Finite temperature DMRG and the Drude weight of spin 1/2 Heisenberg chains

We propose an easy-to-implement approach to study time-dependent correlation functions of one dimensional systems at finite temperature $T$ using the the density matrix renormalization group (DMRG). If the auxiliary degrees of freedom which purify the statistical operator are time-evolved with the physical Hamiltonian but reversed time, the entanglement blow-up inherent to any time-dependent DMRG calculation is dramatically reduced. The numerical effort of finite temperature DMRG becomes comparable to that at $T=0$, and thus significantly longer timescales can be reached. We exploit this to investigate current correlation functions of the XXZ spin $1/2$ Heisenberg chain. At intermediate to large $T$, we can explicitly extract the Drude weight $D$ from the long-time asymptotics. For the isotropic chain, $D$ is finite. At low temperatures, we establish an upper bound for the Drude weight.