Efficient mapping for Anderson impurity problems with matrix product states

Abstract

We propose an efficient algorithm to numerically solve Anderson impurity problems using matrix product states. By introducing a modified chain mapping we obtain significantly lower entanglement, as compared to all previous attempts, while keeping the short-range nature of the couplings. Our approach naturally extends to finite temperatures, with applications to dynamical mean field theory, non-equilibrium dynamics and quantum transport.

Quenching the Anderson impurity model at finite temperature: Entanglement and bath dynamics using matrix product states

“…When employing the thermofield method we need to simulate two independent baths of free fermions -one being empty ( f1k ) and one being filled ( f2k ) -, both interacting with the impurity only. While in principle a direct simulation in the star geometry -using artificial long range interactions -would be possible, we focus here on the chain geometry, following [29]. In particular, we apply two independent chain mappings for the empty and filled fermions f1k and f2k , respectively.…”

“…This is the crucial advantage of the chain mapping introduced in Ref. [29] as compared to the original T = 0 chain mapping, where both empty and filled modes are transformed into a single chain, leading to an entangled state with partially filled chain sites. To carry out the transformation of the Hamiltonian we need the following property of orthogonal polynomials: The monic polynomials {π c,n }, obtained by rescaling the normalized polynomials {p c,n } such that the coefficient of the leading degree term is one, satisfy the recurrence relation [35][36][37]39] (15) with recurrence coefficients {α c,n } and {β c,n }, uniquely defined by the weighting function V 2 c (x).…”

“…The standard chain mapping, on the contrary, is mixing-up all modes, leading to partially occupied chain sites, with nonzero entanglement even in the decoupled ground state. In a recent paper we have shown that the entanglement in the chain structure is significantly reduced by separating filled and empty modes [29], mapping them into two independent chains. This approach shows low entanglement in the MPS, and also neatly generalizes to finite temperatures, by using the thermofield transformation: in contrast to the original matrix product density-operator-based approach [30,31], it does not require imaginary-time evolution to deal with nonzero temperatures.…”

“…The present paper discusses several different orderings of the sites of the improved chain mapping presented in [29], which one can still arbitrarily choose in setting-up an MPS encoding of the resulting nearestneighbor Hamiltonian. We will show that the entanglement growth can be significantly reduced at finite temperature, allowing for much longer simulations, by appropriately alternating filled and empty chain sites in the MPS.…”

“…The paper is organized as follows. In Section II we introduce the model, and summarize our approach based on an improved chain mapping [29], including the thermofield transformation to work at finite temperature. We further discuss the different possible ordering of sites in the MPS that we have considered.…”