FiniteMPS.jl

This package provides the basic MPS operations and also some high-level algorithms.

I write this package aim to study some heavy quantum many-body problems (e.g. high-$T_c$ superconductivity, typical bond dimension $D \sim 10^4$ ), therefore the performance is the the most important metric.

• Non-abelian symmetries (e.g. $U_1\times SU_2$ symmetry for spin-1/2 fermions) can significantly accelerate the computation, therefore we choose TensorKit.jl as the backend to perform the basic tensor operations.
• The hamiltonian MPO can be represented as a sparse one with abstract horizontal bonds so that the interaction terms can be distributed to different threads or workers, together with nested multi-threaded BLAS (with the help of MKL.jl).
• The local operators are classified and wrapped as parametric types for further optimizations via multiple dispatch.
• Controlled bond expansion (CBE) techniques are implemented (modified for compatibility with parallelism) to reduce the complexity, e.g. $O(D^3d^2)$ to $O(D^3d)$ for DMRG and TDVP.

• The MPS-level operations are separated with the lower tensor-level operations so modifying codes with different symmetries, lattices and models is quite simple.
• We provide a hamiltonian generator via automata (like OpSum in ITensors.jl) so that the interactions can be added in a simple and general way, details please see the tutorials.
• We also provide some similar interfaces so that you can measure the observales and correlations conveniently. Note these computations can also be multi-threaded.

Our group works on purification-based finite-temperature simulations therefore we provide more MPO supports than other MPS packages.

• Almost all methods for MPS are also implemented for MPO.
• Some finite-temperature algorithms such as series-expansion thermal tensor network (SETTN) and tangent space tensor renormalization group (tanTRG) are provided.

Now this package contains

• DMRG for searching ground states.
• SETTN and tanTRG for finite-temperature simulations. Both thermal quantites and observables can be obtained.
• Ground state and finite-temperature dynamics can be studied with real-time evolution based on TDVP. Note this is still under development.

To install this package, you can press "]" in REPL and type

pkg> add FiniteMPS

• Only multi-threading is stable in current version, please make sure Julia is started with multiple threads. It will throw a warning to remind you when loading FiniteMPS with a single thread.

  TODO: I will add single-thread mode and try to support multi-processing in the future.

• If the linear algebra backend is OpenBlas, nested multi-threading is forbidden (a related discussion here). So just close the parallelism of BLAS via

  BLAS.set_num_threads(1)

• If using MKL as the linear algebra backend, you can set the number of blas threads similarly and nested multi-threading can be supported in this case. Note MKL is invalid for some cpus and you should also close the BLAS parallelism in such case. You can follow MKL.jl to check if MKL is loaded successfully or not.

  [!WARNING] Warning: Please make sure the total threads number is not larger than the cpu cores (physical, without hyper-threading). Otherwise, the performance will become much worse due to confliction. You can estimate the total threads via

  Threads.nthreads() * BLAS.get_num_threads()

The first step is to choose a local space, i.e. the local 1-site Hilbert space and some 1-site operators according to the model and symmetry used. We predefine some in folder LocalSpace and you can use the documentation to see which operators are provided in it, e.g

help?> U₁Spin

[!NOTE] Note: It is impossible to predefine all local spaces so you must write a new one if that you need is not in it. Moreover, some no method errors may occur when using a new defined one due to the multiple-dispatch-based implementations may not cover all kinds of operators. Please feel free to submit an issue if this happens and I will deal with it as soon as possible.

After that, you need to tell the program the interactions in the hamiltonian. We use a tree-like struct InteractionTree to store them and provide an interface named addIntr! to add terms. The following is a demo code to generate the hamiltonian of an AFM Heisenberg chain.

# generate hamiltonian MPO
L = 32
Root = InteractionTreeNode()
for i in 1:L-1
     # SzSz
     addIntr!(Root,
          (U₁Spin.Sz, U₁Spin.Sz),
          (i, i+1),
          1.0;
          name = (:Sz, :Sz))
     # (S+S- + h.c.)/2
     addIntr!(Root, 
          U₁Spin.S₊₋,
          (i, i+1),
          0.5;
          name = (:Sp, :Sm))
     addIntr!(Root,
          U₁Spin.S₋₊,
          (i, i+1),
          0.5;
          name = (:Sm, :Sp))
end
Ham = AutomataMPO(Root)

Although the FiniteMPS package is written in an object-oriented style, a procedure-oriented script is still required for a concrete question. You can freely design it with the following sweep methods such as

• DMRGSweep2!, DMRGSweep1!
  The standard 2-site/1-site DMRG sweep.
• TDVPSweep2!, TDVPSweep1!
  The standard 2-site/1-site TDVP sweep.

being the usually used building blocks.

Note a trilayer tensor network (namely environment), e.g. $\langle \Psi|H|\Psi\rangle$ where $\ket{\Psi}$ the MPS and $H$ the hamiltonian MPO, instead of the state $\ket\Psi$, should be passed in. The following code will initialize a random MPS and then generate the environment.

# initialize
Ψ = randMPS(L, U₁Spin.pspace, Rep[U₁](i => 1 for i in -1:1//2:1))
Env = Environment(Ψ', Ham, Ψ)

More detailed usages of the above functions can be found in the documentation.

[!NOTE] Note: Symmetries may limit the choice of auxiliary space, e.g. the spin quantum number must be integers or half integers alternately along the sites for spin-1/2 systems as fusing the physical space will exactly shift quantum number by 1/2. If the auxiliary space (Rep[U₁](i => 1 for i in -1:1//2:1) here) is not chosen appropriately to be compatible with the symmetry, some errors will occur when generating the MPS or later.

For example, you can use DMRGSweep2! to obtain the ground state MPS

# 2-DMRG sweeps
D = 256
for i in 1:5
     t = @elapsed info, timer = DMRGSweep2!(Env; trunc = truncdim(D))
     println("Sweep $(i), En = $(info[2][1].Eg), wall time = $(t)s")
end

Then, you can use a similar interface addObs! to generate the tree that stores the observables to be calculated.

# calculate observables
Tree = ObservableTree()
# all to all spin correlation
for i in 1:L, j in i + 1:L
     addObs!(Tree, (U₁Spin.Sz, U₁Spin.Sz), (i, j); name = (:Sz, :Sz))
end
# local moment
for i in 1:L
     addObs!(Tree, U₁Spin.Sz, i; name = :Sz)
end

calObs!(Tree, Ψ)
# collect the results from Tree
Obs = convert(NamedTuple, Tree)

Now you can obtain the spin correlation $\langle S_i^z S_j^z\rangle$ via Obs.SzSz[(i,j)] and local moment $\langle S_i^z \rangle$ via Obs.Sz[(i,)] (should be zero up to a noise due to the $SU_2$ symmetry of the Heisenberg model).

[!WARNING] Warning: There may exist undiscovered bugs, therefore, please use it at your own risk. Benchmark with exact diagonalization (ED) is strongly recommended before using it to simulate any new models.

• Qiaoyi Li build the main architecture and the original version.
• Junsen Wang will maintain the algorithms for spin systems.

• If you benefit from this package, please star it.
• If you find a bug, you can submit an issue or a PR if you have fixed it.
• Add a new instance to LocalSpace folder and submit a PR. Note you should add some comments and document it so that others can easily use it.
• If you are familiar with multi-processing parallelism in Julia and want to join us to improve the package, please contact us!

• This package uses TensorKit.jl to implement basic tensor operations.
• This package uses KrylovKit.jl as Lanczos solver in DMRG and TDVP.
• I benefit a lot from reading the source code of MPSKit.jl and ITensors.jl.