HofstadterBosons¶
full name: tenpy.models.hofstadter.HofstadterBosons
parent module:
tenpy.models.hofstadter
type: class
Inheritance Diagram
Methods

Initialize self. 

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

Add a twosite coupling term on given MPS sites. 

Add a single term to self. 

Add onsite terms to 

Add an onsite term on a given MPS site. 
Sum of all 

Sum of all 


Calculate MPO representation of the Hamiltonian. 

calculate H_bond from 

Calculate the bond Hamiltonian from the MPO Hamiltonian. 

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. 

Load instance from a HDF5 file. 

Modify self in place to group sites. 

Initialize a lattice for the given model parameters. 

Define the local Hilbert space and operators; needs to be implemented in subclasses. 

Add the onsite and coupling terms to the model; subclasses should implement this. 

Export self into a HDF5 file. 
Sanity check, raises ValueErrors, if something is wrong. 

class
tenpy.models.hofstadter.
HofstadterBosons
(model_params)[source]¶ Bases:
tenpy.models.model.CouplingMPOModel
Bosons on a square lattice with magnetic flux.
For now, the Hamiltonian reads:
\[\begin{split}H =  \sum_{x, y} \mathtt{Jx} (e^{i \mathtt{phi}_{x,y} } a^\dagger_{x+1,y} a_{x,y} + h.c.) \\  \sum_{x, y} \mathtt{Jy} (e^{i \mathtt{phi}_{x,y} } a^\dagger_{x,y+1} a_{x,y} + h.c.) \\ + \sum_{x, y} \frac{\mathtt{U}}{2} n_{x,y} (n_{x,y}  1)  \mathtt{mu} n_{x,y}\end{split}\]where \(e^{i \mathtt{phi}_{x,y} }\) is a complex AharonovBohm hopping phase, depending on lattice coordinates and gauge choice (see
tenpy.models.hofstadter.gauge_hopping()
).All parameters are collected in a single dictionary model_params, which is turned into a
Config
object. Parameters
model_params (
Config
) – Parameters for the model. SeeHofstadterBosons
below.
Options

config
HofstadterBosons
¶ option summary bc_MPS (from CouplingMPOModel) in FermionChain.init_lattice
Boundary conditions for the MPS.
bc_x (from CouplingMPOModel) in FermionChain.init_lattice
``"open"  "periodic"``. [...]
bc_y (from CouplingMPOModel) in FermionChain.init_lattice
``"cylinder"  "ladder"``; only read out for 2D lattices. [...]
What quantum number to conserve.
explicit_plus_hc (from CouplingMPOModel) in CouplingMPOModel
Whether the Hermitian conjugate of the MPO is computed at runtime, [...]
Average number of fermions per site, defined as a fraction (numerator, deno [...]
Choice of the gauge used for the magnetic field. This changes the [...]
Hamiltonian parameter as defined above.
Hamiltonian parameter as defined above.
L (from CouplingMPOModel) in FermionChain.init_lattice
The length in xdirection; only read out for 1D lattices. [...]
lattice (from CouplingMPOModel) in FermionChain.init_lattice
The name of a lattice predefined in TeNPy to be initialized. [...]
Length of the lattice in x and ydirection.
Length of the lattice in x and ydirection.
Hamiltonian parameter as defined above.
Size of the magnetic unit cell along x and y, in terms of lattice sites.
Size of the magnetic unit cell along x and y, in terms of lattice sites.
Maximum number of bosons per site.
order (from CouplingMPOModel) in FermionChain.init_lattice
The order of sites within the lattice for nontrivial lattices, [...]
Magnetic flux density, defined as a fraction (numerator, denominator)
External magnetic flux 'threaded' through the cylinder.
sort_mpo_legs (from CouplingMPOModel) in CouplingMPOModel
Whether the virtual legs of the MPO should be sorted by charges, [...]
Hamiltonian parameter as defined above.
How much to print what's being done; higher means print more. [...]

option
mx
: int¶ 
option
my
: int¶ Size of the magnetic unit cell along x and y, in terms of lattice sites.

option
Nmax
: int¶ Maximum number of bosons per site.

option
filling
: tuple¶ Average number of fermions per site, defined as a fraction (numerator, denominator) Changes the definition of
'dN'
in theBosonSite
.

option
Jx
: float¶ 
option
Jy
: float¶ 
option
mu
: float¶ 
option
U
: float¶ Hamiltonian parameter as defined above.

option
conserve
: {'N'  'parity'  None}¶ What quantum number to conserve.

option
phi
: tuple¶ Magnetic flux density, defined as a fraction (numerator, denominator)

option
phi_ext
: float¶ External magnetic flux ‘threaded’ through the cylinder.

option
gauge
: 'landau_x'  'landau_y'  'symmetric'¶ Choice of the gauge used for the magnetic field. This changes the magnetic unit cell.

option

init_sites
(model_params)[source]¶ Define the local Hilbert space and operators; needs to be implemented in subclasses.
This function gets called by
init_lattice()
to get theSite
for the lattice unit cell.Note
Initializing the sites requires to define the conserved quantum numbers. All predefined sites accept
conserve=None
to disable using quantum numbers. Many models in TeNPy read out the conserve model parameter, which can be set to"best"
to indicate the optimal parameters.

init_lattice
(model_params)[source]¶ Initialize a lattice for the given model parameters.
This function reads out the model parameter lattice. This can be a full
Lattice
instance, in which case it is just returned without further action. Alternatively, the lattice parameter can be a string giving the name of one of the predefined lattices, which then gets initialized. Depending on the dimensionality of the lattice, this requires different model parameters. Parameters
model_params (dict) – The model parameters given to
__init__
. Returns
lat – An initialized lattice.
 Return type
Options

option
CouplingMPOModel
.
lattice
: str  Lattice¶ The name of a lattice predefined in TeNPy to be initialized. Alternatively, a (possibly selfdefined) Lattice instance. In the latter case, no further parameters are read out.

option
CouplingMPOModel
.
bc_MPS
: str¶ Boundary conditions for the MPS.

option
CouplingMPOModel
.
order
: str¶ The order of sites within the lattice for nontrivial lattices, e.g,
'default', 'snake'
, seeordering()
. Only used if lattice is a string.

option
CouplingMPOModel
.
L
: int¶ The length in xdirection; only read out for 1D lattices. For an infinite system the length of the unit cell.

option
CouplingMPOModel
.
Lx
: int¶ 
option
CouplingMPOModel
.
Ly
: int¶ The length in x and ydirection; only read out for 2D lattices. For
"infinite"
bc_MPS, the system is infinite in xdirection and Lx is the number of “rings” in the infinite MPS unit cell, while Ly gives the circumference around the cylinder or width of th the rung for a ladder (depending on bc_y).

option
CouplingMPOModel
.
bc_y
: str¶ "cylinder"  "ladder"
; only read out for 2D lattices. The boundary conditions in ydirection.

option
CouplingMPOModel
.
bc_x
: str¶ "open"  "periodic"
. Can be used to force “periodic” boundaries for the lattice, i.e., for the couplings in the Hamiltonian, even if the MPS is finite. Defaults to"open"
forbc_MPS="finite"
and"periodic"
forbc_MPS="infinite
. If you are not aware of the consequences, you should probably not use “periodic” boundary conditions. (The MPS is still “open”, so this will introduce longrange couplings between the first and last sites of the MPS!)

init_terms
(model_params)[source]¶ Add the onsite and coupling terms to the model; subclasses should implement this.

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 \(\sum_{x_0, ..., x_{dim1}} strength[shift(\vec{x})] * OP0 * OP1\), where
OP0 := lat.unit_cell[u0].get_op(op0)
acts on the site(x_0, ..., x_{dim1}, u1)
, andOP1 := lat.unit_cell[u1].get_op(op1)
acts on the site(x_0+dx[0], ..., x_{dim1}+dx[dim1], u1)
. Possible combinationsx_0, ..., x_{dim1}
are determined from the boundary conditions inpossible_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. Theshift(...)
depends on dx, and is chosen such that the first entrystrength[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 JordanWigner transformation. The operator should be defined on all sites in the unit cell. If
None
, autodetermine whether a JordanWigner string is needed, usingop_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 JordanWigner strings. When handling JordanWigner 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 welldefined 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 nearestneighbor 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. ofadd_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, seeget_hc_op_name()
. For spinless 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 spinfull 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 JordanWigner strings for the fermions are added automatically!
See also
add_onsite()
Add terms acting on one site only.
MultiCouplingModel.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 twosite coupling term on given MPS sites.
Wrapper for
self.coupling_terms[category].add_coupling_term(...)
.Warning
This function does not handle JordanWigner strings! You might want to use
add_local_term()
instead. Parameters
strength (float) – The strength of the coupling term.
j (i,) – The MPS indices of the two sites on which the operator acts. We require
0 <= i < N_sites
andi < j
, i.e., op_i acts “left” of op_j. If j >= N_sites, it indicates couplings between unit cells of an infinite MPS.op2 (op1,) – 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.

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
orcoupling_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
orcoupling_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_{dim1}, 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.
See also
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.

calc_H_MPO
(tol_zero=1e15)[source]¶ Calculate MPO representation of the Hamiltonian.
Uses
onsite_terms
andcoupling_terms
to build an MPO graph (and then an MPO).

calc_H_bond
(tol_zero=1e15)[source]¶ calculate H_bond from
coupling_terms
andonsite_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 longerrange terms.:

calc_H_bond_from_MPO
(tol_zero=1e15)[source]¶ Calculate the bond Hamiltonian from the MPO Hamiltonian.
 Parameters
tol_zero (float) – Arrays with norm < 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 longerrange terms.:

calc_H_onsite
(tol_zero=1e15)[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 takeexplicit_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.

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 [Resta1997]. 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 nearestneighbor hopping of fermions with the
FermionSite
. The cylinder axis is the xdirection 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 xdirection 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()
.

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__
withsave_dict_content()
, storing the format under the attribute'format'
.