DCorrelators.jl

A convenient frontend for calculating dynamical correlation functions and related observables based on matrix-product states time evolution methods.

• The symbolic operator representation of a quantum lattice system in condensed matter physics is based on the package QuantumLattices

• The matrix-product states time evolution methods such as TEBD, MPO $W^{II}$ and TDVP are based on packages ITensors and MPSKit

• The bechmark of dynamical correlation functions and related observables is the result from exact diagonalization method based on the packages ExactDiagonalization

Please type ] in the REPL to use the package mode, then type this command:

pkg>add DCorrelators

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If the $x$ variable has only discrete values ($x=na$, for $n=1,2,3,...,N$) and finite length $L$ ($L=Na$), the expansion of the function is

$$f_n=\sum_{m=1}^{N} A_{q} e^{iq_m x_n},\quad q=\frac{2\pi}{L}m,$$

where

$$A_{q}=\frac{1}{N}\sum^N_{n=1}f_n e^{-iq_mx_n}. $$

Dividing $A_{q}$ by the mode $\frac{2\pi}{L}$, the Fourier amplitudes with a per unit spacial interval is

$$ A(q) = \frac{L}{2\pi}A_q=\frac{a}{2\pi}\sum^N_{n=1}f_ne^{-iqx_n}.$$

If the times $t$ are discrete times ($t=l\Delta t$, for $l=0,1,2,...,N$) and the final evolutionary time $t_{\mathrm{end}}=N\Delta t$, the expansion of the function is

$$ f_l=\sum_{p=1}^N A_{\omega} e^{-i\omega_p t_l},\quad \omega=\frac{2\pi}{t_{\mathrm{end}}}p,$$

where

$$A_{\omega}=\frac{1}{N}\sum_{l=1}^N f_l e^{i\omega_p t_l}.$$

To make it a per unit frequency interval, one need to divide by the spacing of the discrete frequency mode and the Fourier amlitudes are given by,

$$A(\omega)=\frac{t_{\mathrm{end}}}{2\pi}A_{\omega}=\frac{\Delta t}{2\pi}\sum_{l=1}^N f_l e^{i\omega t_l}.$$

Although a Fourier series is designed to represent functions that are periodic, one can assume that the finite data sequence can be periodically repeated, which leads to the time at index $l=N$ is identified with the time at $l=0$. However, the small errors made at the end of a period will be irrelevant as long as the primary correlations decay in less time than $t_{\mathrm{end}}$.

By use of double Fourier transforms, one can obtain the $k-\omega$ space correlation function $G(k,\omega)$,

$$G(k,\omega)=\frac{1}{(2\pi)^2}\Delta t\sum_{l=1}^{N_t} a\sum_{n=1}^{N_L} G(x, t) e^{-i(kx-\omega t)}.$$

With

$$\Delta t \sum_{l=1}^{N_t}\to \int_0^{t_{\mathrm{end}}}dt,\quad a\sum_{n=1}^{N_L}\to \int_0^L dx,$$

the continuous form is as follows

$$ G(k,\omega) = \frac{1}{(2\pi)^2} \int_0^{t_{\mathrm{end}}}dt \int_0^L dx G(x, t) e^{-i(kx-\omega t)}. $$

The real-space and real-time correlation function $G(x, t)$ is given by,

$$ \begin{aligned}G\left( x_{n},t\right) &=\frac{1}{N_t}\sum_{l=1}^{N_t}\frac{1}{N_{L}}\sum_{m=1}^{N_{L}}\langle 0 | C\left( x_{m}+x_{n},t_{l}+t\right) C^{\dagger}\left( x_{m},t_{l}\right) | 0\rangle \\ &=\frac{1}{N_t}\sum_{l=1}^{N_t}\frac{1}{N_{L}}\sum_{m=1}^{N_L}\langle 0| e^{iH(t_{l}+t)}C\left( x_{m}+x_{n}\right) e^{-iH(t_l+t) }e^{iHt_{l}}C^{\dagger}\left( x_{m}\right) e^{-iHt_{l}}| 0\rangle \\ &=\frac{1}{N_{L}}\sum_{m=1}^{N_{L}}e^{iE_{0}t}\langle 0| C\left( x_{m}+x_{n}\right) e^{-iHt}C^{\dagger}\left( x_{m}\right) | 0\rangle. \end{aligned}$$

Finally, one gets,

$$ G(k,\omega)=\frac{1}{(2\pi)^2}\Delta t\sum_{l=1}^{N_t} a\sum_{n=1}^{N_L} \frac{1}{N_{L}}\sum_{m=1}^{N_{L}}e^{iE_{0}t}\langle 0| C\left( x_{m}+x_{n}\right) e^{-iHt}C^{\dagger}\left( x_{m}\right) | 0\rangle e^{-i(kx_n-\omega t)}. $$

Here, the matrix-product states time evolution methods are implemented to solve the state $e^{-iHt}C^{\dagger}\left( x_{m}\right) | 0\rangle$.

• Wysin G M. Magnetic Excitations and Geometric Confinement[M]. Philadelphia, USA: IOP, 2015.

• Paeckel S, Köhler T, Swoboda A, et al. Time-evolution methods for matrix-product states[J]. Annals of Physics, 2019, 411: 167998.

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Due to the fast development of this package, releases with different minor version numbers are not guaranteed to be compatible with previous ones before the release of v1.0.0. Comments are welcomed in the issues.

zongyy_phy@smail.nju.edu.cn