Background

Imaginary time formalism

The imaginary time formalism is a tool used in many-body quantum theory to determine the equilibrium properties of systems at finite temperature. In this case the response functions of the system are functions of imaginary time \(\tau\) that is defined on \(\tau \in [0, \beta]\), where \(\beta\) is the inverse temperature.

In particular the single-particle Green’s function \(G\) of a system is a function in imaginary time, and can be written as the time ordered expectation value

\[G_{ab}(\tau) = - \langle \mathcal{T} c_a(\tau) c_b^\dagger(0) \rangle,\]

where \(c^\dagger_b(0)\) creates a particle in state \(b\) at time \(0\) and \(c_a(\tau)\) destroys a particle in state \(a\) at time \(\tau\). The Green’s function – and other derived quantities – can be extended to the interval \(\tau \in [-\beta, \beta]\) using the (anti-)periodicity property

\[G_{ab}(-\tau) = \xi G_{ab}(\beta - \tau),\]

where \(\xi = \pm 1\) for bosonic and fermionic particles, respectively.

Discrete Lehmann Representation

The discrete Lehmann representation (DLR) provides and efficient method of representing imaginary time functions numerically. The theory and methodology behind is based on the analysis of an integral kernel (the analytic continuation kernel) combined with a clever linear algebra factorization (the interpolative decomposition).

In imaginary time the kernel

\[K(\tau, \omega) = \frac{e^{-\omega \tau}}{1 + e^{-\omega}}\]

is used to represent all kinds of response functions, and in Matsubara frequency the corresponding kernels are

\[K_F(i \nu_n, \omega) = -\frac{1}{i\nu_n - \omega} \, , \quad K_B(i\omega_n, \omega) = - \frac{\tanh (\omega/2)}{i\omega_n - \omega}\]

for Fermions and Bosons respectively.

For an in depth background on the discrete Lehmann representation (DLR), we suggest looking at the two papers cited just below.

Citation information

If this library helps you to create software or publications, please let us know, and cite

• our repository: https://github.com/jasonkaye/libdlr

• libdlr: Efficient imaginary time calculations using the discrete Lehmann representation, Jason Kaye, Kun Chen, and Hugo U.R. Strand, Comput. Phys. Commun. 280, 108458, 2022. [arXiv:2110.06765]

• Discrete Lehmann representation of imaginary time Green’s functions, Jason Kaye, Kun Chen, and Olivier Parcollet, Phys. Rev. B 105, 235115, 2022. [arXiv:2107.13094]

libdlr imaginary time point format

libdlr works with scaled imaginary time and real frequency coordinates, in which an imaginary time \(\tau\) is in \(\tau \in [0,1]\) and a real frequency \(\omega\) is in \(\omega \in [-\Lambda,\Lambda]\), for \(\Lambda\) the dimensionless energy cutoff parameter.

Throughout the code, imaginary time grid points in \((0.5,1)\) are stored in a peculiar manner. Namely, suppose \(\tau \in (0.5,1)\). Then instead of storing \(\tau\) directly, we store the number \(\tau^* = \tau-1\). In other words, we store the negative distance of \(\tau\) to 1, rather than tau itself. We call this the relative format. To recover the correct tau points, one simply leaves any points in \([0,1/2] \cup {1}\) alone, and adds 1 to any points in \((1/2,1)\); the subroutine rel2abs performs this conversion.

The reason for this has to do with maintaining full relative accuracy in floating point arithmetic. To evaluate the kernel \(K(\tau,\omega)\), we sometimes need to compute the value \(1-\tau\) for \(\tau\) very close to 1. If we work with tau directly, there is a loss of accuracy due to catastrophic cancellation, which begins to appear in extreme physical regimes and at very high requested accuracies. If we instead compute \(\tau^*\) to full relative accuracy and work with it directly rather than with \(\tau\), for example by exploiting symmetries of \(K(\tau,\omega)\) to avoid ever evaluating \(1-\tau\), we can maintain full relative accuracy.

This apparently annoying characteristic of libdlr is simply the price of maintaining all the accuracy that is possible to maintain in floating point arithmic. But it is largely ignoreable if the loss of accuracy is not noticeable in your application (this will be the case for the vast majority of users). Just use the provided demos to get started, and follow these guidelines:

1. Use provided subroutines, which hide this technical complexity, to carry out all operations.

2. In a situation in which you want to provide a point \(\tau \in (1/2,1)\) at which to evaluate a DLR, there are two options:

   • compute \(\tau^*\) to full relative accuracy, and provide this according to the instructions in the relevant subroutines, thereby maintaining full relative accuracy in calculations, or

   • if you don’t care about the (usually minor) digit loss which comes from ignoring this subtlety, simply convert the provided \(\tau\)-point to \(\tau^*\) using the subroutine abs2rel, which carries out the conversion. Since the point will have started its life in the absolute format, converting it to relative format cannot recover full relative accuracy, but it still needs to be converted in order to be compatible with libdlr subroutines.

3. If you happen to want to evaluate a Green’s function on an equispaced grid on \([0,1]\) in imaginary time, use the subroutine eqpts_rel to generate this grid in the relative format.

A more in-depth discussion of the relative format is given in Appendix C of our paper on libdlr.