Matrix product states and correlation functions

Matrix product states (MPS) and density-matrix renormalization group theory (DMRG, [White, 1992, 1993; Östlund and Rommer, 1995; Schollwöck, 2005; Schollwöck, 2011]¹) have been shown to be a powerful tools for quantum chemistry [Chan and Sharma, 2011; Marti and Reiher, 2011; Kurashige, 2014]. For the discussion of matrix product states, we follow the discussions in [Eisert, 2013] and [Schuch, 2013]. We focus on the case of open boundary conditions. We assume that the system consists of L sites with a local basis |0i = |00i, | ↑i = |10i, | ↓i = |01i and | ↑↓i = |11i. Thus, the local dimen-sion d is assumed to be equal to four and the total number of one-particle basis states is N_χ = 2L.

The central idea of matrix product states is to write the coefficientsc_j1,...,jL of a many-particle wave function

|Ψi= ^X

j1,...,jL∈{0,↑,↓,↑↓}

c_j1,...,jL|j₁, ..., j_Li (8.28) as a product [Fannes et al., 1992; Östlund and Rommer, 1995; Rommer and Östlund, 1997]

cj1,...,j_L = ^Xd1

i1=1 d2

X

i2=1

...

dL−1

X

iL−1=1

M_i^(1),j₁ ¹M_i^(2),j_1,i2²· · ·M_i^(L−1),j_L−2,iL−1^L−1M_i^(L),j_L−1^L (8.29)

=M^(1),j1M^(2),j2· · ·M^(L),jL (8.30)

with the vectors M^(1),j1 ∈ C^1×d1 and M^(L),jL ∈C^dL−1×1 as well as the matrices M^(i),ji ∈ C^di×di+1 for 1< i < L. d_i are called bond dimensions. The maximal bond dimensions are d1 =d, d2 =d², d3 =d³, ... dL−2 =d² dL−1 =d, (8.31) whered is the local dimension. The maximal bond dimensiond_l of the bond between site l and l+ 1 is the number of Schmidt values q_j in the Schmidt-decomposition

|Ψi=

d_l

X

k=1

qk|ΨA,ki ⊗ |ΨB,ki (8.32)

1A more complete list of references can be found in the reviews [Schollwöck, 2005], [Verstraete et al., 2008] and [Schollwöck, 2011].

with

|ΨA,ki= ^X

j1,...,jl

cA,k,j1,...,j_l|j1, ..., jli (8.33)

|ΨB,ki= ^X

jl+1,...,jL

c_{B,k,jl+1,...,jL}|j_l+1, ..., j_Li. (8.34) The von Neumann entropy of entanglement between the subsystem A consisting of the sites 1, ..., l and the subsystem B consisting of the sites l+ 1, ..., L is defined as

SA|B =−Tr (ρ_Alog2ρ_A), (8.35) whereρ_A = TrB|ΨihΨ|is the reduced density matrix and TrB is the partial trace over the basis of the system B. The reduced density matrix ρA can be written with the Schmidt decomposition of the state |Ψi as

ρ_A=^X

k

q_k²|ΨA,kihΨA,k| (8.36) and the entanglement entropy as

SA|B=−^X

k

q_k²log2q²_k. (8.37)

Thus, the entanglement entropySA|Bis directly related to the bond dimensions of a matrix product state. A matrix product state with the maximal bond dimensions can represent any many-particle wave function. Matrix product states can efficiently represent states with small numbers of non-zero Schmidt values q_k because this situation corresponds to a small bond dimension and hence a small size of matrices M^(i),ji.

A matrix product state is a representative of the much more general class of tensor wave functions. A graphical representation of tensors and operations drastically simplifies the notation. We define a scalar as a box

x= ∈C (8.38)

and a vector as a box with a leg

~x= ∈C^D. (8.39)

A vector is a tensor of rank one and has one leg. An n-th order tensor has n legs. The leg can also have an explicitly given index j to select a specific element, i.e.

~x_j = . (8.40)

A matrix as a tensor of rank two is represented as

A= ∈C^D×D. (8.41)

A contraction of an index is a summation over all possible values an index (a shared leg) can take. Thus, a matrix-matrix product can be written as

(A·B)ab =^X

j

A_ajB_jb = = =C_ab. (8.42)

8.8. Adaptive-cluster transformation for reduction of entanglement

The coefficients c_j1,...,jL represent a tensor of rankL and can be drawn as

c_j1,...,jL = . (8.43)

We have oriented the legs vertically to indicate that the indexes correspond to local basis states. These indexes are called physical indexes. The matrix-product-state representa-tion in Eq. (8.29) can be drawn as

c_j1,...,jL = . (8.44)

The indexes that correspond to horizontal legs are called bond indexes.

For a variational minimization of the total energy, i.e., the expectation value of the Hamiltonian ˆH, we also need to be able to compute expectation values hΨ|O|ˆ Ψiof opera-tors ˆO. We first discuss the case where the operator ˆO is supported on neighboring sites, l and l+ 1. With this restriction, ˆO has the general form

Oˆ = ^Xd

jl=1 d

X

jl+1=1 d

X

kl=1 d

X

kl+1=1

O_k^jl_l^,j_,k^j+1_l+1|jj, jl+1ihkl, kl+1|. (8.45) The tensor O can be represented graphically as a tensor with four physical indexes as

O_k^jl,jj+1

l,kl+1 = . (8.46)

Matrix product operators will be discussed in detail in section 8.8.3. The expectation value hΨ|O|ˆ Ψi can be drawn in the graphical representation as

hΨ|O|ˆ Ψi= . (8.47)

As every shared leg represents a summation, a naive evaluation of the contractions would result in an exponential complexity. Thus, a more efficient way of contracting the network is needed. For this purpose, the tensor network is contracted starting from the rightmost and the leftmost end. At first, the two ends are contracted. The tensor

L_α,β = =^Xd

j=1

M_α^(1),jM_β^(1),j∗ = (8.48)

and the tensor

Rα,β = =^Xd

j=1

M_α^(L),jM_β^(L),j∗ = (8.49)

are defined and represent the ends of the network. The analogous contraction of the physical index can be performed for every site. This leads to the definition of the transfer operator

E^(k)

α,β,γ,δ = =^Xd

j=1

M_α,β^(k),j(M_γ,δ^(k),j)^∗ = . (8.50)

Similarly, a two-site transfer operator for the operator ˆO can be defined as

(E_O)_α,β,γ,δ = = . (8.51)

With these definitions the expectation value can be written as

hΨ|O|ˆ Ψi=LE⁽²⁾· · ·E^(l−1)E_OE^(l+2)· · ·E^(L−1)R. (8.52) To estimate the computational complexity, we assume, that the bond dimensions d_i are bounded by a maximal bond dimension D. Rcan be viewed as anD²-dimensional vector, because it has two bond indexes. Therefore, E^(L−1)R is a matrix-vector multiplication with a computational complexity ∈ O(D⁴) and hence the computation of the expectation value has a complexity ∈ O(LD⁴). The transfer operator can be used to discuss the spatial decay of correlations functions of matrix product states. Following [Schuch, 2013;

Eisert, 2013], we take an infinite translational invariant matrix product state

c_j1,...,jL = TrM^j1· · ·M^jL = . (8.53)

and consider the thermodynamic limit L→ ∞. The transfer operator is E_α,β,γ,δ =^Xd

j=1

M_α,β^j (M_γ,δ^j )^∗ = (E_γ,δ,α,β)^∗ (8.54) and site-independent. We estimate the correlation functionhOˆaOˆbi, where ˆOais supported on site a and ˆO_b is supported on site b. We assume a < b. The corresponding transfer operators are

(E_a)_α,β,γ,δ = = (8.55)

(E_b)_α,β,γ,δ = = . (8.56)

8.8. Adaptive-cluster transformation for reduction of entanglement

Thus, the expectation value hOˆ_aOˆ_bi can be written as

hOˆ_aOˆ_bi= TrE_aE^b−a−1E_bE^L−1−b+a

TrE^L . (8.57)

The transfer operator E can be represented as a D² ×D² hermitian matrix with the eigenvalues λ₁ ≥λ₂ ≥...≥λ_D². E can be written in its eigendecomposition as

We assume that the maximal eigenvalue λ1 is unique. E can be chosen such that the maximal eigenvalue λ₁ is equal to one and hence

m→∞lim E^m =|e₁ihe₁|. (8.59)

Thus, in the thermodynamic limitL→ ∞and for finiteb−a, we obtain for the correlation function in Eq. (8.57) the form

hOˆ_aOˆ_bi=he₁|E_aE^b−a−1E_b|e₁i (8.60)

The second terms governs the spatial decay of the correlation function. The correlation length ζ that quantifies the decay for |b−a| 1 of the correlation function as

|hOˆ_aOˆ_bi − hOˆ_aihOˆ_bi| ∝e^{−ζ−1|b−a|} (8.64) is given by the second eigenvalue as

ζ⁻¹ =−log|λ₂|. (8.65)

This shows that correlation functions of matrix product states decay exponentially in case the largest eigenvalue of the transfer operator is unique. This fact also indicates that matrix product states can efficiently represent ground states of gapped models because in gapped models the correlations functions decay exponentially. In case the largest eigenvalue of the transfer operator is not unique, i.e., λ1 =λ2, we have

m→∞lim E^m =|e₁ihe₁|+|e₂ihe₂| (8.66)

and for the correlation function Thus, the correlation function has a constant long-range contribution Olong−range that is independent of the distance of a and b. The spatial decay for large |b −a| is again exponential

|hOˆ_aOˆ_bi − hOˆ_aihOˆ_bi −O_long−range| ∝e^{−ζ−1|b−a|} (8.70) and the correlations length ζ is determined by the third eigenvalue λ₃ as

ζ⁻¹ =−log|λ₃|. (8.71)

The generalization to n-times degeneracy of the largest eigenvalue, i.e. λ₁ = ...= λ_n, is straight forward and also leads to constant long-range contribution. Thus, matrix product states can describe exponentially decaying and constant long-range correlation functions efficiently. However, matrix product states cannot describe algebraically decaying corre-lation functions, i.e.

|hOˆ_aOˆ_bi − hOˆ_aihOˆ_bi| ∝ |b−a|^−α, (8.72) efficiently.

Im Dokument New methods for the ab-initio simulation of correlated systems (Seite 137-142)