tkwantoperator.EnergyCurrent(syst: kwant.system.FiniteSystem, where: Union[List[Tuple[int, int]], List[Tuple[kwant.builder.Site, kwant.builder.Site]]], unicode baseOperator: str = u'Kinetic+', customOnsite: Optional[Callable[[kwant.builder.Site, dict], complex]] = None, sum: bool = False)¶An operator for calculating the local energy current between two sites.
The value of the energy current between two sites has no physical meaning as it is not uniquely defined. However, the total outgoing current from a site (i.e. the sum of the energy currents
from a site \(i\) to all its neighbors \(j\)), calculated by the class
EnergyCurrentDivergence, is uniquely defined.
syst (FiniteSystem) – The finalized Kwant system for which energy currents will be calculated.
where (list [(int, int)] or list [(Site, Site)]) – The hoppings on which the current will be evaluated. It is
saved in a normalized way, with the same order, in the where
attribute of this class.
baseOperator (str) – Can be Hamiltonian, Kinetic+ or Custom (case unsensitive).
It tells the operator which onsite term to use when calculating the
energy current. Saved as an integer in operatorType
customOnsite (None (default) or (Site, dict) \(\rightarrow\) complex matrix-like object) – Overwrites the value of the energy operator \(ε\) on sites:
\(\forall i \, \forall t \; ~ ε_{ii}(t) = \text{customOnsite}(i, t)\),
but only when baseOperator is set to Custom. In practice, its signature
should be customOnsite(site, params) where site is a Site
instance and params is a dict containing the parameters
values (including the current time). The method must return a matrix-like
object (e.g. a tinyarray instance) of the right dimensions at each site
(the same as the onsite Hamiltonian). It is saved in the customOnsite
attribute of this class.
sum (bool) – Whether to sum the currents over each hoppings of the where list and return a
single float result when called. Saved as an integer in the sum attribute of
this class.
Notes
This operator is used to calculate the energy current between each pair of sites \((j, i)\)
in where. At a given time \(t\), the energy current through a hopping \((j, i)\)
(from site \(i\) to site \(j\)) writes:
The above formula has been derived in Reference article for single-orbital Hamiltonians. In practice, the operator EnergyCurrent has been implemented for generic multi-orbital Hamiltonian matrices but the physical meaning of the multi-orbital formula is not clear in general. It has not yet been investigated.
\(ε\) is the energy operator derived from the system’s Hamiltonian:
Its values on hoppings coincide with the Hamiltonian \(\forall i \! \neq \! j \; ε_{ij}(t) = H_{ij}(t)\)
Its value on sites depends on what the user chooses for baseOperator:
Hamiltonian: \(\forall i \, \forall t \; ~ ε_{ii}(t) = H_{ii}(t)\)
Kinetic+: \(\forall i \, \forall t \; ~ ε_{ii}(t) = H_{ii}(t_0 - 1)\) where
\(t_0\) is the instant at which the Hamiltonian starts being time-dependent,
given by time_start (the \(-1\) has been added to avoid use-case problems
with heaviside-like time-dependence).
Custom: \(\forall i \, \forall t \; ~ ε_{ii}(t) = W_{i}(t)\) where
\(W\) is given by the user through customOnsite.
Note that it is necessary to impose \(ε_{ii}(t) = H_{ii}(t_0 - 1)\) for all sites \(i\) (belonging to added_lead_sites)
that were part of the leads and have been added into the central system.
Be cautious that there is currently no implemented check to verify this condition and if it is not satisfied, the output result of Current might be wrong.
The index \(k\) runs formally over all sites in the whole system, but effectively only the sites connected to \(i\) and \(j\) (including \(i\) and \(j\)) give a non zero result.
Attributes
sum¶Whether to sum the currents over each hoppings of the where list and return a
single float result.
int, \(0\) (no) or \(1\) (yes)
operatorType¶Translation to an integer of the baseOperator parameter of the class
constructor: “Hamiltonian” \(\leftrightarrow 0\),
“kinetic+” \(\leftrightarrow 1\) and “custom” \(\leftrightarrow 2\).
syst¶The finalized Kwant system on which energy currents will be calculated.
customOnsite¶Used as the value of the energy operator \(ε\) on sites:
\(\forall i \, \forall t \; ~ ε_{ii}(t) = \text{customOnsite}(i, t)\), when
operatorType = 2.
time_name¶The name the time parameter has in the params dictionary given to __call__. It
is initialised with tkwant’s default name, which is time.
time_start¶The instant \(t_0\) at which time dependence starts in the Hamiltonian.
Initialized with tKwant’s default value, which is 0.
where¶List of hoppings \((j, i)\) (accessed as j, i = where[n, 0], where[n, 1])
on which the energy current is calculated at each call.
2D array of gint (a python int compatible type) of shape \(n \times 2\)
_neighbors_i¶List of lists of the neighbors of each second site \(i\) of each hopping \((j, i)\) in where
_neighbors_j¶List of lists of the neighbors of each first site \(j\) of each hopping \((j, i)\) in where
Methods
__call__()¶Returns the contribution \(I_{ji}^ε(ψ, t)\) of psi to the expectation value
of the current flowing through the hoppings given in where.
This method enables calling an instance A of this class like a function, with a wave function and the “params” dictionary as parameters :
>>> A(psi, params)
For each hopping \((j,i)\) it calculates the following sum, with psi
written as “\(ψ\)” :
psi (array of complex values) – The wave function to use for calculation. Must have the same length as the total number of orbitals in the system.
params (dict, optional) – Dictionary of parameter names and their values.
List of the values fo the contribution of psi to the currents on each hopping \(\left [ I_{ji}^ε(ψ, t) \right ]_{(j,i) \in \text{where}}\), when sum \(= 0\).
Otherwise the sum of these values \(\sum_{(j,i) \in \text{where}} I_{ji}^ε(ψ, t)\).