iTEBD

A single header file for itebd calculations based on the ITensor library.

Installation and Linking

Refer to ITensor's instruction for using iTensor. You just need to add

#include "itebd.h"

and make sure the header file is within path.

Examples

I have created a few examples to reproduce figures of some arXiv papers that utilized iTEBD:

• arXiv:cond-mat/0605597 (figs. 5 and 6)

• arXiv:0907.3206 (fig.3b)

• arXiv:1302.4460 (fig.2a)

• arXiv:1503.02010 (fig.1)

• arXiv:1604.03571 (fig.2a)

Please make the corresponding file's name you want to compile. For example, make 1503.02010 generates executable 1503.02010 that produces figure 1 of arXiv:1503.02010.

By default, it is now set to reproduce figures 5 and 6 of Vidal's original paper.

Basic Usage

Henceforth I assume we have using namespace itensor;

Instantiation

itebd<IndexType,z> sys(ampo,QNA,QNB,Chi0);

• IndexType could be Index or IQIndex, depending on whether you want to use symmetries.

• z optional and defaults to 2. The coordination number. Could be useful for exotic system with >2 neighbors e.g. arXiv:0712.1806 (I have not tried).

• ampo is a two-site AutoMPO object of ITensor. Hamiltonian is specified by ampo. The sites must be of type IQIndex.

• QNA and QNB are optional and defaults to 1. The initial quantum numbers of the two sites (site A and site B). Physical meanings of these numbers are determined by the sites which are IQIndex .

• Chi0 is optional and defaults to 1. The initial bond dimension of each set of quantum numbers.

Propagation (step)

double itebd::step(std::complex<double> dt, size_t steps = 1, double thres = 1E-10, int maxm = 0);

• dt is the timestep on real axis. When it is real, it propagates in real time.

• steps number of steps

• thres is the Cutoff value used for SVD.

• maxm is the Maxm value used for SVD.

• returns: Energy estimated based on the log of the expection value of the evolution operator

double itebd::step_imag(std::complex<double> dt, size_t steps = 1, double thres = 1E-10, int maxm = 0);

• identical to above but in the imaginary axis.

Local Measurements

std::complex<double> itebd::measure(const T &op)

• op is a two-neighboring sites operator with indices the same as the site used to construct ampo.

• returns: Expectation value of the operator at the current state

Long-Range two-opeartor correlation measurements

std::vector<std::complex<double>> itebd::measure(const T &op1A, const T &op2, const std::vector<int> &rv);

• op1A is the operator at the origin and must have the same Index as the first site in the sites used to contruct ampo

• opt2 is the opeartor at the second site. Can have either one of the site indices.

• rv is the list of integer distances between sites 1 and 2 to compute <opt1A*opt2>

• returns: a vector of complex values of <opt1A*opt2>

Helpful member variables / functions

• .tv (vector of doubles): the times (in absolute values) traversed by the system since it starts

• .Sv (vector of doubles): the entanglement entropies the system had for each time in .tv

• .bonddim (int): current bond dimension

• .GA, .GB, .L[0], .L[1]: the Gammas and Lambdas of the state as used in Vidal's paper

• .resetTime() (void): resets both .tv and .Sv

• .setH(const AutoMPO &ampo) (void): sets a new Hamiltonian while keeping the current states unchanged.

• .backup() and .restore() (void): backup and restore the state

• .randomize() (void): randomize all the tensors (e.g. before imaginary time propagation)

Higher Dimensions?

By instantiating with z=4 you can try to use this for 2-dimensional systems where the "environment" is never computed. Some call this "simplified update" of iPEPS and saw good results if the environment is at least computed for measurements. I have not implemented this yet so it should never be used for serious work in 2D. cond-mat.0605597 implements simplified update without calculation of environment.