I have just started to read about DMRG and MPS.

It is said that in case of simple 1D chain with spins states $|\uparrow\rangle$; $|\downarrow\rangle$ and any state in the complete Hilbert space of such a system could be written as :

$|\Psi\rangle=\sum\limits_{i_1,...,i_n} C^{i_1,...,i_n}|i_1,...,i_n\rangle$

Where each index runs over the local basis at each site. This presents a complexity of the order $2^N$ and by writing it in the form of matrix product states we would making a simplification a mean field simplification and reduce the complexity to $2N$.

$C^{i_1,...,i_n}=C^{i_1}C^{i_2}...C^{i_n}$

EDIT:

A General case:

Suppose I have Heisenberg chain kind of a 1D Lattice problem with N sites, where each site lives on a Hilbert space of dimensionality D. Given the fact that the dimensionality of Hilbert space of the entire system scales exponentially as $D^N$ because of the entanglement, it becomes a very complex problem to solve.

Questions:

1. How does writing the system in terms of MPS reduce the complexity of the problem and bring it down to $ND$ ?
2. What is the link between using MPS ansatz and mean field approximation?

I have just started to read about DMRG and MPS.

It is said that in case of simple 1D chain with spins states $|\uparrow\rangle$; $|\downarrow\rangle$ and any state in the complete Hilbert space of such a system could be written as :

$|\Psi\rangle=\sum\limits_{i_1,...,i_n} C^{i_1,...,i_n}|i_1,...,i_n\rangle$

Where each index runs over the local basis at each site. This presents a complexity of the order $2^N$ and by writing it in the form of matrix product states we would making a simplification a mean field simplification and reduce the complexity to $2N$.

$C^{i_1,...,i_n}=C^{i_1}C^{i_2}...C^{i_n}$

EDIT:

A General case:

Suppose I have Heisenberg chain kind of a 1D Lattice problem with N sites, where each site lives on a Hilbert space of dimensionality D. Given the fact that the dimensionality of Hilbert space of the entire system scales exponentially as $D^N$ because of the entanglement, it becomes a very complex problem to solve.

Questions:

1. How does writing the system in terms of MPS reduce the complexity of the problem and bring it down to $ND$ ?
2. What is the link between using MPS ansatz and mean field approximation?

I have just started to read about DMRG and MPS.

It is said that in case of simple 1D chain with spins states $|\uparrow\rangle$; $|\downarrow\rangle$ and any state in the complete Hilbert space of such a system could be written as :

$|\Psi\rangle=\sum\limits_{i_1,...,i_n} C^{i_1,...,i_n}|i_1,...,i_n\rangle$

Where each index runs over the local basis at each site. This presents a complexity of the order $2^N$ and by writing it in the form of matrix product states we would making a simplification a mean field simplification and reduce the complexity to $2N$.

$C^{i_1,...,i_n}=C^{i_1}C^{i_2}...C^{i_n}$

How does the complexity come down? And what is the link between MPS ansatz and mean-field theories?

I have just started to read about DMRG and MPS.

It is said that in case of simple 1D chain with spins states $|\uparrow\rangle$; $|\downarrow\rangle$ and any state in the complete Hilbert space of such a system could be written as :

$|\Psi\rangle=\sum\limits_{i_1,...,i_n} C^{i_1,...,i_n}|i_1,...,i_n\rangle$

Where each index runs over the local basis at each site. This presents a complexity of the order $2^N$ and by writing it in the form of matrix product states we would making a simplification a mean field simplification and reduce the complexity to $2N$.

$C^{i_1,...,i_n}=C^{i_1}C^{i_2}...C^{i_n}$

EDIT:

A General case:

Suppose I have Heisenberg chain kind of a 1D Lattice problem with N sites, where each site lives on a Hilbert space of dimensionality D. Given the fact that the dimensionality of Hilbert space of the entire system scales exponentially as $D^N$ because of the entanglement, it becomes a very complex problem to solve.

Questions:

1. How does writing the system in terms of MPS reduce the complexity of the problem and bring it down to $ND$ ?
2. What is the link between using MPS ansatz and mean field approximation?