3 Figure 2: (Color online) (a). The A [ σ$_{k}$ ] ij has three indices. σ$_{k}$ is the physical index run from 1 to d = 2 for SIAM. i ( j ) is the visual index runs from 1 to χ$_{k}$ ( χ$_{k}$$_{+1}$ ). They control the precision of the VMPS calculation. (b). An operator acts on a single site (c). A MPS where all of the connected bonds have been contracted. Figure 3: (Color online) (a). The expectation value of the operator H over the MPS. (b). Tracing out all the indices on the sites other than k gives the effective Hamiltonian H$_{eff}$ on the k -th site. (c). The minimization problem on site k is equivalent to the eigen-problem of the effective Hamiltonian JW sign, one could unfold the SIAM chain into two spin- 1 2 XY chains. They are coupled at the end point by the longitude Ising coupling due to the Comlomb interaction. The onsite energy of each site is mapped into the local magnetic filed along the z -direction. Since the hopping is between the nearest neighbors, the JWT sign does not shows up in the calculation of the ground state. A simple division of the Hamiltonian into terms acting on the odd and even bonds are possible, see Sec.III. In terms of 1 / 2 spins, the Hamiltonian is where $_{s}$± i = σ$_{x}$ ± iσ$_{y}$ 2 are the ladder operators for spin one half. s and τ denote the spin up/ down fermion operators. The matrix product state (MPS) | ψ$_{MP S}$ 〉 = ∑ σ$_{1}$,σ$_{2}$,...,σ$_{N}$ Tr( A [ σ$_{1}$ ] 1 A [ σ$_{2}$ ] 2 ...A [ σ$_{N}$ ] N ) | σ$_{1}$, σ$_{2}$, ..., σ$_{N}$ 〉 . σ$_{k}$ = 1 , ..., d is the physical index, where d = 2 is the dimen- sion of the local Hilbert space. A [ σ$_{k}$ ] is a matrix with the visual dimension χ$_{k}$ × χ$_{k}$$_{+1}$ . There is an efficient scheme based on the transfer matrix to handle them in low dimensions, i.e. to calculate the overlap between two MPSs, and the expectation value of an operator over two MPSs, the operation of a local operator on a given MPS. They are shown schematically in Fig.3 and see Verstraete et al. 25 for a reference. We minimize the expectation value Where H = ∑ m h m can be parallelized over each term in the Hamiltonian. 〈 ψ$_{MP S}$ | H$_{SIAM}$ | ψ$_{MP S}$ 〉 within the space spanned by the normalized MPS. Such a problem is solved very efficiently by the alternating least squares scheme, in which we perform the optimization of the matrices site by site. In each step, we fix all but the matrix on the current site, see Figure 3. By contract all of these indices we define the effective Hamiltonian on the current site. The minimization problem becomes quadratic and is equivalent to an eigen-problem of the size χ$_{k}$ × χ$_{k}$$_{+1}$ × d . Since only the ground state is needed, we solved it by the Lanczos method. The key step in the Lanczos step is the computation of the matrix-vector multiplication: ∑ l,σ,r $^{H$_{eff}$}${ l $^{′}$,σ $^{′}$,r$^{′}$}{ l,σ,r } A [ σ ] l,r = ∑ m h m { l $^{′}$,σ $^{′}$,r$^{′}$}{ l,σ,r } A [ σ ] l,r H$_{SIAM}$ = H$_{imp}$ + H$_{bath}$ + H$_{hyb}$ H$_{imp}$ = − µ ( s + 0 $_{s}$− 0 + τ + 0 $_{τ}$− 0 ) + U ( s + 0 $_{s}$− $_{0}$)( τ + 0 $_{τ}$− 0 ) H$_{bath}$ = N$_{bath}$ ∑ i =1 ε$_{i}$ ( s + i $_{s}$− i + τ + i $_{τ}$− i ) + N$_{bath}$ − 1 ∑ i =1 γ$_{i}$$_{+1}$ ( s + i $_{s}$− i $_{+1}$+ τ + i $_{τ}$− i $_{+1}$+ h.c. ) H$_{hyb}$ = γ$_{1}$ ( s + 0 $_{s}$− 1 + τ + 0 $_{τ}$− 1 + h.c. ) (2) H k A 1 A2 A3 A4 A k σ i j ( a ) ( b ) ( c ) A 1 A 2 A 3 A 4 H A 1* A 2* A 3* A 4* Ak Heff = λ Ak Ak Heff Ak* Lk Rk Hk Ak Ak* = A k Hk A k* Rk Rk-1 = ( a ) ( b ) ( c )