Matrix Product States Thomas E. Baker—August 17, 2015 The Hilbert space is large. Scanning through the entire space would be take longer than is reasonable. Writing down a good guess for the wavefunction can be a great way to get close enough to the right ground state. A very good guess for the ground state of a gapped system in one dimension is the matrix product state (MPS). DMRG and many tensor network methods rely on the MPS ansatz to represent ground states and low-lying excited states. Each bulk tensor of an MPS has three indices, @@A^{\sigma\_i}\_{a\_i a\_{i+1}}@@. The subscripts convey that indices connecting different bonds may vary in size. The size of a typical index @@a\_i@@ is known as the bond dimension of the MPS, and corresponds to the number of states kept in a DMRG calculation. The index @@\sigma_i@@ is the physical index. This has two values for a spin@@-1/2@@ chain. An MPS diagram for a five site system looks like

We will discuss some characteristics of MPSs and introduce some concepts that are essential for working with MPS in ITensor. ## Using the ITensor MPS Class To make an MPS, one typically uses the following code pattern: SpinHalf sites(5); auto psi = MPS(sites);//construct a random product state with indices //based on the SiteSet used (here SpinHalf) which automatically makes a random product-state MPS. Here we chose the Hilbert space to be spin 1/2 sites, but other types of sites are available (see [[the list of site sets on this page|classes]]). For example, to make a generic five-site MPS where each site has a local dimension (physical index dimension) of two: SiteSet sites(5,2); auto psi = MPS(sites); For read-only access to a particular tensor of the MPS, use the ``psi.A(j)`` method. This method is called "A" because it is very common in the MPS literature to use the letter A to denote MPS tensors. For example, if we want to print the fourth tensor of the above MPS, we could do Print(psi.A(4)); For read and write access to an MPS tensor, use the ``psi.Aref(j)`` method. This method is called "Aref" because it returns a reference to the j'th "A" tensor. Here is an example of using this method to multiply the third MPS tensor by a scalar: psi.Aref(3) *= 0.1827; Another way to modify the tensors of an MPS is to use the ``.setA`` method. ## Working with MPS Gauges The gauge of an MPS can be chosen to make computing local properties very efficient. The gauge used in standard DMRG calculations chooses a certain site to be the orthogonality center. All sites to the left and right of the orthogonality center will be left and right orthogonal, respectively. To set site 4 of the 5-site MPS above to be the orthogonality center, call psi.position(4); ITensor will compute a series of [[singular value decompositions|tutorials/SVD]] (in this case, only one) until the orthogonality center reaches site 4. The MPS diagram now looks like:

To illustrate one advantage of setting the MPS gauge, consider the diagram below where we have drawn purple squares to be left-normalized while yellow squares are right-normalized.

First call ``psi.position(2)`` to center the gauge at site 2 as in the above diagram. Then to compute the expectation value of the "Sz" operator on site 2, we can use the fact that in the diagram above all of the left-orthogonal and right-orthogonal tensors contract to the identity, so can be left out of the calculation (see [[an article on Correlation functions|tutorials/correlations]] for more details). So the only code needed to obtain the expectation value is: auto Mz2 = (dag(prime(psi.A(2),Site))*sites.op("Sz",2)*psi.A(2)).real(); where notice that we only accessed the tensor on site 2 and no other MPS tensors. We need not bother with the other sites since we know they contract to the identity.