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Finite Temperature Auxiliary Field Quantum Monte Carlo in the Canonical Ensemble Tong Shen,1 Yuan Liu,2 Yang Yu,3 and Brenda M

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Finite Temperature Auxiliary Field Quantum Monte Carlo in the Canonical Ensemble Tong Shen,1 Yuan Liu,2 Yang Yu,3 and Brenda M Finite Temperature Auxiliary Field Quantum Monte Carlo in the Canonical Ensemble Tong Shen,1 Yuan Liu,2 Yang Yu,3 and Brenda M. Rubenstein1, a) 1)Department of Chemistry, Brown University, Providence, RI 02912 2)Center for Ultracold Atoms, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 3)Department of Physics, University of Michigan, Ann Arbor, MI 48109 (Dated: 21 October 2020) Finite temperature auxiliary field-based Quantum Monte Carlo methods, including Determinant Quantum Monte Carlo (DQMC) and Auxiliary Field Quantum Monte Carlo (AFQMC), have historically assumed pivotal roles in the inves- tigation of the finite temperature phase diagrams of a wide variety of multidimensional lattice models and materials. Despite their utility, however, these techniques are typically formulated in the grand canonical ensemble, which makes them difficult to apply to condensates like superfluids and difficult to benchmark against alternative methods that are formulated in the canonical ensemble. Working in the grand canonical ensemble is furthermore accompanied by the increased overhead associated with having to determine the chemical potentials that produce desired fillings. Given this backdrop, in this work, we present a new recursive approach for performing AFQMC simulations in the canonical ensemble that does not require knowledge of chemical potentials. To derive this approach, we exploit the convenient fact that AFQMC solves the many-body problem by decoupling many-body propagators into integrals over one-body problems to which non-interacting theories can be applied. We benchmark the accuracy of our technique on illustrative Bose and Fermi Hubbard models and demonstrate that it can converge more quickly to the ground state than grand canonical AFQMC simulations. We believe that our novel use of HS-transformed operators to implement algorithms originally derived for non-interacting systems will motivate the development of a variety of other methods and antici- pate that our technique will enable direct performance comparisons against other many-body approaches formulated in the canonical ensemble. Keywords: Auxiliary Field Quantum Monte Carlo, Determinant Quantum Monte Carlo, Canonical Ensemble, Finite Temperature, Hubbard Model, bosons, fermions I. INTRODUCTION eigenvalues and weighting them to compute its partition func- tion and related observables in a process termed exact diago- 13 For decades, chemical physicists have focused on develop- nalization (ED). While this technique is numerically exact, ing a constellation of electronic structure methods, from mean as its name implies, its computational cost scales exponen- field1,2 to perturbation1,2 to coupled cluster theories,2,3 to de- tially with system size. On the other end of the spectrum, finite scribe the ground and low-lying excited states of molecules temperature mean field theories, including finite temperature 14,15 16,17 and solids. While it is undoubtedly true that many phenomena Hartree Fock and Density Functional (DFT) theories, involve electrons that reside in their ground states, it is becom- trade accuracy for computational expediency by approximat- ing increasingly apparent that there are a wealth of phenomena ing the many-body problem as a one-body problem in which for which this assumption does not hold: atoms and molecules an electron is coupled to an average or “mean” field of the in the centers of large planets and stars can experience GPa other electrons. While such mean field theories continue to of pressure and temperatures of over 10,000 K,4,5 and lasers improve and hold great promise, they still struggle to achieve can be used to heat and thereby steer chemical reactions along the accuracy often needed to correctly capture many-body different mechanistic pathways to facilitate processes such as physics. Between these two poles lie a variety of techniques catalysis.6–8 Temperature is moreover one of the key param- that are generalizations of their ground state counterparts. The past few years, for example, have seen a resurgence of inter- arXiv:2010.09813v1 [quant-ph] 19 Oct 2020 eters that can be used to tune the electronic properties of the 18–22 materials that make up many of modern society’s most impor- est in finite temperature coupled cluster techniques and 23–28 tant technologies9–11 and is responsible for the pernicious loss perturbation theories. Closer to the mean field end of the of coherence in physical realizations of qubits.12 spectrum, finite temperature embedding theories that partition Reflecting this growing list of applications, a growing num- systems into correlated regions embedded within uncorrelated 29 30,31 ber of methods have recently been developed to study them. baths such as Dynamical Mean Field Theory (DMFT) 32–34 One of the most straightforward ways of determining the fi- and the finite temperature SEET and GF2 methods are nite temperature properties of quantum systems is by diago- distinctively capable of directly obtaining the full frequency- nalizing the system’s Hamiltonian to determine all of its many dependent spectra of the systems they model. Nevertheless, all of these methods struggle to balance computational cost with the need to account for the numerous electronic states that contribute to finite temperature expectation values. a)Electronic mail: Author to whom correspondence should be addressed: Finite temperature Quantum Monte Carlo (FT-QMC) meth- [email protected]. ods are particularly advantageous in this regard because they 2 have the exceptional ability to access numerous states with- This makes comparing results from grand canonical ensem- out the exponential or high polynomial costs of other tech- ble approaches to results from canonical ensemble approaches niques by randomly sampling multidimensional state spaces challenging and can also introduce additional statistical noise for the most important states.35–38 One of the most success- which can make arriving at meaningful statistical averages ful of these techniques is Determinant Quantum Monte Carlo more challenging than in the canonical ensemble. Because (DQMC)39–41 and its more recent extension, FT Auxiliary of the differences between these ensembles, properties mea- Field Quantum Monte Carlo (FT-AFQMC).42–44 DQMC has sured in these ensembles may also converge to their limits in a long history of being employed to study a wide variety of different manors, meaning that one ensemble may converge lattice models. One of the key reasons why DQMC has been certain properties more rapidly than the other. so widely adopted is because, for certain important classes of problems,39,45 it does not suffer from the infamous sign Given this context, in this work, we derive a new recursive or phase problems46 that limit the practical utility of many formulation for performing finite temperature AFQMC simu- other stochastic methods. AFQMC methods grew out of lations in the canonical ensemble. Unlike past canonical en- DQMC methods and were developed with the aim of being semble formalisms that relied upon Fourier extracting canon- able to accurately model systems with clear sign problems ical results from grand canonical simulations and thus also 63–67 by leveraging its more sophisticated sampling techniques and depended upon a costly tuning of chemical potentials, phaseless approximation.38,47–49 These methods have since our technique does not require knowledge of the chemical shed light on such sign- and phaseful systems as the Hub- potential. To accomplish this, our method exploits one of bard model off of half-filling46,50,51 and many different ab the key, yet often overlooked features of the AFQMC for- initio molecular44,52–54 and solid state systems.55–58 Further- malism: by decoupling two-body operators into an inte- more, these QMC techniques are versatile – they can be ap- gral over one-body operators, the Hubbard-Stratonovich (HS) 38,68,69 plied to virtually any second-quantized Hamiltonian – and, by Transformation used in AFQMC produces an ensem- sampling the overcomplete space of non-orthogonal determi- ble of non-interacting systems to which theories developed for nants, they are able to sample large portions of Fock space non-interacting systems can be applied. In particular, to derive with a high accuracy, yet comparatively mild computational our approach, we apply a recursive formalism for obtaining cost of O(N3)-O(N4), where N denotes the number of basis the partition functions of ideal gases to the one-body opera- functions.38 tors in AFQMC to ultimately yield a many-body recursive the- ory. In the following, we derive this formalism for systems of This said, one of the Janus-faced features of these meth- bosons, spinless fermions, and spinful fermions, and demon- ods is that they are formulated in the grand canonical ensem- strate its accuracy and practical advantages for several bench- ble, in which a system’s internal energy and particle number mark Hubbard models. Interestingly, we show that energies are allowed to fluctuate according to its fixed temperature and computed in the canonical ensemble converge more quickly to 59 chemical potential. In many systems, it is more natural to fix the ground state than energies computed in the grand canoni- intensive properties rather than extensive quantities (imagine cal ensemble. Ultimately, we believe that this formalism will the inherent difficulty involved with fixing the number of elec- be most useful for studying condensates as well as systems of trons
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