Wick's Theorem Research Articles

Braided Scalar Quantum Field Theory

AbstractWe formulate scalar field theories in a curved braided ‐algebra formalism and analyse their correlation functions using Batalin–Vilkovisky quantization. We perform detailed calculations in cubic braided scalar field theory up to two‐loop order and three‐point multiplicity. The divergent tadpole contributions are eliminated by a suitable choice of central curvature for the ‐structure, and we confirm the absence of UV/IR mixing. The calculations of higher loop and higher multiplicity correlators in homological perturbation theory are facilitated by the introduction of a novel diagrammatic calculus. We derive an algebraic version of the Schwinger–Dyson equations based on the homological perturbation lemma, and use them to prove the braided Wick theorem.

Separation of metric in Wick's theorem.

In quantum chemistry, Wick's theorem is an important tool to reduce products of fermionic creation and annihilation operators. It is especially useful in computations employing reference states. The original theorem has been generalized to tackle multiconfigurational wave functions or nonorthogonal orbitals. One particular issue of the resulting structure is that the metric and density matrices are intertwined despite their different origin. Here, an alternative, rather general tensorial formulation of Wick's theorem is proposed. The main difference is the separation of the metric-the coefficients at normal-ordered operators become products of an n-electron density matrix element and the Pfaffian of a matrix formed by orbital overlaps. Different properties of the formalism are discussed, including the use of density cumulants, the particle-hole symmetry, and applications to transition density matrices, i.e., the case of different bra and ket reference states. The metric-separated version of Wick's theorem provides a platform for the derivation of various quantum chemical methods, especially those complicated by non-trivial reference states and nonorthogonality issues.

Finite-temperature many-body perturbation theory for anharmonic vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm.

A unified theory is presented for finite-temperature many-body perturbation expansions of the anharmonic vibrational contributions to thermodynamic functions, i.e., the free energy, internal energy, and entropy. The theory is diagrammatically size-consistent at any order, as ensured by the linked-diagram theorem proved in this study, and, thus, applicable to molecular gases and solids on an equal footing. It is also a basis-set-free formalism, just like its underlying Bose-Einstein theory, capable of summing anharmonic effects over an infinite number of states analytically. It is formulated by the Rayleigh-Schrödinger-style recursions, generating sum-over-states formulas for the perturbation series, which unambiguously converges at the finite-temperature vibrational full-configuration-interaction limits. Two strategies are introduced to reduce these sum-over-states formulas into compact sum-over-modes analytical formulas. One is a purely algebraic method that factorizes each many-mode thermal average into a product of one-mode thermal averages, which are then evaluated by the thermal Born-Huang rules. Canonical forms of these rules are proposed, dramatically expediting the reduction process. The other is finite-temperature normal-ordered second quantization, which is fully developed in this study, including a proof of thermal Wick's theorem and the derivation of a normal-ordered vibrational Hamiltonian at finite temperature. The latter naturally defines a finite-temperature extension of size-extensive vibrational self-consistent field theory. These reduced formulas can be represented graphically as Feynman diagrams with resolvent lines, which include anomalous and renormalization diagrams. Two order-by-order and one general-order algorithms of computing these perturbation corrections are implemented and applied up to the eighth order. The results show no signs of Kohn-Luttinger-type nonconvergence.

Equation generator for equation-of-motion coupled cluster assisted by computer algebra system

We present an equation generator algorithm that utilizes second-quantized operators in normal order with respect to a correlated or non-correlated reference and the corresponding Wick theorem. The algorithm proposed here, written with Mathematica, enables the generation of non-redundant strings of second-quantized operators that, after classification, are directly assigned to many-body term quantities used to construct the many-body Hamiltonian. We demonstrate the capabilities of the algorithm by computing the coupled-cluster amplitude equations and various blocks of the equation-of-motion many-body Hamiltonian. A comprehensive description of this four-step algorithm is provided alongside concrete examples.

Robust formulation of Wick's theorem for computing matrix elements between Hartree-Fock-Bogoliubov wavefunctions.

Numerical difficulties associated with computing matrix elements of operators between Hartree-Fock-Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree-Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan-Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states.

General ordering theorem

The problem of ordering operators has afflicted quantum mechanics since its foundation. Several orderings have been devised, but a systematic procedure to move from one ordering to another is still missing. The importance of establishing relations among different orderings is demonstrated by Wick's theorem (which relates time ordering to normal ordering), which played a crucial role in the development of quantum field theory. We prove the General Ordering Theorem (GOT), which establishes a relation among any pair of orderings, that act on operators satisfying generic (i.e. operatorial) commutation relations. We expose the working principles of the GOT by simple examples, and we demonstrate its potential by recovering two famous algebraic theorems as special instances: the Magnus expansion and the Baker-Campbell-Hausdorff formula. Remarkably, the GOT establishes a formal relation between these two theorems, and it provides compact expressions for them, unlike the notoriously complicated ones currently known.

Wick Theorem and Hopf Algebra Structure in Causal Perturbative Quantum Field Theory

We consider the general framework of perturbative quantum field theory for the pure Yang–Mills model. We give a more precise version of the Wick theorem using Hopf algebra notations for chronological products and not for Feynman graphs. Next, we prove that the Wick expansion property can be preserved for all cases in order n=2. However, gauge invariance is broken for chronological products of Wick submonomials.

Spatial correlations of dark energy from quantum fluctuations during inflation

This paper contains a detailed study of the properties of a simple model attempting to explain dark energy as originated from quantum fluctuations of a light spectator scalar field in inflation. In [1] we recently outlined how Starobinsky's stochastic formalism can be used to study the spatial correlations imprinted on dark energy by its quantum origin in this model and we studied their possible role in relieving the Hubble tension. Here we provide a more comprehensive derivation of the results in [1] and we refine some of our estimates, comparing to the approximate results obtained previously. Among the main results, we analyze the non-coincident correlators predicted by a full field theoretical treatment and their relation with those computed within the stochastic formalism. We find that in the region where stochastic theory predicts significant sub-Hubble correlators it is in disagreement with field theoretical predictions. However, agreement can be restored by introducing a reduced speed of sound for the scalar field. We also discuss an alternative approach to the problem of studying correlators within the stochastic formalism based directly on the evolution of probability distributions. We find that the two approaches give the same answer for 2-point functions of the field, but not for 4-point functions relevant to density correlators and we discuss the behaviour of the two methods with respect to Wick's theorem.

On the off-diagonal Wick’s theorem and Onishi formula

The projected generator coordinate method based on the configuration mixing of non-orthogonal Bogoliubov product states, along with more advanced methods based on it, require the computation of off-diagonal Hamiltonian and norm kernels. While the Hamiltonian kernel is efficiently computed via the off-diagonal Wick theorem of Balian and Brezin, the norm kernel relies on the Onishi formula (or equivalently the Pfaffian formula by Robledo or the integral formula by Bally and Duguet). Traditionally, the derivation of these two categories of formulae relies on different formal schemes. In the present work, the formulae for the operator and norm kernels are computed consistently from the same diagrammatic method. The approach further offers the possibility to address kernels involving more general states in the future.

Automatic derivation of many-body theories based on general Fermi vacua.

This paper describes Wick&d, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated references and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. Wick&d employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since Wick&d can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of Wick&d, we use it to evaluate the single-reference coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multireference perturbation theory.

Field-theoretical approach to open quantum systems and the Lindblad equation

We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovian master equation for the reduced density matrix, incorporating secular effects and independent of the initial preparation. The kernel is determined by a systematic diagrammatic expansion in powers of the interaction. We consider the Born approximation for the kernel. Applying a condensed-matter pole or, equivalently, a quasiparticle-type approximation, equivalent to the usual assumption of a timescale separation, we derive a master equation of the Markov type. Furthermore, imposing the rotating-wave approximation, we obtain a Markov master equation of the Lindblad form. To illustrate the method, we consider the standard example of a single qubit coupled to a thermal heat bath.

Coupled Bogoliubov equations for electrons and phonons

The interaction between fermions and bosons is central to modern physics. To describe systems with such interactions, the authors apply Wick's theorem to the square of the Fr\"ohlich Hamiltonian and keep all terms up to quadratic order. What emerges are two coupled Bogoliubov equations: one for electrons and one for phonons. Remarkably, the self-consistent solution to these equations accurately describes both the renormalized gap in insulators and the superconducting gap in metals.

Beyond Bogoliubov dynamics

We consider a system of N interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to arbitrary precision. The N-independent corrections are given in terms of the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick's theorem. We determine the n-point correlation functions of the excitations around the condensate, as well as the reduced densities of the N-body system, to arbitrary accuracy, given only the knowledge of the two-point functions of a quasi-free state and the solution of the Hartree equation. In this way, the complex problem of computing all n-point correlation functions for an interacting N-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs for the Bogoliubov two-point functions.

Emergence of gaussianity in the thermodynamic limit of interacting fermions

Systems of interacting fermions can give rise to ground states whose correlations become effectively free-fermion-like in the thermodynamic limit, as shown by Baxter for a class of integrable models that include the one-dimensional XYZ spin-$\frac{1}{2}$ chain. Here, we quantitatively analyse this behaviour by establishing the relation between system size and correlation length required for the fermionic Gaussianity to emerge. Importantly, we demonstrate that this behaviour can be observed through the applicability of Wick's theorem and thus it is experimentally accessible. To establish the relevance of our results to possible experimental realisations of XYZ-related models, we demonstrate that the emergent Gaussianity is insensitive to weak variations in the range of interactions, coupling inhomogeneities and local random potentials.

General Wick's theorem for bosonic and fermionic operators

Wick's theorem provides a connection between time ordered products of bosonic or fermionic fields, and their normal ordered counterparts. We consider a generic pair of operator orderings and we prove, by induction, the theorem that relates them. We name this the General Wick's Theorem (GWT) because it carries Wick's theorem as special instance, when one applies the GWT to time and normal orderings. We establish the GWT both for bosonic and fermionic operators, i.e. operators that satisfy c-number commutation and anticommutation relations respectively. We remarkably show that the GWT is the same, independently from the type of operator involved. By means of a few examples, we show how the GWT helps treating demanding problems by reducing the amount of calculations required.

Dressing of Vertices by Cumulants in Multi-Reference Coupled Cluster.

A new scheme is introduced in Multi-Reference (MR) Coupled Cluster (CC) based on the MR Generalized Normal Ordering (MRGNO) and the corresponding MR Generalized Wick Theorem (MRGWT) of Kutzelnigg and Mukherjee. The key element is the identification of a structure in MRGWT generated terms, facilitated by Goldstone diagram techniques. This allows for bundling the many terms of the MRGWT expansion and introduces a hierarchy in the equations that can be harnessed in devising approximations. One- and two-particle interaction vertices are found to be uniformly substituted for their counterpart dressed by density cumulants. This allows for a straightforward rewriting of the ordinary energy expression of CC with interaction dressed (id) one- and two-particle terms and reveals the presence of three- and higher-rank dressed interaction vertices too. Cumulants appearing out of dressed interaction vertices contribute to the amplitude equations and can be interpreted to have an amplitude dressing role. Dressing of one- and two-particle interaction vertices is the most straightforward to implement and does not hinder computational feasibility, provided that the reference function involves strictly limited active space sizes. The Generalized Valence Bond wave function, underlying pilot numerical tests, fulfills this criterion. Results on multiple bond breaking scenarios point to the need of stepping beyond one- and two-particle id. An extremely simple version of incorporating amplitude dressing in addition to one- and two-particle id is seen to cure the potential energy curves remarkably, stimulating further investigations along this line.

N-Body Reduced Density Matrix-Based Valence Bond Theory and Its Applications in Diabatic Electronic-Structure Computations

Valence bond (VB) theory, as a helpful complement to the more popular molecular orbital theory, is a fundamental electronic-structure theory that aims at interpreting molecular structure and chemical reactions in a lucid way. Both theoretical and experimental chemists have shown great interest in VB theory because of its capability of providing intuitive insight into the nature of chemical bonding and the mechanism of chemical reaction in a clear and comprehensible language rooted in Lewis structure. Therefore, there is a great call for the renaissance of VB theory. Nevertheless, this is possible only after a series of methods and algorithms were developed and efficiently implemented in user-friendly programs so as to serve computational chemists for general applications. In the past three decades, we have devoted a great amount of scientific enthusiasm toward this goal. In this Account, we will concisely summarize and briefly but insightfully discuss recent developments in ab initio VB theory, especially the N-body reduced density matrices (RDM)-based approach and its applications in diabatic electronic-structure computations, which is very useful for the vivid interpretation of many fundamental chemical processes such as electron and energy transfers. Furthermore, because of the fundamentally important role that the diabatic state plays in electron and energy transfers, which are two frontier research topics in both molecular and biochemical sciences, there are a broad range of applications that VB theory can handle.We start by briefly reviewing the general feature of ab initio VB wave functions. In particular, we focus on the multistructural ab initio VB theory that uses strictly localized orbitals, including the fundamental VB self-consistent field (VBSCF) and two post-SCF methods, VBCI and VBPT2, that use the VBSCF wave function as reference. We then allot a section to describing the recent developments of the RDM-based VB approach in the second quantization language. In this section, the enhanced Wick theorem is first outlined, followed by a brief discussion of its applications in evaluating VBSCF energy gradients and a Hessian with respect to the orbital expansion coefficients, together with a short review of the implementation of an automatic formula and code generator (AFCG) designed for many-body methods with nonorthogonal orbitals. Then, we introduce the application of the RDM-based approach in implementing the post-SCF method that addresses dynamic electronic correlation via perturbation theory, viz., the icVBPT2 method that adopts an internal contraction technique naturally. We finish this section by incorporating VB theory with the concept of seniority number, in which the tensor analysis technique is carefully exploited with the RDM-based approach, resulting in significant improvements in both the number of the active electrons/orbitals and in the speedup of the computational efficiency, thus pushing VB theory to its new limit. With these achievements available, we present the applications of VB theory in diabatic electronic-structure computations by using the intuitive insight rendered by VB theory. Therefore, we believe that there is a bright future in VB theory with true opportunities and new challenges coexisting both for theoretical developments and computational applications.

Normal ordered exponential approach to thermal properties and time-correlation functions: general theory and simple examples

A normal ordered exponential parametrisation is used to obtain equations for thermal one-and two-particle reduced density matrices, as well as free energies, partition functions and entropy for both Fermionic (electronic) and Bosonic (vibrational) Hamiltonians. A first principles derivation of the equations, relying only on a simple Wick's theorem and starting from the differential equation , is presented that yields a differential equation for the amplitudes representing density cumulants, as well as the grand potential. In contrast to other approaches reported in the literature, the theory does not use the interaction picture and an integral formulation as a starting point but rather requires a propagation of the resulting differential equation for the amplitudes in the Schrödinger representation. While the theory is applicable to general classes of many-body problems in principle, here, the theory is illustrated using simple model systems. As an application of the related time-dependent formulation numerically exact time-autocorrelation functions and absorption spectra are obtained for harmonic Franck–Condon problems. These examples illustrate the basic soundness of the scheme and are used for pedagogical purposes. Other approaches in the literature are only discussed briefly and no detailed comparative discussion is attempted.

Strong-coupling diagrammatic Monte Carlo technique for correlated fermions and frustrated spins

We describe a controllable and unbiased strong-coupling diagrammatic Monte Carlo technique that is applicable to a wide range of fermionic systems and spin models. Unlike previous strong coupling methods that generally rely on the Grassmannian Hubbard-Stratonovich transformation, our construction is based on Wick's theorem and a recursive procedure to group contractions into effective connected vertices that are non-perturbative in all local physics and can be calculated exactly. The resulting expansion is described by simple diagrammatic rules that make it suitable for systematic treatment via stochastic sampling. Benchmarks against numerical linked cluster expansion display excellent agreement.

Polynomial KP and BKP $$\tau $$-Functions and Correlators

Lattices of polynomial KP and BKP $\tau$-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions are expressed via generalizations of Jacobi's bialternant formula for Schur functions and Nimmo's Pfaffian ratio formula for Schur $Q$-functions. These are obtained by applying Wick's theorem to fermionic vacuum expectation value representations in which the infinite group element acting on the lattice of basis states stabilizes the vacuum.