Free energy evaluation in molecular simulations of both classical and quantum systems is computationally intensive and requires sophisticated algorithms. This is because free energy depends on the volume of accessible phase space, a quantity that is inextricably linked to the integration measure in a coordinate representation of a many-body problem. In contrast, the same problem expressed as a field theory (auxiliary field or coherent states) isolates the particle number as a simple parameter in the Hamiltonian or action functional and enables the identification of a chemical potential field operator. We show that this feature leads a “direct” method of free energy evaluation, in which a particle model is converted to a field theory and appropriate field operators are averaged using a field-theoretic simulation conducted with complex Langevin sampling. These averages provide an immediate estimate of the Helmholtz free energy in the canonical ensemble and the entropy in the microcanonical ensemble. The method is illustrated for a classical polymer solution, a block copolymer melt exhibiting liquid crystalline and solid mesophases, and a quantum fluid of interacting bosons. The averaging is performed using a “field-theoretic” computer simulation that employs fluctuating fields rather than particles. Keywords: molecular simulation, free energy, field-theoretic simulation, polymers, quantum fluids Abstract Free energy evaluation in molecular simulations of both classical and quantum systems is computationally intensive and requires sophisticated algorithms. This is because free energy depends on the volume of accessible phase space, a quantity that is inextricably linked to the integration measure in a coordinate representation of a many-body problem. This proceeds by separating attractive and repulsive nonbonded interactions and applying Hubbard–Stratonovich transforms ( 13 , 14 ). Such field theories contain one or more auxiliary fields (AFs) that serve to decouple the nonbonded interactions in the system, facilitating a reduction to a single-molecule statistical mechanics problem. As a simple example, a monatomic fluid with interactions described by a pair potential u ( r ) has a canonical partition function given by ( 15 ) [1] Z ( n , V , T ) = 1 n ! 2 and 3 , but the functional Q [ i w ] is now the partition function of a single polymer in the purely imaginary field i w ( r ) . Since a polymer is a one-dimensional chain of bonded segments, Q [ i w ] can be efficiently computed for a prescribed field w ( r ) by a transfer matrix approach ( 14 ). Quantum Fluids Quantum many-body systems can also be given either a coordinate or field-theoretic representation. For a collection of n bosons in the canonical ensemble, the partition function can be expressed in a coordinate basis as ( 23 ) [5] Z ( n , V , T ) = 1 n ! As in the classical partition function of Eq. 1 , the n dependence in Eq. 5 cannot be isolated, so there is no simple chemical potential operator. There is also a well-established route to expressing an equilibrium quantum many-body system as a field theory. The method involves reframing the problem in second quantization using a complete basis of abstract, single-particle occupation number states in which Bose or Fermi statistics are embedded ( 28 ). 8 explore complex-valued w configurations near constant phase paths that pass through saddle points w s ( r ) of the model. These saddle points satisfy δ H / δ w ( r ) ∣ w s = 0 and correspond to mean-field configurations. Indeed, with η = 0, Eq. 8 is a gradient-descent scheme for finding mean-field solutions ( 14 ). Access to mean-field solutions is an important advantage of the field-theoretic representation that we shall see also aids in free energy estimation. If the stochastic dynamics of Eq. However, surfactant molecules, liquid crystals, and block copolymers can form larger-scale periodic mesophases and be described by soft-core models, for which FTS is ideally suited ( 14 ). The diblock copolymer melt model considered here has six known ordered mesophases at the mean-field (SCFT) level: body-centered cubic spheres, face-centered cubic spheres, hexagonally packed cylinders (HEX), lamellae (LAM), bicontinuous cubic double gyroid (GYR), and bicontinuous orthorhombic ( 51 , 52 ). LAM and HEX are liquid crystalline phases with at least one homogeneous direction in the unit cell; the remaining mesophases listed are solids. Results Our first example of direct free energy evaluation is for the homopolymer solution model, which exhibits only a single homogeneous fluid phase. Fig. 1 reports the intensive free energy in excess of the mean-field (SCFT) value (i.e., the fluctuation contribution) across five decades of dimensionless chain concentration C . A cubic cell of side length L = 6.4 R g was Beyond the ease and efficiency of free energy evaluation, FTSs of classical systems have a number of advantages over traditional particle-based Monte Carlo and molecular dynamics methods ( 1 – 3 ), including a computational cost that is nearly independent of density or polymer chain length ( 62 ), more straightforward and efficient treatment of long-range electrostatic interactions ( 18 , 21 , 63 ), and direct access to mean-field solutions for homogeneous and inhomogeneous systems that become increasingly accurate at high concentration ( 14 ). Moreover, soft-core models are a common starting point in soft-matter simulations using tools such as dissipative particle dynamics ( 66 – 68 ). Such models can be analytically converted to a field theory, allowing for efficient phase diagram construction via SCFT or FTS-CL ( 69 ), the latter utilizing the free energy method presented here. Nonetheless, the extra step of coarse graining represents a barrier if the starting model is atomistic. For quantum many-body systems, the most significant limitation is that the approach advocated here is inapplicable to particles with Fermi statistics.

Collective fields in the functional renormalization group for fermions, Ward identities, and the exact solution of the Tomonaga-Luttinger model We develop a new formulation of the functional renormalization group (RG) for interacting fermions. Our approach unifies the purely fermionic formulation based on the Grassmannian functional integral, which has been used in recent years by many authors, with the traditional Wilsonian RG approach to quantum systems pioneered by Hertz [Phys. Rev. B 14, 1165 (1976)], which attempts to describe the infrared behavior of the system in terms of an effective bosonic theory associated with the soft modes of the underlying fermionic problem. In our approach, we decouple the interaction by means of a suitable Hubbard-Stratonovich transformation (following the Hertz-approach), but do not eliminate the fermions; instead, we derive an exact hierarchy of RG flow equations for the irreducible vertices of the resulting coupled field theory involving both fermionic and bosonic fields. The freedom of choosing a momentum transfer cutoff for the bosonic soft modes in addition to the usual band cutoff for the fermions opens the possibility of new RG schemes. In our approach, we decouple the interaction by means of a suitable Hubbard-Stratonovich transformation (following the Hertz-approach), but do not eliminate the fermions; instead, we derive an exact hierarchy of RG flow equations for the irreducible vertices of the resulting coupled field theory involving both fermionic and bosonic fields. The freedom of choosing a momentum transfer cutoff for the bosonic soft modes in addition to the usual band cutoff for the fermions opens the possibility of new RG schemes. In a pioneering paper, Hertz Hertz76 showed how the powerful machinery of the Wilsonian RG can be generalized to study quantum critical phenomena in Fermi systems. Technically, this is achieved with the help of so-called Hubbard-Stratonovich transformations, which replace the fermionic two-particle interaction by a suitable bosonic field that couples to a quadratic form in the fermion operators. Negele88 The fermions can then be integrated out in a formally exact way, resulting in an effective action for the bosonic field. Of course, there are many possible ways of decoupling fermionic two-body interactions by means of Hubbard-Stratonovich transformations. In this case, it is better to construct an effective action involving all soft modes explicitly. However, for a given problem the nature of the soft modes is not known a priori , so that the explicit introduction of the corresponding degrees of freedom by means of a suitable Hubbard-Stratonovich transformation is always based on some prejudice about the nature of the ground state and the low-lying excitations of the system. Recently, the breakdown of simple Ginzburg-Landau-Wilson theory has also been discussed in the context of quantum antiferromagnets by Senthil et al. This is done by explicitly decoupling the interaction via a Hubbard-Stratonovich transformation in the spirit of Hertz Hertz76 and then considering the functional renormalization group equations for the mixed field theory involving both fermionic and bosonic fields. This type of approach has been suggested previously by Correia, Polonyi, and Richert, Correia01 who studied the homogeneous electron gas by means of a gradient expansion of a functional version of a Callan-Symanzik equation. Here, we follow the more standard approach and derive a hierarchy of flow equations for the vertex functions of our coupled Fermi-Bose theory. Appendix B contains a derivation of the skeleton diagrams for the first few irreducible vertices of our theory using the Dyson-Schwinger equations of motion, which follow from the invariance of the functional integral with respect to infinitesimal shift transformations. Finally, in Appendix C we use the gauge invariance of the mixed Fermi-Bose action to derive a cascade of infinitely many Ward identities involving vertices with two fermion legs and an arbitrary number of boson legs. II Interacting Fermions as coupled Fermi-Bose systems In this section we discuss the Hubbard-Stratonovich transformation and set up a condensed notation to treat fermionic and bosonic fields on the same footing. This will allow us to keep track of the rather complicated diagrammatic structure of the flow equations associated with our coupled Fermi-Bose system in a very efficient way. A similar notation has been used previously in Refs. Salmhofer01, and Baier03, . II.1 Hubbard-Stratonovich transformation We consider a normal fermionic many-body system with two-particle density-density interactions. This is why a dependence of the dispersion ξ 𝐤 ​ σ subscript 𝜉 𝐤 𝜎 \xi_{{\bf{k}}\sigma} on σ 𝜎 \sigma has been kept. The interaction is bilinear in the densities and can be decoupled by means of a Hubbard-Stratonovich transformation. One should keep in mind that a boson line represents the two-body electron-electron interaction which is screened by zero-sound bubbles for small momentum transfers. This means that in fermionic language our vertices are not only one-particle irreducible but are also approximately two-particle irreducible in the zero-sound channel in the sense that particle-hole bubbles are eliminated in favor of the effective bosonic propagator. In order to obtain the generating functional of the corresponding irreducible vertices, we perform a Legendre transformation with respect to all field components, introducing the classical field footnotefield Φ α = δ ​ 𝒢 c δ ​ J α . subscript ¯ 𝐷 ¯ 𝐾 ¯ 𝐤 \bar{D}_{\bar{K}}=|\bar{\bf{k}}|\;. (48) Keeping in mind that the bosonic field mediates the effective interaction, it is clear that Λ Λ \Lambda is a cutoff for the momentum transfer of the interaction. This is precisely the same cutoff scheme employed in the seminal work by Hertz, Hertz76 who discussed also more general frequency-dependent cutoffs for the labels of the bosonic

Auxiliary-field quantum Monte Carlo study of TiO and MnO molecules Calculations of the binding energy of the transition metal oxide molecules TiO and MnO are presented, using a recently developed phaseless auxiliary field quantum Monte Carlo approach. This method maps the interacting many-body problem onto a linear combination of non-interacting problems by a complex Hubbard-Stratonovich transformation, and controls the phase/sign problem with a phaseless approximation relying on a trial wave function. It employs random walks in Slater determinant space to project the ground state of the system, and allows use of much of the same machinery as in standard density functional theory calculations, such as planewave basis and non-local pseudopotentials. The calculations used a single Slater determinant trial wave function obtained from a density functional calculation, with no further optimization. The calculated binding energies are in good agreement with experiment and with recent diffusion Monte Carlo results. [cond-mat/0510791] Auxiliary-field quantum Monte Carlo study of TiO and MnO molecules Auxiliary-field quantum Monte Carlo study of TiO and MnO molecules W. A. Al-Saidi, Henry Krakauer, and Shiwei Zhang Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795 Abstract Calculations of the binding energy of the transition metal oxide molecules TiO and MnO are presented, using a recently developed phaseless auxiliary field quantum Monte Carlo approach. This method maps the interacting many-body problem onto a linear combination of non-interacting problems by a complex Hubbard-Stratonovich transformation, and controls the phase/sign problem with a phaseless approximation relying on a trial wave function. It employs random walks in Slater determinant space to project the ground state of the system, and allows use of much of the same machinery as in standard density functional theory calculations, such as planewave basis and non-local pseudopotentials. The calculations used a single Slater determinant trial wave function obtained from a density functional calculation, with no further optimization. The central idea in standard AF QMC methods BSS ; Koonin is the mapping of the interacting many-body problem into a linear combination of non-interacting problems in external auxiliary fields. Averaging over different AF configurations is then performed by Monte Carlo (MC) techniques. However, except for special cases (e.g., the Hubbard model with on-site interactions), the two-body interactions will require auxiliary fields that are complex . As a result, the single-particle orbitals become complex, and the MC averaging over AF configurations becomes an integration over complex variables in many dimensions, and a phase problem occurs. Both the one-body ( T i ​ j subscript 𝑇 𝑖 𝑗 T_{ij} ) and two-body matrix elements ( V i ​ j ​ k ​ l subscript 𝑉 𝑖 𝑗 𝑘 𝑙 V_{ijkl} ) are known. As in other QMC methods, the auxiliary field quantum Monte Carlo obtains the ground state | Ψ G ⟩ ket subscript Ψ 𝐺 \left|\Psi_{G}\right\rangle of H ^ ^ 𝐻 {\hat{H}} by projecting from a trial wave function | Ψ T ⟩ ket subscript Ψ 𝑇 \left|\Psi_{T}\right\rangle , using the imaginary-time propagator e − τ ​ H ^ superscript 𝑒 𝜏 ^ 𝐻 e^{-\tau{\hat{H}}} : | Ψ G ⟩ ∝ lim n → ∞ ( e − τ ​ H ^ ) n ​ | Ψ T ⟩ . (3) The two-body part of the propagator can be written as an integral of one-body operators by a Hubbard-Stratonovich transformation HS : e − τ ​ H ^ 2 = ∏ α ( 1 2 ​ π ​ ∫ − ∞ ∞ e − 1 2 ​ σ α 2 ​ e τ ​ σ α ​ λ α ​ v ^ α ​ 𝑑 σ α ) , superscript 𝑒 𝜏 subscript ^ 𝐻 2 subscript product 𝛼 1 2 𝜋 superscript subscript superscript 𝑒 1 2 superscript subscript 𝜎 𝛼 2 superscript 𝑒 𝜏 subscript 𝜎 𝛼 subscript 𝜆 𝛼 subscript ^ 𝑣 𝛼 differential-d subscript 𝜎 𝛼 e^{-\tau{\hat{H}_{2}}}=\prod_{\alpha}\Bigg{(}{1\over\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{1}{2}\sigma_{\alpha}^{2}}e^{\sqrt{\tau}\,\sigma_{\alpha}\,\sqrt{\lambda_{\alpha}}\,{\hat{v}_{\alpha}}}d\sigma_{\alpha}\Bigg{)}, (4) after H ^ 2 subscript ^ 𝐻 2 {\hat{H}_{2}} is turned into a sum of squares of one-body operators: H ^ 2 = − 1 2 ​ ∑ α λ α ​ v ^ α 2 subscript ^ 𝐻 2 1 2 subscript 𝛼 subscript 𝜆 𝛼 superscript subscript ^ 𝑣 𝛼 2 {\hat{H}_{2}}=-{1\over 2}\sum_{\alpha}\lambda_{\alpha}{\hat{v}_{\alpha}}^{2} , with λ α subscript 𝜆 𝛼 \lambda_{\alpha} a real number. zhang_krakauer the phaseless auxiliary field QMC method was presented to control the phase problem. The first ingredient of this method is an importance-sampling transformation using a complex importance function, ⟨ Ψ T | ϕ ⟩ inner-product subscript Ψ 𝑇 italic-ϕ \langle\Psi_{T}|\phi\rangle , where | Ψ T ⟩ ket subscript Ψ 𝑇 |\Psi_{T}\rangle is a trial wave function. In the resulting random walk, a walker | ϕ ⟩ ket italic-ϕ |\phi\rangle is propagated to a new position | ϕ ′ ⟩ ket superscript italic-ϕ ′ |\phi^{\prime}\rangle in each step by | ϕ ′ ​ ( σ ) ⟩ = ℬ ​ ( σ − σ ¯ ) ​ | ϕ ⟩ . Ω Ω \Omega is the super-cell volume, 𝐤 𝐤 {\bf{k}} and 𝐤 ′ superscript 𝐤 ′ {\bf{k^{\prime}}} are planewaves within the cutoff radius, and the 𝐪 𝐪 {\bf{q}} -vectors satisfy | 𝐤 + 𝐪 | 2 / 2 < E cut superscript 𝐤 𝐪 2 2 subscript 𝐸 cut |{\bf{k}}+{\bf{q}}|^{2}/2<E_{\rm{cut}} . A Hubbard-Stratonovich transformation is applied to decouple the electron-electron interaction H ^ 2 subscript ^ 𝐻 2 \hat{H}_{2} into a linear combination of one-body operators. To obtain the trial wave function | Ψ T ⟩ ket subscript Ψ 𝑇 |\Psi_{T}\rangle for each QMC calculation, a DFT calculation with the generalized gradient approximation (GGA) is carried out with the ABINIT Abinit program, using the same pseudopotentials and planewave basis. | Ψ T ⟩ ket subscript Ψ 𝑇 |\Psi_{T}\rangle is then taken as the single Slater determinant formed from the occupied single-particle orbitals obtained from this DFT calculation, with no further optimization . The random walkers are all initialized to | Ψ T ⟩ ket subscript Ψ 𝑇 |\Psi_{T}\rangle , so the many-body

Resummation of Feynman Diagrams and the Inversion of Matrices In many field theoretical models one has to resum two- and four-legged subdiagrams in order to determine their behaviour. In this article we present a novel formalism which does this in a nice way. It is based on the central limit theorem of probability and an inversion formula for matrices which is obtained by repeated application of the Feshbach projection method. We discuss applications to the Anderson model, to the many-electron system and to the phi^4-model. In particular, for the many-electron system with attractive delta-interaction, we find that the existence of a BCS gap and a macroscopic value of the Hubbard-Stratonovich field for zero momentum enforce each other. Published as: J.Phys.A34:281-304,2001 DOI: 10.1088/0305-4470/34/2/306 arXiv categories: cond-mat.stat-mech cond-mat.supr-con hep-th math-ph math.MP

Mott insulator to superfluid transition in the Bose-Hubbard model: a strong-coupling approach We present a strong-coupling expansion of the Bose-Hubbard model which describes both the superfluid and the Mott phases of ultracold bosonic atoms in an optical lattice. By performing two successive Hubbard-Stratonovich transformations of the intersite hopping term, we derive an effective action which provides a suitable starting point to study the strong-coupling limit of the Bose-Hubbard model. This action can be analyzed by taking into account Gaussian fluctuations about the mean-field approximation as in the Bogoliubov theory of the weakly interacting Bose gas. In the Mott phase, we reproduce results of previous mean-field theories and also calculate the momentum distribution function. In the superfluid phase, we find a gapless spectrum and compare our results with the Bogoliubov theory. Published as: Phys. Rev. A 71, 033629 (2005) DOI: 10.1103/PhysRevA.71.033629 arXiv categories: cond-mat.str-el