1. Matrix Product States

1.1. Construction of MPS

Any quantum many-body states of N sites can be written as

(1)\[| \psi \rangle = \sum\limits_{\sigma_1, \cdots. \sigma_N} c_{\sigma_1, \cdots, \sigma_N} | \sigma_1 \rangle \otimes | \sigma_2 \rangle \otimes \cdots \otimes | \sigma_N \rangle\]

where \(\sigma_i\) denotes the indices of electronic state of each sites. Each \(\sigma_i\) has \(d\) choices, and we call \(d\) as Physical Dimension, and \(i\) are physical indices.

Exact singular value decomposition will give N blocks for each site, and the many-body state can be written as:

(2)\[\sum\limits_{a_1, \cdots, a_{N-1}} \sum\limits_{\sigma_1, \cdots, \sigma_{N}} M_{1, a_1}^{\sigma_1} | \sigma_1 \rangle \otimes M_{a_1, a_2}^{\sigma_2} | \sigma_2 \rangle \otimes \cdots \otimes M_{a_{N-1}, 1}^{\sigma_N} | \sigma_N \rangle\]

If SVD takes place in sequence, the dimension of each \(M\) matrix will be at least \(d, d^2, \cdots, d^{L/2}\), which increases exponentially! So approximation should be taken that the dimension of each \(M\) are truncated to a fixed value \(D\), the bond dimension or virtual dimension. Correspondetly, :math:`a_1, a_2 cdots ` are called virtual indices.

Generally, an arbitrary MPS can be generated randomly.

1.2. Gauge Freedom and Canonicalization

If we insert arbitrary \(X X^{-1}\) between two \(M\), then the product \(MX\) and \(X^{-1}M\) lead to a different construction of MPS which gives the same result of \(| \psi \rangle\). Hence to remove the gauge freedom, we can add restrictions such as left-canonical or right-canonical:

• Left Canonical Gauge:

  \[\sum\limits_\sigma A^{\sigma\dagger} A^{\sigma} = I\]

• Right Canonical Gauge:

  \[\sum\limits_\sigma B^{\sigma} B^{\sigma \dagger} = I\]

Then using SVD, the MPS can be written as:

\[| \Psi \rangle = \sum\limits_\sigma A^{\sigma_1} \cdots A^{\sigma_l} M^{\sigma_{l+1}} B^{\sigma_{l+2}} \cdots B^{\sigma_N} | \sigma_1 \cdots \sigma_N \rangle\]

in which the freedom can be restricted only to the choice of \(l\). Also this gauge restriction will bring out numerical convenience.

1.3. Matrix Product Operator (MPO)

For a general operator \(\hat O\) on the hilbert space of \(N\) sites, it can be written as:

\[\begin{split}\hat O = \sum\limits_{\sigma_1, \cdots, \sigma_N \\ \sigma'_1, \cdots, \sigma'_N} c^{(\sigma_1, \cdots, \sigma_N)(\sigma'_1, \cdots \sigma'_N)} |\sigma_1, \cdots \sigma_N \rangle \langle \sigma'_1, \cdots, \sigma'_N |\end{split}\]

As the construction of MPS, MPO can also be written as the form:

(3)\[\hat O = \sum\limits_{\sigma, \sigma'} W^{\sigma_1, \sigma'_1} \cdots W^{\sigma_L, \sigma'_L} | \sigma \rangle \langle \sigma' |\]

in which the \(W^{\sigma \sigma'}\) is a matrix with two bond dimensions. The representation of MPO can be simplified as:

(4)\[\hat O = \hat W^{[1]} \hat W^{[2]} \cdots \hat W^{[N]}\]

in which each matrix element of operator matrix \(\hat W^{[l]}\) is defined as:

\[\hat W^{[l]}_{bb'} = \sum\limits_{\sigma\sigma'}{W^{\sigma\sigma'}_{bb'}} | \sigma_l \rangle \langle \sigma_l' |\]

For the neighbor-interaction Hamiltonians, the operator matrices are usually sparse. For instance, in the Heisenberg model:

\[\hat H = - J_x \sum \hat X_j \hat X_{j+1} - J_Y \sum \hat Y_j \hat Y_{j+1} - J_Z \sum \hat Z_j \hat Z_{j+1} - h \sum Z_j\]

then for the MPO matrix not on the left or right

\[\begin{split}\hat W = \begin{bmatrix} \hat I \\ \hat X \\ \hat Y \\ \hat Z \\ -h\hat Z & - J_X \hat X & J_Y \hat Y & - \hat J_Z \hat Z & I \end{bmatrix}\end{split}\]

and for left and right:

\[ \begin{align}\begin{aligned}\begin{split}\hat W_{L} = \begin{bmatrix} -h \hat Z & -J_X \hat X & - J_Y \hat Y & - J_Z \hat Z & \hat I \end{bmatrix} \\\end{split}\\\begin{split}\hat W_{R} = \begin{bmatrix} \hat I \\ \hat X \\ \hat Y \\ \hat Z \\ - h \hat Z \end{bmatrix}\end{split}\end{aligned}\end{align} \]

1.4. Contraction and Property Calculation

References:

Reviews: [1] , [2] , [3]