# CouplingModel¶

Inheritance Diagram

Methods

 CouplingModel.__init__(lattice[, …]) Initialize self. CouplingModel.add_coupling(strength, u1, …) Add twosite coupling terms to the Hamiltonian, summing over lattice sites. CouplingModel.add_coupling_term(strength, i, …) Add a two-site coupling term on given MPS sites. Add an exponentially decaying long-range coupling. CouplingModel.add_local_term(strength, term) Add a single term to self. CouplingModel.add_multi_coupling(strength, ops) Add multi-site coupling terms to the Hamiltonian, summing over lattice sites. Add a general M-site coupling term on given MPS sites. CouplingModel.add_onsite(strength, u, opname) Add onsite terms to onsite_terms. CouplingModel.add_onsite_term(strength, i, op) Add an onsite term on a given MPS site. Sum of all coupling_terms. Sum of all onsite_terms. CouplingModel.calc_H_MPO([tol_zero]) Calculate MPO representation of the Hamiltonian. CouplingModel.calc_H_bond([tol_zero]) calculate H_bond from coupling_terms and onsite_terms. CouplingModel.calc_H_onsite([tol_zero]) Calculate H_onsite from self.onsite_terms. Add an external flux to the coupling strength. Repeat the unit cell for infinite MPS boundary conditions; in place. CouplingModel.from_hdf5(hdf5_loader, h5gr, …) Load instance from a HDF5 file. CouplingModel.group_sites([n, grouped_sites]) Modify self in place to group sites. CouplingModel.save_hdf5(hdf5_saver, h5gr, …) Export self into a HDF5 file. Sanity check, raises ValueErrors, if something is wrong.
class tenpy.models.model.CouplingModel(lattice, bc_coupling=None, explicit_plus_hc=False)[source]

Base class for a general model of a Hamiltonian consisting of two-site couplings.

In this class, the terms of the Hamiltonian are specified explicitly as OnsiteTerms or CouplingTerms.

Deprecated since version 0.4.0: bc_coupling will be removed in 1.0.0. To specify the full geometry in the lattice, use the bc parameter of the Lattice.

Parameters
• lattice (Lattice) – The lattice defining the geometry and the local Hilbert space(s).

• bc_coupling ((iterable of) {'open' | 'periodic' | int}) – Boundary conditions of the couplings in each direction of the lattice. Defines how the couplings are added in add_coupling(). A single string holds for all directions. An integer shift means that we have periodic boundary conditions along this direction, but shift/tilt by -shift*lattice.basis[0] (~cylinder axis for bc_MPS='infinite') when going around the boundary along this direction.

• explicit_plus_hc (bool) – If True, the Hermitian conjugate of the MPO is computed at runtime, rather than saved in the MPO.

onsite_terms

The OnsiteTerms ordered by category.

Type

{‘category’: OnsiteTerms}

coupling_terms

The CouplingTerms ordered by category. In case we’ve added terms with more than 2 operators, e.g. with add_multi_coupling(), the values of the dictionary may also be MultiCouplingTerms.

Type

{‘category’: CouplingTerms}

exp_decaying_terms

Collection of coupling terms with exponentially decaying long-range interactions. Filled by add_exponentially_decaying_coupling().

Type

ExponentiallyDecayingTerms

explicit_plus_hc

If True, self represents the terms in onsite_terms, coupling_terms and exp_decaying_terms plus their hermitian conjugate added. The flag will be carried on to the MPO, which will have a reduced bond dimension if self.add_coupling(..., plus_hc=True) was used. Note that add_onsite(), add_coupling(), add_multi_coupling() and add_exponentially_decaying_coupling() respect this flag, ensuring that the represented Hamiltonian is indepentent of the explicit_plus_hc flag.

Type

bool

test_sanity()[source]

Sanity check, raises ValueErrors, if something is wrong.

add_local_term(strength, term, category=None, plus_hc=False)[source]

Add a single term to self.

The repesented term is strength times the product of the operators given in terms. Each operator is specified by the name and the site it acts on; the latter given by a lattice index, see Lattice.

Depending on the length of term, it can add an onsite term or a coupling term to onsite_terms or coupling_terms, respectively.

Parameters
• strength (float/complex) – The prefactor of the term.

• term (list of (str, array_like)) – List of tuples (opname, lat_idx) where opname is a string describing the operator acting on the site given by the lattice index lat_idx. Here, lat_idx is for example [x, y, u] for a 2D lattice, with u being the index within the unit cell.

• category – Descriptive name used as key for onsite_terms or coupling_terms.

• plus_hc (bool) – If True, the hermitian conjugate of the terms is added automatically.

add_onsite(strength, u, opname, category=None, plus_hc=False)[source]

Add onsite terms to onsite_terms.

Adds $$\sum_{\vec{x}} strength[\vec{x}] * OP$$ to the represented Hamiltonian, where the operator OP=lat.unit_cell[u].get_op(opname) acts on the site given by a lattice index (x_0, ..., x_{dim-1}, u),

The necessary terms are just added to onsite_terms; doesn’t rebuild the MPO.

Parameters
• strength (scalar | array) – Prefactor of the onsite term. May vary spatially. If an array of smaller size is provided, it gets tiled to the required shape.

• u (int) – Picks a Site lat.unit_cell[u] out of the unit cell.

• opname (str) – valid operator name of an onsite operator in lat.unit_cell[u].

• category (str) – Descriptive name used as key for onsite_terms. Defaults to opname.

• plus_hc (bool) – If True, the hermitian conjugate of the terms is added automatically.

add_coupling()

Add a terms acting on two sites.

add_onsite_term()

Add a single term without summing over $$vec{x}$$.

add_onsite_term(strength, i, op, category=None, plus_hc=False)[source]

Add an onsite term on a given MPS site.

Wrapper for self.onsite_terms[category].add_onsite_term(...).

Parameters
• strength (float) – The strength of the term.

• i (int) – The MPS index of the site on which the operator acts. We require 0 <= i < L.

• op (str) – Name of the involved operator.

• category (str) – Descriptive name used as key for onsite_terms. Defaults to op.

• plus_hc (bool) – If True, the hermitian conjugate of the term is added automatically.

all_onsite_terms()[source]

Sum of all onsite_terms.

add_coupling(strength, u1, op1, u2, op2, dx, op_string=None, str_on_first=True, raise_op2_left=False, category=None, plus_hc=False)[source]

Add twosite coupling terms to the Hamiltonian, summing over lattice sites.

Represents couplings of the form $$\sum_{x_0, ..., x_{dim-1}} strength[shift(\vec{x})] * OP0 * OP1$$, where OP0 := lat.unit_cell[u0].get_op(op0) acts on the site (x_0, ..., x_{dim-1}, u1), and OP1 := lat.unit_cell[u1].get_op(op1) acts on the site (x_0+dx[0], ..., x_{dim-1}+dx[dim-1], u1). Possible combinations x_0, ..., x_{dim-1} are determined from the boundary conditions in possible_couplings().

The coupling strength may vary spatially if the given strength is a numpy array. The correct shape of this array is the coupling_shape returned by tenpy.models.lattice.possible_couplings() and depends on the boundary conditions. The shift(...) depends on dx, and is chosen such that the first entry strength[0, 0, ...] of strength is the prefactor for the first possible coupling fitting into the lattice if you imagine open boundary conditions.

The necessary terms are just added to coupling_terms; this function does not rebuild the MPO.

Deprecated since version 0.4.0: The arguments str_on_first and raise_op2_left will be removed in version 1.0.0.

Parameters
• strength (scalar | array) – Prefactor of the coupling. May vary spatially (see above). If an array of smaller size is provided, it gets tiled to the required shape.

• u1 (int) – Picks the site lat.unit_cell[u1] for OP1.

• op1 (str) – Valid operator name of an onsite operator in lat.unit_cell[u1] for OP1.

• u2 (int) – Picks the site lat.unit_cell[u2] for OP2.

• op2 (str) – Valid operator name of an onsite operator in lat.unit_cell[u2] for OP2.

• dx (iterable of int) – Translation vector (of the unit cell) between OP1 and OP2. For a 1D lattice, a single int is also fine.

• op_string (str | None) – Name of an operator to be used between the OP1 and OP2 sites. Typical use case is the phase for a Jordan-Wigner transformation. The operator should be defined on all sites in the unit cell. If None, auto-determine whether a Jordan-Wigner string is needed, using op_needs_JW().

• str_on_first (bool) – Whether the provided op_string should also act on the first site. This option should be chosen as True for Jordan-Wigner strings. When handling Jordan-Wigner strings we need to extend the op_string to also act on the ‘left’, first site (in the sense of the MPS ordering of the sites given by the lattice). In this case, there is a well-defined ordering of the operators in the physical sense (i.e. which of op1 or op2 acts first on a given state). We follow the convention that op2 acts first (in the physical sense), independent of the MPS ordering. Deprecated.

• raise_op2_left (bool) – Raise an error when op2 appears left of op1 (in the sense of the MPS ordering given by the lattice). Deprecated.

• category (str) – Descriptive name used as key for coupling_terms. Defaults to a string of the form "{op1}_i {op2}_j".

• plus_hc (bool) – If True, the hermitian conjugate of the terms is added automatically.

Examples

When initializing a model, you can add a term $$J \sum_{<i,j>} S^z_i S^z_j$$ on all nearest-neighbor bonds of the lattice like this:

>>> J = 1.  # the strength
>>> for u1, u2, dx in self.lat.pairs['nearest_neighbors']:
...     self.add_coupling(J, u1, 'Sz', u2, 'Sz', dx)


The strength can be an array, which gets tiled to the correct shape. For example, in a 1D Chain with an even number of sites and periodic (or infinite) boundary conditions, you can add alternating strong and weak couplings with a line like:

>>> self.add_coupling([1.5, 1.], 0, 'Sz', 0, 'Sz', dx)


Make sure to use the plus_hc argument if necessary, e.g. for hoppings:

>>> for u1, u2, dx in self.lat.pairs['nearest_neighbors']:
...     self.add_coupling(t, u1, 'Cd', u2, 'C', dx, plus_hc=True)


Alternatively, you can add the hermitian conjugate terms explictly. The correct way is to complex conjugate the strength, take the hermitian conjugate of the operators and swap the order (including a swap u1 <-> u2), and use the opposite direction -dx, i.e. the h.c. of add_coupling(t, u1, 'A', u2, 'B', dx) is add_coupling(np.conj(t), u2, hc('B'), u1, hc('A'), -dx), where hc takes the hermitian conjugate of the operator names, see get_hc_op_name(). For spin-less fermions (FermionSite), this would be

>>> t = 1.  # hopping strength
>>> for u1, u2, dx in self.lat.pairs['nearest_neighbors']:
...     self.add_coupling(t, u1, 'Cd', u2, 'C', dx)
...     self.add_coupling(np.conj(t), u2, 'Cd', u1, 'C', -dx)  # h.c.


With spin-full fermions (SpinHalfFermions), it could be:

>>> for u1, u2, dx in self.lat.pairs['nearest_neighbors']:
...     self.add_coupling(t, u1, 'Cdu', u2, 'Cd', dx)  # Cdagger_up C_down
...     self.add_coupling(np.conj(t), u2, 'Cdd', u1, 'Cu', -dx)  # h.c. Cdagger_down C_up


Note that the Jordan-Wigner strings for the fermions are added automatically!

add_onsite()

Add terms acting on one site only.

add_multi_coupling_term()

for terms on more than two sites.

add_coupling_term()

Add a single term without summing over $$vec{x}$$.

add_coupling_term(strength, i, j, op_i, op_j, op_string='Id', category=None, plus_hc=False)[source]

Add a two-site coupling term on given MPS sites.

Wrapper for self.coupling_terms[category].add_coupling_term(...).

Warning

This function does not handle Jordan-Wigner strings! You might want to use add_local_term() instead.

Parameters
• strength (float) – The strength of the coupling term.

• i (int) – The MPS indices of the two sites on which the operator acts. We require 0 <= i < N_sites and i < j, i.e., op_i acts “left” of op_j. If j >= N_sites, it indicates couplings between unit cells of an infinite MPS.

• j (int) – The MPS indices of the two sites on which the operator acts. We require 0 <= i < N_sites and i < j, i.e., op_i acts “left” of op_j. If j >= N_sites, it indicates couplings between unit cells of an infinite MPS.

• op1 (str) – Names of the involved operators.

• op2 (str) – Names of the involved operators.

• op_string (str) – The operator to be inserted between i and j.

• category (str) – Descriptive name used as key for coupling_terms. Defaults to a string of the form "{op1}_i {op2}_j".

• plus_hc (bool) – If True, the hermitian conjugate of the term is added automatically.

all_coupling_terms()[source]

Sum of all coupling_terms.

add_multi_coupling(strength, ops, _deprecate_1='DEPRECATED', _deprecate_2='DEPRECATED', op_string=None, category=None, plus_hc=False)[source]

Add multi-site coupling terms to the Hamiltonian, summing over lattice sites.

Represents couplings of the form $$sum_{\vec{x}} strength[shift(\vec{x})] * OP_0 * OP_1 * ... * OP_{M-1}$$, involving M operators. Here, $$OP_m$$ stands for the operator defined by the m-th tuple (opname, dx, u) given in the argument ops, which determines the position $$\vec{x} + \vec{dx}$$ and unit-cell index u of the site it acts on; the actual operator is given by self.lat.unit_cell[u].get_op(opname).

The coupling strength may vary spatially if the given strength is a numpy array. The correct shape of this array is the coupling_shape returned by tenpy.models.lattice.possible_multi_couplings() and depends on the boundary conditions. The shift(...) depends on the dx entries of ops and is chosen such that the first entry strength[0, 0, ...] of strength is the prefactor for the first possible coupling fitting into the lattice if you imagine open boundary conditions.

The necessary terms are just added to coupling_terms; this function does not rebuild the MPO.

Deprecated since version 0.6.0: We switched from the three arguments u0, op0 and other_op with other_ops=[(u1, op1, dx1), (op2, u2, dx2), ...] to a single, equivalent argment ops which should now read ops=[(op0, dx0, u0), (op1, dx1, u1), (op2, dx2, u2), ...], where dx0 = [0]*self.lat.dim. Note the changed order inside the tuples!

Parameters
• strength (scalar | array) – Prefactor of the coupling. May vary spatially, and is tiled to the required shape.

• ops (list of (opname, dx, u)) – Each tuple determines one operator of the coupling, see the description above. opname (str) is the name of the operator, dx (list of length lat.dim) is a translation vector, and u (int) is the index of lat.unit_cell on which the operator acts. The first entry of ops corresponds to $$OP_0$$ and acts last in the physical sense.

• op_string (str | None) –

If a string is given, we use this as the name of an operator to be used inbetween the operators, excluding the sites on which any operators act. This operator should be defined on all sites in the unit cell.

If None, auto-determine whether a Jordan-Wigner string is needed (using op_needs_JW()) for each of the segments inbetween the operators and also on the sites of the left operators.

• category (str) – Descriptive name used as key for coupling_terms. Defaults to a string of the form "{op0}_i {other_ops[0]}_j {other_ops[1]}_k ...".

• plus_hc (bool) – If True, the hermitian conjugate of the terms is added automatically.

Examples

A call to add_coupling() with arguments add_coupling(strength, u1, 'A', u2, 'B', dx) is equivalent to the following:

>>> dx_0 = [0] * self.lat.dim  # = [0] for a 1D lattice, [0, 0] in 2D
>>> self.add_multi_coupling(strength, [('A', dx_0, u1), ('B', dx, u2)])


To explicitly add the hermitian conjugate, you need to take the complex conjugate of the strength, reverse the order of the operators and take the hermitian conjugates of the individual operator names:

>>> self.add_multi_coupling(np.conj(strength), [(hc('B'), dx, u2), (hc('A'), dx_0, u1)])  # h.c.


add_onsite()

Add terms acting on one site only.

add_coupling()

Add terms acting on two sites.

add_multi_coupling_term()

Add a single term, not summing over the possible $$\vec{x}$$.

add_multi_coupling_term(strength, ijkl, ops_ijkl, op_string, category=None, plus_hc=False)[source]

Add a general M-site coupling term on given MPS sites.

Wrapper for self.coupling_terms[category].add_multi_coupling_term(...).

Warning

This function does not handle Jordan-Wigner strings! You might want to use add_local_term() instead.

Parameters
• strength (float) – The strength of the coupling term.

• ijkl (list of int) – The MPS indices of the sites on which the operators acts. With i, j, k, … = ijkl, we require that they are ordered ascending, i < j < k < ... and that 0 <= i < N_sites. Inidces >= N_sites indicate couplings between different unit cells of an infinite MPS.

• ops_ijkl (list of str) – Names of the involved operators on sites i, j, k, ….

• op_string (list of str) – Names of the operator to be inserted between the operators, e.g., op_string[0] is inserted between i and j.

• category (str) – Descriptive name used as key for coupling_terms. Defaults to a string of the form "{op0}_i {op1}_j {op2}_k ...".

• plus_hc (bool) – If True, the hermitian conjugate of the term is added automatically.

add_exponentially_decaying_coupling(strength, lambda_, op_i, op_j, subsites=None, op_string=None, plus_hc=False)[source]

Add an exponentially decaying long-range coupling.

$strength \sum_{i < j} \lambda^{|i-j|} A_{subsites[i]} B_{subsites[j]}$

Where the operator A is given by op_i, and B is given by op_j. Note that the sum over i,j is long-range, for infinite systems going beyond the MPS unit cell. Moreover, note that the distance in the exponent is the distance within subsites.

Parameters
• strength (float) – Overall prefactor.

• lambda (float) – Decay-rate

• op_i (string) – Names for the operators.

• op_j (string) – Names for the operators.

• subsites (None | 1D array) – Selects a subset of sites within the MPS unit cell on which the operators act. Needs to be sorted. None selects all sites.

• op_string (None | str) – The operator to be inserted between A and B; If None, this function checks whether a fermionic "JW" string is needed for the given operators; in this case the right op_j acts first.

• plus_hc (bool) – If True, the hermitian conjugate of the term is added automatically.

Examples

At least for simple enough 1D chains (or ladders), you can use fit_with_sum_of_exp() to approximate a long-range function with a few sum of exponentials and then add them with this function.

>>> def decay(x):
...     return np.exp(-0.1*x) / x**2
>>> from tenpy.tools.fit import fit_with_sum_of_exp, sum_of_exp
>>> n_exp = 5
>>> fit_range = 50
>>> lam, pref = fit_with_sum_of_exp(decay, n_exp, fit_range)
>>> x = np.arange(1, fit_range + 1)
>>> print('error in fit:', np.sum(np.abs(decay(x) - sum_of_exp(lam, pref, x))))
error in fit: 0.00010731105234095347
>>> for pr, la in zip(pref, lam):

calc_H_onsite(tol_zero=1e-15)[source]

Calculate H_onsite from self.onsite_terms.

Deprecated since version 0.4.0: This function will be removed in 1.0.0. Replace calls to this function by self.all_onsite_terms().remove_zeros(tol_zero).to_Arrays(self.lat.mps_sites()). You might also want to take explicit_plus_hc into account.

Parameters

tol_zero (float) – prefactors with abs(strength) < tol_zero are considered to be zero.

Returns

• H_onsite (list of npc.Array)

• onsite terms of the Hamiltonian. If explicit_plus_hc is True, – Hermitian conjugates of the onsite terms will be included.

calc_H_bond(tol_zero=1e-15)[source]

calculate H_bond from coupling_terms and onsite_terms.

Parameters

tol_zero (float) – prefactors with abs(strength) < tol_zero are considered to be zero.

Returns

H_bond – Bond terms as required by the constructor of NearestNeighborModel. Legs are ['p0', 'p0*', 'p1', 'p1*']

Return type

list of Array

:raises ValueError : if the Hamiltonian contains longer-range terms.:

calc_H_MPO(tol_zero=1e-15)[source]

Calculate MPO representation of the Hamiltonian.

Uses onsite_terms and coupling_terms to build an MPOGraph (and then an MPO).

Parameters

tol_zero (float) – Prefactors with abs(strength) < tol_zero are considered to be zero.

Returns

H_MPO – MPO representation of the Hamiltonian.

Return type

MPO

coupling_strength_add_ext_flux(strength, dx, phase)[source]

Add an external flux to the coupling strength.

When performing DMRG on a “cylinder” geometry, it might be useful to put an “external flux” through the cylinder. This means that a particle hopping around the cylinder should pick up a phase given by the external flux [resta1998]. This is also called “twisted boundary conditions” in literature. This function adds a complex phase to the strength array on some bonds, such that particles hopping in positive direction around the cylinder pick up exp(+i phase).

Warning

For the sign of phase it is important that you consistently use the creation operator as op1 and the annihilation operator as op2 in add_coupling().

Parameters
• strength (scalar | array) – The strength to be used in add_coupling(), when no external flux would be present.

• dx (iterable of int) – Translation vector (of the unit cell) between op1 and op2 in add_coupling().

• phase (iterable of float) – The phase of the external flux for hopping in each direction of the lattice. E.g., if you want flux through the cylinder on which you have an infinite MPS, you should give phase=[0, phi] souch that particles pick up a phase phi when hopping around the cylinder.

Returns

strength – The strength array to be used as strength in add_coupling() with the given dx.

Return type

complex array

Examples

Let’s say you have an infinite MPS on a cylinder, and want to add nearest-neighbor hopping of fermions with the FermionSite. The cylinder axis is the x-direction of the lattice, so to put a flux through the cylinder, you want particles hopping around the cylinder to pick up a phase phi given by the external flux.

>>> strength = 1. # hopping strength without external flux
>>> phi = np.pi/4 # determines the external flux strength
>>> strength_with_flux = self.coupling_strength_add_ext_flux(strength, dx, [0, phi])
>>> for u1, u2, dx in self.lat.pairs['nearest_neighbors']:
...     self.add_coupling(strength_with_flux, u1, 'Cd', u2, 'C', dx)
...     self.add_coupling(np.conj(strength_with_flux), u2, 'Cd', u1, 'C', -dx)

enlarge_mps_unit_cell(factor=2)[source]

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

This has to be done after finishing initialization and can not be reverted.

Parameters

factor (int) – The new number of sites in the MPS unit cell will be increased from N_sites to factor*N_sites_per_ring. Since MPS unit cells are repeated in the x-direction in our convetion, the lattice shape goes from (Lx, Ly, ..., Lu) to (Lx*factor, Ly, ..., Lu).

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

group_sites(n=2, grouped_sites=None)[source]

Modify self in place 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).

This has to be done after finishing initialization and can not be reverted.

Parameters
• n (int) – Number of sites to be grouped together.

• grouped_sites (None | list of GroupedSite) – The sites grouped together.

Returns

grouped_sites – The sites grouped together.

Return type

list of GroupedSite

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().

This implementation saves the content of __dict__ with save_dict_content(), storing the format under the attribute 'format'.

Parameters
• hdf5_saver (Hdf5Saver) – Instance of the saving engine.

• h5gr (:classGroup) – HDF5 group which is supposed to represent self.

• subpath (str) – The name of h5gr with a '/' in the end.