Time-reversal symmetry adaptation in relativistic density matrix renormalization group algorithm

In the nonrelativistic Schr\"{o}dinger equation, the total spin $S$ and spin projection $M$ are good quantum numbers. In contrast, spin symmetry is lost in the presence of spin-dependent interactions such as spin-orbit couplings in relativistic Hamiltonians. Previous implementations of relativistic density matrix renormalization group algorithm (R-DMRG) only employing particle number symmetry are much more expensive than nonrelativistic DMRG. Besides, artificial breaking of Kramers degeneracy can happen in the treatment of systems with odd number of electrons. To overcome these issues, we introduce time-reversal symmetry adaptation for R-DMRG. Since the time-reversal operator is antiunitary, this cannot be simply achieved in the usual way. We define a time-reversal symmetry-adapted renormalized basis and present strategies to maintain the structure of basis functions during the sweep optimization. With time-reversal symmetry adaptation, only half of the renormalized operators are needed and the computational costs of Hamiltonian-wavefunction multiplication and renormalization are reduced by half. The present construction of time-reversal symmetry-adapted basis also directly applies to other tensor network states without loops.