A diagrammatic formalism for lattices of 1/2 is developed. It is based on an unconstrained mapping between spin and Majorana operators. This allows the use of standard tools of diagrammatic quantum many-body theory without requiring projections. We derive, in particular, the Feynman rules for the expansion around a color-preserving mean-field theory. We then present the numerical results obtained by computing the corrections up to second order for the Heisenberg model in one and two dimensions, showing that perturbative corrections are not only numerically important, but also qualitatively improve the results of mean-field theory. These results pave the way for the use of Majorana diagrammatic tools in theoretical and numerical studies of quantum spin systems.
A diagrammatic formalism for lattices of 1/2 is developed. It is based on an unconstrained mapping between spin and Majorana operators. This allows the use of standard tools of diagrammatic quantum many-body theory without requiring projections. We derive, in particular, the Feynman rules for the expansion around a color-preserving mean-field theory. We then present the numerical results obtained by computing the corrections up to second order for the Heisenberg model in one and two dimensions, showing that perturbative corrections are not only numerically important, but also qualitatively improve the results of mean-field theory. These results pave the way for the use of Majorana diagrammatic tools in theoretical and numerical studies of quantum spin systems.
Quantum spin models with a large number of interaction partners per spin are frequently used to describe modern many-body quantum optical systems such as arrays of Rydberg atoms, atom-cavity systems, or trapped ion crystals. For theoretical analysis the mean-field (MF) ansatz is routinely applied. However, besides special cases of all-to-all or strong long-range interactions, the MF ansatz provides only approximate results. Here we present a systematic correction to MF theory based on diagrammatic perturbation theory for quantum spin correlators in thermal equilibrium. Our analytic results are universally applicable for any lattice geometry and spin-length S. We provide precomputed and easy-to-use building blocks for Ising, Heisenberg and transverse field Ising models in the symmetry-unbroken regime. We showcase the quality and simplicity of the method by computing magnetic phase boundaries and excitations gaps. We also treat the Dicke-Ising model of ground-state superradiance where we show that corrections to the MF phase boundary vanish.
This thesis is devoted to diagrammatic frameworks for computing the dynamic structure factor (DSF) and the static susceptibility of spin systems at finite temperature in the thermodynamic limit. We consider two complementary diagrammatic frameworks. First, we study functional renormalization group (FRG) approaches based on pseudo-fermion (pf-FRG) and pseudo-Majorana (pm-FRG) representations. Second, we develop high-order strong-coupling and high-temperature expansions (HTE) using free and restricted-graph techniques. To enable finite-temperature calculations in pf-FRG, we eliminate unphysical states by incorporating the Popov–Fedotov trick. Within pm-FRG, we introduce a temperature-flow scheme that significantly improves the efficiency and accuracy of finite-size scaling analyses of thermal phase transitions. We further extend pm-FRG to include external magnetic fields, allowing treatment of XXZ models with a field along the z direction. While the pseudo-Majorana representation maps local spin-1/2 operators to free Majorana fermions, higher spin values S>1/2 can instead be represented as strongly interacting real bosons, an idea underlying the recently proposed spin-FRG. Rather than implementing the spin-FRG itself, we analyze its perturbation series using a strong-coupling expansion. To evaluate the high-dimensional imaginary-time integrals arising in this expansion analytically with efficient bookkeeping, we introduce the kernel trick, which rewrites any Matsubara diagram in terms of the partial spectral functions of its constituent Green’s functions. The resulting fourth-order strong-coupling expansion for general spin S provides systematic beyond-mean-field corrections to critical temperatures and fields in both 1/d and large-interaction-range expansions. More importantly, it provides an analytic explanation of the long-standing quantum-to-classical correspondence observed in diagrammatic Monte Carlo simulations: the normalized static susceptibilities of quantum and classical Heisenberg models coincide after a simple temperature rescaling. Our analysis shows that this arises from the fact that the static susceptibility is well approximated by a simple functional form akin to a mean-field susceptibility, which we call the renormalized-mean-field form. To reach higher orders, we generalize the HTE to Matsubara-frequency-resolved Green’s functions (Dyn-HTE). This includes the static susceptibility (at zero Matsubara frequency) as a special case and validates the renormalized-mean-field form up to 12th order in perturbation theory. By analytically continuing the frequency dependence obtained from Dyn-HTE using a continued-fraction expansion, we compute DSFs for one-parameter Heisenberg models on arbitrary lattices, which can be directly compared with experimental inelastic neutron-scattering spectra. All Dyn-HTE calculations are implemented in a publicly available, well-documented open-source code. Finally, we clarify the relation between pseudo-Majorana and spin-based diagrammatic approaches by deriving the exact pseudo-Majorana action directly from the spin Hamiltonian, without invoking any parton representation. This derivation paves the way for extending Majorana-based methods to higher-spin systems.
The diagrammatic method is one of the most powerful tools of theoretical physics. It allows us to efficiently organize perturbative computations in statistical physics, quantum many-body theory and quantum field theory. The main feature of this method is a representation of individual terms of a perturbative expansion as diagrams ( graphs ). Diagrams consist of vertices representing terms in the perturbation, lines representing pairings between vertices and, possibly, external legs . There exist several kinds of diagrams. We will try to present them in a systematic way. In Sect. 20.1 we present a diagrammatic formalism whose goal is to organize integration of polynomials with respect to a Gaussian measure. This formalism is used extensively in classical statistical physics. It also plays an important role in quantum physics, especially in the Euclidean approach, since many quantum quantities can be expressed in terms of Gaussian integrals over classical variables. We use the term “Gaussian integration” in a rather broad sense. Beside commuting “bosonic” variables, we also consider anti-commuting “fermionic” variables, where we use the Berezin integral with respect to a Gaussian weight. Even in the case of commuting variables, the “Gaussian integral” is not necessarily meant in the sense of measure theory. It denotes an algebraic operation performed on polynomials (or formal power series), which in the case of a positive definite covariance coincides with the usual integral with a Gaussian weight. However, we allow the covariance to be complex, or even negative definite, and do not insist that the operation have a measure theoretic meaning.
We study a rotation invariant Majorana fermion model in one dimension using diagrammatic perturbation theory and numerical diagonalization of small systems. The model is inspired by a Majorana representation of the antiferromagnetic spin-1/2 chain, and it is similar in form to the t-J model of electrons, except that the Majorana fermions carry spin-1 and Z_2 charge. We discuss the implications of our results for the low-energy excitations of the spin-1/2 chain. We also discuss a generalization of our model from 3 species of Majorana fermions to N species; the SO(4) symmetric model is particularly interesting.
Abstract Modeling complex systems, like neural networks, simple liquids or flocks of birds, often works in reverse to textbook approaches: given data for which averages and correlations are known, we try to find the parameters of a given model consistent with it. In general, no exact calculation directly from the model is available and we are left with expensive numerical approaches. A particular situation is that of a perturbed Gaussian model with polynomial corrections for continuous degrees of freedom. Indeed perturbation expansions for this case have been implemented in the last 60 years. However, there are models for which the exactly solvable part is non-Gaussian, such as independent Ising spins in a field, or an ideal gas of particles. We implement a diagrammatic perturbative scheme in weak correlations around a non-Gaussian yet solvable probability weight. This applies in particular to spin models (Ising, Potts, Heisenberg) with weak couplings, or to a simple liquid with a weak interaction potential. Our method casts systems with discrete degrees of freedom and those with continuous ones within the same theoretical framework. When the core theory is Gaussian it reduces to the well-known Feynman diagrammatics.
We present diagrammatic approximations to the spin dynamics of the 2d Heisenberg antiferromagnet for all temperatures, employing an auxiliary-fermion representation. The projection onto the physical subspace is effected by introducing an imaginary-valued chemical potential as proposed by Popov and Fedotov. The method requires that the fermion number at any lattice site is strictly conserved. We compare results obtained within a self-consistent approximation using two different auxiliary-particle projection schemes, (1) exact and (2) on average. Significant differences between the two are found at higher temperatures, whereas in the limit of zero temperature (approaching the magnetically ordered ground state) identical results emerge from (1) and (2), providing the qualitatively correct dynamical scaling behavior. An interpretation of these findings is given. We also present in some detail the derivation of the approximation, which goes far beyond mean-field theory and is formulated in terms of complex-valued spectral functions of auxiliary fermions.