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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.
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Local three- and four-point correlators yield important insight into strongly correlated systems and have many applications. However, the non-perturbative, accurate computation of multipoint correlators is challenging, particularly in the real-frequency domain for systems at low temperatures. In an accompanying paper, we introduce generalized spectral representations for multipoint correlators. Here, we develop a numerical renormalization group (NRG) approach, capable of efficiently evaluating these spectral representations, to compute local three- and four-point correlators of quantum impurity models. The key objects in our scheme are partial spectral functions, encoding the systems dynamical information. Their computation via NRG allows us to simultaneously resolve various multiparticle excitations down to the lowest energies. By subsequently convolving the partial spectral functions with appropriate kernels, we obtain multipoint correlators in the imaginary-frequency Matsubara, the real-frequency zero-temperature, and the real-frequency Keldysh formalisms. We present exemplary results for the connected four-point correlators of the Anderson impurity model, and for resonant inelastic x-ray scattering (RIXS) spectra of related impurity models. Our method can treat temperatures and frequencies -- imaginary or real -- of all magnitudes, from large to arbitrarily small ones.
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.
In the first part of this paper, we study the spin-S Kitaev model using spin wave theory. We discover a remarkable geometry of the minimum energy surface in the N-spin space. The classical ground states, called Cartesian or CN-ground states, whose number grows exponentially with the number of spins N, form a set of points in the N-spin space. These points are connected by a network of flat valleys in the N-spin space, giving rise to a continuous family of classical ground states. Further, the CN-ground states have a correspondence with dimer coverings and with self avoiding walks on a honeycomb lattice. The zero point energy of our spin wave theory picks out a subset from a continuous family of classically degenerate states as the quantum ground states; the number of these states also grows exponentially with N. In the second part, we present some exact results. For arbitrary spin-S, we show that localized Z_2 flux excitations are present by constructing plaquette operators with eigenvalues pm 1 which commute with the Hamiltonian. This set of commuting plaquette operators leads to an exact vanishing of the spin-spin correlation functions, beyond nearest neighbor separation, found earlier for the spin-1/2 model [G. Baskaran, S. Mandal and R. Shankar, Phys. Rev. Lett. 98, 247201 (2007)]. We introduce a generalized Jordan-Wigner transformation for the case of general spin-S, and find a complete set of commuting link operators, similar to the spin-1/2 model, thereby making the Z_2 gauge structure more manifest. The Jordan-Wigner construction also leads, in a natural fashion, to Majorana fermion operators for half-integer spin cases and hard-core boson operators for integer spin cases, strongly suggesting the presence of Majorana fermion and boson excitations in the respective low energy sectors.
We numerically assess model wave functions for the recently proposed particle-hole-symmetric Pfaffian (`PH-Pfaffian) topological order, a phase consistent with the recently reported thermal Hall conductance [Banerjee et al., Nature 559, 205 (2018)] at the ever enigmatic $ u=5/2$ quantum-Hall plateau. We find that the most natural Moore-Read-inspired trial state for the PH-Pfaffian, when projected into the lowest Landau level, exhibits a remarkable numerical similarity on accessible system sizes with the corresponding (compressible) composite Fermi liquid. Consequently, this PH-Pfaffian trial state performs reasonably well energetically in the half-filled lowest Landau level, but is likely not a good starting point for understanding the $ u=5/2$ ground state. Our results suggest that the PH-Pfaffian model wave function either encodes anomalously weak $p$-wave pairing of composite fermions or fails to represent a gapped, incompressible phase altogether.
We consider near-critical two-dimensional statistical systems with boundary conditions inducing phase separation on the strip. By exploiting low-energy properties of two-dimensional field theories, we compute arbitrary $n$-point correlation of the order parameter field. Finite-size corrections and mixed correlations involving the stress tensor trace are also discussed. As an explicit illustration of the technique, we provide a closed-form expression for a three-point correlation function and illustrate the explicit form of the long-ranged interfacial fluctuations as well as their confinement within the interfacial region.
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