Imaginary Time Evolution Method for Finding the Ground State

Last Updated on September 16, 2024 by Max

Imaginary time evolution is a powerful computational technique widely used in quantum mechanics to find the ground state of a quantum system. The idea stems from a mathematical transformation where time is treated as an imaginary quantity, allowing the evolution of a quantum state to “filter out” its ground state. This method is useful for systems where the ground state is not analytically obtainable, such as complex many-body systems or quantum fields.

In this article, we will explore the concept of imaginary time evolution, its mathematical formulation, and the steps to apply this technique to find the ground state of a given Hamiltonian.

Mathematical Background

To understand imaginary time evolution, let’s start by recalling the time-dependent Schrödinger equation (TDSE),

\[i\hbar \frac{\partial \psi(x, t)}{\partial t} = \hat{H} \psi(x, t),\]

where, \(\psi(x, t)\) is the wave function of the quantum system that describes its state, \(\hbar\) is the reduced Planck’s constant, and \(\hat{H}\) is the Hamiltonian operator representing the total energy of the system.

The general solution to the Schrödinger equation can be written as a linear combination of the eigenstates \(\phi_n(x)\) of the Hamiltonian:

\[\psi(x, t) = \sum_{n} c_n \phi_n(x) e^{-i E_n t / \hbar},\]

where \(c_n\) are coefficients determined by the initial conditions of the system, and \(E_n\) are the eigenenergies corresponding to the eigenstates \(\phi_n(x)\).

Imaginary Time Evolution

To find the ground state, we consider a transformation from real time \(t\) to imaginary time \(\tau = it\). Substituting \(t = -i\tau\) into the TDSE, we obtain:

\[-\hbar \frac{\partial \psi(x, \tau)}{\partial \tau} = \hat{H} \psi(x, \tau),\]

or equivalently:

\[\frac{\partial \psi(x, \tau)}{\partial \tau} = -\frac{1}{\hbar} \hat{H} \psi(x, \tau).\]

This is known as the imaginary time Schrödinger equation. The solution to this equation takes the form:

\[\psi(x, \tau) = \sum_{n} c_n \phi_n(x) e^{-E_n \tau / \hbar}.\]

Ground State Filtering Mechanism

In the expression for \(\psi(x, \tau)\), each term involves an exponential factor \(e^{-E_n \tau / \hbar}\). As \(\tau \to \infty\), the term corresponding to the smallest eigenvalue \(E_0\) (the ground state energy) will dominate because it decays slower than all other terms. Hence, in the limit of large \(\tau\):

\[\psi(x, \tau) \approx c_0 \phi_0(x) e^{-E_0 \tau / \hbar}.\]

Therefore, by evolving a trial wave function \(\psi(x, 0)\) in imaginary time, it converges to the ground state \(\phi_0(x)\) up to a normalization factor.

Normalization and Repetition

To ensure numerical stability and prevent the wave function from becoming arbitrarily small, we normalize the wave function after each small step in \(\tau\). The normalized wave function at each step can be written as:

\[\psi(x, \tau + \Delta \tau) = \frac{\psi(x, \tau) – \frac{\Delta \tau}{\hbar} \hat{H} \psi(x, \tau)}{\|\psi(x, \tau) – \frac{\Delta \tau}{\hbar} \hat{H} \psi(x, \tau)\|}.\]

This process is repeated iteratively until convergence is achieved, where the state \(\psi(x, \tau)\) no longer changes significantly, indicating it has reached the ground state.

Algorithm for Imaginary Time Evolution

To implement the imaginary time evolution numerically, we use the following steps:

1. Initialize the Wave Function:
   Start with a trial wave function \(\psi(x, 0)\) that has non-zero overlap with the ground state \(\phi_0(x)\). This could be a random function or a physically motivated guess.
2. Discretize Time and Space:
   Discretize the time evolution parameter \(\tau\) into small steps \(\Delta \tau\). Similarly, discretize the spatial domain \(x\) if working in a finite difference scheme.
3. Iterative Evolution:
   Evolve the wave function in small imaginary time steps,
   \[ \psi(x, \tau + \Delta \tau) = \frac{\psi(x, \tau) – \frac{\Delta \tau}{\hbar} \hat{H} \psi(x, \tau)}{\|\psi(x, \tau) – \frac{\Delta \tau}{\hbar} \hat{H} \psi(x, \tau)\|}. \]
4. Normalize the Wave Function:
   Normalize \(\psi(x, \tau)\) after each step to avoid divergence or numerical overflow.
5. Check Convergence:
   Continue the evolution until the change in \(\psi(x, \tau)\) between successive iterations is below a chosen threshold, indicating convergence to the ground state.

Practical Applications

Imaginary time evolution is widely used in:

• Quantum Monte Carlo Simulations: To find ground states of complex many-body systems.
• Density Functional Theory (DFT): For minimizing the energy functional to find the ground state electron density.
• Condensed Matter Physics: To study properties of strongly correlated systems, like Bose-Einstein condensates.

Conclusion

Imaginary time evolution offers a straightforward and effective approach to finding the ground state of a quantum system, particularly in cases where direct analytical solutions are not possible. By evolving a trial wave function in imaginary time, the technique systematically suppresses higher-energy components, isolating the ground state. It is a critical tool in computational quantum mechanics and is fundamental for research in areas ranging from condensed matter physics to quantum chemistry.

By implementing the steps outlined in this article, you can use imaginary time evolution to find the ground state of any quantum system numerically, provided you have a suitable Hamiltonian and initial trial wave function.