-partite entanglement7. Four-Component DMRG (4c-DMRG)8. Example for geometrical optimization using DMRG9. MPS based on-the-fly basis optimization

10. MPS and TNS on kilo-processor architectures: smart hybrid CPU-GPUimplementation

Efficient simulation of quantum systems on classical computers?

◮ R. Feynman (1985): simulating quantum systems on classical computerstakes an amount of time that scales exponentially with size of system,while quantum simulations can scale in polynomial time with system size.

◮ Tensor network states as low-rank aprroximations of high-dimensionaltensor spaces → can provide efficient simulations on classical computers

◮ Major aim: no need to build expensive laboratories and experimentalenvironments to design new materials and system just simulate them oncomputers

◮ In certain cases we can already reproduce and simulate experimental results

Motivations to utilize concepts of quantum information theory

◮ Properties of subsystems give information about the whole system

◮ Entanglement describes the quantum correlation between subsystems

◮ Mutual information: correlation between two selected subsystemsembedded in a larger system

↓

1. Multiply connected 2-dimensional entanglement network which describesphysical properties of the system

2. Development of better numerical methods that reflect the entanglementpattern of the problem

Applications motivated by recent experiments

Cold atoms◮ Fermionic alkaline earth atoms (AEA) in optical lattice potentials →

quantum simulators of strongly correlated materialsGorshkov et al, Nat. Phys. 6, 289(2010)

◮ In some cold atom systems the simplest realizations of strong correlationphysics may have no solid state analog,for example SU(n), n > 2 Hubbard model→ d and f band electrons Hermele et al, PRL 103, 135301 (2009)

◮ Exotic magnetic phases, i.e., chiral spin liquidTaie et al PRL 105, 190401 (2010).

Disorder-free materials (PDA):Polymerization induced by temperature or ultraviolet light:

Heeger, MacDiarmid, Shirakawa (1977)

Quantum chemistry:◮ Open d-shell properties → quantum chemistry of transition metal oxides

Reiher, Chimia 63, 140 (2009)

Interacting 1D system

Brief historical overview (incomplete list)

◮ Renormalization Group (Kadanoff transformation) (1966)

◮ Block Renormalization Group (BRG) method Pfeuty(1978)

◮ Numerical Renormalization Group (NRG): Wilson(1975)

◮ Density Matrix Renormalization Group (DMRG): White(1992)

◮ Dynamical DMRG (DDMRG):

Jeckelmann(2002), Hallberg(1995),...

◮ 2D-DMRG:

Noack, White, and Scalapino (1994,1996), Xiang(2000(2000)), McCulloch(2001)...

◮ Transfer Matrix DMRG:

Batista (1998), Hallberg (1999), Nishino(1995), Moukouri(1996), Peschel(1998),...

◮ Momentum space version of DMRG (k-DMRG):

Xiang(1996), Nishimoto, Noack(2002), Legeza, Solyom(2003),...

◮ Quantum chemistry version of DMRG (QC-DMRG):

White(1999), Mitrushenkov(2001), Chan(2002), Legeza(2002),

Reiher(2005), Zgid(2006), Yanai(2008), Xiang(2010),...

◮ Time-dependent DMRG (t-DMRG):

Schmitteckert(2004), Cirac, Verstraete(2004), Vidal, Schollwoeck (2004), Daley, White

(2004), Manmana, Noack(2005), ...

Brief historical overview

◮ Matrix Product State (MPS):

Ostlund, Rommer(1995), Cirac, Verstraete(2004), Vidal (2003), Oseledets(2009), ...

◮ Tensor Network Sates(TNS):

Cirac, Verstraete(2004), Vidal (2003), Orus(2005), Eisert(2005), Hackbush(2010),...

◮ Projected Entangled-Pair State (PEPS):Cirac, Verstraete(2004)

◮ Infinite Projected Entangled-Pair State (iPEPS):Jordan, Orus, Vidal, Verstraete, Cirac(2009) Corboz, White, Vidal, Troyer (2011), ...

◮ Multiscale entanglement renormalization algorithm (MERA):

Vidal (2007), Coborz (2009)

◮ 2D-MERA:

Evenbly, Vidal (2009)

◮ ...

Some useful reviews, also see additional references therein

◮ Density Matrix Renormalization: A New Numerical Method in Physics, Eds. I. Peschel, X.

Wang, M. Kaulke, and K. Hallberg, ,Springer, 1999.

◮ U. Schollwock, Rev. Mod. Phys. 77, 259 (2005).

◮ R. M. Noack and S. R. Manmana, in Diagonalization- and Numerical

Renormalization-Group-Based Methods for Interacting Quantum Systems, (AIP, 2005), vol.

789, pp. 93–163.

◮ F. Verstraete, J.I. Cirac, V. Murg, Adv. Phys. 57 (2), 143 (2008).

◮ O. Legeza, R. Noack, J. Solyom, and L. Tincani, in Computational Many-Particle Physics,

eds. H. Fehske, R. Schneider, and A. Weisse 739, 653–664 (2008).

◮ K. Hallberg, Advances in Physics 55, pp 477-526 (2006).

◮ U. Schollwock, Ann. Phys. (NY) 326, 96 (2011).

◮ G. K.-L. Chan and S. Sharma, Annu. Rev. Phys. Chem. 62, 465–481 (2011).

◮ K. H. Marti and M. Reiher, Z. Phys. Chem. 224, 583-599 (2010). K. H. Marti, M. Reiher,

Phys. Chem. Chem. Phys. 13 6750-6759 (2011).

◮ G. K.-L. Chan, Wiley Inter-disciplinary Reviews: Computational Molecular Science 2 (6)

(2012) 907–920.

Some useful reviews, also see additional references therein

◮ O. Legeza, T. Rohwedder, R. Schneider, Numerical approaches for high-dimensional PDE’s

for quantum chemistry, in Encyclopedia of Applied and Computational Mathematics,

Editor-in-chief: Engquist, Bjorn ; Chan, T.; Cook, W.J.; Hairer, E.; Hastad, J.; Iserles, A.;

Langtangen, H.P.; Le Bris, C.; Lions, P.L.; Lubich, C.; Majda, A.J.; McLaughlin, J.;

Nieminen, R.M.; ODEN, J.; Souganidis, P.; Tveito, A. (Eds.) Springer 2013 ISBN

978-3-540-70530-7

◮ O. Legeza, T. Rohwedder, R. Schneider, Sz. Szalay, arxiv:1310.2736 (2013).

◮ W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, SSCM Vol. 42, Springer,

2012.

◮ R. Orus, Annals of Physics 349, 117 (2014).

◮ L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).

◮ J. Eisert, M. Cramer, M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010).

◮ S. Wouters and D. Van Neck, European Phys. J. D 68, 272 (2014)

◮ Sz. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, O. Legeza. Int. J.

Quantum Chem. 2015, DOI: 10.1002/qua.24898

◮ http://tagung-theoretische-chemie.uni-graz.at/en/past-workshops/workshop-2014/

◮ http://ldqs.iyte.edu.tr/program/

Tensor product methods and entanglement optimization for ab initioquantum chemistry

◮ We will focus only on the the quantum chemistry version of density matrixrenormalization group (QC-DMRG) method

◮ A very tutorial presentation will be given using a concrete example

◮ Connection to matrix product and tensor network states will be discussedbriefly

◮ Concept of entanglement will be discussed in respect of:Network optimization proceduresTwo-dimensional entanglement structure of the systemCorrelations among the molecular orbitals

References:

◮ New wavefunction methods and entanglement optimizations in quantumchemistry, Mariapfarr, Austria (2013)http://tagung-theoretische-chemie.uni-graz.at/en/past-workshops/workshop-2014/

◮ II. International Summer School on Exact and Numerical Methods forLow-Dimensional Quantum Structures, Izmir, Turkey (2014)http://ldqs.iyte.edu.tr/program/

◮ Sz. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider,O. Legeza. Int. J. Quantum Chem. 2015, DOI: 10.1002/qua.24898

Hamiltonian of the interacting system

◮ The system is described by the Hamiltonian, for example,

H =∑

ijαβ

Tαβij c

†iαcjβ +

1

2

∑

ijklαβγδ

Vαβγδijkl c

†iαc

†jβckγclδ , (1)

◮ Tij denotes the kinetic and onsite terms

◮ Vijkl stands for two-particle scatterings

◮ We consider usually lattice models in real space (DMRG)

◮ In quantum chemistry sites are electron orbitals (QC-DMRG)

◮ In UHF QC spin-dependent inetractions (UHF-QCDMRG)

◮ In relativistic quantum chemistry sites are spinors (4c-DMRG)

◮ In nuclear problems sites are proton/neutron orbitals (JDMRG)

◮ In k-space representation sites are momentum eigenstates (k-DMRG)

◮ Major aim: to obtain the desired eigenstates of H.

◮ In ab initio calculations M atoms, Ne electrons, Coulomb-interaction with~ = e = m = 1

H =

Ne∑

i=1

−▽2i

2−

Ne ,M∑

i=1,α=1

Zα

|ri − rα|+

1

2

Ne∑

i 6=j=1

1

|ri − rj |

Hamilton-operator, where Zα is the charge of nucleus α, and r denotes thepositions of the nuclei and electrons.

◮ The ground state solution is often approximated in a mean-field manner,where the explicit electron-electron interaction is interchanged by aneffective single-particle term. The so called Hartree-Fock (HF) solution ofthe resulting model often recovers the major part (99 %) of the totalenergy and provides a good starting point for post-HF computations.

◮ The correlation energy can be defined as the difference of the exact andthe HF energy. The electron correlation, i.e., the beyond mean-fieldbehavior of the electrons, plays an important role in the quantitativedescription of the chemical properties.

Hamiltonian of the interacting electron system

◮ The system is described by the Hamiltonian, for example,

H =∑

ijσ

Tijσc†iσcjσ +

∑

ijklσσ′

Vijklσσ′c†iσc

†jσ′ckσ′clσ

◮ Tij denotes the one-electron integral comprising the kinetic energy of theelectrons and the external electric field of the nuclei.

◮ Vijkl stands for the two-electron integrals and contains the e-e repulsionoperator.

Vijkl =

∫

d3x1d

3x2Φ

∗i (~x1)Φ

∗j (~x2)

1

~x1 − ~x2Φk(~x2)Φl(~x1)

◮ Molecular integrals are calculated via one-electron basis of atom-centeredGaussians

◮ Major aim: to obtain the desired eigenstates of H.

Tutorial examples will be presented for the LiF molecule

◮ Atomic orbital (AO) basis was adopted from the work of C. W. Bauschlicher and

S. R. Langhoff J. Chem. Phys., 89, 4246-425 (1988).

◮ The AO basis consists of 9s and 4p functions contracted to 4s and 2pfunctions on the Li atom and 9s, 6p and 1d functions contracted to 4s, 3pand 1d on the F atom. This basis set is suitable to describe the ionic andcovalent LiF states as well.

◮ The two lowest 1Σ+ states of LiF around the equilibrium bond length canbe qualitatively described by the 1σ22σ23σ24σ21π4 and1σ22σ23σ24σ15σ11π4 configurations.

◮ The MO basis was obtained by CASSCF optimizations, with two activeelectrons on two active orbitals (4σ and 5σ) (CAS(2,2)). MO’s wereoptimized simultaneously for both 1Σ+ states. Six of the valence electronswere excited to all orbitals in the active space (i.e. to 12 orbitals in thisexample).

◮ The three lowest lying occupied and 13 highest virtual orbitals wereexcluded from the DMRG computations.

Molecular orbitals, configuration space

Molecular orbitals are obtained, e.g., in a suitable mean-field or MCSCFcalculation (for example using the MOLPRO program package).Example.: LiF (6/12)

◮ Two major approximations:(1) selection of the finite number of basis states(2) restricted number of configurations is taken into account

Corrections to the Hartree-Fock state

◮ The full configuration interaction (full-CI) wavefunction can be expressedin terms of Slater determinants by removing one (S), two (D), three (T)or four (Q) electrons form the HF orb.:

ΨFCI = a0ΦSCF +∑

S

aSΦS +∑

D

aDΦD +∑

T

aTΦT + . . . .

◮ Correlation energy: ECorr = EFCI − EHF.

◮ Other expansions are also possible, e.g., Coupled Cluster (see Schneider’stalk).

The problem in the language of tensor factorization

◮ Local tensor space Λ, with dimΛ = q, is denoted by •◮ Operators for basis |n, σ〉 with n = {0, 1} and σ = {↓, ↑}

c† =

(0 01 0

)

, I2 =

(1 00 1

)

, Ph2 =

(1 00 −1

)

◮ In a C4 representation • represents an orbital basis with q = 4. Relevant

orbital operators in the |−〉, | ↓〉, | ↑〉, | ↓↑〉 basis:

c†↓ = c

† ⊗ I2 =

0 0 0 00 0 0 01 0 0 00 1 0 0

, c†↑ = Ph2 ⊗ c

† =

0 0 0 01 0 0 00 0 0 00 0 −1 0

I = I2 ⊗ I2 =

1 0 0 00 1 0 00 0 1 00 0 0 1

, Ph = Ph2 ⊗ Ph2 =

1 0 0 00 −1 0 00 0 −1 00 0 0 1


◮ We can put together two C4 tensor spaces, i.e., form a two-orbital system

(••)◮ Λ(1,2) = Λ(1) ⊗ Λ(2) with dimΛ(1,2) = dimΛ(1)dimΛ(2) = q2

◮ Basis of the •• system: |φ(1,2)

{α1,α2}〉 = |φ(1)

α1 〉 ⊗ |φ(2)α2 〉

{α1, α2} = (α1 − 1)q + α2

- -- ↓- ↑- ↓↑↓ −↓ ↓...

...↓↑ ↓↑

α1, α2 ∈ {1, 2, 3, 4} = {|−〉, | ↓〉, | ↑〉, | ↓↑〉}

{α1α2} ∈ {1, 2, 3, 4 . . . 16}


◮ Relevant operators for the •• system (dimΛ(1,2) = 4× 4):

c†1,↓ = c

†↓ ⊗ I , c

†2,↓ = Ph ⊗ c

†↓ , I ⇐ I ⊗ I ,

c†1,↑ = c

†↑ ⊗ I , c

†2,↑ = Ph ⊗ c

†↑ , Ph ⇐ Ph ⊗ Ph,

H =

h1,1 . . . h1,16... . . .

...h16,1 . . . h16,16

◮ Full diagonalization of H gives exact solution (full-CI).

◮ The kth eigenstate of a two-orbital Hamiltonian is

Ψ(1,2)k =

∑

α1,α2U

(1,2)α1,α2,k

|φ(1)α1 〉 ⊗ |φ(2)

α2 〉;α1, α2 = 1 . . . q; k = 1 . . . q2.

Change of basis and truncation

◮ We can represent the Hamiltonian in the eigenbasis of the •• system by abasis state transformation

◮ (Order eigenstates in a descending order, and) form an operator from thecorresponding eigenstates as:

O =

Ψ(1,2)1

Ψ(1,2)2

...

Ψ(1,2)

q2

, OO† = I

◮ Transform operators to the new basis as H ⇒ OHO†

c†1,↓ ⇒ Oc

†1,↓O

†, c†2,↓ ⇒ Oc

†2,↓O

†, etc

◮ Idea of truncation: select M < q2 states only so OO† 6= I.

Example: two half-spins on ••, |φ↓〉 ≡ | ↓〉, |φ↑〉 ≡ | ↑〉

Ψm =∑

α1,α2

Om,2(α1−1)+α2|φα1〉 ⊗ |φα2〉, α1, α2 ∈ {1, 2} ≡ {↓, ↑}

O ↓ ↓ ↓ ↑ ↑ ↓ ↑ ↑ Sz S

Ψ1 1 0 0 0 -1 1

Ψ2 0 1/√2 1/

√2 0 0 1

Ψ3 0 1/√2 −1/

√2 0 0 0

Ψ4 0 0 0 1 1 1

Take column 1 and 3 for ↓ Take column 2 and 4 for ↑

1 0

0 1/√2

0 −1/√2

0 0

= (B (2)[↓])m,α1

0 0

1/√2 0

1/√2 0

0 1

= (B (2)[↑])m,α1

In the literature A ≡ BT is used.

Ψm =∑

α1,α2

(A(2)[α2])α1,m|φα1〉 ⊗ |φα2〉

Quantum numbers for N orbitals

◮ Symmetry operators can be used to cast the Hilbert space to smallerindependent subspaces

◮ The eigenstates of the Hamiltonian can be labeled by the eigenvalues of asymmetry operator (quantum number, Q)

◮ In QC, for finite systems, the number of electrons with down and up spins,Q = {N↓, N↑} is fixed. The size of the related Hilbert space is:(2N)!/((2N − Ne)!Ne !), with Ne = N↓ + N↑.

• basis N↓ N↑

|−〉 0 0| ↓〉 1 0| ↑〉 0 1| ↓↑〉 1 1

•• basis N↓ N↑

|−,−〉 0 0|−, ↓〉 1 0... ... ...

| ↓↑, ↓↑〉 2 2

◮ For two orbitals: Q{α,β} = f (Qα,Qβ) (Abelian, non-Abelian quantumnumbers, etc)


◮ For a system with N molecular orbitals: • • • . . . •

◮ Λ(1,2,...,N) = ⊗Ni=1Λ

(i) with dimΛ(1,2,...,N) =∏N

i dimΛ(i) = qN

◮ Ψ(1,2,...,N)k =

∑

α1...αNU

(1,2,...,N)α1,α2,...αN ,k |φ

(1)α1 〉 ⊗ |φ(2)

α2 〉 ⊗ . . .⊗ |φ(N)αN

〉;

◮ U(1,2,...,N)α1,α2,...αN ,k is a tensor of order N corresponding to the kth eigenstate of

the N-orbital Hamiltonian

Example: N = 8

◮ Problem: dimension of U scales exponentially with N

→ We need approximative methods

Idea of renormalization: Block Hamiltonians

◮ First we form two-orbital Hamiltonians (blocks) from every two-orbitalsand diagonalize them to obtain their eigenstates:

Ψ(1,2)k =

∑

α1,α2U

(1,2)α1,α2,k

|φ(1)α1 〉 ⊗ |φ(2)

α2 〉

Ψ(3,4)k =

∑

α3,α4U

(3,4)α3,α4,k

|φ(3)α3 〉 ⊗ |φ(4)

α4 〉

Ψ(5,6)k =

∑

α5,α6U

(5,6)α5,α6,k

|φ(5)α5 〉 ⊗ |φ(6)

α6 〉

Ψ(7,8)k =

∑

α7,α8U

(7,8)α7,α8,k

|φ(7)α7 〉 ⊗ |φ(8)

α8 〉

α = 1 . . . q and k = 1 . . . q2

Idea of renormalization: Block Hamiltonians

◮ Form two-block Hamiltonians from every two blocks again, anddiagonalize them to obtain their eigenstates:

Ψ((1,2),(3,4))k =

∑

α1,α2U

((1,2),(3,4))α1,α2,k

|Ψ(1,2)α1 〉 ⊗ |Ψ(3,4)

α2 〉

Ψ((5,6),(7,8))k =

∑

α1,α2U

((5,6),(7,8))α1,α2,k

|Ψ(5,6)α1 〉 ⊗ |Ψ(7,8)

α2 〉

α = 1 . . . q2 and k = 1 . . . q4

◮ Form two-block Hamiltonians from every two blocks again, anddiagonalize them to obtain their eigenstates:

Ψ(((1,2),(3,4)),((5,6)(7,8)))k =

∑

α1,α2U

(((1,2),(3,4)),((5,6),(7,8)))α1,α2,k

|Ψ((1,2),(3,4))α1 〉 ⊗ |Ψ((5,6),(7,8))

α2 〉

α = 1 . . . q4 and k = 1 . . . q8

◮ So far we have carried out only a change of basis.

Idea of renormalization: Truncation

◮ The original idea was that in each RG step we truncate the Hilbert spaceof the blocks by keeping only q new states out of the q2 eigenstates states.

◮ The original form of the Hamiltonian is retained, for lattice modelsanalytic solution is possible (flow equations, fixed-point, etc (Example:ITF model))

◮ we can keep q < M ≪ qN states but we loose analytic solution, newoperators appear (NRG)

◮ In general we keep dimension of the Hilbert space under control but weloose information → approximate solution of the problem

◮ Questions:– how to choose which states to keep at each RG step?– how many states to keep at each RG step?– how accurate will be the truncated solution compared to the full-CIsolution?


◮ For a system with N molecular orbitals: • • • . . . •

◮ Λ(1,2,...,N) = ⊗Ni=1Λ

(i) with dimΛ(1,2,...,N) =∏N

i dimΛ(i) = qN

◮ Ψ(1,2,...,N)k =

∑

α1...αNU

(1,2,...,N)α1,α2,...αN ,k |φ

(1)α1 〉 ⊗ |φ(2)

α2 〉 ⊗ . . .⊗ |φ(N)αN

〉;

◮ U(1,2,...,N)α1,α2,...αN ,k is a tensor of order N corresponding to the kth eigenstate of

the N-orbital Hamiltonian

Example: N = 8

◮ Problem: dimension of U scales exponentially with N

→ We need approximative methods

Tensor product approximations:

◮ Matrix Product State (MPS) representation:

|Ψ〉 =q∑

α1,α2,...,αN

A1α1A

2α2

· · ·ANαN

|α1〉|α2〉 · · · |αN〉

Ai [m,m]qi

◮ We can call this a network. Matrix product state (MPS) / Tensor Train(TT): Ostlund, Rommer (1995), Verstraete, Cirac (2004), Oseledets (2009), Hackbusch (2009)

DMRG provides MPS wavefunction:

Density matrix renormalization group wavefunction: (White, 1992)

|ΨTG〉 =∑

αlαl+1αl+2αr

ψαlαl+1αl+2αr |φ(l)αl〉 ⊗ |φ(sl )

αl+1〉 ⊗ |φ(sr )

αl+2〉 ⊗ |φ(r)

αr〉

where ψαlαl+1αl+2αr coefficients (4-index tensor) are determined by an iterativediagonalization of the superblock Hamiltonian.DMRG algorithm provides the optimized set of Ai matrices.

DMRG wavefunction in MPS form for the l − • • −r superblock

Ψ =∑

{α}

∑

ml

∑

mr

ψmlαl+1αl+2mr

×(Bl [αl ] . . .B2[α2])ml ;α1

×(Bl+3[αl+3] . . .BN−1[αN−1])mr ;αN× |α1 . . . αN〉,

Connection to the CI-type wavefunction (Marti, Reiher 2011)

Ψ =∑

{α}

C{α}Φ{α},

Therefore, the CI-coefficients in MPS form:

C{α} =

Ml∑

ml

Mr∑

mr

ψmlαl+1αl+2mr

×(Bl [αl ] . . .B2[α2])ml ;α1× (Bl+3[αl+3] . . .BN−1[αN−1])mr ;αN

Two-dimensional network

A tree tensor network in which all sites in the tree represent physical orbitals(red lines) and in which entanglement is transferred via the virtual bonds thatconnect the sites (black lines).

Maximal distance between two orbitals, 2∆, scales logarithmically with N forz > 2.

Schmidt-decomposition for a bipartite system

◮ For a bipartite system: |ΨT 〉 =∑

ij ψij |φLi 〉 ⊗ |φR

j 〉

◮ Reduced density matrix: ρ(L,R)

i,i′ =∑

j ψijψ∗i′j

◮ If |Ψ〉 pure state then for |Ψ〉 ∈ Λ = ΛL ⊗ ΛR

|Ψ〉 =r≤min(ML,MR )∑

i=1

ωi |ei 〉 ⊗ |fi 〉 .

◮ |ei 〉, |fi 〉 biorthogonal basis, and r is the Schmidt number

◮ If r = 1 ⇒product state, for example, | ↓↑〉| ↓↑〉◮ If r > 1 ⇒entangled state: non-local property of quantum mechanics.

Example: 1/√2(| ↓〉| ↑〉 − | ↑〉| ↓〉)

◮ Neumann entropy: s(ργ) = −Tr(ργ ln ργ) , γ ≡ L,R

◮ |ΨT 〉 pure state → s(ρL) = s(ρR)

◮ In general, ρL = TrR and ρR = TrL are in mixed state

◮ If ωi = 1/M maximally mixed state

◮ In general ρ(N) 6= ρL ⊗ ρR

Entanglement

◮ Renyi entropy:s(ρi )α = 11−α

lnTrραifull spectrum of ρα (Calabrese) related to scaling of algorithms based onmatrix product states(Schuch)

◮ Neumann entropy: sα→1(ρi ) = −Tr(ρi ln ρi ) , i ≡ L,R

◮ Single copy entanglement: S∞(ρi ) ≡ lnmax(ρi )

◮ |ΨT 〉 pure state → s(ρL) = s(ρR)

◮ Kullback-Leibler-relative entropy:

K (ρ||σ) ≡ Tr(ρ ln ρ− ρ lnσ),

where ρ and σ reduced density matrix for the left block corresponding totwo different environment.

◮ For more than two subsystems:Mutual entropy: s(ρAB) = s(ρA) + s(ρB)− I

where I describes correlation between A and B.

◮ I = 0 → A and B are uncorrelated.

Density-matrix spectra for integrable models

◮ Density matrix: ρ(σ, σ′) = Ψ(σ)Ψ(σ)◮ Cut system into two parts: σL = {σ1, σ2, . . . σM}, σR = {σM+1, . . . σN}◮ ρL(σL, σ

′L) =

∑

σRΨ(σL, σR)Ψ(σ′

L, σR)◮ relate the quantum chains to two-dimensional classical systems

T ≡ Trotter decomposition of exp(−βH), [H,T ] = 0.◮ density matrices ρL then become partition functions of strips with a cut

Nishino, 1996

◮ can be expressed as products of corner transfer matrices (CTM’s) ’a laBaxter: ρL = ABCD

◮ for integrable models the spectra of such CTM’s are known in thethermodynamic limit

The spectra which occur in numerical density-matrix renormalization group(DMRG) calculations for quantum chains can be obtained analytically forintegrable models via corner transfer matrices. Peschel, Kaulke, Legeza, 1998

Analytical results: 1d-Ising model in transverse field

◮ square lattice with Ising spins at the lattice points and an isotropiccoupling K.

◮ The transfer matrix (indicated by shading in the lower part)commutes withthe Hamiltonian Peschel, 1985; Igloi, Lajko, 1996

H = −N−1∑

n=1

σxn − δσx

N − λ

N−1∑

n=1

σznσ

zn+1, δ = cosh2K, λ = sinh

22K

HCTM = conts ×

∑

n≥1

(2n − 1)σxn + λ

∑

n≥1

2nσznσ

zn+1

HCTM =∞∑

j=0

ǫjnj , nj = c†j cj = 0, 1

ǫj = (2j + 1)ǫ, k = λ : λ < 1(T > Tc)

ǫj = 2jǫ, k = 1/λ : λ > 1(T < Tc)

where ǫ = πI (k ′)/I (k), I (k) complete elliptic integral of the first kind,k ′ =

√1− k2 and 0 ≤ k ≤ 1.

Analytical results: 1d-Ising model in transverse field

λ < 1 : The ground state has E = 0.Excited states with one fermion leadto the odd levels E = ǫ, 3ǫ, 5ǫ, . . ..With two fermions one obtains inaddition all even levels except 2ǫ.The level 8ǫ is two-fold degenerate.The three-fermion excitations startat 9ǫ and lead to further degenera-cies etc. The spacing of the E ’s,except at 2ǫ, is therefore ǫ.

DMRG with λ = 0.8,N = 120

λ > 1 : The ground state has againE = 0, but it is now twice degen-erate due to the long range order.This leads to a corresponding two-fold degeneracy of all excited levels.Formally, this follows from the factthat ǫ0 = 0. The excited states areinteger multiples of 2ǫ, i.e. the spac-ing of the E ’s is 2ǫ.Peschel, Kaulke, Legeza, 1998 DMRG with λ = 0.8,N = 120

Behavior of Block Entropy and area ”law”

◮ 1D◮ Noncritical Systems: correlation functions decay exponentially

Entropy constant: S(l) ∼ const.

◮ Critical Systems: entropy diverges logarithmically (Vidal et al., 2003)

S(l) ∼ logl fo a subsystem of length lconformally invariant system:

S(l) = (c/6) ln

[

2L

πasin(

πl

L)

]

+ g + c ′1 (openBCs)

→ conformal charge (Calabrese, 2004)

◮ 2D◮ Entropy area “law” (Srednicki, 1993)

S(l) & Ld−1 Ld−1: area of boundary

◮ Critical systems:

S(l) & Ld−1 ln l → required m :∼ eS(l) ∼ eLd−1

Error sources and data sparsity

◮ In each DMRG step, the basis states of the system block are transformedto a new truncated basis set by a unitary transformation based on thepreceding SVD

◮ This transformation depends on how accurately the environment isrepresented and on the level of truncation

◮ Environmental error, δεsweep, is minimized by a successive application ofthe sweepings

◮ Truncation error: δεTR = 1−∑M

α=1 ωα

◮ For δεsweep → 0, δErel = Const × δεTR (O.L and G. Fath, PRB 1996).

◮ DMRG is a variational method

◮ DMRG is a data-sparse representation of the wavefunction

sparsity ≡ dim(ΛSB)/ dim(ΛFCI)

Example: LiF (6/12) with fixed M = 16 block states

0 5 10 15 20 25 30 35

−106.965

−106.96

−106.955

−106.95

E

E

GS

EFCI

0 5 10 15 20 25 30 3510

−1010

−810

−610

−4

100

∆ E

rel

0 5 10 15 20 25 30 350

5

10

15

Blo

ck s

tate

s

M

l

Mr

0 5 10 15 20 25 30 3510

−16

10−12

10−8

10−4

100

δεT

r

0 5 10 15 20 25 30 350

200

400 dim(ΛFCI

)=134596

dim

(ΛS

B)

Iteration step

Example: optimized ordering, CI-DEAS and fixed M = 16

0 5 10 15 20 25 30 35

−106.96

−106.95

−106.94

−106.93

E

E

GS

EFCI

0 5 10 15 20 25 30 3510

−1010

−810

−610

−4

100

∆ E

rel

0 5 10 15 20 25 30 350

5

10

15

Blo

ck s

tate

s

M

l

Mr

0 5 10 15 20 25 30 3510

−16

10−12

10−8

10−4

100

δεT

r

0 5 10 15 20 25 30 350

200

400dim(Λ

FCI)=134596

dim

(ΛS

B)

Iteration step

Example: optimized ordering, CI-DEAS and fixed M = 64

0 5 10 15 20 25 30 35

−106.96

−106.95

−106.94

E

E

GS

EFCI

0 5 10 15 20 25 30 3510

−1010

−810

−610

−4

100

∆ E

rel

0 5 10 15 20 25 30 350

20

40

60

Blo

ck s

tate

s

M

l

Mr

0 5 10 15 20 25 30 3510

−16

10−12

10−8

10−4

100

δεT

r

0 5 10 15 20 25 30 350

2000

4000 dim(ΛFCI

)=134596

dim

(ΛS

B)

Iteration step

Target state

◮ In the MPS-based approaches, several eigenstates can be calculated withina single calculation.

◮ Reduced density matrix of the target state, ρ, can be formed from thereduced density matrices of the lowest n eigenstates as

ρ =∑

γ

pγργ

with γ = 1 . . . n,∑

γ pγ = 1 and Trργ = 1.

◮ Excited states corresponding to the action of given operators can also bemixed (see Noack’s talk).

Example: targeting several states together

0 5 10 15 20 25 30 35

−106.95−106.9

−106.85−106.8

−106.75

E

GS1XSFCI

0 5 10 15 20 25 30 3510

−1010

−810

−610

−4

100

∆ E

rel

GS1XS

0 5 10 15 20 25 30 350

20

40

60

Blo

ck s

tate

s

M

l

Mr

0 5 10 15 20 25 30 3510

−16

10−12

10−8

10−4

100

δεT

r

0 5 10 15 20 25 30 350

2000

4000 dim(ΛFCI

)=134596

dim

(ΛS

B)

Iteration step

Dynamic Block State Selection (DBSS) procedure

Optimal truncation scheme: δεTR < ǫ fixed in advance→ M is chosen accordingly in every stepExample from 2002: DBSS approach applied on F2 (D2h) (18/18)ΛFCI = (2N)!/[(2N − Ne)!Ne !] = 9075135300ΛDMRG = Ml × 4× 4×Mr = 28800000 (Ml = 1200,Mr = 1500) sparsity≃ 315ΛDMRG = Ml × 4× 4×Mr = 7680 (Ml = 120,Mr = 4)sparsity≃ 1181658 O.L. , Roder, Hess, Phys Rev B (2002)

0 10 20 30 40 50 60 700

500

1000

1500

2000

Blo

ck s

tate

s

F2, L=18, M

min=256, TRE

max=10−8

Ml

Mr

0 10 20 30 40 50 60 7010

−8

10−6

10−4

10−2

Rel

ativ

e er

ror Ord = [15 10 7 6 16 5 1 3 12 14 13 2 11 4 17 8 18 9]

0

0.2

0.4

0.6

Ent

ropy

S


Example 1d-Hubbard model:

H =∑

iσ

t(c†iσci+1σ + c†i+1σciσ) + U

∑

i

c†i↑ci↑c

†i↓ci↓ .

ΛSB = Λ(L) ⊗ Λ(R),

0 10 20 30 40 50 60 70 800

2

4

6

8

10

12

Iteration step in the 6th sweep

ln(dim Λ)ln( dim Λ

SB)

S

Block entropy, S , and ln(dimΛ),ln(dimΛSB) as a function of itera-tion steps for the half-filled 1-d Hub-bard model with N = 80 for U = 1with fixed number of block states(M = 512) using PBC.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

βScaled iteration step

N=80N=60N=40

Parameter β as a function ofrescaled iteration step using DBSSapproach for TREmax = 10−4,Mmin = 16, U = 1, N = 40, 60, 80.

β ≡ ln(dimΛ)− S .


◮ To control the weight of retained information during the RG procedure:

ρ(L) = pkeptρ(L)kept + (1− pkept)ρ

(L)lost ,

where ρ(L)kept is formed from the M largest eigenvalues of ρ(L) and ρ

(L)lost

from the remaining eigenvalues with Trρ(L)kept = Trρ

(L)lost = 1.

◮ The accessible information for such a binary channel would be less thanthe Kholevo bound

χ ≤ S(ρ)− pkeptS(ρkept)− (1− pkept)S(ρlost) ,

◮ In DMRG the atypical subspace is neglected → loss of information.

◮ truncation scheme: χ ≡ S(ρ)− S(ρkept) < ǫ fixed in advance → M ischosen accordingly in every step.O. L. and Solyom, PRB(2004)

◮ Entropy reduction by forming an enlarged system:

sL(l) + sl+1 + IL(l) = sL(l + 1), with IL(l) ≤ 0

◮ Obtained correlation in one RG step:

IL(l) = sL(l + 1)− sL(l)− sl+1 1 ≤ l < N − 1

◮ Entropy sum rule:N−1∑

l=1

IL(l) = −N∑

l=1

sl .

◮ In case of truncation:

N−1∑

l=1

IL(l)+N∑

l=1

sl < (N−1)ǫ .

10−8

10−6

10−4

10−2

10−6

10−5

10−4

10−3

10−2

10−1

100

χ(ε)

U=1U=10U=100

10−8

10−6

10−4

10−2

10−6

10−5

10−4

10−3

10−2

10−1

100

TRE

Rel

ativ

e er

ror

U=1U=10U=100

◮ For L = N − 2 → r = ML = MR = 1, correlation functions can becalculated very quickly

A priory defined error margin for ground and excited states

One can define the required accuracy prior to the calculations

All parameters of the algorithm are adjusted dynamically based on the strengthof entanglement encoded in the wavefunction

O.L., Solyom, PRB (2003)

Stability and performance of the QC-Dmrg method

◮ Example: Ionic–neutral curve crossing of LiF → continuous curvesO.L., Roder, Hess, Mol. Phys. (2002)

◮ The wave function character changes as a function of bond length (forexample in the region of an avoided crossing)

7 8 9 10 11 12 13 14−107

−106.95

−106.9

Bond length in a.u.

Ene

rgy

in a

.u.

X1Σ+

(2)1Σ+

2 4 6 8 10 12 140

2

4

6

8

10

12

14

Bond length in a.u.D

ipol

e m

omen

t in

a.u.

X1Σ+

(2)1Σ+

◮ Using the DBSS the a priory set error margin is reached for all bondlengths but close to the avoided crossing more block states are selected.

◮ Dipole moment is a continuous function of the bond length.

Example: Full tensor components LiF (6/12)

0 200 400 600 800 1000 1200 1400 1600 1800 200010

−12

10−10

10−8

10−6

10−4

10−2

100

i

a i2

ψ = Σi a

i |n

i>

Σi a

i2 = 1

HF12345>=6

0 50 100 150 200 250 300 350 40010

−12

10−10

10−8

10−6

10−4

10−2

100

i

a i2

ψ = Σi a

Σi a

i2 = 1

ψ = Σi a

i |n

i>

Σi a

i2 = 1

HF12345>=6

◮ The weight of the N-orbital basis states corresponding to the variousCI-levels are shown by different colors in a descending order in a log-linscale. (HF state (red), SCI (blue), DCI (green), etc.)

CI-based Dynamically Extended Active Space (CI-DEAS) [O.L, 2003]

Effect of the environment block

◮ New block states after SVD in each RG step depends on how accuratelythe environment is represented.

◮ ρ(N) 6= ρL ⊗ ρR

◮ For a pure ΨT: sL = sR

◮ Kullback-Leibler relative quantum entropy:

K (ρL||σL) ≡ Tr (ρL ln ρL − ρL lnσL) ,

where ρL and σL denote reduced density matrices of the left blockcorresponding to two different right (environment) blocks.

◮ K measures how the system (left) block reduced density matrix changes aswe change the environment.

◮ Optimize environment block based on Kullback-Leibler entropy.

Example: LiF 6/12, orbitals 1,2,3 are the HF orbitals.

Consider three dif-ferent Mr basis sets:

− − − − − − − − −

− − − − − − − − ↓− − − − − − − − ↑− − − − − − − − ↓↑− − − − − − − ↓ −

− − − − − − − ↑ −

− − − − − − − ↓ ↓− − − − − − − ↑ ↑− − − − − − − ↓ ↑− − − − − − − ↑ ↓− − − − − − − ↓↑ −

− − − − − − ↓ − −

− − − − − − ↑ − −

− − − − − − ↓ − ↑− − − − − − ↑ − ↓− − − − − − ↓ ↓ −

− − − − − − ↑ ↑ −

− − − − − − − − −

− − − − − ↓ − − −

− − − − − ↑ − − −

− − − − − ↓↑ − − −

− − − − − − ↓ − −

− − − − − − ↑ − −

− − − − − ↓ ↓ − −

− − − − − ↑ ↑ − −

− − − − − ↑ ↓ − −

− − − − − ↓ ↑ − −

− − − − − − ↓↑ − −

− − − − − − − − ↓− − − − − − − − ↑− − − − − ↓ − − ↓− − − − − ↑ − − ↑− − − − − ↑ − − ↓− − − − − ↓ − − ↑

− − − − − − − − −

− − − ↓ − − − − −

− − − ↑ − − − − −

− − − ↓↑ − − − − −

− − − − ↓ − − − −

− − − − ↑ − − − −

− − − ↑ ↓ − − − −

− − − ↓ ↑ − − − −

− − − − ↓↑ − − − −

↓ − − − − − − − −

↑ − − − − − − − −

↓ − − ↑ − − − − −

↑ − − ↓ − − − − −

↓ − − − ↑ − − − −

↑ − − − ↓ − − − −

↓↑ − − − − − − − −

The reduced density matrix of theL = l +1 subsystem depends on thebasis states used for the R = 1 + r

environment:

2 4 6 8 10 12 14 16

10−15

10−10

10−5

100

➤

✦�

s✭▲✮

❂✵✳✷✽

s✭▲✮

❂✵✳✷✷

s✭▲✮

❂✵✳✶✹

s✭▲✮

❂✹✳✺❡✲✵✺

case−1case−2case−3exact

Efficient construction of the active space

The larger the entropy value for a given orbital the larger its contribution to thetotal correlation energy. Example LiF (6/25):

1 5 10 15 20 250

0.05

0.1

0.15

0.2

Orbital index

s(1)

A

1

A2

B1

B2

1 5 10 15 20 250

0.2

0.4

0.6

0.8

Orbital index

s(1)

A

1

A2

B1

B2

r=3.05 r=13.7

CAS-vector ≡ ordering sites with decreasing site entropy values.

CAS-vector(r=3.05): [ 2 1 3 16 17 15 9 10 13 12 11 6 5 4 21 20 19 14 23 2218 8 7 24 25]Include orbitals with largest entropies in the expansion of the active space.

CI-based Dynamically Extended Active Space procedure

ΨCI = a0ΦSCF +∑

S

aSΦS +∑

D

aDΦD +∑

T

aTΦT + . . . .

→CAS-vector ≡ ordering sites with decreasing site entropy values.

Block entanglement

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B = TrA

SB = −Tr(BlnB)

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For critical 1− d systems : sN(l) =c

6ln

[2N

πsin

(πl

N

)]

+ g ,

Vidal, Latorre, Rico, Kitaev, PRL (2003), Calabrese, Cardy, JSM (2004),

Laflorencie, Sørensen, Chang, Affleck, PRL (2006), O.L., Solyom, Tincani, Noack, PRL (2007)

Spatial structures: Block entropy

In Fourier space:

s(k) =1

N

N∑

l=0

e−ikl

sN(l)

for discrete wave numbers, k = 2πj/N, k ∈ (−π, π).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−15

−10

−5

0

q/π

Ns(

q)

(a)

~ 306090

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−5

0

q/π

Ns(

q)

(b)

~ 306090

0 0.005 0.01 0.015 0.02 0.025 0.030

0.05

0.1

0.15

0.2

0.25

1/N

s(k)

θ = −π/2, k=πθ = −π/4, k=πθ = 0, k=πθ = .15π, k=.53πθ = π/4, k=2π/3

Peaks in |s(k)| { position of soft modes in critical systemsspatial inhomogeneity of the ground state

O.L., Solyom, Tincani, Noack (2007)

Mutual information: entanglement correlation

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A subsystem B subsystem

B = TrA

SB = −Tr(BlnB)

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pp ⇒ Sp

Sp describes the entanglement of site p with the rest of the system.

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p qp,q ⇒ Sp,q

Sp,q describes the entanglement of orbital p and q with the rest of the system.

I p,q describes the mutual information between orbital p and q

Ip,q = S

p + Sq − S

p,q

O.L., Solyom, PRB (2003): Quantum Chemistry,

O.L., Solyom, PRL (2005): quantum phase transitions (QPT) with q = p + 1.

Rissler, White, Noack, ECP (2005): Quantum chemistry, arbitrary p and q.

One- (ρi ) and two-orbital (ρi ,j) reduced density matrix

|ψ〉 =∑

α1,...,αN

Cα1,...,αN|α1...αN〉 ,

◮ ρi,j is calculated by taking the trace of |Ψ〉〈Ψ| over all local bases exceptfor αi and αj , the bases of sites i and j , i.e.,

ρi,j([αi , αj ], [α′i , α

′j ]) =

∑

α1,...,✚αi ,...,

✚αj ,...,αN

Cα1,...,αi ,...,αj ,...,αNC

∗α1,...,α

′i,...,α′

j,...,,αN

.

◮ In the MPS representation, calculation of ρij corresponds to thecontraction of the network except at sites i and j . The computational costscales as (N − 2)m3q3

◮ This can be decomposed as a sum of projector operators based on the freevariables αi and αj .

◮ ρi and ρi,j can be constructed from operators describing transitionsbetween single-site basis states.

2-site density matrix and generalized correlation functions

Transitions between states of a q-dimensional local Hilbert space:

T (m)i =

i−1⊗

j=1

I⊗ T (m) ⊗L⊗

j=i+1

I.

where (T (m))k,l = δ(l+q[k−1]),m for m = 1 . . . q2.Example spin-1/2 boson (qbit):

↓ ↑↓ T (1)

i T (2)i

↑ T (3)i T (4)

i

T (1)i −Sz

i + 12I

T (2)i S−

i

T (3)i S+

i

T (4)i Sz

i + 12I

ρij ↓ ↓ ↓ ↑ ↑ ↓ ↑ ↑↓ ↓ T (1)T (1)

↓ ↑ T (1)T (4) T (2)T (3)

↑ ↓ T (3)T (2) T (4)T (1)

↑ ↑ T (4)T (4)

The two-site reduced density matrix for spin-1/2 fermions

n=0,

sz=0n=1, sz=- 1

2n=1, sz=

12

n=2,

sz=-1n=2, sz=0

n=2,

sz=1n=3, sz=- 1

2n=3, sz=

12

n=4,

sz=0

ρi,j –– ––↓ –↓ – ––↑ –↑ – –↓ –↓ ––↓↑ –↓ –↑ –↑ –↓ –↓↑ – –↑ –↑ –↓ –↓↑ –↓↑ –↓ –↑ –↓↑ –↓↑ –↑ –↓↑ –↓↑–– 1/1

––↓ 1/6 2/5–↓ – 5/2 6/1

––↑ 1/11 3/9–↑ – 9/3 1/11

–↓ –↓ 6/6

––↓↑ 1/16 2/15 3/14 4/13–↓ –↑ 5/12 6/11 7/10 8/9–↑ –↓ 9/8 10/7 11/6 12/5–↓↑ – 13/4 14/3 15/2 16/1

–↑ –↑11/11

–↓ –↓↑ 6/16 8/14–↓↑ –↓ 14/8 16/6

–↑ –↓↑11/16 15/12

–↓↑ –↑15/12 16/11

–↓↑ –↓↑16/16

Expressing the two-site reduced density matrix, ρi,j , in terms of single-site operators, T(m)i

with m = 1 · · · 16.

For better readability only the operator number indices, m are shown, thus m/n corresponds to

〈Ψ| T(m)i

T(n)j

|Ψ〉. n and sz , denote the quantum numbers of two sites.

T (m) operators in terms of standard site operators

T (1) 1− n↑ − n↓ + n↑n↓

T (2) c↓ − n↑c↓

T (3) c↑ − n↓c↑

T (4) c↓c↑

T (5) c†↓ − n↑c

†↓

T (6) n↓ − n↑n↓

T (7) c†↓c↑

T (8) −n↓c↑

T (9) c†↑ − n↓c

†↑

T (10) c↓c†↑

T (11) n↑ − n↑n↓

T (12) n↑c↓

T (13) c†↓c

†↑

T (14) −n↓c†↑

T (15) n↑c†↓

T (16) n↑n↓

Generalized Correlation Functions

◮ Elements of the reduced density matrix

[ρi ]α′i,αi

= 〈α′i |ρi |αi 〉 = Tr ρi |αi 〉〈α′

i |︸︷︷︸

T (mi )

= Tr |Ψ〉〈Ψ|T (mi )i

︸︷︷︸

I⊗···⊗T (mi )⊗···⊗I

= 〈T (mi )i 〉

◮ Elements of the two-site reduced density matrix expressed as generalizedcorrelation functions

[ρij ]α′i,α′

j,αi ,αj

= · · · = 〈T (mi )i T (mj )

j 〉

Unconnected part:

〈T (mi )i T (mj )

j 〉C = 〈T (mi )i T (mj )

j 〉 − 〈T (mi )i 〉〈T (mj )

j 〉

◮ asymptotic behavior for large |i − j | determines the asymptotic behavior ofthe two-site mutual information Iij

◮ can be related to “usual” physical correlation functions

Calculating operator expectation values

◮ We transform the operators to the final basis for a given superblockconfiguration using the transformation matrices

Ol,l−1 · · ·O3,2O2,1TiTjO†2,1O

†3,2 · · ·Ol,l−1, i , j ∈ Blockl

Or,r−1 · · ·O3,2O2,1TiTjO†2,1O

†3,2 · · ·Or,r−1, i , j ∈ Blockr

Ol,l−1 · · ·O3,2O2,1TiO†2,1O

†3,2 · · ·Ol,l−1, i ∈ Blockl

Or,r−1 · · ·O3,2O2,1TjO†2,1O

†3,2 · · ·Or,r−1, j ∈ Blockr

where O is a matrix of size [m′,m × q] where m′ was determined by theDBSSprocedure based on the predefined accuracy threshold χ andl + 2 + r = N for the two-site DMRG.

◮ This is equivalent to the network contraction procedure in the MPSlanguage.

SU(4) Hubbard model, 1/3 Filling

(a)

1

3

5

7

9

1113

15

17

19

21

23

2

4

6

8

1012

14

16

18

20

2224

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1/N

Ent

angl

emen

t bon

d st

reng

th

(b) N/2,N/2−3

N/2,N/2−4

N/2

N/2−2

N/2+1

N/2−3

N/2−1

(c)

trimerized phase

◮ for multipartite measure seeposter of Sz. Szalay

◮ for other exotic phases inSU(4) see poster of G. Barczaand E. Szirmai

Quantum chemistry (some 40 electrons on 40 orbitals)

Example: Task to determine the electronic structure of the binuclearoxo-bridged copper clusters

bis(µ-oxo) µ− η2 : η2 peroxo

◮ CASSCF calculations yield a qualitative wrong interpretation of the energydifference between different isomers.

◮ Too large active space required to get qualitatively correct picture forstandard QC methods.

◮ open d shell problem◮ K.H. Marti, I.Malkin Ondik, G. Moritz, and M. Reiher, J. Chem. Phys. 128, 014104 (2008).

Y. Kurashige, and T. Yanai, J. Chem. Phys. 130, 234114 (2009).T. Yanai, Y. Kurashige, E. Neuscamman, and G.K.-L. Chan, J.Chem.Phys. 132, 024105(2010).

G. Barcza, O. Legeza, K. H. Marti, and M. Reiher, Phys. Rev. A. 83, 012508 (2011).

Site entropy profile→ highly entangled orbitals


0 5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

Orbital index

s(1)

A

g

B3u

B2u

B1g

B1u

B2g

B3g

Au

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Orbital index

s(1)

A

g

B3u

B2u

B1g

B1u

B2g

B3g

Au

Entanglement picture of the two isomers


1

3

5

7

9

11

13

15

17

192123

25

27

29

31

33

35

37

39

4143

24

6

8

10

12

14

1618

20222426

28

30

32

34

36

3840

42 44

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10−1

10−2

10−3

1

3

5

7

9

11

13

15

17

192123

25

27

29

31

33

35

37

39

4143

24

6

8

10

12

14

1618

20222426

28

30

32

34

36

3840

42 44

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10−1

10−2

10−3

◮ peroxo: orbital pairs 3–14 and 13–35 are highly entangled→ bonding and anti-bonding orbitals → the O–O bond is intact

◮ bis(µ-oxo): all five orbitals 3, 13, 14, 34, and 35 are entangled→ four equivalent Cu–O bonds

◮ O–O bond breaking process → transition from the peroxo to the bisoxo isomer

◮ Mutual information + DMRG → relative energy of the two-isomers

Entanglement localization, example for bis(µ-oxo)

energetical ordering optimized ordering

1

3

5

7

9

11

13

15

17

192123

25

27

29

31

33

35

37

39

4143

24

6

8

10

12

14

1618

20222426

28

30

32

34

36

3840

42 44

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10−1

10−2

10−3

44

43

42

30

32

11

9

7

21

1815

4

12

16

14

35

36

37

25

2428

4140

31

33

29

10

8

2019

22263

17

15

13

34

38

3926

27 23

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

100

10−1

10−2

10−3

◮ Reordering orbitals by minimizing the entanglement distance:Idist =

∑

i,j Ii,j × |i − j |η ,◮ Apply spectral graph theory: Fiedler vector x = (x1, . . . xN) is the solution that

minimizes F (x) = x†Lx =∑

ij Ii,j(xi − xj)2, with

∑

i xi = 0 and∑

i x2i = 1, and the graph Laplacian is

L = D − I with Di,i =∑

j Ii,j .The second eigenvector of the Laplacian is the Fiedler vector.

Entanglement localization

0 10 20 30 400

0.2

0.4

0.6

0.8

Orbital index

Orb

ital e

ntro

py (a)

5 10 15 20 25 30 35 400

0.5

1

Left block length

Blo

ck e

ntro

py (b)

5 10 15 20 25 30 35 40−1.5

−1

−0.5

0

Left block length

Mut

ual i

nfor

mat

ion

(c)010203040506

0 10 20 30 400

0.2

0.4

0.6

0.8

Orbital index

Orb

ital e

ntro

py (a)

5 10 15 20 25 30 35 400

0.5

1

Left block length

Blo

ck e

ntro

py (b)

5 10 15 20 25 30 35 40

−1

−0.5

0

Left block length

Mut

ual i

nfor

mat

ion

(c)010203040506

Relative energy of the isomers: a notebook calculation

0 50 100 150 200 250 300

−541.54

−541.52

−541.5

−541.48

−541.46

−541.44

−541.42

−541.4

−541.38

−541.36

Ene

rgy

(a)M

min=64, CI=3, opt ORD, opt CAS, M

start=64

Mmin

=64, χ=10−4, CI=3, opt ORD, opt CAS, Mstart

=256

Mmin

=256, χ=10−5, CI=3, opt ORD, opt CAS, Mstart

=512

M=800 in Ref [?]

Fixed number of block states vs DBSSDynamic Block State Selection

sL(l + 1)− sTruncL (l + 1) < χ

◮ DBSS guarantees that the number of block states areadjusted according to the entanglement between theDMRG blocks and the a priori defined accuracy can bereached.

◮ Dynamically Extended Active Space (CI-DEAS) procedure

method ∆E

Reference energies

CASSCF(16,14) 1CASPT2(16,14) 6bs-B3LYP 221RASPT2(24,28) 120

Previously published DMRG energies

[1],DMRG(26,44)[m=800] 78[2],DMRG(32,62)[m=2400] 149[3],DMRG(28,32)[m=2048]-SCF 107[3],DMRG(28,32)[m=2048]SCF/CT 113

DMRG energies from this work

DMRG(26,44)[64/256/10−4] 111

DMRG(26,44)[256/512/10−4] 115

DMRG(26,44)[256/1024/10−4] 113

DMRG(26,44)[256/512/10−5] 113

DMRG(26,44)[256/1024/10−5] 113

M. Reiher et al, J. Chem. Phys. 128, (2008).T. Yanai et al, J. Chem. Phys. 130, (2009).G.K.-L. Chan et al, J.Chem.Phys. 132, (2010).

Ab initio spin densities: [Fe(NO)]2+

DMRG(7,7)[16] DMRG(7,7)[32] DMRG(7,7)[48]

(a) dpc = 1.131A

CAS(7,7)SCF

(b) dpc = 1.131A

DMRG(7,7)[32] DMRG(7,7)[48]

(c) dpc = 0.598 A

CAS(7,7)SCF

(d) dpc = 0.598 A

zy

x

Fe

N

O

�

dpcdpc

dpc

dpc

(e) [FeNO]2+

(Boguslawski, Marti, Legeza, Reiher, 2012)

For large active space

m = 128 m = 256 m = 512 m = 1024 m = 2048

(a) DMRG(13,20)[m]–DMRG(13,29)[2048] spin density differences

m = 128 m = 256 m = 512 m = 1024 m = 2048

(b) DMRG(13,24)[m]–DMRG(13,29)[2048] spin density differences

m = 128 m = 256 m = 512 m = 1024

(c) DMRG(13,29)[m]–DMRG(13,29)[2048] spin density differences

m = 2048

(d) DMRG(13,29)

Entanglement Measures for Single- and Multireference Correlation Effects

Boguslawski, Tecmer, Legeza, Reiher (2012)

(a) [Fe(NO)2+]

O

NFe O

N

N

O

(b) FeL(NO)

[Fe(NO)2+]

1

3

5

7

9

11

1315

17

19

21

23

25

27

29

2

4

6

8

10

12

1416

18

20

22

24

26

28

A

10−1

10−2

10−3

(a) Mutual information

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1)

(b) Single orbital entropy

FeLNO

1

3

5

7

9

11

13

15

1719

21

23

25

27

29

31

3335

2

4

6

8

10

12

14

161820

22

24

26

28

30

32

34

A’

A’’

10−1

10−2

10−3

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

(a) Mutual information

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1)

(b) Single orbital entropy

Entanglement effect of the basis (active space)

0 50 100 150 200 250 300 350 40010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Sor

ted

valu

es o

f the

mut

ual i

nfor

mat

ion

Static correlation

Dynamic correlation

FeLNO(21,35)FeNO(13,29)FeNO(11,14)FeNO(11,11)FeNO(11,9)

0 20 40 60 80 10010

−6

10−5

10−4

10−3

10−2

10−1

100

Sor

ted

valu

es o

f the

mut

ual i

nfor

mat

ion

FeLNO(21,35)FeNO(13,29)FeNO(11,14)FeNO(11,11)FeNO(11,9)

1

3

5

7

9

2

4

6

8

A

10−1

10−2

10−3

Mutual information

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1

)

Single orbital entropy

(a) CAS(11,9)

1

3

5

7

9

11

2

4

6

8

10

A

10−1

10−2

10−3

Mutual information

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1

)


(b) CAS(11,11)

1

3

5

7

9

11

13

2

4

68

10

12

14

A

10−1

10−2

10−3

Mutual information

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1

)


(c) CAS(11,14)

Chemical bond forming and breaking vs Entanglement

Boguslawski, Tecmer, Barcza, Legeza, Reiher, 2013

Four-Component Density Matrix Renormalization Group

◮ TlH with double group C2v symmetry (14 e / 94 spinors),

◮ DIRAC12(2012), a relativistic ab initio electronic structure program,

◮ including spin-orbit coupling,

◮ 4c-DMRG with max(M) ≃ 4500− 5000, χ = 10−5

500 1000 1500−0.84

−0.835

−0.83

−0.825

−0.82

−0.815

−0.81

−0.805

−0.8

−0.795

r=1.8720(exp)

Iteration

Ene

rgy

shift

ed b

y 20

275

800 1000 1200

500 1000 1500−0.84

−0.835

−0.83

−0.825

−0.82

−0.815

−0.81

−0.805

−0.8

−0.795

Iteration

Ene

rgy

shift

ed b

y 20

275

r=1.9220

4c−DMRGCISDCISDTCISDTQCCSDCCSDTCCSDTQ

500 1000

0 1 2 3 4 5 6 7

x 10−3

−0.84

−0.83

−0.82

−0.81

−0.8

−0.79

1/M

Ene

rgy

shift

ed b

y 20

275

Mmax

=4500

512

256

128DMRG fixed MCC−SDTQDMRG χ=10−5

S. Knecht, M. Reiher, O Legeza (2013)

Four-Component Density Matrix Renormalization Group

1 10 20 30 40 50 60 70 80 900

0.02

0.04

0.06

0.08

0.1

0.12

Orbital index

s(1)

1 3 5 7

911

1315

171921232527

2931

3335

3739

4143454749515355

5759

6163

65676971737577

7981

8385

878991 93

2 4 6 810121416182022242628

3032

34363840424446485052545658

6062

64666870727476788082848688909294

Cost1=39.8598

Cost2=1795.9568

Itot

=2.2298

100

10−1

10−2

10−3

Nuclear shell models Ex.: 64Ge pf+g9/2, α = (n, l , j ,m, τz).

O.L, L. Veis, J. Dukelsky, A. Poves (2015)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.89/

27/

25/

23/

21/

25/

23/

21/

21/

23/

21/

27/

25/

23/

21/

21/

23/

25/

27/

21/

23/

21/

21/

23/

25/

21/

23/

25/

27/

29/

2

g9/2 g9/2

f5/2 f5/2

p1/2 p1/2p3/2 p3/2

f7/2 f7/2

a)

g9/2

9/2

g9/2

7/2

g9/2

5/2

g9/2

3/2

g9/2 1/2

f5/2 5/2

f5/2 3/2

f5/2 1/2

p1/2 1/2p3/2 3/2p3/2 1/2

f7/2 7/2

f7/2 5/2

f7/2 3/2

f7/2 1/2f7/2

1/2

f7/2

3/2

f7/2

5/2

f7/2

7/2p3/2 1/2p3/2 3/2

p1/2 1/2

f5/2 1/2

f5/2 3/2

f5/2 5/2

g9/2 1/2

g9/2 3/2g9/2 5/2

g9/2 7/2g9/2 9/2b)

0 0.02 0.04 0.06 0.08 0.1 0.12

0 0.5 1 1.5

x 10−3

−78.46

−78.44

−78.42

−78.4

−78.38

−78.36

−78.34

−78.32

−78.3

−78.28

−78.26

1/M

EG

S

1024

2048

2500

3500

512

1024

2048

4096

56Ni (spdf)

0 0.5 1 1.5

x 10−3

−304.1

−304

−303.9

−303.8

−303.7

−303.6

−303.5

−303.4

64Ge (spdf)

1/M

512

1024

2048

3072

4096

C2 rep.

C4 rep.

Exact

0 0.5 1 1.5

x 10−3

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1/M

∆/M

eV

512

1024

2048

0 0.5 1 1.5

x 10−3

0.85

0.9

0.95

1

1.05

1.1

1/M

512

1024

2048

3072

DMRGExactMCSM−extrap

Metal-Insulator like Transition (MIT) in pseudo-onedimensional systems

Fertitta, Paulus, Barcza, O.L. (PRB, 2014)

◮ MIT in pseudo-onedimensional systems i.e. lithium and beryllium rings,through the ab initio Density Matrix Renormalization Group (DMRG)

◮ transition from the metallic to the insulating regime, i.e. till thedissociation in atomic species, reveals a high complexity due to strongcorrelation effects which are hard to be described using traditionalmulticonfigurational approaches.

◮ besides modelling the Hubbard system with a Li ring using only s

functions, we can study the effect of p orbital basis on the MIT

◮ electron correlation from predominantly dynamic correlation in the metalliccase to static correlation in the dissociation limit

◮ quantum entanglement, which is crucial in the DMRG routine, stronglydepends on this choice

Metal-Insulator like Transition (MIT) in pseudo-onedimensional systems

◮ in the case of beryllium rings two main HF configurations are important,depending on the distance regime, in order to describe the PotentialEnergy Surface (PES) till dissociation

◮ Around the minimum, the Hartree-Fock wavefunction presents a high p

character while at larger distances, the HF configuration consists of doublyoccupied linear combinations of (almost) pure 2s orbitals.

◮ Orbital localization will then yield different LOs. In the first case σ-likeorbitals with a sp character will emerge, while from configuration 2, onewill obtain atom-like 2s and 2p basis

−0.05

−0.04

−0.03

−0.02

−0.01

0.00

2.0 2.5 3.0 3.5 4.0

Coh

esiv

e E

nerg

y (E

h)

Be−Be (Å)

A1

1g

A1

1g

Figure : (Color online) Hartree-Fock ground state potential energy curves for

◮ Full ab initio Hamiltonian in which the hopping and Coulomb matrixelements are not restricted to nearest neighbors only

◮ Hartree-Fock calculations, Foster-Boys localization procedure

◮ The atomic basis sets consisted of 1s, 2s and 2p functions only, and 1sorbitals were always kept frozen. A STO-3G was used for lithium[?] whilefor beryllium we employed a basis set relying on cc-pVDZ[?] where onlytwo contracted s functions and one contracted radial p function were used.

H =∑

i,j,σ

Tijc†iσcjσ +

∑

i,j,k,lσ,σ′

Vijklc†iσ′c

†jσckσ′clσ

Convergence issues

2 2.5 3 3.5−87.5

−87.45

−87.4

−87.35

−87.3

−87.25

−87.2

−87.15

−87.1

Be−Be (A)

Gro

und

stat

e en

ergy

(E

h)

Configuration 1

o

M=128M=256M=512χ=10−4

2 2.5 3 3.5−87.5

−87.45

−87.4

−87.35

−87.3

−87.25

−87.2

−87.15

−87.1

Be−Be (A)

Configuration 2

o

M=128M=256M=512χ=10−4

Figure : Ground state energy of Be6 as a function of Be-Be distance for various fixednumber of block states, M, and for χ = 10−4 using configuration 1 (left) andconfiguration 2 (right) and localized orbital basis.

◮ We also utilize various concepts inherited from quantum informationtheory (QIT)which allows us to use DMRG as a black box method

◮ Our black-box QC-DMRG is composed of two phases: the preprocessingphase in which the ordering and CAS-vector are optimized using fixedsmall number of block states and the production phase in which anaccurate calculation is performed using the DBSS procedure in order toreach an a priory set error margin

Dependence of entanglement on orbital basis and ordering: Ioverall = Iij |i − j |η

Orbital index

Orb

ital i

ndex

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Orbital index

Orb

ital i

ndex

0.05

0.1

0.15

0.2

0.25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Orbital index

s(1)

A

g

B3u

B2u

B1g

B1u

B2g

B3g

Au

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Orbital index

s(1)

Figure : (Color online) Orbital ordering optimization using the Fiedler vector for theground state for Be6 for a stretched structure, dBe−Be = 3.30A, using the DMRGmethod with canonical (left) and local (right) orbitals using configuration 2.Colorscaled plot of two-orbital mutual information (upper) and single-orbital entropyprofile (lower). I = 7.81, I = 332.38 with the canonical basis and

For the ground state for Be6 close to the equilibrium structure

Orbital index

Orb

ital i

ndex

0.01

0.02

0.03

0.04

0.05

0.06

Orbital index

Orb

ital i

ndex

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Orbital index

s(1)

A

g

B3u

B2u

B1g

B1u

B2g

B3g

Au

0

0.05

0.1

0.15

0.2

0.25

0.3

Orbital index

s(1)

Figure : (Color online) Orbital ordering optimization using the Fiedler vector for theground state for Be6 close to the equilibrium structure, dBe−Be = 2.15A, using theDMRG method with canonical (left) and local (right) orbitals using configuration 1.Colorscaled plot of two-orbital mutual information (upper) and single-orbital entropyprofile (lower). Itot = 4.35, Idist = 149.70 with the canonical basis and

Li ring with only 2s functions: Hubbard-like model

0 2 4 6 8 10 12 1410

−12

10−10

10−8

10−6

10−4

10−2

100

102

l = |i − j|

T1,l

d = 3.05A, N = 18d = 3.05A, N = 26d = 6.00A, N = 18d = 6.00A, N = 26

5 10 15 20 25

0.8

1

1.2

1.4

1.6

1.8

2

2.2

l

SN(l)

d = 3.05A, N = 18

d = 6.00A, N = 18

d = 3.05A, N = 26

d = 6.00A, N = 26

◮ mettalic region: spin gap ∆s =≃ 10−8 Eh

◮ charge gap shows a finite-size dependence, ∆c = 0.09Eh for N → ∞◮ setting all Vijkl = 0 c = 2.04 as expected result for a Hubbard model with

U = 0.◮ at the dissociation limit insulating situation, spin gap zero◮ charge gap shows no finite-size dependence ∆c = 0.26Ehfor

dLi−Li = 4.00A◮ The orbital entropy s(1) = 0.720(2) being close to ln 2 = 0.693 → only

two basis states gain finite weight ( weight for the empty and doublyoccupied orbital 0.002(1) and 0.500(4) )

◮ c = 1.03

Metallic regime

dLi−Li = 3.05A

Insulating regime

dLi−Li = 6.00AHoppings

T(9)

ix T

(3)

j: −

i,↑

j==> ↑

i,−

j: ρ

i,j(5,4)

1

3

5

7

9

11

13

15

17

2

4

6

810

12

14

16

18

(a)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

T(9)

ix T

(3)

j: −

i,↑

j==> ↑

i,−

j: ρ

i,j(5,4)

1

3

5

7

9

11

13

15

17

2

4

6

810

12

14

16

18

(b)

0.0

0.1

0.2

0.3

0.4

0.5

T(9)

ix T

(8)

j: −

i,↑↓

j==> ↑

i,↓

j: ρ

i,j(9,7)

1

3

5

7

9

11

13

15

17

2

4

6

810

12

14

16

18

(c)

0.00

0.02

0.04

0.06

0.08

0.10

T(9)

ix T

(8)

j: −

i,↑↓

j==> ↑

i,↓

j: ρ

i,j(9,7)

1

3

5

7

9

11

13

15

17

2

4

6

810

12

14

16

18

(d)

0.000

0.005

0.010

0.015

T(14)

ix T

(8)

j: ↓

i,↑↓

j==> ↑↓

i,↓

j: ρ

i,j(13,12)

1

3

5

7

9

11

13

15

17

2

4

6

810

12

14

16

18

(e)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

T(14)

ix T

(8)

j: ↓

i,↑↓

j==> ↑↓

i,↓

j: ρ

i,j(13,12)

1

3

5

7

9

11

13

15

17

2

4

6

810

12

14

16

18

(f)

0.0

0.5

1.0

1.5

2.0

x 10−3

T(13)

ix T

(4)

j: −

i,↑↓

j==> ↑↓

i,−

j: ρ

i,j(10,7)

1

3

5

7

9

11

13

15

17

2

4

6

810

12

14

16

18

(g)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

T(13)

ix T

(4)

j: −

i,↑↓

j==> ↑↓

i,−

j: ρ

i,j(10,7)

1

3

5

7

9

11

13

15

17

2

4

6

810

12

14

16

18

(h)

0.0

0.5

1.0

1.5

2.0

x 10−3

FlippingT

(10)

ix T

(7)

j: ↓

i,↑

j==> ↑

i,↓

j: ρ

i,j(9,8)

1

3

5

711

13

15

17

2

4

6

810

12

14

16

18

(i)

0.05

0.10

0.15

0.20

0.25

0.30

0.35

T(10)

ix T

(7)

j: ↓

i,↑

j==> ↑

i,↓

j: ρ

i,j(9,8)

1

3

5

711

13

15

17

2

4

6

810

12

14

16

18

(j)

0.1

0.2

0.3

0.4

0.5

Li ring with 2s and 2p functions

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3

p23

s4

p24

s5

p25

s6

p26

0

0.1

0.2

0.3

0.4

0.5

0.6

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3

p23

s4

p24

s5

p25

s6

p26

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.5

1

1.5

p1

s

pzp2 p1

s

pzp2 p1

s

pzp2 p1

s

pzp2 p1

s

pzp2 p1

s

pzp2

Orbital index

s(1)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

p1

s

pzp2 p1

s

pzp2 p1

s

pzp2 p1

s

pzp2 p1

s

pzp2 p1

s

pzp2

Orbital index

s(1)

Figure : (Color online) Schematic plot of the two-orbital mutual information (upperpanels) and one-orbital entropy (lower panels) for Li6 ring calculated using 2s and 2patomic functions for bond lengths dLi−Li = 3.05A(left) and for dLi−Li = 6.00A(right).

Be ring with 2s and 2p functions

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3

p23

s4

p24

s5

p25

s6

p26

0.05

0.1

0.15

0.2

0.25

0.3

0.35

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p1

s

pz

p2 p1

s

pz

p2 p1

s

pz

p2 p1

s

pz

p2 p1

s

pz

p2 p1

s

pz

p2

Orbital index

s(1)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

p1

s

pz

p2 p1

s

pz

p2 p1

s

pz

p2 p1

s

pz

p2 p1

s

pz

p2 p1

s

pz

p2

Orbital index

s(1)

Figure : (Color online) Schematic plot of the two-orbital mutual information (upperpanels) and one-orbital entropy (lower panels) for Be6 ring calculated using 2s and 2patomic functions for bond lengths dBe−Be = 2.15A(left) and for

dBe−Be = 3.30A(right) using configuration 2.

Metallic regime

dLi−Li = 3.05A

Insulating regime

dLi−Li = 6.00A

Metallic regime

dBe−Be = 2.15A

Insulating regime

dBe−Be = 3.30AHoppings

T(9)

ix T

(3)

j: −

i,↑

j==> ↑

i,−

j: ρ

i,j(5,4)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(a)

0.05

0.10

0.15

0.20

0.00

T(9)

ix T

(3)

j: −

i,↑

j==> ↑

i,−

j: ρ

i,j(5,4)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(b)

0.0

0.1

0.2

0.3

0.4

T(9)

ix T

(3)

j: −

i,↑

j==> ↑

i,−

j: ρ

i,j(5,4)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(c)

0.00

0.05

0.10

0.15

T(9)

ix T

(3)

j: −

i,↑

j==> ↑

i,−

j: ρ

i,j(5,4)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(d)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5x 10

−3

T(9)

ix T

(8)

j: −

i,↑↓

j==> ↑

i,↓

j: ρ

i,j(9,7)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(e)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

T(9)

ix T

(8)

j: −

i,↑↓

j==> ↑

i,↓

j: ρ

i,j(9,7)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(f)

0.000

0.005

0.010

0.015

0.020

T(9)

ix T

(8)

j: −

i,↑↓

j==> ↑

i,↓

j: ρ

i,j(9,7)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(g)

0.000

0.005

0.010

0.015

0.020

0.025

T(9)

ix T

(8)

j: −

i,↑↓

j==> ↑

i,↓

j: ρ

i,j(9,7)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(h)

0.000

0.002

0.004

0.006

0.008

0.010

T(14)

ix T

(8)

j: ↓

i,↑↓

j==> ↑↓

i,↓

j: ρ

i,j(13,12)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(i)

0.05

0.10

0.15

0.20

0.00

T(14)

ix T

(8)

j: ↓

i,↑↓

j==> ↑↓

i,↓

j: ρ

i,j(13,12)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(j)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x 10−3 T

(14)

ix T

(8)

j: ↓

i,↑↓

j==> ↑↓

i,↓

j: ρ

i,j(13,12)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(k)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

T(14)

ix T

(8)

j: ↓

i,↑↓

j==> ↑↓

i,↓

j: ρ

i,j(13,12)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(l)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

T(13)

ix T

(4)

j: −

i,↑↓

j==> ↑↓

i,−

j: ρ

i,j(10,7)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(m)

0.05

0.10

0.15

0.20

0.00

T(13)

ix T

(4)

j: −

i,↑↓

j==> ↑↓

i,−

j: ρ

i,j(10,7)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(n)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x 10−3 T

(13)

ix T

(4)

j: −

i,↑↓

j==> ↑↓

i,−

j: ρ

i,j(10,7)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(o)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

T(13)

ix T

(4)

j: −

i,↑↓

j==> ↑↓

i,−

j: ρ

i,j(10,7)

p11

pz1

p12

pz2

p13

pz3

p14

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(p)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

FlippingT

(10)

ix T

(7)

j: ↓

i,↑

j==> ↑

i,↓

j: ρ

i,j(9,8)

p11

pz1

p12

pz2

p13

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(q)

0.05

0.10

0.15

0.20

T(10)

ix T

(7)

j: ↓

i,↑

j==> ↑

i,↓

j: ρ

i,j(9,8)

p11

pz1

p12

pz2

p13

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(r)

0.1

0.2

0.3

0.4

T(10)

ix T

(7)

j: ↓

i,↑

j==> ↑

i,↓

j: ρ

i,j(9,8)

p11

pz1

p12

pz2

p13

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(s)

0.05

0.10

0.15

T(10)

ix T

(7)

j: ↓

i,↑

j==> ↑

i,↓

j: ρ

i,j(9,8)

p11

pz1

p12

pz2

p13

pz4

p15

pz5

p16

pz6

s1

p21

s2

p22

s3p2

3

s4

p24

s5

p25

s6

p26

(t)

0.5

1.0

1.5

2.0

2.5

3.0

3.5x 10

−3

2.5 2.6 2.7 2.8 2.9−87.4

−87.3

−87.2

−87.1

Ene

rgy

(Eh)

GSXS

2.5 2.6 2.7 2.8 2.9

5

10

15

Be−Be (A)o

I Tot

Configuration 1

Configuration 2

2.5 2.6 2.7 2.8 2.9 3 3.15

10

15

20

25

Be−Be (A)o

I Tot

Configuration 1

Configuration 2

Figure : Energies of the two lowest lying 1Ag singlet states and total quantuminformation Itot for Be6 and Be10 (in this case only Itot is considered) as a function ofdBe−Be, calculated using localized orbitals for configuration 1 and configuration 2.The sudden change in Itot indicating the position of the avoided crossing, occurs atdBe−Be ≃ 2.55A for Be6 and 2.72A for Be10.

Part II

Higher dimensional networks

◮ Tree Tensor Network State (TTNS) algorithm• Multiply connected networks• Structure of the network• Optimization of the network topology• TTNS study of the avoided crossing in LiF

Entanglement → Multiply connected networks

LiF at r=3.05 LiF at r=13.7Rissler, White, Noack, ECP (2006)

Murg, Verstraete, Schneider, Nagy, Legeza (2013)

◮ DMRG → Matrix product states, i.e. optimization along one-spatialdimension

◮ Need for an algorithm that reflects the entanglement topology of theproblem → Tensor Network State (TNS) methods

◮ Use tensors Ai [α]m1...mz where z is the coordination number

Tree Tensor Network State (TTNS)

|Ψ〉 =∑

α1,...,αN

Cα1...αN|α1, ..., αN〉.

Cα1...αNdescribe a tree tensor network, i.e., they emerge from contractions of a

set of tensors {A1, . . . ,AN}, where

Ai [α]m1...mz

,

is a tensor at each vertex i of the network, with z virtual indices m1 . . .mz ofdimension D and one physical index α of dimension q, with z being the

coordination number of that orbital.

A

A

(a)

(d) (b)

(c)1m

2m

3m

1m

2m

3m

4m

a

a

Vidal, Corboz (2009);

Murg, Verstraete, O.L., Noack (2010,2014);

Nakatani, Chan (2013)

Two-dimensional network

A tree tensor network in which all sites in the tree represent physical orbitals(red lines) and in which entanglement is transferred via the virtual bonds thatconnect the sites (black lines).

Maximal distance between two orbitals, 2∆, scales logarithmically with N forz > 2.

Decomposition of the Hamiltonian as TTNO

(a) expectation value 〈Ψ|H|Ψ〉 with respect to the TTNS(b) The Hamiltonian H, represented as TTNO of component tensors hi in themiddle.(b1) decomposition of the Hamiltonian as MPO(b2) decomposition of the Hamiltonian as TTNO

Structure of the network

• The coefficients Cα1...αNare obtained by contracting the virtual indices of the

tensors.

• The structure of the network can be arbitrary.

• The coordination number can vary from orbital to orbital.

• The only condition is that the network is bipartite, i.e., by cutting one bond,the network separates into two disjoint parts.

• For z = 2, the one-dimensional MPS-ansatz used in DMRG is recovered.

• Entanglement is transferred via the virtual bonds that connect the orbitals.

• For z > 2 the number of virtual bonds required to connect two arbitraryorbitals scales logarithmically with the number of orbitals N, whereas thescaling is linear in N for z = 2.

• The maximal distance between two orbitals, 2∆, scales logarithmicallywith N for z > 2.

Tensor topology optimization:∑

ij Iij × dηij

Iij is model dependent:

◮ depends on Tij and Vijkl interaction strengths

◮ depends on the choice of basis

◮ major aim: could we optimize basis on-the-fly (different approaches areunder investigations, Chan, Murg, Verstraete, O.L., Krumnow, Eisert,Schneider); unsolved problem

dij depends on the tensor topology

◮ a possible solution: TTNS with orbital dependent coordination number zi .

A

A

(a)

(d) (b)

(c)1m

2m

3m

1m

2m

3m

4m

a

a

Number of orbitals in the tree:

N = 1+z

∆∑

j=1

(z−1)j−1 =z(z − 1)∆ − 2

z − 2

The maximal distance between twoorbitals, 2∆, scales logarithmicallywith N for z > 2.

Tensor topology optimization:∑

ij Iij × dηij (Ex. LiF 6/25)

Energetical ordering (MPS) dij = |i − j | Entanglement localization (MPS)

Tree Tensor Network State (TTNS)

Variable tensor orders and convergence properties

(a) Network-1

1 234 5 67 8 91011 12 1314 15 161718

(b) Network-2 (c) Network-3

12 3

4 56

7 8

9

10

11

12

13 14

15

16 17

18

1

2

3

45

6

7

8

9

1011

12

13

14

15

16

17

18

(d) Convergence rate

0 50 100 150 200 250 300 350

10−5

10−4

Iteration step

∆ E

GS

Network−1Network−2Network−3

Optimization of the sweeping

In case of the tree-network, there is more freedom to choose the optimalsweeping procedure, i.e., to choose the optimal path through which thenetwork is traversed.

We sweep through the network by going recursively back and forth througheach branch. Therefore, according to the labeling of the orbitals on the latticeshown in the figure one sweep goes through the orbitals:1 2 3 4 5 4 6 4 3 7 8 7 3 2 9 10 9 11 9 2 1 12 13 14 13 15 13 12 16 17 16 1816 12 1 19 20 21 20 22 20 19 23 24 23 25 23 19.

TTNS study of the avoided crossing in LiF (6/25)

2 4 6 8 10 12 14−107.15

−107.1

−107.05

−107

−106.95

−106.9

−106.85

−106.8

Bond length(r)

Sin

glet

ene

rgie

s

GS, S=01XS, S=02XS, S=03XS, S=0

2 4 6 8 10 12 1410

−6

10−5

10−4

10−3

Bond length(r)

∆ E

GS, z=2

GS, z=3

3XS, z=2

3XS, z=3

0 50 100 15010

−6

10−5

10−4

10−3

10−2

Iteration step

∆ E

GS

r=3.05r=5.5r=11.5r=13.7

0 50 100 15010

−6

10−5

10−4

10−3

10−2

Iteration step

∆ E

3XS

r=3.05r=5.5r=11.5r=13.7

2 4 6 8 10 12 14

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Bond length(r)

I Tot

V. Murg, F. Verstraete, R. Schneider, P. Nagy, O. L. (2014)

Basis states transformation applied to the Hamiltonian

There are two ways to implement the basis transformation: one based on thestate and the other based on the Hamiltonian.

H =∑

ij

Tijc†i cj +

∑

ijkl

Vijklc†i c

†j ckcl ,

The function E (U) can be expressed as

E (U) =∑

ij

T (U)ij〈c†i cj〉+∑

ijkl

V (U)ijkl〈c†i c†j ckcl〉 with

T (U) = UTU†

V (U) = (U ⊗ U)V (U ⊗ U)†.

The correlation functions 〈c†i cj〉 and 〈c†i c†j ckcl〉 are calculated with respect to

the original state and are not dependent on the parameters in U. With thefunction E (U) in this form, its gradient can be calculated explicitly. Bothquantities can be evaluated efficiently for different parameter sets U.Murg, Verstraete, O.L., Noack (2010)

Local mode transformation: black-box tool to improve basis (Krumnow,

Schneider, Legeza, Eisert, 2014)

Concept of chemical bond and aromaticity based on quantum informationtheory Szilvasi, Barcza, O.L. (2105)

Graphical representation of mutual information for benzene The color of the bonds indicates the mutual

information of the connected two sites. The shape of the molecular orbitals (isovalue 0.05) are shown directly near

the sites. The blue and red colors of the orbitals refer to the sign of the wavefunction. Atoms are labeled with their

usual colors; H, B, C, N, and O atoms marked with white, brown, tan, blue, and red colors, respectively.

◮ Black lines correspond to C-C and C-H bonds

◮ Delocalized pz -orbitals → aromaticity


Graphical representation of mutual information for pirol The color of the bonds indicates the mutual information of

the connected two sites. The shape of the molecular orbitals (isovalue 0.05) are shown directly near the sites. The

blue and red colors of the orbitals refer to the sign of the wavefunction. Atoms are labeled with their usual colors;

H, B, C, N, and O atoms marked with white, brown, tan, blue, and red colors, respectively.

◮ Black lines correspond to C-C and C-H bonds◮ Five delocalized pz -orbitals but there are two electrons on N◮ These + Huckel rule→ aromaticity


Graphical representation of mutual information for borole The color of the bonds indicates the mutual information

of the connected two sites. The shape of the molecular orbitals (isovalue 0.05) are shown directly near the sites.

The blue and red colors of the orbitals refer to the sign of the wavefunction. Atoms are labeled with their usual

colors; H, B, C, N, and O atoms marked with white, brown, tan, blue, and red colors, respectively.

◮ Black lines correspond to C-C and C-H bonds◮ Five delocalized pz -orbitals and two electrons less due to B◮ → antiaromaticity, no-cyclic delocalization

Extensions to three sites

◮ Structure of correlations as well as that of the entanglement becomesmuch more complicated if more than two sites are involved

◮ Generalizations of the two-site mutual information

◮ Three-site correlation can be defined as a Venn-diagram-based three-sitegeneralization

Iijk = si + sj + sk − sij − sik − sjk + sijk . (1)

↓↓↓ ↓↓↑ ↓↑↓ ↑↓↓ ↓↑↑ ↑↓↑ ↑↑↓ ↑↑↑↓↓↓ 1/1/1

↓↓↑ 1/1/4 1/2/3 2/1/3↓↑↓ 1/3/2 1/4/1 2/3/1↑↓↓ 3/1/2 3/2/1 4/1/1

↓↑↑ 1/4/4 2/3/4 2/4/3↑↓↑ 3/2/4 4/1/4 4/2/3↑↑↓ 3/4/2 4/3/2 4/4/1

↑↑↑ 4/4/4

Block-diagonal form of the three-site density matrix expressed in terms of correlation functions. Here ↓↓↓denotes

the state |↓〉i |↓〉j |↓〉k and m/n/l denotes 〈Ψ| T(m)i

T(n)j

T(l)k

|Ψ〉.

Entanglement in multipartite mixed states

◮ Entanglement =⇒ manybody system is in pure state,multipartite subsystems are in mixed state. How entangled?

◮ Entanglement (partial separability) for multipartite mixed states, theory forarbitrary number of parts:

[Szalay, arXiv:1503.06071 [quant-ph] (2015)]

◮ Classification: the classes shows a hierarchic structure,also related to LOCC convertibility.

◮ Quantificaiton: the most painless generalization ofEntanglement Entropy (for pure states) andEntanglement of Formation (for mixed states).

http://arxiv.org/abs/1503.06071

Tripartite systems – Class hierarchy of mixed states[Szalay,

arXiv:1503.06071[quan

t-ph](2015)]

P■

❂✮

P■■ � ❖★✭P■✁ ♥ ❢✂✄

☎✆

P■■■ � ❖✧✭P■■✁ ♥ ❢✂✄


Tripartite systems – Entanglement measures[Szalay,

arXiv:1503.06071[quan

t-ph](2015)] Multipartite Entanglement Entropy (pure π = |ψ〉〈ψ|)

E1|23,2|13,3|12(π) = min{E1|23(π),E2|13(π),E3|12(π)

}

Eb|ac,c|ab(π) = min{Eb|ac(π),Ec|ab(π)

}

Ea|bc(π) = 1/2(S(πa) + S(πbc)

)≡ S(πa)

E1|2|3(π) = 1/2(S(π1) + S(π2) + S(π3)

)

◮ indicator functions X

◮ entanglement monotone X

◮ multipartite monotone: quantity higher in the hierarchy takeslower value X

◮ meaning: info-geometric measure of correlation X

Multipartite Entanglement of Formation (mixed ): convex roofextensions


Geometrical optimization using DMRG

Barcza, Gebhard, Barford, Legeza (PRB 2010, PRB 2013)

Experimentalists prepare single crystals of diacetylene monomersMonomer = R-C≡C-C≡C-R

nBCMU family: R = (CH2)n-OCONH-CH2-COO-(CH2)3CH3

Polymerization induced by temperature or ultraviolet light:

◮ We have almost perfect chains: several micrometer long

◮ Disorder plays no role! (like ultracold atoms)

◮ The exciton state S lies at ES = 1.8 eV

◮ The band edge is located at Egap = 2.4 eV.

◮ The exciton binding energy of ∆sex = 0.6 eV is about 25% of the

single-particle gap. The Coulomb interaction is substantial and effective→failure of previous attempts(DFT,LDA).

◮ For excitation energies ~ω < 3 eV, only the π-electron system of thecarbon atoms on the chain is relevant.

◮ There is one pz electron per C atom, the system is half filled.

Experimental data

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R

nR

rt = 1.20 A, (1.239)

rd = 1.36 A, (1.373)

rs = 1.43 A, (1.425)

Energy/eV 3BCMU 4BCMU

EX11.5 1.4

EX21.7 1.6

ES = ∆sopt 1.896 1.810

EX32.0 1.9

Egap 2.482 2.378∆sex = Egap − ES 0.586 0.568

ET = ∆st 1.0 ± 0.05 0.95 ± 0.05

ET∗ = ∆st + ∆topt 2.36 ± 0.05 2.30 ± 0.05

∆topt 1.360 1.345

∆tex = Egap − ET 1.5 ± 0.05 1.4 ± 0.05

directly measured data, approximated

◮ measured data: electron absorption/ one-photon absorption

◮ approximated data: two-phonon absorbtion/non-radiative decay processes

Parametrization of PPP: Peierls–Hubbard-Ohno model

H = T+V+Hlattice

T = −∑

l ;σ

(t0+δel /2+δl/2)

(

c†l,σ cl+1,σ + h.c.

)

V = U

N∑

l=1

(

nl,↑ − 1

2

)(

nl,↓ − 1

2

)

+1

2

N∑

l 6=m=1

VOhnol,m (nl − 1) (nm − 1)

Hlattice =

N−1∑

l=1

δ2l4πt0λl

, δrl = rl−Rl = − δl2α

0 1 2 3 40

0.5

1

1.5

2

|z|/R

V(z

)/V

VOhno

VScreened

VOhnol,m =

V

ǫd

√

1 +

(

rl−rmR

)2

R =

√

14.397

V

Control parameters

t0, U, V , α

Peierls-distrortion

Geometrical optimization

According to the Hellmann–Feynman theorem, the negative derivatives of theenergy functional with respect to δrl define the force fields fl ,

− fl

2α= − δl

2πt0λl

− Γ + Fl [δp] , (2)

Fl [δp] =1

2

⟨∑

σ

(c+l,σ cl+1,σ + c

+l+1,σ cl,σ

)⟩

+1

2ǫd

N∑

i 6=j=1

βV /A2

[1 + β(|~ri −~rj | /A)2 ]3/2(3)

×[

xi,j∂xi,j∂δl

+ yi,j∂yi,j∂δl

]

〈(ni − 1) (nj − 1)〉 ,

where δp (p = 1, . . . ,N − 1) are the Peierls modulations of the electron transferamplitudes and xi,j = xi − xj , yi,j = yi − yj . The force fields fl are zero at theoptimal values δoptl for a chosen state |Ψ〉. The minimization of the force fieldshas to be done self-consistently.

Numerical simulations up to 102 electrons on 102 atoms

S=0 excitations

0 0.05 0.11.5

2

2.5

3

3.5

4

4.5

1/N

Ei

X1

X2a

X2b

S

1 1.5 2 2.50

1

2

3

4

5

6

Ei

dipo

le o

verla

p

ext.int.

Egap3BCMU

Energy/eV 3BCMU DMRG

EX1 1.5 1.74 [1.94]EX2 1.7 1.85 [1.94]a

ES = ∆sopt 1.896 2.00 [2.05]

EX3 2.0

S=1 excitations

spinflip density waves with q momen-tum: ǫsf(q) = csfq

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

10

12

1/N(N

/4

+1)(

EY

l−

ET)

l = 1

l = 2

l = 3

l = 4

(

N

4+ 1

)

(EYl− ET) = csf

lπ

d+

αl

N+ . . .

quantized quasi-momentum, ql = πl/[(N/4 + 1)d ]

MPS and TNS on kilo-processor architectures:

Nemes, Barcza, Nagy, Legeza, Szolgay, 2014

◮ The most time-dominant step of the diagonalization can be expressed as alist of dense matrix operations

◮ A smart hybrid CPU-GPU implementation, which exploits the power ofboth CPU and GPU and tolerates problems exceeding the GPU memorysize.

◮ A new CUDA kernel has been designed for asymmetric matrix-vectormultiplication to accelerate the diagonalization

◮ Example: Hubbard model on Intel Xeon E5-2640 2.5GH CPU + NVidiaK20 GPU:

1071 GFlops and×3.5 speedup is reached.(Theoretical maximum is1.17 TFlops)

tensor product factorization and entanglement properties ... · florian gebhard, reinhard m. noack,...

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