The Density-Matrix Renormalisation Group, Disorder and
the Heisenberg Chain
A.M.Goldsborough and R. A. Römer
1
2
Department of Physics, University of Warwick, Coventry, UK
2
Department of Physics and Centre for Scientific Computing, University of Warwick, Coventry, UK
1
Abstract
Matrix Product States (MPS)
We have developed a density-matrix renormalisation group (DMRG) program to
• Matrix Product States (MPS) provide the mathematical underpinning to DMRG and
calculate the ground state and first excited states of a spin-1/2, one-dimensional
Heisenberg lattice. The program can model the XXX and XXZ Hamiltonians, has
Results
• The model under consideration is the spin-1/2 Heisenberg Hamiltonian:
enables far greater control of the algorithm.
• An MPS [2] has the form:
the ability to vary parameters such as interaction strength, Z-strength, external
magnetic field and chain length. Disorder is introduced into the system via a
disordered on-site potential and a disordered inter-particle interaction strength.
Using the program we can calculate the properties of systems with much greater
Where Ji is the inter-site interaction strength, α is the relative Z strength, hi is the
• In graphical notation:
external field strength, S± are spin raising and lowering operators defined as
length than is possible with an exact diagonalisation routine and to a high
accuracy.
where Sx,y,z are spin-1/2 operators in the form of Pauli matrices.
• Accuracy of the DMRG algorithm for the XXX model (Ji = 1, α =1, hi = 0) for 14
Motivation
• The norm is:
sites compared to an exact diagonalisation routine:
• The size of the Hilbert space of quantum spin systems scales exponentially with
chain length L.
• Exact diagonalisation (ED) techniques can be used for a few tens of states for a
spin-1/2 Heisenberg chain.
• The density-matrix renormalisation group (DMRG) algorithm allows the
• A matrix product operator (MPO) has the form:
calculation of low energy properties of a chain to high accuracy for many
thousands or more sites with far less computational cost.
• Using matrix product states and tensor networks, simulation of time evolution and
higher dimensional lattices is now possible.
• The anisotropic spin-1/2 Heisenberg Hamiltonian (XXZ) is currently under study,
but in principle the DMRG techniques can be used on any lattice Hamiltonian.
DMRG
• Therefore an expectation value is:
• Execution time as a function of chain length for the XXX model (Ji = 1, α =1, hi =
0) for m = 20:
The DMRG algorithm [1] is split into two sections:
•
The Infinite Algorithm
1. Start with a left and right block of size l where a matrix of order 2(l+2) (in the
spin-1/2 case) is diagonalisable. Insert two sites in the middle:
2. Form the superblock from L, R and the two sites:
MPS DMRG
• With MPS the truncation of the basis comes from setting a maximum size of the
indices ai in the MPS.
3. Diagonalise to find the target (ground) state energy and eigenvector.
4. Create the density matrix and diagonalise, keeping the m eigenvectors
• Then the finite DMRG algorithm with MPS is as follows:
1. Identify the active site (dotted), where we are using a Hamiltonian MPO.
• Execution time as a function of the number of states kept m for XXX (Ji = 1, α =1,
hi = 0) for 20 sites.
corresponding to eigenvalues of highest magnitude.
5. Build new left and right blocks that each include one of the centre sites and
use the m eigenvectors to project them into a new basis, truncating their size
to m×m.
6. Insert two sites in the middle and repeat the algorithm from step 2 until the
desired chain length is reached.
In this manner a chain of any length can be approximated by a Hamiltonian
2. Build left and right blocks by contracting the indices of the tensors to the left
and right respectively.
matrix of size (2m+2)×(2m+2).
•
The Finite Algorithm
This keeps the chain length constant but increases the accuracy of the
approximation.
1. For a chain the desired length ld. Start with the left block size ld /2 , right
Future Work
block size ld /2 – 2 and two sites in between:
3. Contract the left and right blocks with the MPO of the active site.
• Implement a DMRG for the Hubbard model.
• Simulate two dimensional systems using projected entangled pair states (PEPS) [3].
• Analyse the effect of disorder for one and two dimensional systems.
2. Repeat steps 2 to 5 of the finite algorithm.
3. Increase the size of the left block at the expense of the right in this way until
the right block is size 1. This is a right sweep.
Acknowledgements
I would like to thank the following people and organisations:
• EPSRC for financial support.
4. Now grow the right block at the expense of the left block until the left block
is size 1. This is a left sweep.
4. Reshape this tensor into a matrix. This is the equivalent of the superblock
Hamiltonian of the standard DMRG algorithm.
5. Diagonalise the matrix to find the target (ground) state energy and
• The organisers and participants of the Networking tensor networks: many-body
systems and simulations conference at the Centro de Ciencias de Benasque Pedro
Pascual for valuable discussions regarding the MPS formalism of DMRG.
eigenvector.
6. Reshape the eigenvector to an MPS tensor and use this as the new active site
tensor.
7. Move one site to the right and repeat steps 2-6.
5. Sweep back and forth across the chain until the target state has converged.
8. In this way sweep back and forth across the chain as before until the energy
has converged.
Contact Details
• Email: [email protected]
• Website: http://go.warwick.ac.uk/ep/pg/phrfbk
References
[1] R.M. Noack and S.R. White. “The Density Matrix Renormalization Group” in “Density Matrix Renormalization: A New Numerical Method in Physics”, Eds. I. Peschel, X. Wang, M. Kaulke, and K. Hallberg, Springer Verlag, Berlin, June 1999.
[2] U. Schollwöck. “The Density-Matrix Renormalization Group in the Age of Matrix Product States”. Annals of Physics 326(1): 96–192 (2011).
[3] F. Verstraete, V. Murg and J.I. Cirac. “Matrix Product States, Projected Entangled Pair States, and Variational Renormalization Group Methods for Quantum Spin Systems”. Advances in Physics 57(2): 143-224 (2008).