Unitary circuits for strongly correlated fermions

We introduce a scheme for efficiently describing pure states of strongly correlated fermions in higher dimensions using unitary circuits featuring a causal cone. A local way of computing local expectation values is presented. We formulate a dynamical reordering scheme, corresponding to time-adaptive Jordan-Wigner transformation, that avoids nonlocal string operators. Primitives of such a reordering scheme are highlighted. Fermionic unitary circuits can be contracted with the same complexity as in the spin case. The scheme gives rise to a variational description of fermionic models not suffering from a sign problem. We present numerical examples in a $9\times 9$ and $6\times 6$ fermionic lattice model to show the functioning of the approach.

Figure 1

Example for the evaluation of a local observable (black box) with respect to a fermionic MERA (gray boxes). Each gray box represents a single fermionic unitary. Dark gray boxes make up the causal cone of the observable; all others lie outside of it and cancel each other. Only for conceptual clarity, a 1D MERA is depicted, not a 2D one, for which dynamical reordering becomes relevant.

Figure 2

The basic primitives of (a) representing local operators, (b) reordering fermionic modes, (c) adding fermionic modes, (d) conjugating with unitaries, (e) doing partial traces, and (f) generating partial projections.

Figure 3

Numerical examples of the ground-state energy $E_0$ for several $9\times 9$ and $6\times 6$ (boxes and triangles) instances of strongly correlated fermionic models, $H=\mathfrak{t}{\sum }_{\langle j,k\rangle }(f_j^{†}{f}_k+\mathrm{H}.\mathrm{c}.)+{\sum }_j{f}_j^{†}{f}_j+\mathfrak{u}{\sum }_{\langle j,k\rangle }{f}_j^{†}{f}_j{f}_k^{†}{f}_k$; the free models $\mathfrak{u}=0$ are marked. As the MERA ansatz is a truly variational ansatz, providing upper bounds to the energy, success is guaranteed by giving lower bounds as provided by Anderson bounds (lines). This is based on an exact diagonalization of $5\times 5$ fermionic models, taking care of arising string operators. Even for this lowest order 2D MERA, in which, in each renormalization step, nine fermionic modes are mapped to one, we find an impressive precision. The inset shows the convergence in time for a $9\times 9$ free model ($\mathfrak{u}=0,\mathfrak{t}=0.3$) in a minimization using imaginary time evolution. In the same way, correlators and on-site properties can be efficiently computed.