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Register a new userNumerical evaluation of two-time correlation functions in open quantum systems with matrix product state methods: a comparison
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We compare the efficiency of different matrix product state (MPS) based methods for the calculation of two-time correlation functions in open quantum systems. The methods are the purification approach [1] and two approaches [2,3] based on the Monte-Carlo wave function (MCWF) sampling of stochastic quantum trajectories using MPS techniques. We consider a XXZ spin chain either exposed to dephasing noise or to a dissipative local spin flip. We find that the preference for one of the approaches in terms of numerical efficiency depends strongly on the specific form of dissipation.
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The many-body problem is usually approached from one of two perspectives: the first originates from an action and is based on Feynman diagrams, the second is centered around a Hamiltonian and deals with quantum states and operators. The connection between results obtained in either way is made through spectral (or Lehmann) representations, well known for two-point correlation functions. Here, we complete this picture by deriving generalized spectral representations for multipoint correlation functions that apply in all of the commonly used many-body frameworks: the imaginary-frequency Mastubara and the real-frequency zero-temperature and Keldysh formalisms. Our approach is based on separating spectral from time-ordering properties and thereby elucidates the relation between the three formalisms. The spectral representations of multipoint correlation functions consist of partial spectral functions and convolution kernels. The former are formalism independent but system specific; the latter are system independent but formalism specific. Using a numerical renormalization group (NRG) method described in an accompanying paper, we present numerical results for selected quantum impurity models. We focus on the four-point vertex (effective interaction) obtained for the single-impurity Anderson model and for the dynamical mean-field theory (DMFT) solution of the one-band Hubbard model. In the Matsubara formalism, we analyze the evolution of the vertex down to very low temperatures and describe the crossover from strongly interacting particles to weakly interacting quasiparticles. In the Keldysh formalism, we first benchmark our results at weak and infinitely strong interaction and then reveal the rich real-frequency structure of the DMFT vertex in the coexistence regime of a metallic and insulating solution.
We prove that ground states of gapped local Hamiltonians on an infinite spin chain can be efficiently approximated by matrix product states with a bond dimension which scales as D~(L-1)/epsilon, where any local quantity on L consecutive spins is approximated to accuracy epsilon.
The time-dependent matrix-product-state (TDMPS) simulation method has been used for numerically simulating quantum computing for a decade. We introduce our C++ library ZKCM_QC developed for multiprecision TDMPS simulations of quantum circuits. Besides its practical usability, the library is useful for evaluation of the method itself. With the library, we can capture two types of numerical errors in the TDMPS simulations: one due to rounding errors caused by the shortage in mantissa portions of floating-point numbers; the other due to truncations of nonnegligible Schmidt coefficients and their corresponding Schmidt vectors. We numerically analyze these errors in TDMPS simulations of quantum computing.
We present a method for simulating the time evolution of one-dimensional correlated electron-phonon systems which combines the time-evolving block decimation algorithm with a dynamical optimization of the local basis. This approach can reduce the computational cost by orders of magnitude when boson fluctuations are large. The method is demonstrated on the nonequilibrium Holstein polaron by comparison with exact simulations in a limited functional space and on the scattering of an electronic wave packet by local phonon modes. Our study of the scattering problem reveals a rich physics including transient self-trapping and dissipation.
Lattice models consisting of high-dimensional local degrees of freedom without global particle-number conservation constitute an important problem class in the field of strongly correlated quantum many-body systems. For instance, they are realized in electron-phonon models, cavities, atom-molecule resonance models, or superconductors. In general, these systems elude a complete analytical treatment and need to be studied using numerical methods where matrix-product states (MPS) provide a flexible and generic ansatz class. Typically, MPS algorithms scale at least quadratic in the dimension of the local Hilbert spaces. Hence, tailored methods, which truncate this dimension, are required to allow for efficient simulations. Here, we describe and compare three state-of-the-art MPS methods each of which exploits a different approach to tackle the computational complexity. We analyze the properties of these methods for the example of the Holstein model, performing high-precision calculations as well as a finite-size-scaling analysis of relevant ground-state obervables. The calculations are performed at different points in the phase diagram yielding a comprehensive picture of the different approaches.
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