5.1. Input file for tense_simple

• File format is TOML format.

• The input file has five sections : model, parameter, lattice, observable, correlation .

• In future, the file will be split by specifying a file name.

5.1.1. parameter section

The contents of the parameter section are copied directly to the parameter section of the input file of tenes. You can also specify the imaginary time step size for the imaginary time evolution operator in simple and full updates in the subsections simple_update, full_update.

Name	Description	Type	Default
tau	imaginary time step in the imaginary time evolution operator	Real	0.01

The following parameters are common to the tenes input file.

parameter.tensor

Name	Description	Type	Default
D	The virtual bond dimensions of the central tensor	Integer	2
CHI	The virtual bond dimensions of the angular transfer matrix	Integer	4
save_dir	Directory to write optimized tensors	Str	“”
load_dir	Directory to read initial tensor	Str	“”

• save_dir - Store optimized tensors below this directory. - When it is empty, the tensors are not saved.

• load_dir - Read various tensors from below this directory. - Must be the same degree of parallelism as when saved. - Not read if it is empty.

parameter.simple_update

Name	Description	Type	Default
num_step	Number of simple updates	Integer	0
lambda_cutoff	cutoff of the mean field to be considered zero in the simple update	Real	1e-12

parameter.full_update

Name	Description	Type	Default
num_step	Number of full updates	Integer	0
env_cutoff	Cutoff of singular values to be considered as zero when computing environment through full updates	Real	1e-12
inverse_precision	Cutoff of singular values to be considered as zero when computing the pseudoinverse matrix with full update	Real	1e-12
convergence_epsilon	Convergence criteria for truncation optimization with full update	Real	1e-12
iteration_max	Maximum iteration number for truncation optimization on full updates	Integer	1000
gauge_fix	Whether the tensor gauge is fixed	Boolean	true
fastfullupdate	Whether the Fast full update is adopted	Boolean	true

parameter.ctm

Name	Description	Type	Default
projector_cutoff	Cutoff of singular values to be considered as zero when computing CTM projectors	Real	1e-12
convergence_epsilon	CTM convergence criteria	Real	1e-10
iteration_max	Maximum iteration number of convergence for CTM	Integer	100
projector_corner	Whether to use only the 1/4 corner tensor in the CTM projector calculation	Boolean	true
use_rsvd	Whether to replace SVD with Random SVD	Boolean	false
rsvd_oversampling_factor	Ratio of the number of the oversampled elements to that of the obtained elements in the Random SVD method	Real	2.0

parameter.random

Name	Description	Type	Default
seed	Seed of the pseudo-random number generator used to initialize the tensor	Integer	11

Each MPI process has the own seed as seed plus the process ID (MPI rank).

Example

[parameter]
[parameter.tensor]
D  = 4     # tensor_dim
CHI  = 16  # env_dim

[parameter.simple_update]
num_step = 1000

[parameter.full_update]
num_step = 1

[parameter.ctm]
iteration_max = 5

5.1.2. lattice section

Specify the lattice information. Square lattice and honeycomb lattice are defined as lattice types.

Name	Description	Type
type	Lattice name (square lattice or honeycomb lattice)	Str
L_sub	Unit cell size	An integer or a list of two integers

When a list of two integers is passed as L_sub, the first element gives the value of Lx and the second one does Ly. If L_sub is an integer, Both Lx and Ly will have the same value. A list of three or more elements causes an error.

Sites in a unit cell are indexed starting from 0. These are arranged in order from the x direction.

Sites in a unit cell of L_sub = [2,3] are arranged as follows:

y
^     4 5
|     2 3
.->x  0 1

Square lattice

There are two types of bond, horizontal (0) and vertical (1) (corresponding to - and | in the below figure).

The unit cell for L_sub = 2 is given as follows:

0   1
|   |
2 - 3 - 2
|   |
0 - 1 - 0

Honeycomb lattice

Unit cell size (Each element of L_sub) must be an even number.

There are 3 types of bonds: x, y, and z (corresponding to -, ~, | in the below figure). Each site with an even index has a rightward (x), a leftward (y), and an upward (z) bonds and each site with an odd index has a leftward (x), a rightward (y), and a bottomward (z) bonds.

The unit cell for L_sub = 2 is given as follows:

0   1
    |
2 ~ 3 - 2
|
0 - 1 ~ 0

5.1.3. model section

Specify the type of the model. Only the Spin system can be spcified in ver. 0.1.

Name	Description	Type
type	The type of the model	Str

Spin system

Spin system

\[\mathcal{H} = \sum_{\langle ij \rangle}\left[\sum_\alpha^{x,y,z} J^\alpha_{ij} S^\alpha_i S^\alpha_j + B \left(\vec{S}_i\cdot\vec{S}_j\right)^2 \right] - \sum_i \left[ h S^z_i + \Gamma S^x_i - D \left(S^z_i\right)^2 \right]\]

:header: “Name”, “Description”, “Type”, “Default” :widths: 30, 30, 10, 10

S	Magnituide of the local spin	Real	0.5
Jx	The x component of the exchange interaction J	Real or a list of Real	1.0
Jy	The y component of the exchange interaction J	Real or a list of Real	1.0
Jz	The z component of the exchange interaction J	Real or a list of Real	1.0
BQ	Biquadratic interaction B	Real or a list of Real	0.0
h	longitudinal magnetic field h	Real	0.0
G	Transverse magnetic field \(\Gamma\)	Real	0.0
D	On-site spin anisotropy D	Real	0.0

By providing a list of exchange and biquadratic interactions, we can vary the magnitude of the interaction for each type of lattice bond. If the number of elements in the list is less than the type of lattice bond, the remainder is filled in with the last element specified.

5.1.4. observable section

By default, the local physical quantities used for physical quantities measurements: \(S^z\) and \(S^x\) . Measurements of more detailed physical quantities can be made by overwriting the format common to the input file of tenes. For details, See observable section Input file for tenes.

5.1.5. correlation section

For tenes_simple , correlation functions \(C = \langle A(0)B(r)\rangle\) are not calculated by default. For calculating correlation functions, they have to be specified in the same file format as the input file of tenes. For details, See correlation section Input file for tenes.