# Overview¶

## Repository¶

The root directory of the git repository contains the following folders:

- tenpy
The actual source code of the library. Every subfolder contains an

`__init__.py`

file with a summary what the modules in it are good for. (This file is also necessary to mark the folder as part of the python package. Consequently, other subfolders of the git repo should not include a`__init__.py`

file.)- toycodes
Simple toy codes completely independet of the remaining library (i.e., codes in

`tenpy/`

). These codes should be quite readable and intend to give a flavor of how (some of) the algorithms work.- examples
Some example files demonstrating the usage and interface of the library.

- doc
A folder containing the documentation: the user guide is contained in the

`*.rst`

files. The online documentation is autogenerated from these files and the docstrings of the library. This folder contains a make file for building the documentation, run`make help`

for the different options. The necessary files for the reference in`doc/reference`

can be auto-generated/updated with`make src2html`

.- tests
Contains files with test routines, to be used with pytest. If you are set up correctly and have pytest installed, you can run the test suite with

`pytest`

from within the`tests/`

folder.- build
This folder is not distributed with the code, but is generated by

`setup.py`

(or`compile.sh`

, respectively). It contains compiled versions of the Cython files, and can be ignored (and even removed without loosing functionality).

## Code structure: getting started¶

There are several layers of abstraction in TeNPy. While there is a certain hierarchy of how the concepts build up on each other, the user can decide to utilize only some of them. A maximal flexibility is provided by an object oriented style based on classes, which can be inherited and adjusted to individual demands.

The following figure gives an overview of the most important modules, classes and functions in TeNPy.
Gray backgrounds indicate (sub)modules, yellow backgrounds indicate classes.
Red arrows indicate inheritance relations, dashed black arrows indicate a direct use.
(The individual models might be derived from the `NearestNeighborModel`

depending on the geometry of the lattice.)
There is a clear hierarchy from high-level algorithms in the `tenpy.algorithms`

module down to basic
operations from linear algebra in the `tenpy.linalg`

module.

### Most basic level: linear algebra¶

Note

See Charge conservation with np_conserved for more information on defining charges for arrays.

The most basic layer is given by in the `linalg`

module, which provides basic features of linear algebra.
In particular, the `np_conserved`

submodule implements an `Array`

class which is used to represent
the tensors. The basic interface of `np_conserved`

is very similar to that of the NumPy and SciPy libraries.
However, the `Array`

class implements abelian charge conservation.
If no charges are to be used, one can use ‘trivial’ arrays, as shown in the following example code.

```
"""Basic use of the `Array` class with trivial arrays."""
# Copyright 2019-2020 TeNPy Developers, GNU GPLv3
import tenpy.linalg.np_conserved as npc
M = npc.Array.from_ndarray_trivial([[0., 1.], [1., 0.]])
v = npc.Array.from_ndarray_trivial([2., 4. + 1.j])
v[0] = 3. # set indiviual entries like in numpy
print("|v> =", v.to_ndarray())
# |v> = [ 3.+0.j 4.+1.j]
M_v = npc.tensordot(M, v, axes=[1, 0])
print("M|v> =", M_v.to_ndarray())
# M|v> = [ 4.+1.j 3.+0.j]
print("<v|M|v> =", npc.inner(v.conj(), M_v, axes='range'))
# <v|M|v> = (24+0j)
```

The number and types of symmetries are specified in a `ChargeInfo`

class.
An `Array`

instance represents a tensor satisfying a charge rule specifying which blocks of it are nonzero.
Internally, it stores only the non-zero blocks of the tensor, along with one `LegCharge`

instance for each
leg, which contains the charges and sign qconj for each leg.
We can combine multiple legs into a single larger `LegPipe`

,
which is derived from the `LegCharge`

and stores all the information necessary to later split the pipe.

The following code explicitly defines the spin-1/2 \(S^+, S^-, S^z\) operators and uses them to generate and diagonalize the two-site Hamiltonian \(H = \vec{S} \cdot \vec{S}\). It prints the charge values (by default sorted ascending) and the eigenvalues of H.

```
"""Explicit definition of charges and spin-1/2 operators."""
# Copyright 2019-2020 TeNPy Developers, GNU GPLv3
import tenpy.linalg.np_conserved as npc
# consider spin-1/2 with Sz-conservation
chinfo = npc.ChargeInfo([1]) # just a U(1) charge
# charges for up, down state
p_leg = npc.LegCharge.from_qflat(chinfo, [[1], [-1]])
Sz = npc.Array.from_ndarray([[0.5, 0.], [0., -0.5]], [p_leg, p_leg.conj()])
Sp = npc.Array.from_ndarray([[0., 1.], [0., 0.]], [p_leg, p_leg.conj()])
Sm = npc.Array.from_ndarray([[0., 0.], [1., 0.]], [p_leg, p_leg.conj()])
Hxy = 0.5 * (npc.outer(Sp, Sm) + npc.outer(Sm, Sp))
Hz = npc.outer(Sz, Sz)
H = Hxy + Hz
# here, H has 4 legs
H.iset_leg_labels(["s1", "t1", "s2", "t2"])
H = H.combine_legs([["s1", "s2"], ["t1", "t2"]], qconj=[+1, -1])
# here, H has 2 legs
print(H.legs[0].to_qflat().flatten())
# prints [-2 0 0 2]
E, U = npc.eigh(H) # diagonalize blocks individually
print(E)
# [ 0.25 -0.75 0.25 0.25]
```

### Sites for the local Hilbert space and tensor networks¶

The next basic concept is that of a local Hilbert space, which is represented by a `Site`

in TeNPy.
This class does not only label the local states and define the charges, but also
provides onsite operators. For example, the `SpinHalfSite`

provides the
\(S^+, S^-, S^z\) operators under the names `'Sp', 'Sm', 'Sz'`

, defined as `Array`

instances similarly as
in the code above.
Since the most common sites like for example the `SpinSite`

(for general spin S=0.5, 1, 1.5,…), `BosonSite`

and
`FermionSite`

are predefined, a user of TeNPy usually does not need to define the local charges and operators explicitly.
The total Hilbert space, i.e, the tensor product of the local Hilbert spaces, is then just given by a
list of `Site`

instances. If desired, different kinds of `Site`

can be combined in that list.
This list is then given to classes representing tensor networks like the `MPS`

and
`MPO`

.
The tensor network classes also use `Array`

instances for the tensors of the represented network.

The following example illustrates the initialization of a spin-1/2 site, an `MPS`

representing the Neel state, and
an `MPO`

representing the Heisenberg model by explicitly defining the W tensor.

```
"""Initialization of sites, MPS and MPO."""
# Copyright 2019-2020 TeNPy Developers, GNU GPLv3
from tenpy.networks.site import SpinHalfSite
from tenpy.networks.mps import MPS
from tenpy.networks.mpo import MPO
spin = SpinHalfSite(conserve="Sz")
print(spin.Sz.to_ndarray())
# [[ 0.5 0. ]
# [ 0. -0.5]]
N = 6 # number of sites
sites = [spin] * N # repeat entry of list N times
pstate = ["up", "down"] * (N // 2) # Neel state
psi = MPS.from_product_state(sites, pstate, bc="finite")
print("<Sz> =", psi.expectation_value("Sz"))
# <Sz> = [ 0.5 -0.5 0.5 -0.5]
print("<Sp_i Sm_j> =", psi.correlation_function("Sp", "Sm"), sep="\n")
# <Sp_i Sm_j> =
# [[1. 0. 0. 0. 0. 0.]
# [0. 0. 0. 0. 0. 0.]
# [0. 0. 1. 0. 0. 0.]
# [0. 0. 0. 0. 0. 0.]
# [0. 0. 0. 0. 1. 0.]
# [0. 0. 0. 0. 0. 0.]]
# define an MPO
Id, Sp, Sm, Sz = spin.Id, spin.Sp, spin.Sm, spin.Sz
J, Delta, hz = 1., 1., 0.2
W_bulk = [[Id, Sp, Sm, Sz, -hz * Sz], [None, None, None, None, 0.5 * J * Sm],
[None, None, None, None, 0.5 * J * Sp], [None, None, None, None, J * Delta * Sz],
[None, None, None, None, Id]]
W_first = [W_bulk[0]] # first row
W_last = [[row[-1]] for row in W_bulk] # last column
Ws = [W_first] + [W_bulk] * (N - 2) + [W_last]
H = MPO.from_grids([spin] * N, Ws, bc='finite', IdL=0, IdR=-1)
print("<psi|H|psi> =", H.expectation_value(psi))
# <psi|H|psi> = -1.25
```

### Models¶

Note

See Models for more information on sites and how to define and extend models on your own.

Technically, the explicit definition of an `MPO`

is already enough to call an algorithm like DMRG in `dmrg`

.
However, writing down the W tensors is cumbersome especially for more complicated models.
Hence, TeNPy provides another layer of abstraction for the definition of models, which we discuss first.
Different kinds of algorithms require different representations of the Hamiltonian.
Therefore, the library offers to specify the model abstractly by the individual onsite terms and coupling terms of the Hamiltonian.
The following example illustrates this, again for the Heisenberg model.

```
"""Definition of a model: the XXZ chain."""
# Copyright 2019-2020 TeNPy Developers, GNU GPLv3
from tenpy.networks.site import SpinSite
from tenpy.models.lattice import Chain
from tenpy.models.model import CouplingModel, NearestNeighborModel, MPOModel
class XXZChain(CouplingModel, NearestNeighborModel, MPOModel):
def __init__(self, L=2, S=0.5, J=1., Delta=1., hz=0.):
spin = SpinSite(S=S, conserve="Sz")
# the lattice defines the geometry
lattice = Chain(L, spin, bc="open", bc_MPS="finite")
CouplingModel.__init__(self, lattice)
# add terms of the Hamiltonian
self.add_coupling(J * 0.5, 0, "Sp", 0, "Sm", 1) # Sp_i Sm_{i+1}
self.add_coupling(J * 0.5, 0, "Sp", 0, "Sm", -1) # Sp_i Sm_{i-1}
self.add_coupling(J * Delta, 0, "Sz", 0, "Sz", 1)
# (for site dependent prefactors, the strength can be an array)
self.add_onsite(-hz, 0, "Sz")
# finish initialization
# generate MPO for DMRG
MPOModel.__init__(self, lat, self.calc_H_MPO())
# generate H_bond for TEBD
NearestNeighborModel.__init__(self, lat, self.calc_H_bond())
```

While this generates the same MPO as in the previous code, this example can easily be adjusted and generalized, for
example to a higher dimensional lattice by just specifying a different lattice.
Internally, the MPO is generated using a finite state machine picture.
This allows not only to translate more complicated Hamiltonians into their corresponding MPOs,
but also to automate the mapping from a higher dimensional lattice to the 1D chain along which the MPS
winds.
Note that this mapping introduces longer-range couplings, so the model can no longer be defined to be a
`NearestNeighborModel`

suited for TEBD if another lattice than the `Chain`

is to be used.
Of course, many commonly studied models are also predefined.
For example, the following code initializes the Heisenberg model on a kagome lattice;
the spin liquid nature of the ground state of this model is highly debated in the current literature.

```
"""Initialization of the Heisenberg model on a kagome lattice."""
# Copyright 2019-2020 TeNPy Developers, GNU GPLv3
from tenpy.models.spins import SpinModel
model_params = {
"S": 0.5, # Spin 1/2
"lattice": "Kagome",
"bc_MPS": "infinite",
"bc_y": "cylinder",
"Ly": 2, # defines cylinder circumference
"conserve": "Sz", # use Sz conservation
"Jx": 1.,
"Jy": 1.,
"Jz": 1. # Heisenberg coupling
}
model = SpinModel(model_params)
```

### Algorithms¶

The highest level in TeNPy is given by algorithms like DMRG and TEBD.
Using the previous concepts, setting up a simulation running those algorithms is a matter of just a few lines of code.
The following example runs a DMRG simulation, see `dmrg`

, exemplary for the transverse field Ising model at the critical point.

```
"""Call of (finite) DMRG."""
# Copyright 2019-2020 TeNPy Developers, GNU GPLv3
from tenpy.networks.mps import MPS
from tenpy.models.tf_ising import TFIChain
from tenpy.algorithms import dmrg
N = 16 # number of sites
model = TFIChain({"L": N, "J": 1., "g": 1., "bc_MPS": "finite"})
sites = model.lat.mps_sites()
psi = MPS.from_product_state(sites, ['up'] * N, "finite")
dmrg_params = {"trunc_params": {"chi_max": 100, "svd_min": 1.e-10}, "mixer": True}
info = dmrg.run(psi, model, dmrg_params)
print("E =", info['E'])
# E = -20.01638790048513
print("max. bond dimension =", max(psi.chi))
# max. bond dimension = 27
```

The switch from DMRG to gls{iDMRG} in TeNPy is simply accomplished by a change of the parameter
`"bc_MPS"`

from `"finite"`

to `"infinite"`

, both for the model and the state.
The returned `E`

is then the energy density per site.
Due to the translation invariance, one can also evaluate the correlation length, here slightly away from the critical point.

```
"""Call of infinite DMRG."""
# Copyright 2019-2020 TeNPy Developers, GNU GPLv3
from tenpy.networks.mps import MPS
from tenpy.models.tf_ising import TFIChain
from tenpy.algorithms import dmrg
N = 2 # number of sites in unit cell
model = TFIChain({"L": N, "J": 1., "g": 1.1, "bc_MPS": "infinite"})
sites = model.lat.mps_sites()
psi = MPS.from_product_state(sites, ['up'] * N, "infinite")
dmrg_params = {"trunc_params": {"chi_max": 100, "svd_min": 1.e-10}, "mixer": True}
info = dmrg.run(psi, model, dmrg_params)
print("E =", info['E'])
# E = -1.342864022725017
print("max. bond dimension =", max(psi.chi))
# max. bond dimension = 56
print("corr. length =", psi.correlation_length())
# corr. length = 4.915809146764157
```

Running time evolution with TEBD requires an additional loop, during which the desired observables have to be measured. The following code shows this directly for the infinite version of TEBD.

```
"""Call of (infinite) TEBD."""
# Copyright 2019-2020 TeNPy Developers, GNU GPLv3
from tenpy.networks.mps import MPS
from tenpy.models.tf_ising import TFIChain
from tenpy.algorithms import tebd
M = TFIChain({"L": 2, "J": 1., "g": 1.5, "bc_MPS": "infinite"})
psi = MPS.from_product_state(M.lat.mps_sites(), [0] * 2, "infinite")
tebd_params = {
"order": 2,
"delta_tau_list": [0.1, 0.001, 1.e-5],
"max_error_E": 1.e-6,
"trunc_params": {
"chi_max": 30,
"svd_min": 1.e-10
}
}
eng = tebd.Engine(psi, M, tebd_params)
eng.run_GS() # imaginary time evolution with TEBD
print("E =", sum(psi.expectation_value(M.H_bond)) / psi.L)
print("final bond dimensions: ", psi.chi)
```