Basic ideas of DMRG & MPS

The most difficult problem while dealing with numerics for quantum many-body system is the exponential growth of the Hilbert space of the system with its size.

If we take for instance a spin-1/2 chain ofLsites, the corresponding Hilbert space would be of size2^L. Luckily, the first assumption on which DMRG relies is that one can always find a reduced Hilbert space, often extremely small if compared to the full one, which contains the relevant physics and which can be parametrized efficiently by the MPS formalism. In fact, as one would guess from the name of the formalism, quantum many-body states can be represented as products of matrices.

Also the operators we use to measure the relevant quantities can be re-interpreted in this matrix language, in this case we speak aboutmatrix product operators(MPO).

The ground state search is then performed, as we will see, as a variational procedure in the MPS space.

Before entering into details, let us introduce one of the most useful results of linear algebra: the so calledsingular value decomposition(SVD). SVD guarantees that for any arbitrary rectangular matrixM of dimensionsm×nthere exists a de-composition such that

M =U SV^†. (3.1)

S is diagonal with non-negative elementsSaa = sa calledsingular valuesand it has dimensions(min(m, n)×min(m, n)). Singular values are organized such that sa ≤ sa⁰ ifa > a⁰. The numberrof singular values which are non-zero gives the rank of M.Uhas dimensionsm×min(m, n)andU^†U =I, whileV^†has dimensions min(m, n)×nandV V^†=I. It is particularly important for the following to stress that it is possible to optimally approximate a matrix M of rankrby a matrix M’ of rankr⁰ < rproperly chosen. This approximation consists in setting to zero all but the firstr⁰singular values of S to zero (and to shrink the column dimension of U and the row dimension ofV^†accordingly).

3.1 DMRG & MPS

The basic idea of the DMRG techinque consists in splitting the Hilbert space in two blocks which we callleft(L) andright(R). Any pure state on a bipartite lattice LRcan be written as

|Ψi=X

are in the set of the orthonormal basis of the two blocks, of dimensions NLandNR, in which we have divided the system. The coefficientsCαβcan be read as the elements of a matrixC. As previously said, the size of each block basis grows exponentially with the lattice size. In order to make the computation accessible one needs at some point to reduce the size of the bases keeping onlyN < N_L, N_Rstates to approximate each of them.

The SVD procedure represents the basis of a very useful representation of quan-tum states of a bipartite systemLR, theSchmidt decomposition. It also provides a powerful optimization of the truncated bases. Let us now reconsider the state|Ψiin Eq. (3.2) and let’s perform an SVD of the matrixC. We get

|Ψi=X

wheresr=Srrare the elements on the diagonal of the matrixS. If this last sum is performed over only theN ≤min(N_L, N_R)positive and nonzero singular values, what we get is theSchmidt decompositionof our quantum state. This decomposition is related to the reduced density matrices by

ρ^L=TrR|Ψi hΨ|=X

are eigenvectors ofρ^L,R, ands²_rare the eigenvalues. An approximate description of the state|Ψican be obtained by truncating its Schmidt decomposition retaining only the states

r^L and

r^R

corresponding to the D largest valuess_r:

|Ψ⁰i ≈

3. METHODS

withs1 ≥ s2 ≥ s3 ≥ · · · ≥ 0. If normalization is desired, the retained singular values need to be rescaled. The accuracy of the approximation can be evaluated by thetruncation errorεDdefined as the difference in norm between the exact state and the approximated one using the optimized basis:

D= X

r>N

s²_r=k|Ψi − |Ψ⁰ik² (3.7) For a givenDthis error is minimized by the truncation procedure adopted in Eq. (3.6) and it clearly depends on the distribution of thes_r.

Let’s go back now to our chain ofLsites with d-dimensional local state spaces {σ_i}on each sitei = 1, . . . , L. It can be shown that any representative state of a spin-1/2 chain (and more in general of any quantum state on a lattice) can be written as:

|Ψi= X

σ₁,...,σ_L

cσ₁...σ_L|σ1. . . σLi. (3.8) Let’s see now how it can be represented exactly in a matrix product form. Additional details can be found in Ref. [21], here we follow the same procedure described there.

We start by reshaping the vectorcσ₁...σ_L with dimensiond^L into a matrix M with dimensions d×d^L−1 where we have reshaped back into a vector the product between matrices S and V^†, andr1 ≤ d. Now we can decompose the matrix U into a collection ofdrow Now this new matrixM can undergo a new SVD following the same idea as before to give we continue to SVD until the end of the chain, what we get is

cσ₁...σ_L = X

3.1 DMRG & MPS

where the last compact form implies the matrix product induced by the sums over the`iindexes. The quantum state can be now represented in the matrix product state form as:

|Ψi= X

σ1...σL

L^σ1L^σ2. . .L^σL−1L^σL|σ1. . . σ_Li. (3.14) These matricesLare such thatP

σ_iL^σi†L^σ=I and they are calledleft-normalized, and the corresponding MPSleft-canonical. The MPS formalism is quite versatile, therefore one can easily get for exampleright-canonicalMPS, if one starts the series of SVDs from the right, or evenmixed-canonicalif one starts from both ends of the chain. In this last case the decomposition gives:

|Ψi= X

σ₁...σ_L

L^σ1. . .L^σlSR^σl+1. . .R^σL|σ₁. . . σ_Li, (3.15)

with singular values at the bond(l+ 1, l). Within this notation the Schmidt decom-position presented above into two blocks (left block from 1 tol, right block from l+ 1toL) is easy to reintroduce. If we reconsider Eq. (3.3) one immediately notes the parallelism

r^R

= X

σ1...σ_l

(L^σ1. . .L^σl)|σ1. . . σli (3.16) r^L

= X

σl+1...σL

(R^σl+1. . .R^σL)|σl+1. . . σLi (3.17)

From a simple dimensional analysis it turns out immediately that in practical calcu-lations such decompositions cannot be performed exactly given the limited computa-tional resources. Luckily it turns out that in most cases the state can be nevertheless described accurately using matrices of reduced dimensions. If we consider for in-stance the mixed-canonical representation we have just introduced, it is possible to cut the spectra of the reduced density operators and to optimally approximate (in the 2-norm) the state following the same philosophy of Eq. (3.6). This argument can be generalized from the approximation incurred by a single truncation to that caused by L−1truncations, one at each bond. If we make no assumptions on the normalization of the matrices, we can write a general MPS for open boundary conditions as:

|Ψi= X

σ₁,...,σ_L

M^σ1M^σ2. . .M^σL−1M^σL|σ1. . . σLi. (3.18)