Parameters

Please note that throughout this documentation we use the older short syntax for the lattice_builder input. We recommend to use the the newer, explicit syntax as described in the lattice_validator documentation, see Schema definitions.

Unit-Cell and System Parameters

A unit cell is the smallest building block of a Bravais lattice. It is composed of sites and bonds, which represent the physical system. The complete Bravais lattice is generated by repeating the unit cell along integer multiples of the lattice vectors.

For the site type Spins, the representation of the spin operators can be controlled using the Spin Representation input variable. Spin Representation has to be a positive integer \(n\), which determines the spin quantum number \(s = n/2\).

NOTE: In presence of transverse magnetic fields (\(B_x \neq 0\) and/or \(B_y \neq 0\)) \(S_z\) is no longer a good quantum number. In order to allow for states to break this symmetry please specify the input parameter mod Sz to be non-zero. mod Sz has to be a positive even integer. For example, mod Sz: 2 allows eigenstates to be a superposition of states differing by single or multiple spin flips.

Site Parameters

General

• ID:

  Automatically given by the system - no input necessary.

• Name:

  Name of the site, which can be chosen freely and can be repeated.

• Position:

  Location of the site in cartesian coordinates.

Spinless fermion sites

• Onsite Energy:

  Energy associated with creating one electron at this site.

Spinful fermions sites

• Onsite Energy:

  Energy associated with creating one electron at this site.

• Bx, By, Bz:

  The three components of the onsite magnetic field.

• Onsite U:

  "Hubbard U", i.e. the interaction energy between electrons on the same site.

• Onsite BCS:

  The onsite BCS pairing (complex value, input of real and imaginary parts needed).

For spinful fermions we also provide a spin channel notation, i.e., a parametrization of the spin-dependent onsite energy. You can toggle between magnetic field notation and spin channel notation at your convenience. The conversion of your parameters is done automatically. For spinful electrons the magnetic field notation corresponds to the Onsite Energy and the magnetic field given in terms of Bx, By and Bz. Alternatively the spin channel notation provides spin-dependent energies for the ↑↑, ↓↓, and ↑↓ spin channel.

• Spin channel parameters ↑↑, ↑↓, ↓↓:

  ↑↓ is a complex values, so the input of real and imaginary parts is required.

Note: The value of ↓↑ energy is omitted here, since it is the complex conjugate of ↑↓ energy.

Spins

• Bx, By, Bz:

  The three components or directions of the onsite magnetic field.

Lattice Vectors

Lattice vectors describe the translation between unit cells in space.

• Lattice Vector a:

  Vector describing the translation of the unit-cell in the the first dimension.

• Lattice Vector b:

  Vector describing the translation of the unit-cell in the the second dimension.

Note: Lattice vectors are not unique. Lattice vectors have to be linearly independent.

Bond Parameters

Bonds describe connections between sites, either within a unit-cell or between unit-cells.

General

• From (Site ID):

  ID of the first site the bond connects.

• To (Site ID):

  ID of the second site the bond connects.

Bonds between unit-cells

• Translation x, y: Additional shift for the second site by integer multiples of the lattice vectors.

Note: A bond configured as bond between unit-cells, but given a translation of x = 0 and y = 0 will automatically be transfered to a bond within unit-cells, since bonds within unit-cells are defined as have a translation of [0, 0].

Spinless fermion bonds

• U:

  Energy cost of having one electron each on the sites connected by the bond.

• t:

  Energy cost associated with moving one electron along the bond.

• D:

  The anomalos BCS-like pairing across a bond.

Spinful fermion bonds

• Jz, Jperp, Jcross: These define a spin-spin inter-site interactions.

• U: Energy cost of having one electron each on the atoms connected by the bond, i.e., the inter-site density-density interaction for spinful fermions.

• t-Values ↑↑, ↑↓, ↓↑, ↓↓: Energy cost associated with moving one electron along the bond. For spinful electrons the hopping terms can be spin-dependent.

  These values can be complex (the hermitian conjugate terms in the Hamiltonian are automatically generated).

• D-values ↑↑, ↑↓, ↓↑, ↓↓: The anomalous BCS-like pairing on a bond. For spinful electrons the anomalous pairing terms can be spin-dependent.

  These values can be complex (the hermitian conjugate terms in the Hamiltonian are automatically generated).

Note: There are several ways to define physically identical bonds between unit cells.

System Parameters

General

A lattice system in the Lattice Application is assembled from clusters, which in turn are constructed from repetitions of unit cells.

• Cluster Size x, y:

  These values describe the number of repetitions of unit-cells along the corresponding lattice vectors.

For example, cluster_size : [3, 2, 1] implies that three unit cells in the direction of the first lattice vector and two unit cells in the direction of the second lattice vector define the cluster. Similarly, a system_size of [4, 4, 1] describes a lattice system with four clusters in the direction of the first lattice vector (12 unit cells) and four clusters in the direction of the second lattice vector (9 unit cells).

• Boundary Conditions Vector a: This value describes the boundary condition on the side of lattice vector a

• Boundary Conditions Vector b: This value describes the boundary condition on the side of lattice vector b

Spinless Fermion System

• N:

  Fixes the number of fermions in the system. For spinless fermions it must be less or equal to the total number of sites.

• mod N:

  In the presence of anomalous pairing, N is no longer a good quantum number. In order to allow for states to break this symmetry please specify the input parameter mod N to be non-zero, usually mod_N = 2. mod_N has to be a positive even integer.

Spinful Fermion System

• N:

  Fixes the number of fermions in the system. For spinful fermions it must be less or equal to twice the total number of sites.

• mod N:

  In the presence of anomalous pairing, N is no longer a good quantum number. In order to allow for states to break this symmetry please specify the input parameter mod N to be non-zero, usually mod_N = 2. mod_N has to be a positive even integer.

• Sz:

  Fixes the z-component of the total spin where -N <= Sz <= N and ** 0 <= N <= M**, where "M" is the number of sites and "N" the number of fermions, and Sz <= 2 * M - N for for N > M. Note that the Sz component is twice the physical Sz expectation number in order to classify the quantum numbers with integers.

• mod Sz:

  In the presence of a transverse magnetic field (\(B_x \neq 0\) and/or \(B_y \neq 0\)) or spin-orbit coupling (non-zero t-Values for ↑↓, ↓↑ channel) \(S_z\) is no longer a good quantum number. In order to allow for breaking this symmetry mod Sz to be a positive even integer.

Spin System

• Sz:

  The spin polarization fixes the z-component of the total spin where -M <= Sz <= M, where "M" is the number of sites. Note that the Sz component is twice the physical Sz expectation number in order to classify the quantum numbers with integers.

• mod Sz:

  In the presence of a transverse magnetic field (\(B_x \neq 0\) and/or \(B_y \neq 0\)) \(S_z\) is no longer a good quantum number. In order to allow for states to break this symmetry mod Sz has to be a positive even integer. For example, mod Sz: 2 allows eigenstates to be a superposition of states differing by single or multiple spin flips.

Job Parameters

The job parameters specify the fine-tuning of the DMRG calculation.

• Name:

  Name of the Job, which can be chosen freely.

• Structure Type:

  Choose between the creation of a spins job, a spinless fermions job and a spinful fermions job.

• Structures:

  Choose one of your structures for the job.

• Number of States:

  default: 1, max-values for spinless fermions: \({M\choose N}\), max-value for spinful fermions: \({M\choose n_{\uparrow}}{M\choose n_{\downarrow}}\)

• Inverse Mean Field:

  default iHF. See SCCE/iMF for more information.

• Output Type:

  default hdf5. Specifies the file type of the result file.