Predefined YASTN tensors

In YASTN, we predefined several classes containing sets of standard operators, which can be employed in simulations of many typical physical models.

Spin-1/2 and Pauli matrices

class yastn.operators.Spin12(**kwargs)

A set of standard operators for 2-dimensional Hilbert space. Defines identity \(I\) , spin-1/2 operators \(S^x,\ S^y,\ S^z\), raising and lowering operators \(S^+,\ S^-\), Pauli matrices \(X,\ Y,\ Z\) (if allowed by symmetry). Class contains also eigenvectors of \(S^z\), \(S^x\), \(S^y\), and local Hilbert space that is yastn.Leg.

Parameters:

• sym (str) – Explicit symmetry to be used. Allowed options are 'dense', 'Z2', or 'U1'.

• kwargs – Other YASTN configuration parameters can be provided, see yastn.make_config().

Notes

The following basis ordering and charge conventions are assumed

• For sym='dense', the basis order is (z=+1, z=-1).

• For sym='Z2', charge t=0 \(\rightarrow\) z=1, t=1 \(\rightarrow\) z=-1; i.e., \(z = e^{i \pi t}\).

• For sym='U1', charge t=-1 \(\rightarrow\) z=-1, t=1 \(\rightarrow\) z=1; i.e., z = t.

When using yastn.to_numpy() to recover usual dense representation of the algebra for sym='U1' symmetry, reverse=True is required since by default the charges are ordered in the increasing order.

Default configuration sets fermionic to False.

I() → Tensor

Identity operator.

sm() → Tensor

Spin-1/2 lowering operator \(S^-=S^x - iS^y\).

sp() → Tensor

Spin-1/2 raising operator \(S^+=S^x + iS^y\).

space() → Leg

yastn.Leg object describing local Hilbert space.

sx() → Tensor

Spin-1/2 \(S^x\) operator

sy() → Tensor

Spin-1/2 \(S^y\) operator

sz() → Tensor

Spin-1/2 \(S^z\) operator

vec_x(val=1) → Tensor

Normalized eigenvectors of \(\sigma^x\).

vec_y(val=1) → Tensor

Normalized eigenvectors of \(\sigma^y\).

vec_z(val=1) → Tensor

Normalized eigenvectors of \(\sigma^z\).

x() → Tensor

Pauli \(\sigma^x\) operator.

y() → Tensor

Pauli \(\sigma^y\) operator.

z() → Tensor

Pauli \(\sigma^z\) operator.

Spin-1

class yastn.operators.Spin1(**kwargs)

A set of standard operators for 3-dimensional Hilbert space as Spin-1 representation of su(2) algebra. Defines identity, \(S^z,\ S^x,\ S^y\) operators and \(S^+,\ S^-\) raising and lowering operators (if allowed by symmetry). Define eigenvectors of \(S^z\), \(S^x\), \(S^y\), and local Hilbert space as a yastn.Leg.

Parameters:

• sym (str) – Explicit symmetry to be used. Allowed options are 'dense', 'Z3', or 'U1'.

• kwargs – Other YASTN configuration parameters can be provided, see yastn.make_config().

Notes

The following basis ordering and charge conventions are assumed

• For sym='dense', the basis order is (sz=+1, sz=0, sz=-1).

• For sym='Z3', charge t=0 \(\rightarrow\) sz=+1, t=1 \(\rightarrow\) sz=0; t=2 \(\rightarrow\) sz=-1.

• For sym='U1', charge t=-1 \(\rightarrow\) sz=-1, t=0 \(\rightarrow\) sz=0, t=1 \(\rightarrow\) sz=1; i.e., sz = t.

When using yastn.to_numpy() to recover usual dense representation of the algebra for sym='U1' symmetry, reverse=True is required since by default the charges are ordered in the increasing order.

Default configuration sets fermionic to False.

I() → Tensor

Identity operator.

g() → Tensor

Returns rank-2 metric tensor g, such that the quadratic Casimir in terms of basis \(\vec{S} = [S^z; S^+; S^-]\) can be computed as \(\vec{S}^T g \vec{S}\). The signature of g is

0(+1)--g--1(+1)

sm() → Tensor

Spin-1 lowering operator \(S^-=S^x - iS^y\).

sp() → Tensor

Spin-1 raising operator \(S^+=S^x + iS^y\).

space() → Leg

yastn.Leg object describing local Hilbert space.

sx() → Tensor

Spin-1 \(S^x\) operator.

sy() → Tensor

Spin-1 \(S^y\) operator.

sz() → Tensor

Spin-1 \(S^z\) operator.

vec_s() → Tensor

Returns vector of Spin-1 generators, in order: \(S^z, S^+, S^-\). The generators are indexed by first index of the resulting rank-3 tensors. Signature convention is

       2(-1)
       |
0(-1)--S
       |
       1(+1)

vec_x(val=1) → Tensor

Normalized eigenvectors of \(S^x\).

vec_y(val=1) → Tensor

Normalized eigenvectors of \(S^y\).

vec_z(val=1) → Tensor

Normalized eigenvectors of \(S^z\).

Spinless fermions

class yastn.operators.SpinlessFermions(**kwargs)

Predefine operators for spinless fermions.

Standard operators for single fermionic species and 2-dimensional Hilbert space. Defines identity, creation, annihilation, and density operators. Defines vectors for empty and occupied states, and local Hilbert space as a yastn.Leg.

Parameters:

• sym (str) – Explicit symmetry to be used. Allowed options are 'Z2', or 'U1'.

• kwargs (any) – Passed to yastn.make_config() to change backend, default_device or other config parameters.

Fixes fermionic fields in config to True.

I() → Tensor

Identity operator.

c() → Tensor

Lowering operator.

cp() → Tensor

Raising operator.

n() → Tensor

Particle number operator.

space() → Leg

yastn.Leg object describing local Hilbert space.

vec_n(val=0) → Tensor

State with occupation 0 or 1.

Spinful fermions

class yastn.operators.SpinfulFermions(**kwargs)

Predefine operators for spinful fermions.

Generator of standard operators for local Hilbert space with two fermionic species and 4-dimensional Hilbert space.

Predefine identity, creation, annihilation, density, and spin operators. Defines vectors with possible occupations, and local Hilbert space as a yastn.Leg.

Parameters:

• sym (str) – Explicit symmetry to be used. Allowed options are 'Z2', 'U1', 'U1xU1', or 'U1xU1xZ2'.

• kwargs (any) – Passed to yastn.make_config() to change backend, default_device or other config parameters.

Notes

Fermionic field in config is fixed such that:

• For 'Z2', 'U1' and 'U1xU1xZ2', the two species (spin-up and spin-down) are treated as indistinguishable. In that case, creation and annihilation operators of the two species anti-commute (fermionic statistics is encoded in the Z2 channel of U1xU1xZ2).

• For 'U1xU1' the two species (spin-up and spin-down) are treated as distinguishable. In that case, creation and annihilation operators of the two species commute.

I() → Tensor

Identity operator in 4-dimensional Hilbert space.

c(spin='u') → Tensor

Annihilation operator, with spin='u' for spin-up, and spin='d' for spin-down.

cp(spin='u') → Tensor

Creation operator, with spin='u' for spin-up, and spin='d' for spin-down.

n(spin='u') → Tensor

Particle number operator, with spin='u' for spin-up, and spin='d' for spin-down.

space() → Leg

yastn.Leg object describing local Hilbert space.

vec_n(val=(0, 0)) → Tensor

State with occupation given by tuple (nu, nd).