DMRG and how it opens on to Tensor Network States (Part 1)

Quantum many-body systems are generically difficult to study due to the exponential number of degrees of freedom in terms of the number of particles, requiring high performance algorithms to access the universal physics. For a long time there was no robust way to deal with low energy physics of quantum many body systems. The foremost technique was to diagnolise the Hamiltonians and try to find the lowest energy eigenvector and it's eigenvalues but considering the exponential growth of states it was not feasible. There came many algorithms which tried to deal with quantum many body systems like mean field approach and cluster mean field theory but these were only good enough till there were low correlations in the system. Later a powerful approach called the Numerical Renormalization Group was invented which was based on the strong intuition that the ground state is composed of non interacting low energy states of the bipartition.

In the Numerical Renormalization Group Algorithm we try to put block sites together. We try to take a block, find it's eigenvalues and it's eigenvectors, gather the eigenstates with the lowest energy and create it's projector operator.

\newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\bra}[1]{\left\langle{#1}\right|}

$H = \sum_{i=1}^{d^n} v_i \ket{v_i} \bra{v_i}$

$P = \sum_{i=1}^{m} \ket{v_i} \bra{v_i}$

In the above equation d is the dimension of the spin and n is the number of sites. m is the number of the lowest energy eigenstates. Now using the above projective operator we create a projective Hamiltonian for the current block.

$H_N = P^\dagger H P$

Now we construct the Hamiltonian for 2 blocks which is a system of size 2n using the projected Hamiltonian of each bipartition and the interaction among them.

$H_{2N} = H_N \otimes I + I \otimes H_N$

We again follow the initial steps and continue this procedure until we converge at the desired number of sites.

Picture Credits : imsc.res.in

But unfortunately the NRG algorithm fails for most of the system and only works with the Kondo model. The reason being a large combination of high-energy eigenstates being needed to obtain a state of 2N size falsifying the assumptions on which NRG is based.

Here we introduce the DRMG (Density Matrix Renormalization Group) algorithm which is a great modification over the NRG where we work on slowing down the growth of the system size with a higher precision of the final state. The modification we choose is to cut off the bigger system in basis states of the density matrix rather than eigenstates or energy. Here we discusss the infinite system DMRG.

• For the DMRG algorithm we start with two blocks A and B which have D basis.
• We add a site next to both the blocks with dimesion d basis states and try to create a superblock out of it.
• We try to represent it in the density matrix notation so that we are able to do the partial trace to create new blocks A and B.
• Next we have a truncated D with reduced basis for both the blocks A and B(We will discuss about it later). We repeat the above steps to reach the ground state and the reduced Hilbert space for a larger size system.

Image credits : Adrian Feiguin , University of Wyoming

We have two questions now. What is the ground state of the superblock A..B and how do we find the truncated D-dimension Hilbert space for both the new blocks A and B.

Let the basis for the initial blocks A and B be $\ket{a}_A$ and $\ket{a}_B$ and the d-dimension basis for the added sites next to A and B be $\ket{\sigma}_A$ and $\ket{\sigma}_B$.

The resultant state of the superblock A..B can be written as

$\ket{\psi} = \sum_{a_A {\sigma}_A {\sigma}_B a_B} {\psi} \ket{a}_A \ket{\sigma}_A \ket{\sigma}_B \ket{a}_B$

which can be re written as $\ket{\psi} = \sum_{i_A,i_B} {\psi}_{i_A j_B} \ket{i}_A \ket{j}_B$

where $\ket{i}_A$ and $\ket{j}_B$ are the combined basis states for the new blocks A and B with the dimension (Dd).

The ground state of the combined superblock A..B can be measured by diagonalizing the Hamiltonian A..B in the basis $\ket{\psi}$ and finding the expectation value which minimizes the energy $E = \left\langle \psi \middle | H_{A..B} \middle| \psi \right\rangle$ This is how we can calculate the ground state energy for the superblock.

Now to the next question : How do we truncate the new Blocks A and B with the basis dimensions as Dd without having an exponential growth. The answer is to choose the D most relevant states in both the blocks. We inititate this by first taking a partial trace over density matrix of the superblock state A..B. ${\rho}_A = Tr_B\ket{\psi} \bra{\psi}$ The eigenvalues and the eigenvectors of the partial density matrix A can be calculated by diagonalization method. Next we choose the eigenstates with the maximum eigenvalues, in this way we are able to approximate the operator A with the eigenstates with the most significance in the similar way we do it in approximating a matrix using SVD. We do this because we want to approximate the ground state and these highest value eigenstates conribute mostly to the ground state.

Thus we are able to find the truncated D-dimension Hilbert space for the new blocks A and B and we can extend it to find the ground state and reduced Hilbert space for a very large size.

In the next articles in the series we will be looking deep into the truncation process of the DMRG and how we link the DMRG to Tensor Network states.