MPS¶
full name: tenpy.networks.mps.MPS
parent module:
tenpy.networks.mps
type: class
Inheritance Diagram
Methods



Return an MPS which represents 

Apply a local (one or multisite) operator to self. 

Apply a (global) product of local onsite operators to self. 

Return the average charge for the block on the left of a given bond. 

Bring self into canonical 'B' form, (re)calculate singular values. 

Bring a finite (or segment) MPS into canonical form (in place). 

Deprecated wrapper around 

Bring an infinite MPS into canonical form (in place). 

Convert infinite MPS to canonical form. 

Return the charge variance on the left of a given bond. 

Compresss an MPS. 

Compress self with a single sweep of SVDs; in place. 

Compute the momentum quantum numbers of the entanglement spectrum for 2D states. 

Tranform self into different canonical form (by scaling the legs with singular values). 

Returns a copy of self. 

Correlation function 

Calculate the correlation length by diagonalizing the transfer matrix. 

Artifically enlarge the bond dimension by the specified extra legs/charges. 

Repeat the unit cell for infinite MPS boundary conditions; in place. 

Calculate the (halfchain) entanglement entropy for all nontrivial bonds. 

Calculate entanglement entropy for general geometry of the bipartition. 

Calculate entanglement entropy for general geometry of the bipartition. 

return entanglement energy spectrum. 

Expectation value 

Expectation value 

Expectation value 

Calculate expectation values for a bunch of terms and sum them up. 

Extract an segment from a finite or infinite MPS. 

Construct a matrix product state from a set of numpy arrays Bflat and singular vals. 

Construct an MPS from a single tensor psi with one leg per physical site. 

Load instance from a HDF5 file. 

Construct an MPS from a product state given in lattice coordinates. 

Construct a matrix product state from a given product state. 

Create an MPS of entangled singlets. 

Gauge the legcharges of the virtual bonds such that the MPS has a total qtotal. 

Return (view of) B at site i in canonical form. 

Return singular values on the left of site i 

Return singular values on the right of site i 

Like 

Given a list of operators, select the one corresponding to site i. 

Return reduced density matrix for a segment. 

Calculates the nsite wavefunction on 

Calculate and return the qtotal of the whole MPS (when contracted). 

Modify self inplace to group sites. 

Modify self inplace to split previously grouped sites. 

Modify self inplace to enlarge the MPS unit cell; in place. 

Calculate the twosite mutual information \(I(i:j)\). 
Check that self is in canonical form. 


Compute overlap 

Applies the permutation perm to the state (inplace). 

Locally perturb the state a little bit; in place. 

Return probabilites of charge value on the left of a given bond. 

Shift the section we define as unit cellof an infinite MPS; in place. 

Sample measurement results in the computational basis. 

Export self into a HDF5 file. 

Set B at site i. 

Set singular values on the left of site i 

Set singular values on the right of site i 

SVD a twosite wave function theta and save it in self. 
Perform a spatial inversion along the MPS. 


Swap the two neighboring sites i and i+1 (inplace). 

Correlation function between (multisite) terms, moving the left term, fix right term. 

Correlation function between (multisite) terms, moving the right term, fix left term. 
Correlation function between sums of multisite terms, moving the right sum of term. 

Sanity check, raises ValueErrors, if something is wrong. 
Class Attributes and Properties
Number of physical sites; for an iMPS the len of the MPS unit cell. 

Dimensions of the (nontrivial) virtual bonds. 

List of local physical dimensions. 

Distinguish MPS vs iMPS. 

Slice of the nontrivial bond indices, depending on 
 class tenpy.networks.mps.MPS(sites, Bs, SVs, bc='finite', form='B', norm=1.0)[source]¶
Bases:
object
A Matrix Product State, finite (MPS) or infinite (iMPS).
 Parameters
sites (list of
Site
) – Defines the local Hilbert space for each site.Bs (list of
Array
) – The ‘matrices’ of the MPS. Labels arevL, vR, p
(in any order).SVs (list of 1D array) – The singular values on each bond. Should always have length L+1. Entries out of
nontrivial_bonds
are ignored.bc (
'finite'  'segment'  'infinite'
) – Boundary conditions as described in the tabel of the module docstring.form ((list of) {
'B'  'A'  'C'  'G'  'Th'  None
 tuple(float, float)}) – The form of the stored ‘matrices’, see table in module docstring. A single choice holds for all of the entries.
 bc¶
Boundary conditions as described in above table.
 Type
{‘finite’, ‘segment’, ‘infinite’}
 form¶
Describes the canonical form on each site.
None
means noncanonical form. Forform = (nuL, nuR)
, the stored_B[i]
ares**form[0]  Gamma  s**form[1]
(in Vidal’s notation). Type
list of {
None
, tuple(float, float)}
 chinfo¶
The nature of the charge.
 Type
ChargeInfo
 norm¶
The norm of the state, i.e.
sqrt(<psipsi>)
. Ignored for (normalized)expectation_value()
, but important foroverlap()
. Type
 grouped¶
Number of sites grouped together, see
group_sites()
. Type
 segment_boundaries¶
Only defined for ‘segment’ bc if
canonical_form_finite()
has been called. If defined, it contains the U_L and V_R that would be returned by that function. Type
tuple of
Array
 (None, None)
 _B¶
The ‘matrices’ of the MPS. Labels are
vL, vR, p
(in any order). We recommend usingget_B()
andset_B()
, which will take care of the different canonical forms. Type
list of
npc.Array
 _S¶
The singular values on each virtual bond, length
L+1
. May beNone
if the MPS is not in canonical form. Otherwise,_S[i]
is to the left of_B[i]
. We recommend usingget_SL()
,get_SR()
,set_SL()
,set_SR()
, which takes proper care of the boundary conditions. Sometimes (e.g. during DMRG with an enabled mixer), entries may temporarily be a nondiagonaltenpy.linalg.np_conserved.Array
to be inserted between the left canonical ‘A’ tensors on the left and rigthcanonical _B[i] on the right. Type
list of {
None
, 1D array,Array
}
 _valid_forms¶
Class attribute. Mapping for canonical forms to a tuple
(nuL, nuR)
indicating thatself._B[i] = s[i]**nuL  Gamma[i]  s[i]**nuR
is saved. Type
 _transfermatrix_keep¶
How many states to keep at least when diagonalizing a
TransferMatrix
. Important if the state develops a neardegeneracy. Type
 _p_label, _B_labels
Class attribute. _p_label defines the physical legs of the Btensors, _B_labels lists all the labels of the B tensors. Used by methods like
get_theta()
to avoid the necessity of reimplementations for derived classes like thePurification_MPS
if just the number of physical legs changed.
 copy()[source]¶
Returns a copy of self.
The copy still shares the sites, chinfo, and LegCharges of the B tensors, but the values of B and S are deeply copied.
 save_hdf5(hdf5_saver, h5gr, subpath)[source]¶
Export self into a HDF5 file.
This method saves all the data it needs to reconstruct self with
from_hdf5()
.Specifically, it saves
sites
,chinfo
(under these names),_B
as"tensors"
,_S
as"singular_values"
,bc
as"boundary_condition"
, andform
converted to a single array of shape (L, 2) as"canonical_form"
, Moreover, it savesnorm
,L
,grouped
and_transfermatrix_keep
(as “transfermatrix_keep”) as HDF5 attributes, as well as the maximum ofchi
under the name “max_bond_dimension”.
 classmethod from_hdf5(hdf5_loader, h5gr, subpath)[source]¶
Load instance from a HDF5 file.
This method reconstructs a class instance from the data saved with
save_hdf5()
. Parameters
hdf5_loader (
Hdf5Loader
) – Instance of the loading engine.h5gr (
Group
) – HDF5 group which is represent the object to be constructed.subpath (str) – The name of h5gr with a
'/'
in the end.
 Returns
obj – Newly generated class instance containing the required data.
 Return type
cls
 classmethod from_lat_product_state(lat, p_state, **kwargs)[source]¶
Construct an MPS from a product state given in lattice coordinates.
This is a wrapper around
from_product_state()
. The purpuse is to make the p_state argument independent of the order of the Lattice, and specify it in terms of lattice indices instead. Parameters
lat (
Lattice
) – The underlying lattice defining the geometry and Hilbert Space.p_state (array_like of {int  str  1D array}) – Defines the product state to be represented. Should be of dimension lat.dim`+1, entries are indexed by lattice indices. Entries of the array as for the `p_state argument of
from_product_state()
. It gets tiled to the shapelat.shape
, if it is smaller.**kwargs – Other keyword arguments as definied in
from_product_state()
. bc is set by default fromlat.bc_MPS
.
 Returns
product_mps – An MPS representing the specified product state.
 Return type
Examples
Let’s first consider a
Ladder
composed of aSpinHalfSite
and aFermionSite
.To initialize a state of upspins on the spin sites and halffilled ferions, you can use:
Note that the same p_state works for a finite lattice of even length, say
L=10
, as well. We then just “tile” in xdirection, i.e., repeat the specified state 5 times:You can also easily halffill a
Honeycomb
, for example with only the A sites occupied, or as stripe parallel to the xdirection (stripe_x, alternating along y axis), or as stripes parallel to the ydirection (stripe_y, alternating along x axis).
 classmethod from_product_state(sites, p_state, bc='finite', dtype=<class 'numpy.float64'>, permute=True, form='B', chargeL=None)[source]¶
Construct a matrix product state from a given product state.
 Parameters
sites (list of
Site
) – The sites defining the local Hilbert space.p_state (list of {int  str  1D array}) – Defines the product state to be represented; one entry for each site of the MPS. An entry of str type is translated to an int with the help of
state_labels()
. An entry of int type represents the physical index of the state to be used. An entry which is a 1D array defines the complete wavefunction on that site; this allows to make a (local) superposition.bc ({'infinite', 'finite', 'segmemt'}) – MPS boundary conditions. See docstring of
MPS
.dtype (type or string) – The data type of the array entries.
permute (bool) – The
Site
might permute the local basis states if charge conservation gets enabled. If permute is True (default), we permute the given p_state locally according to each site’sperm
. The p_state entries should then always be given as if conserve=None in the Site.form ((list of) {
'B'  'A'  'C'  'G'  None
 tuple(float, float)}) – Defines the canonical form. See module docstring. A single choice holds for all of the entries.chargeL (charges) – Leg charges at bond 0, which are purely conventional.
 Returns
product_mps – An MPS representing the specified product state.
 Return type
Examples
Example to get a Neel state for a
TFIChain
:>>> from tenpy.networks.mps import MPS >>> L = 10 >>> M = tenpy.models.tf_ising.TFIChain({'L': L}) >>> p_state = ["up", "down"] * (L//2) # repeats entries L/2 times >>> psi = MPS.from_product_state(M.lat.mps_sites(), p_state, bc=M.lat.bc_MPS)
The meaning of the labels
"up","down"
is defined by theSite
, in this example aSpinHalfSite
.Extending the example, we can replace the spin in the center with one with arbitrary angles
theta, phi
in the bloch sphere. However, note that you can not write this bloch state (fortheta != 0, pi
) when conserving symmetries, as the two physical basis states correspond to different symmetry sectors.>>> spin = tenpy.networks.site.SpinHalfSite(conserve=None, sort_charge=False) >>> p_state = ["up", "down"] * (L//2) # repeats entries L/2 times >>> theta, phi = np.pi/4, np.pi/6 >>> bloch_sphere_state = np.array([np.cos(theta/2), np.exp(1.j*phi)*np.sin(theta/2)]) >>> p_state[L//2] = bloch_sphere_state # replace one spin in center >>> psi = MPS.from_product_state([spin]*L, p_state, bc=M.lat.bc_MPS, dtype=complex)
Note that for the more general
SpinChain
, the order of the two entries for thebloch_sphere_state
would be exactly the opposite (when we keep the the northpole of the bloch sphere being the upstate). The reason is that the SpinChain uses the generalSpinSite
, where the states are orderd ascending from'down'
to'up'
. TheSpinHalfSite
on the other hand uses the order'up', 'down'
where that the Pauli matrices look as usual.
 classmethod from_Bflat(sites, Bflat, SVs=None, bc='finite', dtype=None, permute=True, form='B', legL=None)[source]¶
Construct a matrix product state from a set of numpy arrays Bflat and singular vals.
 Parameters
sites (list of
Site
) – The sites defining the local Hilbert space.Bflat (iterable of numpy ndarrays) – The matrix defining the MPS on each site, with legs
'p', 'vL', 'vR'
(physical, virtual left/right).SVs (list of 1D array 
None
) – The singular values on each bond. Should always have length L+1. By default (None
), set all singular values to the same value. Entries out ofnontrivial_bonds
are ignored.bc ({'infinite', 'finite', 'segmemt'}) – MPS boundary conditions. See docstring of
MPS
.dtype (type or string) – The data type of the array entries. Defaults to the common dtype of Bflat.
permute (bool) – The
Site
might permute the local basis states if charge conservation gets enabled. If permute is True (default), we permute the given Bflat locally according to each site’sperm
. The p_state argument should then always be given as if conserve=None in the Site.form ((list of) {
'B'  'A'  'C'  'G'  None
 tuple(float, float)}) – Defines the canonical form of Bflat. See module docstring. A single choice holds for all of the entries.leg_L (LegCharge 
None
) – Leg charges at bond 0, which are purely conventional. IfNone
, use trivial charges.
 Returns
mps – An MPS with the matrices Bflat converted to npc arrays.
 Return type
 classmethod from_full(sites, psi, form=None, cutoff=1e16, normalize=True, bc='finite', outer_S=None)[source]¶
Construct an MPS from a single tensor psi with one leg per physical site.
Performs a sequence of SVDs of psi to split off the B matrices and obtain the singular values, the result will be in canonical form. Obviously, this is only welldefined for finite or segment boundary conditions.
 Parameters
sites (list of
Site
) – The sites defining the local Hilbert space.psi (
Array
) – The full wave function to be represented as an MPS. Should have labels'p0', 'p1', ..., 'p{L1}'
. Additionally, it may have (or must have for ‘segment’ bc) the legs'vL', 'vR'
, which are trivial for ‘finite’ bc.form (
'B'  'A'  'C'  'G'  None
) – The canonical form of the resulting MPS, see module docstring.None
defaults to ‘A’ form on the first site and ‘B’ form on all following sites.cutoff (float) – Cutoff of singular values used in the SVDs.
normalize (bool) – Whether the resulting MPS should have ‘norm’ 1.
bc ('finite'  'segment') – Boundary conditions.
outer_S (None  (array, array)) – For ‘semgent’ bc the singular values on the left and right of the considered segment, None for ‘finite’ boundary conditions.
 Returns
psi_mps – MPS representation of psi, in canonical form and possibly normalized.
 Return type
 classmethod from_singlets(site, L, pairs, up='up', down='down', lonely=[], lonely_state='up', bc='finite')[source]¶
Create an MPS of entangled singlets.
 Parameters
site (
Site
) – The site defining the local Hilbert space, taken uniformly for all sites.L (int) – The number of sites.
pairs (list of (int, int)) – Pairs of sites to be entangled; the returned MPS will have a singlet for each pair in pairs.
up (int  str) – A singlet is defined as
(up down>  down up>)/2**0.5
,up
anddown
give state indices or labels defined on the corresponding site.down (int  str) – A singlet is defined as
(up down>  down up>)/2**0.5
,up
anddown
give state indices or labels defined on the corresponding site.lonely (list of int) – Sites which are not included into a singlet pair.
bc ({'infinite', 'finite', 'segmemt'}) – MPS boundary conditions. See docstring of
MPS
.
 Returns
singlet_mps – An MPS representing singlets on the specified pairs of sites.
 Return type
 property L¶
Number of physical sites; for an iMPS the len of the MPS unit cell.
 property dim¶
List of local physical dimensions.
 property finite¶
Distinguish MPS vs iMPS.
True for an MPS (
bc='finite', 'segment'
), False for an iMPS (bc='infinite'
).
 property chi¶
Dimensions of the (nontrivial) virtual bonds.
 property nontrivial_bonds¶
Slice of the nontrivial bond indices, depending on
self.bc
.
 get_B(i, form='B', copy=False, cutoff=1e16, label_p=None)[source]¶
Return (view of) B at site i in canonical form.
 Parameters
i (int) – Index choosing the site.
form (
'B'  'A'  'C'  'G'  'Th'  None
 tuple(float, float)) – The (canonical) form of the returned B. ForNone
, return the matrix in whatever form it is. If any of the tuple entry is None, also don’t scale on the corresponding axis.copy (bool) – Whether to return a copy even if form matches the current form.
cutoff (float) – During DMRG with a mixer, S may be a matrix for which we need the inverse. This is calculated as the Penrose pseudoinverse, which uses a cutoff for the singular values.
label_p (None  str) – Ignored by default (
None
). Otherwise replace the physical label'p'
with'p'+label_p'
. (For derived classes with more than one “physical” leg, replace all the physical leg labels accordingly.)
 Returns
B – The MPS ‘matrix’ B at site i with leg labels
'vL', 'p', 'vR'
. May be a view of the matrix (ifcopy=False
), or a copy (if the form changed orcopy=True
). Return type
:raises ValueError : if self is not in canoncial form and form is not None.:
 set_B(i, B, form='B')[source]¶
Set B at site i.
 Parameters
i (int) – Index choosing the site.
B (
Array
) – The ‘matrix’ at site i. No copy is made! Should have leg labels'vL', 'p', 'vR'
(not necessarily in that order).form (
'B'  'A'  'C'  'G'  'Th'  None
 tuple(float, float)) – The (canonical) form of the B to set.None
stands for noncanonical form.
 set_svd_theta(i, theta, trunc_par=None, update_norm=False)[source]¶
SVD a twosite wave function theta and save it in self.
 Parameters
i (int) – theta is the wave function on sites i, i + 1.
theta (
Array
) – The twosite wave function with labels combined into"(vL.p0)", "(p1.vR)"
, ready for svd.trunc_par (None  dict) – Parameters for truncation, see
truncation
. IfNone
, no truncation is done.update_norm (bool) – If
True
, multiply the norm of theta intonorm
.
 get_op(op_list, i)[source]¶
Given a list of operators, select the one corresponding to site i.
 Parameters
 Returns
op – One of the entries in op_list, not copied.
 Return type
npc.array
 get_theta(i, n=2, cutoff=1e16, formL=1.0, formR=1.0)[source]¶
Calculates the nsite wavefunction on
sites[i:i+n]
. Parameters
i (int) – Site index.
n (int) – Number of sites. The result lives on
sites[i:i+n]
.cutoff (float) – During DMRG with a mixer, S may be a matrix for which we need the inverse. This is calculated as the Penrose pseudoinverse, which uses a cutoff for the singular values.
formL (float) – Exponent for the singular values to the left.
formR (float) – Exponent for the singular values to the right.
 Returns
theta – The nsite wave function with leg labels
vL, p0, p1, .... p{n1}, vR
. In Vidal’s notation (with s=lambda, G=Gamma):theta = s**form_L G_i s G_{i+1} s ... G_{i+n1} s**form_R
. Return type
 convert_form(new_form='B')[source]¶
Tranform self into different canonical form (by scaling the legs with singular values).
 Parameters
new_form ((list of) {
'B'  'A'  'C'  'G'  'Th'  None
 tuple(float, float)}) – The form the stored ‘matrices’. The table in module docstring. A single choice holds for all of the entries.
:raises ValueError : if trying to convert from a
None
form. Usecanonical_form()
instead!:
 increase_L(new_L=None)[source]¶
Modify self inplace to enlarge the MPS unit cell; in place.
Deprecated since version 0.5.1: This method will be removed in version 1.0.0. Use the equivalent
psi.enlarge_mps_unit_cell(new_L//psi.L)
instead ofpsi.increase_L(new_L)
.
 enlarge_mps_unit_cell(factor=2)[source]¶
Repeat the unit cell for infinite MPS boundary conditions; in place.
 Parameters
factor (int) – The new number of sites in the unit cell will be increased from L to
factor*L
.
 roll_mps_unit_cell(shift=1)[source]¶
Shift the section we define as unit cellof an infinite MPS; in place.
Suppose we have a unit cell with tensors
[A, B, C, D]
(repeated on both sites). Withshift = 1
, the new unit cell will be[D, A, B, C]
, whereasshift = 1
will give[B, C, D, A]
. Parameters
shift (int) – By how many sites to move the tensors to the right.
 enlarge_chi(extra_legs, random_fct=<builtin method normal of numpy.random.mtrand.RandomState object>)[source]¶
Artifically enlarge the bond dimension by the specified extra legs/charges. In place.
First converts the MPS in B form. This function then fills up the ‘vR’ leg with zeros, and then groups physical and vR legs to fill up the ‘vL’ leg with orthogonal rows. In than way, we get an MPS with larger bond dimension, still in rightcanonical form (assuming no “overcomplete” charge block), representing the same state, with the additional singular values being exactly zero.
Note
You should probably choose the extra charges to be sensible, to expand into the space you are interested in, and not just into a random direction!
 Parameters
extra_legs (list of {None,
LegCharge
, int}) – The extra charges to be added on the virtual legs, with qconj=+1. Lenght L +1 for finite, length L for infinite, with entry i left of site i. If an int is given, fill up with a single block of charges like the Schmidt state with highest weight. Note that this might force the resulting state to not be in strict “rightcanonical” B form if that charge block becomes overcomplete.random_fct – Function generating random entries to choose extra orthogonal rows in the B tensors. Should accept a size keyword for the shape, and return numpy arrays.
 Returns
permutations – Permutation performed on each virtual leg, such that
new_S = concatenate(old_S, zeros)[perm]
. Return type
list of 1D array  None
 spatial_inversion()[source]¶
Perform a spatial inversion along the MPS.
Exchanges the first with the last tensor and so on, i.e., exchange site i with site
L1  i
. This is equivalent to a mirror/reflection with the bond left of L/2 (even L) or the site (L1)/2 (odd L) as a fixpoint. For infinite MPS, the bond between MPS unit cells is another fix point.
 group_sites(n=2, grouped_sites=None)[source]¶
Modify self inplace to group sites.
Group each n sites together using the
GroupedSite
. This might allow to do TEBD with a Trotter decomposition, or help the convergence of DMRG (in case of too long range interactions). Parameters
n (int) – Number of sites to be grouped together.
grouped_sites (None  list of
GroupedSite
) – The sites grouped together.
See also
group_split
Reverts the grouping.
 group_split(trunc_par=None)[source]¶
Modify self inplace to split previously grouped sites.
 Parameters
trunc_par (dict) – Parameters for truncation, see
truncation
. Defaults to{'chi_max': max(self.chi)}
. Returns
trunc_err – The error introduced by the truncation for the splitting.
 Return type
See also
group_sites
Should have been used before to combine sites.
 get_grouped_mps(blocklen)[source]¶
Like
group_sites()
, but make a copy. Parameters
blocklen (int) – Number of subsequent sites to be combined; n in
group_sites()
. Returns
New MPS object with bunched sites.
 Return type
grouped_MPS
 get_total_charge(only_physical_legs=False)[source]¶
Calculate and return the qtotal of the whole MPS (when contracted).
If set, the
segment_boundaries
are included (unless only_physical_legs is True). Parameters
only_physical_legs (bool) – For
'finite'
boundary conditions, the total charge can be gauged away by changing the LegCharge of the trivial legs on the left and right of the MPS. This option allows to project out the trivial legs to get the actual “physical” total charge. Returns
qtotal – The sum of the qtotal of the individual B tensors.
 Return type
charges
 gauge_total_charge(qtotal=None, vL_leg=None, vR_leg=None)[source]¶
Gauge the legcharges of the virtual bonds such that the MPS has a total qtotal.
Acts in place, i.e. changes the B tensors. Make a (shallow) copy if needed.
 Parameters
qtotal ((list of) charges) – If a single set of charges is given, it is the desired total charge of the MPS (which
get_total_charge()
will return afterwards). By default (None
), use 0 charges, unless vL_leg and vR_leg are specified, in which case we adjust the total charge to match these legs.vL_leg (None  LegCharge) – Desired new virtual leg on the very left and right. Needs to have the same block strucuture as the current legs, but can have shifted charge entries. For infinite MPS, we need vL_leg to be the conjugate leg of vR_leg. For segment MPS, these legs are the outermost legs, possibly including the
segment_boundaries
.vR_leg (None  LegCharge) – Desired new virtual leg on the very left and right. Needs to have the same block strucuture as the current legs, but can have shifted charge entries. For infinite MPS, we need vL_leg to be the conjugate leg of vR_leg. For segment MPS, these legs are the outermost legs, possibly including the
segment_boundaries
.
 entanglement_entropy(n=1, bonds=None, for_matrix_S=False)[source]¶
Calculate the (halfchain) entanglement entropy for all nontrivial bonds.
Consider a bipartition of the sytem into \(A = \{ j: j <= i_b \}\) and \(B = \{ j: j > i_b\}\) and the reduced density matrix \(\rho_A = tr_B(\rho)\). The vonNeumann entanglement entropy is defined as \(S(A, n=1) = tr(\rho_A \log(\rho_A)) = S(B, n=1)\). The generalization for
n != 1, n>0
are the Renyi entropies: \(S(A, n) = \frac{1}{1n} \log(tr(\rho_A^2)) = S(B, n=1)\)This function calculates the entropy for a cut at different bonds i, for which the the eigenvalues of the reduced density matrix \(\rho_A\) and \(\rho_B\) is given by the squared schmidt values S of the bond.
 Parameters
n (int/float) – Selects which entropy to calculate; n=1 (default) is the ususal vonNeumann entanglement entropy.
bonds (
None
 (iterable of) int) – Selects the bonds at which the entropy should be calculated.None
defaults torange(0, L+1)[self.nontrivial_bonds]
, i.e.,range(1, L)
for ‘finite’ MPS andrange(0, L)
for ‘infinite’ MPS.for_matrix_S (bool) – Switch calculate the entanglement entropy even if the _S are matrices. Since \(O(\chi^3)\) is expensive compared to the ususal \(O(\chi)\), we raise an error by default.
 Returns
entropies – Entanglement entropies for halfcuts. entropies[j] contains the entropy for a cut at bond
bonds[j]
, i.e. between sitesbonds[j]1
andbonds[j]
. For infinite systems with defaultbonds=None
, this means thatentropies[0]
will be a cut left of site 0 and is the one you should look at to e.g. study the scaling of the entanglement with chi or to extract the topological entanglement entropy  don’t take the average over bonds, in particular if you have 2D cylinders or ladders. On the contrary, for finite systems withbonds=None
, take the central value of the returned arrayentropies[len(entropies)//2)] == entropies[(L1)//2]
(and not justentropies[L//2]
) to extract the halfchain entanglement entropy. Return type
1D ndarray
 entanglement_entropy_segment(segment=[0], first_site=None, n=1)[source]¶
Calculate entanglement entropy for general geometry of the bipartition.
This function is similar as
entanglement_entropy()
, but for more general geometry of the region A to be a segment of a few sites.This is acchieved by explicitly calculating the reduced density matrix of A and thus works only for small segments. The alternative
entanglement_entropy_segment2()
might work for larger segments at small enough bond dimensions. Parameters
segment (list of int) – Given a first site i, the region
A_i
is defined to be[i+j for j in segment]
.first_site (
None
 (iterable of) int) – Calculate the entropy for segments starting at these sites.None
defaults torange(Lsegment[1])
for finite or range(L) for infinite boundary conditions.n (int  float) – Selects which entropy to calculate; n=1 (default) is the ususal vonNeumann entanglement entropy, otherwise the nth Renyi entropy.
 Returns
entropies –
entropies[i]
contains the entropy for the the regionA_i
defined above. Return type
1D ndarray
 entanglement_entropy_segment2(segment, n=1)[source]¶
Calculate entanglement entropy for general geometry of the bipartition.
This function is similar to
entanglement_entropy_segment()
, but allows more sites in segment. The trick is to exploit that for a pure state (which the MPS represents) and a bipartition into regions A and B, the entropy is the same in both regions, \(S(A) = S(B)\). Hence we can trace out the specified segment and obtain \(\rho_B = tr_A(rho)\), where A is the specified segment. The price is a huge computation cost of \(O(chi^6 d^{3x})\) where x is the number of physical legs not included into segment between min(segment) and max(segment). Parameters
 Returns
entropy – The entropy for the the region defined by the segment (or equivalently it’s complement).
 Return type
 entanglement_spectrum(by_charge=False)[source]¶
return entanglement energy spectrum.
 Parameters
by_charge (bool) – Whether we should sort the spectrum on each bond by the possible charges.
 Returns
ent_spectrum – For each (nontrivial) bond the entanglement spectrum. If by_charge is
False
, return (for each bond) a sorted 1D ndarray with the convention \(S_i^2 = e^{\xi_i}\), where \(S_i\) labels a Schmidt value and \(\xi_i\) labels the entanglement ‘energy’ in the returned spectrum. If by_charge is True, return a a list of tuples(charge, sub_spectrum)
for each possible charge on that bond. Return type
 get_rho_segment(segment)[source]¶
Return reduced density matrix for a segment.
Note that the dimension of rho_A scales exponentially in the length of the segment.
 probability_per_charge(bond=0)[source]¶
Return probabilites of charge value on the left of a given bond.
For example for particle number conservation, define \(N_b = sum_{i<b} n_i\) for a given bond b. This function returns the possible values of N_b as rows of charge_values, and for each row the probabilty that this combination occurs in the given state.
 Parameters
bond (int) – The bond to be considered. The returned charges are summed on the left of this bond.
 Returns
charge_values (2D array) – Columns correspond to the different charges in self.chinfo. Rows are the different charge fluctuations at this bond
probabilities (1D array) – For each row of charge_values the probablity for these values of charge fluctuations.
 average_charge(bond=0)[source]¶
Return the average charge for the block on the left of a given bond.
For example for particle number conservation, define \(N_b = sum_{i<b} n_i\) for a given bond b. Then this function returns \(<\psi N_b \psi>\).
 charge_variance(bond=0)[source]¶
Return the charge variance on the left of a given bond.
For example for particle number conservation, define \(N_b = sum_{i<b} n_i\) for a given bond b. Then this function returns \(<\psi N_b^2 \psi>  (<\psi N_b \psi>)^2\).
 mutinf_two_site(max_range=None, n=1)[source]¶
Calculate the twosite mutual information \(I(i:j)\).
Calculates \(I(i:j) = S(i) + S(j)  S(i,j)\), where \(S(i)\) is the single site entropy on site \(i\) and \(S(i,j)\) the twosite entropy on sites \(i,j\).
 Parameters
 Returns
coords (2D array) – Coordinates for the mutinf array.
mutinf (1D array) –
mutinf[k]
is the mutual information \(I(i:j)\) between the sitesi, j = coords[k]
.
 overlap(other, charge_sector=None, ignore_form=False, understood_infinite=False, **kwargs)[source]¶
Compute overlap
<selfother>
. Parameters
other (
MPS
) – An MPS with the same physical sites.charge_sector (None  charges 
0
) – Selects the charge sector in which the dominant eigenvector of the TransferMatrix is.None
stands for all sectors,0
stands for the sector of zero charges. If a sector is given, it assumes the dominant eigenvector is in that charge sector.ignore_form (bool) – If
False
(default), take into account the canonical formform
at each site. IfTrue
, we ignore the canonical form (i.e., whether the MPS is in left, right, mixed or no canonical form) and just contract all the_B
as they are. (This can give different results!)understood_infinite (bool) – Raise a warning to make aware of A warning about infinite MPS. Set
understood_infinite=True
to suppress the warning.**kwargs – Further keyword arguments given to
TransferMatrix.eigenvectors()
; only used for infinite boundary conditions.
 Returns
overlap – The contraction
<selfother> * self.norm * other.norm
(i.e., taking into account thenorm
of both MPS). For an infinite MPS,<selfother>
is the overlap per unit cell, i.e., the largest eigenvalue of the TransferMatrix. Return type
dtype.type
 expectation_value(ops, sites=None, axes=None)[source]¶
Expectation value
<psiopspsi>/<psipsi>
of (nsite) operator(s).Given the MPS in canonical form, it calculates nsite expectation values. For example the contraction for a twosite (n = 2) operator on site i would look like:
 .SB[i]B[i+1].             op             .SB*[i]B*[i+1].
 Parameters
ops ((list of) {
Array
 str }) – The operators, for wich the expectation value should be taken, All operators should all have the same number of legs (namely 2 n). If less than self.L operators are given, we repeat them periodically. Strings (like'Id', 'Sz'
) are translated into singlesite operators defined bysites
.sites (None  list of int) – List of site indices. Expectation values are evaluated there. If
None
(default), the entire chain is taken (clipping for finite b.c.)axes (None  (list of str, list of str)) – Two lists of each n leg labels giving the physical legs of the operator used for contraction. The first n legs are contracted with conjugated B, the second n legs with the nonconjugated B.
None
defaults to(['p'], ['p*'])
for single site operators (n = 1), or(['p0', 'p1', ... 'p{n1}'], ['p0*', 'p1*', .... 'p{n1}*'])
for n > 1.
 Returns
exp_vals – Expectation values,
exp_vals[i] = <psiops[i]psi>
, whereops[i]
acts on site(s)j, j+1, ..., j+{n1}
withj=sites[i]
. Return type
1D ndarray
Examples
Let’s prepare a state in alternating
+z>, +x>
states:>>> spin_half = tenpy.networks.site.SpinHalfSite(conserve=None) >>> p_state = ['up', [np.sqrt(0.5), np.sqrt(0.5)]]*3 >>> psi = tenpy.networks.mps.MPS.from_product_state([spin_half]*6, p_state)
One site examples (n=1):
>>> Sz = psi.expectation_value('Sz') >>> print(Sz) [0.5 0. 0.5 0. 0.5 0. ] >>> Sx = psi.expectation_value('Sx') >>> print(Sx) [ 0. 0.5 0. 0.5 0. 0.5] >>> print(psi.expectation_value(['Sz', 'Sx'])) [ 0.5 0.5 0.5 0.5 0.5 0.5] >>> print(psi.expectation_value('Sz', sites=[0, 3, 4])) [0.5 0. 0.5]
Two site example (n=2), assuming homogeneous sites:
>>> SzSx = npc.outer(psi.sites[0].Sz.replace_labels(['p', 'p*'], ['p0', 'p0*']), ... psi.sites[1].Sx.replace_labels(['p', 'p*'], ['p1', 'p1*'])) >>> print(psi.expectation_value(SzSx)) # note: len L1 for finite bc, or L for infinite [0.25 0. 0.25 0. 0.25]
Example measuring <psiSzSxpsi2> on each second site, for inhomogeneous sites:
>>> SzSx_list = [npc.outer(psi.sites[i].Sz.replace_labels(['p', 'p*'], ['p0', 'p0*']), ... psi.sites[i+1].Sx.replace_labels(['p', 'p*'], ['p1', 'p1*'])) ... for i in range(0, psi.L1, 2)] >>> print(psi.expectation_value(SzSx_list, range(0, psi.L1, 2))) [0.25 0.25 0.25]
 expectation_value_term(term, autoJW=True)[source]¶
Expectation value
<psiop_{i0}op_{i1}...op_{iN}psi>/<psipsi>
.Calculates the expectation value of a tensor product of singlesite operators acting on different sites i0, i1, … (not necessarily next to each other). In other words, evaluate the expectation value of a term
op0_i0 op1_i1 op2_i2 ...
.For example the contraction of three onesite operators on sites i0, i1=i0+1, i2=i0+3 would look like:
 .SB[i0]B[i0+1]B[i0+2]B[i0+3].          op1 op2  op3          .SB*[i0]B*[i0+1]B*[i0+2]B*[i0+3].
 Parameters
term (list of (str, int)) – List of tuples
op, i
where i is the MPS index of the site the operator named op acts on. The order inside term determines the order in which they act (in the mathematical convention: the last operator in term is rightmost, so it acts first on a ket).autoJW (bool) – If True (default), automatically insert Jordan Wigner strings for Fermions as needed.
 Returns
exp_val – The expectation value of the tensorproduct of the given onsite operators,
<psiop_i0 op_i1 ... op_iN psi>/<psipsi>
, wherepsi>
is the represented MPS. Return type
float/complex
See also
correlation_function
efficient way to evaluate many correlation functions.
Examples
>>> a = psi.expectation_value_term([('Sx', 2), ('Sz', 4)]) >>> b = psi.expectation_value_term([('Sz', 4), ('Sx', 2)]) >>> c = psi.expectation_value_multi_sites(['Sz', 'Id', 'Sx'], i0=2) >>> assert a == b == c
 expectation_value_multi_sites(operators, i0)[source]¶
Expectation value
<psiop0_{i0}op1_{i0+1}...opN_{i0+N}psi>/<psipsi>
.Calculates the expectation value of a tensor product of singlesite operators acting on different sites next to each other. In other words, evaluate the expectation value of a term
op0_i0 op1_{i0+1} op2_{i0+2} ...
, looking like this (with op short for operators, forlen(operators)=3
):.–S–B[i0]—B[i0+1]–B[i0+2]–B[i0+3]–.      op[0] op[1] op[2] op[3]      .–S–B*[i0]–B*[i0+1]B*[i0+2]B*[i0+3].Warning
This function does not automatically add JordanWigner strings! For correct handling of fermions, use
expectation_value_term()
instead. Parameters
 Returns
exp_val – The expectation value of the tensorproduct of the given onsite operators,
<psioperators[0]_{i0} operators[1]_{i0+1} ... psi>/<psipsi>
, wherepsi>
is the represented MPS. Return type
float/complex
 expectation_value_terms_sum(term_list, prefactors=None)[source]¶
Calculate expectation values for a bunch of terms and sum them up.
This is equivalent to the following expression:
sum([self.expectation_value_term(term)*strength for term, strength in term_list])
However, for effiency, the term_list is converted to an MPO and the expectation value of the MPO is evaluated.
Deprecated since version 0.4.0: prefactor will be removed in version 1.0.0. Instead, directly give just
TermList(term_list, prefactors)
as argument. Parameters
 Returns
terms_sum (list of (complex) float) – Equivalent to the expression
sum([self.expectation_value_term(term)*strength for term, strength in term_list])
._mpo – Intermediate results: the generated MPO. For a finite MPS,
terms_sum = _mpo.expectation_value(self)
, for an infinite MPSterms_sum = _mpo.expectation_value(self) * self.L
See also
expectation_value_term
evaluates a single term.
tenpy.networks.mpo.MPO.expectation_value
expectation value density of an MPO.
 correlation_function(ops1, ops2, sites1=None, sites2=None, opstr=None, str_on_first=True, hermitian=False, autoJW=True)[source]¶
Correlation function
<psiop1_i op2_jpsi>/<psipsi>
of single site operators.Given the MPS in canonical form, it calculates 2site correlation functions. For examples the contraction for a twosite operator on site i would look like:
 .SB[i]B[i+1]...B[j].           op2    op1           .SB*[i]B*[i+1]...B*[j].
Onsite terms are taken in the order
<psi  op1 op2  psi>
.If opstr is given and
str_on_first=True
, it calculates: for i < j for i > j   .SB[i]B[i+1]... B[j]. .SB[j]B[j+1]... B[i].              opstr opstr op2   op2                 op1     opstr opstr op1              .SB*[i]B*[i+1]... B*[j]. .SB*[j]B*[j+1]... B*[i].
For
i==j
, no opstr is included. Forstr_on_first=False
, the opstr on sitemin(i, j)
is always left out.Strings (like
'Id', 'Sz'
) in the arguments are translated into singlesite operators defined by theSite
on which they act. Each operator should have the two legs'p', 'p*'
.Warning
This function is only evaluating correlation functions by moving right, and hence can be inefficient if you try to vary the left end while fixing the right end. In that case, you might be better off (=faster evaluation) by using
term_correlation_function_left()
with a small for loop over the right indices. Parameters
ops1 ((list of) {
Array
 str }) – First operator of the correlation function (acting after ops2). If a list is given,ops1[i]
acts on site i of the MPS. Note that even if a list is given, we still just evaluate twosite correlations!psi.correlation_function(['A','B'], ['C', 'D'])
evaluates<A_i C_j>
for even i and even j,<B_i C_j>
for even i and odd j,<B_i C_j>
for odd i and even j, and<B_i D_j>
for odd i and odd j.ops2 ((list of) {
Array
 str }) – Second operator of the correlation function (acting before ops1). If a list is given,ops2[j]
acts on site j of the MPS.sites1 (None  int  list of int) – List of site indices i; a single int is translated to
range(0, sites1)
.None
defaults to all sitesrange(0, L)
. Is sorted before use, i.e. the order is ignored.sites2 (None  int  list of int) – List of site indices; a single int is translated to
range(0, sites2)
.None
defaults to all sitesrange(0, L)
. Is sorted before use, i.e. the order is ignored.opstr (None  (list of) {
Array
 str }) – Ignored by default (None
). Operator(s) to be inserted betweenops1
andops2
. If less thanL
operators are given, we repeat them periodically. If given as a list,opstr[r]
is inserted at site r (independent of sites1 and sites2).str_on_first (bool) – Whether the opstr is included on the site
min(i, j)
. Note the order, which is chosen that way to handle fermionic JordanWigner strings correctly. (In other words: choosestr_on_first=True
for fermions!)hermitian (bool) – Optimization flag: if
sites1 == sites2
andOps1[i]^\dagger == Ops2[i]
(which is not checked explicitly!), the resultingC[x, y]
will be hermitian. We can use that to avoid calculations, sohermitian=True
will run faster.autoJW (bool) – Ignored if opstr is given. If True, autodetermine if a JordanWigner string is needed. Works only if exclusively strings were used for op1 and op2.
 Returns
C – The correlation function
C[x, y] = <psiops1[i] ops2[j]psi>
, whereops1[i]
acts on sitei=sites1[x]
andops2[j]
on sitej=sites2[y]
. If opstr is given, it gives (forstr_on_first=True
):For
i < j
:C[x, y] = <psiops1[i] prod_{i <= r < j} opstr[r] ops2[j]psi>
.For
i > j
:C[x, y] = <psiprod_{j <= r < i} opstr[r] ops1[i] ops2[j]psi>
.For
i = j
:C[x, y] = <psiops1[i] ops2[j]psi>
.
The condition
<= r
is replaced by a strict< r
, ifstr_on_first=False
. Return type
2D ndarray
Examples
Let’s prepare a state in alternating
+z>, +x>
states:>>> spin_half = tenpy.networks.site.SpinHalfSite(conserve=None) >>> p_state = ['up', [np.sqrt(0.5), np.sqrt(0.5)]]*3 >>> psi = tenpy.networks.mps.MPS.from_product_state([spin_half]*6, p_state, "infinite")
Default arguments calculate correlations for all i and j within the MPS unit cell. To evaluate the correlation function for a single i, you can use
sites1=[i]
. Alternatively, you can useterm_correlation_function_right()
(orterm_correlation_function_left()
):>>> psi.correlation_function("Sz", "Sx") array([[ 0. , 0.25, 0. , 0.25, 0. , 0.25], [ 0. , 0. , 0. , 0. , 0. , 0. ], [ 0. , 0.25, 0. , 0.25, 0. , 0.25], [ 0. , 0. , 0. , 0. , 0. , 0. ], [ 0. , 0.25, 0. , 0.25, 0. , 0.25], [ 0. , 0. , 0. , 0. , 0. , 0. ]]) >>> psi.correlation_function("Sz", "Sx", [0]) array([[ 0. , 0.25, 0. , 0.25, 0. , 0.25]]) >>> corr1 = psi.correlation_function("Sz", "Sx", [0], range(1, 10)) >>> corr2 = psi.term_correlation_function_right([("Sz", 0)], [("Sx", 0)], 0, range(1, 10)) >>> assert np.all(np.abs(corr2  corr1) < 1.e12)
For fermions, it autodetermines that/whether a Jordan Wigner string is needed:
>>> fermion = tenpy.networks.site.FermionSite(conserve='N') >>> p_state = ['empty', 'full'] * 3 >>> psi = tenpy.networks.mps.MPS.from_product_state([fermion]*6, p_state, "finite") >>> CdC = psi.correlation_function("Cd", "C") # optionally: use `hermitian=True` >>> psi.correlation_function("C", "Cd")[1, 2] == CdC[2, 1] True >>> np.all(np.diag(CdC) == psi.expectation_value("Cd C")) # "Cd C" is equivalent to "N" True
See also
expectation_value_term
for a single combination of i and j of
A_i B_j`
.term_correlation_function_right
for correlations between multisite terms, fix left term.
term_correlation_function_left
for correlations between multisite terms, fix right term.
 term_correlation_function_right(term_L, term_R, i_L=0, j_R=None, autoJW=True, opstr=None)[source]¶
Correlation function between (multisite) terms, moving the right term, fix left term.
For
term_L = [('A', 0), ('B', 1)]
andterm_R = [('C', 0), ('D', 1)]
, calculate the correlation function \(A_{i+0} B_{i+1} C_{j+0} D_{j+1}\) for fixed i and varying j according to i_L/j_R. The terms may not overlap. For fermions, the order of the terms is following the usual mathematical convention, where term_R acts first on a physical ket. Parameters
term_L (list of (str, int)) – Each a term representing a sum of operators on different sites, e.g.,
[('Sz', 0), ('Sz', 1)]
or[('Cd', 0), ('C', 1)]
.term_R (list of (str, int)) – Each a term representing a sum of operators on different sites, e.g.,
[('Sz', 0), ('Sz', 1)]
or[('Cd', 0), ('C', 1)]
.i_L (int) – Offset added to the indices of term_L.
j_R (list of int  None) – List of offsets to be added to the indices of term_R. Is sorted before use, i.e. the order is ignored. For finite MPS, None defaults to
range(j0, L)
, where j0 is chosen such that term_R starts one site right of the term_L. For infinite MPS, None defaults torange(L, 11*L, L)
, i.e., one term per MPS unit cell for a distance of up to 10 unit cells.autoJW (bool) – Whether to automatically take care of JordanWigner strings.
opstr (str) – Force an intermediate operator string to used inbetween the terms. Can only be used in combination with
autoJW=False
.
 Returns
corrs – Values of the correlation function, one for each entry in the list j_R.
 Return type
1D array
See also
correlation_function
varying both i and j at once.
term_list_correlation_function_right
generalization to sums of terms on the left/right.
 term_correlation_function_left(term_L, term_R, i_L=None, j_R=0, autoJW=True, opstr=None)[source]¶
Correlation function between (multisite) terms, moving the left term, fix right term.
Same as
term_correlation_function_right()
, but vary index i of the left term instead of the j of the right term.
 term_list_correlation_function_right(term_list_L, term_list_R, i_L=0, j_R=None, autoJW=True, opstr=None)[source]¶
Correlation function between sums of multisite terms, moving the right sum of term.
Generalization of
term_correlation_function_right()
to the case where term_list_L and term_R are sums of terms. This function calculates<psiterm_list_L[i_L] term_list_R[j]psi> for j in j_R
.Assumes that overall terms with an odd number of operators requiring a JordanWigner string don’t contribute. (In systems conserving the fermionic particle number (parity), this is true.)
 Parameters
term_list_L (
TermList
) – Each a TermList representing the sum of terms to be applied.term_list_R (
TermList
) – Each a TermList representing the sum of terms to be applied.i_L (int) – Offset added to all the indices of term_list_L.
j_R (list of int  None) – List of offsets to be added to the indices of term_list_R. Is sorted before use, i.e. the order is ignored. For finite MPS, None defaults to
range(j0, L)
, where j0 is chosen such that term_R starts one site right of the term_L. For infinite MPS, None defaults torange(L, 11*L, L)
, i.e., one term per MPS unit cell for a distance of up to 10 unit cells.autoJW (bool) – Whether to automatically take care of JordanWigner strings.
opstr (str) – Force an intermediate operator string to be used inbetween the terms. (Even used within the term_list_L/R for terms with smallerthan maximal support.) Can only be used in combination with
autoJW=False
.
 Returns
corrs – Values of the correlation function, one for each entry in the list j_R.
 Return type
1D array
See also
term_correlation_function_right
version for a single term in both term_list_L/R.
 sample_measurements(first_site=0, last_site=None, ops=None, rng=None, norm_tol=1e12)[source]¶
Sample measurement results in the computational basis.
This function samples projective measurements on a continguous range of sites, tracing out the remaining sites.
Note that for infinite boundary conditions, the probablility of sampling a set of sigmas is not
psi.overlap(MPS.from_product_state(sigmas, ...))^2
, because the latter would poject to the set sigmas on each (translated) MPS unit cell, while this function is only projecting to them in a single MPS unit cell. Parameters
first_site (int) – Take measurements on the sites in
range(first_site, last_site + 1)
. last_site defaults toL
 1.last_site (int) – Take measurements on the sites in
range(first_site, last_site + 1)
. last_site defaults toL
 1.ops (list of str) – If not None, sample in the eigenbasis of
self.sites[i].get_op(ops[(i  first_site) % len(ops)])
and directly return the corresponding eigenvalue in sigmas.rng (
numpy.random.Generator
) – The random number generator; if None, a new numpy.random.default_rng() is generated.norm_tol (float) – Tolerance
 Returns
sigmas (list of int  list of float) – On each site the index of the local basis that was measured, as specified in the corrsponding
Site
insites
. Note that this can change depending on whether/what charges you conserve! Explicitly specifying the measurement operator will avoid that issue.weight (float) – The weight
sqrt(trace(psi><psisigmas...><sigmas...))
, i.e., the probability of measuringsigmas...>
isweigth**2
. For a finite system where we sample all sites (i.e., the trace over the compliment of the sites is trivial), this is the actual overlap<sigmas...psi>
including the phase.
 norm_test()[source]¶
Check that self is in canonical form.
 Returns
norm_error – For each site the norm error to the left and right. The error
norm_error[i, 0]
is defined as the normdifference between the following networks: theta[i]. s[i].    vs   theta*[i]. s[i].
Similarly,
norm_errror[i, 1]
is the normdifference of: .theta[i] .s[i+1]    vs   .theta*[i] .s[i+1]
 Return type
array, shape (L, 2)
 canonical_form(**kwargs)[source]¶
Bring self into canonical ‘B’ form, (re)calculate singular values.
Simply calls
canonical_form_finite()
orcanonical_form_infinite2()
. Keyword arguments are passed on to the corrsponding specialized versions.
 canonical_form_finite(renormalize=True, cutoff=0.0, envs_to_update=None)[source]¶
Bring a finite (or segment) MPS into canonical form (in place).
If any site is in
form
None
, it does not use any of the singular values S (for ‘finite’ boundary conditions, or only the very left S for ‘segment’ b.c.). If all sites have a form, it respects the form to ensure that one S is included per bond. The final state is always in rightcanonical ‘B’ form.Performs one sweep left to right doing QR decompositions, and one sweep right to left doing SVDs calculating the singular values.
 Parameters
renormalize (bool) – Whether a change in the norm should be discarded or used to update
norm
. Note that even renoramlize=True does not reset thenorm
to 1. To do that, you would rather have to setpsi.norm = 1
explicitly!cutoff (float  None) – Cutoff of singular values used in the SVDs.
envs_to_update (None  list of
MPSEnvironment
) – Clear the environments; for segment also update the left/rightmost LP/RP.
 Returns
U_L, V_R – Only returned for
'segment'
boundary conditions. The unitaries defining the new left and right Schmidt states in terms of the old ones, with legs'vL', 'vR'
. Return type
 canonical_form_infinite(**kwargs)[source]¶
Deprecated wrapper around
canonical_form_infinite1()
.
 canonical_form_infinite1(renormalize=True, tol_xi=1000000.0)[source]¶
Bring an infinite MPS into canonical form (in place).
If any site is in
form
None
, it does not use any of the singular values S. If all sites have a form, it respects the form to ensure that one S is included per bond. The final state is always in rightcanonical ‘B’ form.Proceeds in three steps, namely 1) diagonalize right and left transfermatrix on a given bond to bring that bond into canonical form, and then 2) sweep right to left, and 3) left to right to bringing other bonds into canonical form.
 Parameters
renormalize (bool) – Whether a change in the norm should be discarded or used to update
norm
. Note that even renoramlize=True does not reset thenorm
to 1. To do that, you would rather have to setpsi.norm = 1
explicitly!tol_xi (float) – Raise an error if the correlation length is larger than that (which indicates a degenerate “cat” state, e.g., for spontaneous symmetry breaking).
 canonical_form_infinite2(renormalize=True, tol=1e15, arnoldi_params=None, cutoff=1e15)[source]¶
Convert infinite MPS to canonical form.
Implementation following Algorithm 1,2 in [vanderstraeten2019].
 Parameters
renormalize (bool) – Whether a change in the norm should be discarded or used to update
norm
. Note that even renoramlize=True does not reset thenorm
to 1. To do that, you would rather have to setpsi.norm = 1
explicitly!tol (float) – Precision down to which the state should be in canonical form.
cutoff – Truncation cutoff for small singular values.
 correlation_length(target=1, tol_ev0=1e08, charge_sector=0)[source]¶
Calculate the correlation length by diagonalizing the transfer matrix.
Assumes that self is in canonical form.
Works only for infinite MPS, where the transfer matrix is a useful concept. Assuming a singlesite unit cell, any correlation function splits into \(C(A_i, B_j) = A'_i T^{ji1} B'_j\) with some parts left and right and the \(ji1\)th power of the transfer matrix in between. The largest eigenvalue is 1 (if self is properly normalized) and gives the dominant contribution of \(A'_i E_1 * 1^{ji1} * E_1^T B'_j = <A> <B>\), and the second largest one gives a contribution \(\propto \lambda_2^{ji1}\). Thus \(\lambda_2 = \exp(\frac{1}{\xi})\).
More general for a Lsite unit cell we get \(\lambda_2 = \exp(\frac{L}{\xi})\), where the xi is given in units of 1 lattice spacing in the MPS.
Warning
For a higherdimensional lattice (which the MPS class doesn’t know about), the correct unit is the lattice spacing in xdirection, and the correct formula is \(\lambda_2 = \exp(\frac{L_x}{\xi})\), where L_x is the number of lattice spacings in the infinite direction within the MPS unit cell, e.g. the number of “rings” of a cylinder in the MPS unit cell. To get to these units, divide the returned xi by the number of sites within a “ring”, for a lattice given in
N_sites_per_ring
. Parameters
target (int) – We look for the target + 1 largest eigenvalues.
tol_ev0 (float) – Print warning if largest eigenvalue deviates from 1 by more than tol_ev0.
charge_sector (None  charges 
0
) – Selects the charge sector in which the dominant eigenvector of the TransferMatrix is.None
stands for all sectors,0
stands for the zerocharge sector. Defaults to0
, i.e., assumes the dominant eigenvector is in charge sector 0.
 Returns
xi – If target = 1, return just the correlation length, otherwise an array of the target largest correlation lengths. It is measured in units of a single spacing between sites in the MPS language, see the warning above.
 Return type
float  1D array
 add(other, alpha, beta, cutoff=1e15)[source]¶
Return an MPS which represents
alphaself> + beta others>
.Works only for ‘finite’, ‘segment’ boundary conditions. For ‘segment’ boundary conditions, the virtual legs on the very left/right are assumed to correspond to each other (i.e. self and other have the same state outside of the considered segment). Takes into account
norm
. Parameters
other (
MPS
) – Another MPS of the same length to be added with self.alpha (complex float) – Prefactors for self and other. We calculate
alpha * self> + beta * other>
beta (complex float) – Prefactors for self and other. We calculate
alpha * self> + beta * other>
cutoff (float  None) – Cutoff of singular values used in the SVDs.
 Returns
sum – An MPS representing
alphaself> + beta other>
. Has same total charge as self. Return type
 apply_local_op(i, op, unitary=None, renormalize=False, cutoff=1e13, understood_infinite=False)[source]¶
Apply a local (one or multisite) operator to self.
Note that this destroys the canonical form if the local operator is nonunitary. Therefore, this function calls
canonical_form()
if necessary.For infinite MPS, it applies the operator in parallel within each unit site, i.e., really it applies \(\prod_{u \in \mathbb{Z}} (\mathrm{op}_{i + u *L})\) where
L
is the number of sites in the MPS unit cell. Parameters
i (int) – (Leftmost) index of the site(s) on which the operator should act.
op (str  npc.Array) – A physical operator acting on site i, with legs
'p', 'p*'
for a singlesite operator or with legs['p0', 'p1', ...], ['p0*', 'p1*', ...]
for an operator acting on n>=2 sites. Strings (like'Id', 'Sz'
) are translated into singlesite operators defined bysites
.unitary (None  bool) – Whether op is unitary, i.e., whether the canonical form is preserved (
True
) or whether we should callcanonical_form()
(False
).None
checks whethernorm(op dagger(op)  identity)
is smaller than cutoff.renormalize (bool) – Whether the final state should keep track of the norm (False, default) or be renormalized to have norm 1 (True).
cutoff (float) – Cutoff for singular values if op acts on more than one site (see
from_full()
). (And used as cutoff for a unspecified unitary.)understood_infinite (bool) – Raise a warning to make aware of A warning about infinite MPS. Set
understood_infinite=True
to suppress the warning.
 apply_product_op(ops, unitary=None, renormalize=False)[source]¶
Apply a (global) product of local onsite operators to self.
Note that this destroys the canonical form if any local operator is nonunitary. Therefore, this function calls
canonical_form()
if necessary.The result is equivalent to the following loop, but more efficient by avoiding intermediate calls to
canonical_form()
inside the loop:for i, op in enumerate(ops): self.apply_local_op(i, op, unitary, renormalize, cutoff)
 Parameters
ops ((list of) str  npc.Array) – List of onsite operators to apply on each site, with legs
'p', 'p*'
. Strings (like'Id', 'Sz'
) are translated into singlesite operators defined bysites
.unitary (None  bool) – Whether op is unitary, i.e., whether the canonical form is preserved (
True
) or whether we should callcanonical_form()
(False
).None
checks whethermax(norm(op dagger(op)  identity) for op in ops) < 1.e14
renormalize (bool) – Whether the final state should keep track of the norm (False, default) or be renormalized to have norm 1 (True).
 perturb(randomize_params=None, close_1=True, canonicalize=None)[source]¶
Locally perturb the state a little bit; in place.
 Parameters
randomize_params (dict) – Parameters for the
RandomUnitaryEvolution
.close_1 (bool) – Select the default
RandomUnitaryEvolution.distribution_func
to be used, if close_1 is True, useU_close_1()
for complex andO_close_1()
for real MPS; for close_1 False useCUE()
orCRE()
, respectively.canonicalize (bool) – Wether to call psi.canonical_from in the end. Defaults to ``not close_1`.
 swap_sites(i, swap_op='auto', trunc_par=None)[source]¶
Swap the two neighboring sites i and i+1 (inplace).
Exchange two neighboring sites: form theta, ‘swap’ the physical legs and split with an svd. While the ‘swap’ is just a transposition/relabeling for bosons, one needs to be careful about the signs from JordanWigner strings for fermions.
 Parameters
i (int) – Swap the two sites at positions i and i+1.
swap_op (
None
'auto', 'autoInv'
Array
) – The operator used to swap the phyiscal legs of the twosite wave function theta. ForNone
, just transpose/relabel the legs. Alternative give an npcArray
which represents the full operator used for the swap. Should have legs['p0', 'p1', 'p0*', 'p1*']
whith'p0', 'p1*'
contractible. For'auto'
we try to be smart about fermionic signs, see note below.trunc_par (dict) – Parameters for truncation, see
truncation
.
 Returns
trunc_err – The error of the represented state introduced by the truncation after the swap.
 Return type
Notes
For fermions, it’s crucial to use the correct swap_op. The swap_op is a twosite operator exchanging ‘p0’ and ‘p1’ legs. For bosons, this is really just a relabeling (done for
swap_op=None
). Alternatively, you can construct the operator explicitly like this:siteL, siteR = psi.sites[i], psi.sites[i+1] dL, dR = siteL.dim, siteR.dim legL, legR = siteL.leg, siteR.leg swap_op_dense = np.eye(dL*dR) swap_op = npc.Array.from_ndarray(swap_op_dense.reshape([dL, dR, dL, dR]), [legL, legR, legL.conj(), legR.conj()], labels=['p1', 'p0', 'p0*', 'p1*'])
However, for fermions we need to be very careful about the JordanWigner strings. Let’s derive how the operator should look like.
You can write a state as
\[\psi> = \sum_{[n_j]} \psi_{[n_j]} \prod_j (c^\dagger_j)^{n_j} vac>\]where
[n_j]
denontes a set of \(n_j \in [0, 1]\) for each physical site j and the product over j is taken in increasing order. Let \(P\) be the operator switchingi <> i+1
, with inverse \(P^\dagger\). Then:\[\begin{split}P \psi> = \sum_{[n_j]} \psi_{[n_i]} P \prod_j (c^\dagger_i)^{n_j} vac> \\ = \sum_{[n_j]} \psi_{[n_i]} P \prod_j (c^\dagger_i)^{n_j} vac>\end{split}\]When \(P\) acts on the product of \(c^\dagger_{i}\) operators, it commutes \((c^\dagger_i)^{n_i}\) with \((c^\dagger_{i+1})^{n_{i+1}}\). This gives a a sign \((1)^{n_i * n_{i+1}}\). We must hence include this sign in the swap operator. The n_i in the equations above is given by
JW_exponent
. This leads to the following swap operator used for fermions withswap_op='auto'
, suitable to just permute sites:siteL, siteR = psi.sites[i], psi.sites[i+1] dL, dR = siteL.dim, siteR.dim legL, legR = siteL.leg, siteR.leg n_i_n_j = np.outer(siteL.JW_exponent, siteR.JW_exponent).reshape(dL*dR) swap_op_dense = np.diag((1)**n_i_n_j) swap_op = npc.Array.from_ndarray(swap_op_dense.reshape([dL, dR, dL, dR]), [legL, legR, legL.conj(), legR.conj()], labels=['p1', 'p0', 'p0*', 'p1*'])
In some cases you might want to use a more complicated swap operator. As outlined in (the appendix of) [shapourian2017], a typical hamiltonian of the form \(H = t \sum_i c_i^\dagger c_{i+1} + h.c. + \text{density interaction}\) is invariant under a reflection \(R\) acting as \(R c^e_x R^\dagger = i c^o_{x}\) and \(R c^o_x R^\dagger = i c^e_{x}\) for even/odd fermion sites. The following code includes the factor of \(i\), or rather \(i\) since we have creation operators, into the swap operator and is used with
swap_op='autoInv'
:siteL, siteR = psi.sites[i], psi.sites[i+1] dL, dR = siteL.dim, siteR.dim legL, legR = siteL.leg, siteR.leg n_i = np.outer(siteL.JW_exponent, np.ones(dR)).reshape(dL*dR) n_j = np.outer(np.ones(dL), siteR.JW_exponent).reshape(dL*dR) swap_op_dense = np.diag((1)**(n_i * n_j) * (1.j)**n_i * (1.j)**n_j) swap_op = npc.Array.from_ndarray(swap_op_dense.reshape([dL, dR, dL, dR]), [legL, legR, legL.conj(), legR.conj()], labels=['p1', 'p0', 'p0*', 'p1*'])
 permute_sites(perm, swap_op='auto', trunc_par=None, verbose=None)[source]¶
Applies the permutation perm to the state (inplace).
Deprecated since version 0.8.0: Drop / ignore verbose argument, never print something.
 Parameters
perm (ndarray[ndim=1, int]) – The applied permutation, such that
psi.permute_sites(perm)[i] = psi[perm[i]]
(where[i]
indicates the ith site).swap_op (
None
'auto', 'autoInv'
Array
) – The operator used to swap the phyiscal legs of a twosite wave function theta, seeswap_sites()
.trunc_par (dict) – Parameters for truncation, see
truncation
.
 Returns
trunc_err – The error of the represented state introduced by the truncation after the swaps.
 Return type
 compute_K(perm, swap_op='auto', trunc_par=None, canonicalize=1e06, verbose=None, expected_mean_k=0.0)[source]¶
Compute the momentum quantum numbers of the entanglement spectrum for 2D states.
Works for an infinite MPS living on a cylinder, infinitely long in x direction and with periodic boundary conditions in y directions. If the state is invariant under ‘rotations’ around the cylinder axis, one can find the momentum quantum numbers of it. (The rotation is nothing more than a translation in y.) This function permutes some sites (on a copy of self) to enact the rotation, and then finds the dominant eigenvector of the mixed transfer matrix to get the quantum numbers, along the lines of [pollmann2012], see also (the appendix and Fig. 11 in the arXiv version of) [cincio2013].
Deprecated since version 0.8.0: Drop / ignore verbose argument, never print something.
 Parameters
perm (1D ndarray 
Lattice
) – Permuation to be applied to the physical indices, seepermute_sites()
. If a lattice is given, we use it to read out the lattice structure and shift each site by one latticevector in ydirection (assuming periodic boundary conditions). (If you have aCouplingModel
, give its lat attribute for this argument)swap_op (
None
'auto', 'autoInv'
Array
) – The operator used to swap the phyiscal legs of a twosite wave function theta, seeswap_sites()
.trunc_par (dict) – Parameters for truncation, see
truncation
.canonicalize (float) – Check that self is in canonical form; call
canonical_form()
ifnorm_test()
yieldsnp.linalg.norm(self.norm_test()) > canonicalize
.expected_mean_k (float) – As explained in [cincio2013], the returned W is extracted as eigenvector of a mixed transfer matrix, and hence has an undefined phase. We fix the overall phase such that
sum(s[j]**2 exp(iK[j]) == np.sum(W) = np.exp(1.j*expected_mean_k)
.
 Returns
U (
Array
) – Unitary representation of the applied permutation on left Schmidt states.W (ndarray) – 1D array of the form
S**2 exp(i K)
, where S are the Schmidt values on the left bond. You can usenp.abs()
andnp.angle()
to extract the (squared) Schmidt values S and momenta K from W.q (
LegCharge
) – LegCharge corresponding to W.ov (complex) – The eigenvalue of the mixed transfer matrix <psiTpsi> per
L
sites. An absolute value different smaller than 1 indicates that the state is not invariant under the permutation or that the truncation error trunc_err was too large!trunc_err (
TruncationError
) – The error of the represented state introduced by the truncation after swaps when performing the truncation.
 compress(options)[source]¶
Compresss an MPS.
Options
 config MPS_compress¶
option summary By default (``None``) this feature is disabled. [...]
Whether to combine legs into pipes. This combines the virtual and [...]
Mandatory. [...]
init_env_data (from Sweep) in DMRGEngine.init_env
Dictionary as returned by ``self.env.get_initialization_data()`` from [...]
lanczos_params (from Sweep) in Sweep
Lanczos parameters as described in :cfg:config:`Lanczos`.
max_sweeps (from VariationalCompression) in VariationalCompression
Minimum and maximum number of sweeps to perform for the compression.
min_sweeps (from VariationalCompression) in VariationalCompression
Minimum and maximum number of sweeps to perform for the compression.
orthogonal_to (from Sweep) in DMRGEngine.init_env
Deprecated in favor of the `orthogonal_to` function argument (forwarded fro [...]
Number of sweeps to be performed without optimization to update the environment.
start_env_sites (from VariationalCompression) in VariationalCompression
Number of sites to contract for the inital LP/RP environment in case of inf [...]
tol_theta_diff (from VariationalCompression) in VariationalCompression
Stop after less than `max_sweeps` sweeps if the 1site wave function change [...]
Truncation parameters as described in :cfg:config:`truncation`.

option compression_method:
'SVD'  'variational'
¶ Mandatory. Selects the method to be used for compression. For the SVD compression, trunc_params is the only other option used.
 option trunc_params: dict¶
Truncation parameters as described in
truncation
.

option compression_method:
 compress_svd(trunc_par)[source]¶
Compress self with a single sweep of SVDs; in place.
Perform a single rightsweep of QR/SVD without truncation, followed by a leftsweep with truncation, very much like
canonical_form_finite()
.Warning
In case of a strong compression, this does not find the optimal, global solution.
 Parameters
trunc_par (dict) – Parameters for truncation, see
truncation
.