Gaussian phase-space representations for fermions

Transcription

1 Gaussian phase-space representations for fermions J. F. Corney an P. D. Drummon ARC Centre of Excellence for Quantum-Atom Optics, University of Queenslan, Brisbane 4072, Queenslan, Australia Receive 2 November 2005; revise manuscript receive 23 January 2006; publishe 22 March 2006 We introuce a positive phase-space representation for fermions, using the most general possible multimoe Gaussian operator basis. The representation generalizes previous bosonic quantum phase-space methos to Fermi systems. We erive equivalences between quantum an stochastic moments, as well as operator corresponences that map quantum operator evolution onto stochastic processes in phase space. The representation thus enables first-principles quantum ynamical or equilibrium calculations in many-boy Fermi systems. Potential applications are to strongly interacting an correlate Fermi gases, incluing coherent behavior in open systems an nanostructures escribe by master equations. Examples of an ieal gas an the Hubbar moel are given, as well as a generic open system, in orer to illustrate these ieas. DOI: /PhysRevB PACS numbers: F, Ca, Yz, Ss I. INTRODUCTION The stuy of strongly correlate Fermi gases is one of the most active areas in moern conense matter an atomic, molecular, an optical AMO physics. In quantum egenerate electron gases, improvements in conense matter materials have le to sophisticate experiments, typically in reuce imensional environments. Many interesting quantum phenomena are observe in these systems, incluing such features as the quantum Hall effect, 1 metal-insulator phase-transitions, 2 high T-c superconuctors, 3 an single electron gates in nanostructures. 4 Recently, pioneering experiments in strongly interacting ultracol Fermi gases have opene up novel experiments of unpreceente simplicity an precision, both in the crossover between Bose-Einstein conensate BEC an BCS regimes 5 an in lattices. 6 The unerlying atomic interactions are extremely well-unerstoo, an the ynamics, interactions, an geometry are all highly aaptable. Measurement techniques are also rapily improving, with irect measurements of collective moes, 7 thermoynamic properties, 8 vortices, 9 an even momentum correlations being recently reporte. 10 This situation provies a substantial opportunity to evelop an test first-principles theoretical methos for the investigation of correlations an ynamical effects in quantum egenerate Fermi gases. To this en, we introuce a generalize phase-space representation for correlate fermionic systems. The representation is base on a Gaussian operator basis for fermionic ensity operators. Like the analogous basis for bosons, 11 the fermionic operator basis enables the representation of arbitrary physical ensity operators as a positive istribution over a phase space. This representation allows quantum evolution, either in real time or in inverse temperature, to be viewe as a stochastic evolution of covariances or Green s functions. Phase-space methos base on coherent states 12 have long been use for bosonic systems, with great success. These approaches inclue the Wigner function, 13 the Q function, 14 as well as the well-known Glauber-Suarshan P function, 15 an its generalizations. 16,17 The early methos base on classical phase spaces were later generalize to give the positive- P istribution, 18 which has prove a successful way to simulate quantum many-boy systems from first principles. 19 This metho reuces quantum ynamics to the time evolution of a positive istribution on an overcomplete basis set of coherent-state projection operators, which are special cases of the bosonic Gaussian operators. Applications have been to quantum statistics of lasers, 20 superfluorescence, 21 parametric amplifiers, 18,22 an quantum solitons, 23 as well as quantum ynamics 24 an thermal correlations 25 in Bose-Einstein conensates. Fermionic phase-space representations are relevant to a long-staning problem in theoretical physics, which is the sign problem that occurs in quantum Monte-Carlo QMC calculations of many-boy fermionic physics There are many ifferent approximate techniques that can be use, but the intention of this paper is to establish funamentally exact proceures that avoi the Fermi sign problem. As reporte earlier, the Gaussian metho has been successfully applie to the ifficult case of the repulsive Hubbar moel. 32 Here, we concentrate on the founational issues of the Gaussian representation metho, presenting the general ientities require to apply the metho to a wie range of problems in fermionic many-boy physics, incluing both ultracol atomic an conense matter systems. To procee, we make use of three important results, prove elsewhere, 33 i the Gaussian fermion operators form a complete basis for any physical ensity operator, ii the istribution can always be chosen positive, an iii there are mappings to a secon-orer ifferential form for all twoboy operators. From these properties, we show that positive-efinite Fokker-Planck equations exist for manyboy fermionic systems, provie that the istribution tails remain sufficiently boune. Such Fokker-Planck equations enable first-principles stochastic simulation methos, either in real time or at finite temperature. As is usual in such methos, care must be taken with sampling errors an bounary terms ue to the istribution tails. Due to the nonuniqueness of the representation, there is a type of gauge freeom in the choice of stochastic equation. We show how this stochastic gauge freeom, which has been successfully use to remove bounary terms in bosonic representations, 34 can in principle also be use here /2006/7312/ /$ The American Physical Society

2 J. F. CORNEY AND P. D. DRUMMOND Representations for fermionic ensity operators were introuce by Cahill an Glauber 35 using fermionic coherent states. 36 These provie a means of efining quasiprobabilities for fermionic states analogous to the well-known bosonic istributions. 35,37 However, the resulting quasiprobabilities are functions of noncommuting Grassmann variables an are thus not irectly computationally accessible. Nevertheless, fermion coherent states an Grassmann algebra are useful for eriving analytical results in Fermi systems. The Gaussian metho introuce here overcomes the problems inherent in using Grassmann algebra variables. The Gaussian expansion utilizes an operator basis constructe from pairs of operators, instea of a state-vector basis. Because pairs of fermion operators obey commutation relations rather than anticommutation relations, a natural solution of the anticommutation problem is achieve. The resulting phase space thus exists on a omain of commuting c numbers, rather than anticommuting Grassmann variables. Furthermore, the phase-space equations obviate the nee to evaluate large eterminants in simulations. This metho substantially generalizes an extens earlier phase-space techniques use in quantum optics to treat electronic transitions in atoms. 20,38 It is ifferent than auxiliary fiel quantum Monte Carlo AFQMC methos 39 in conense matter theory, which use Gaussian operators, but involve path integrals rather than positive expansions of the ensity matrix. We begin in Sec. II by efining the Gaussian operator basis on which the representation is base an introucing some convenient notations. In Sec. III, we efine the Gaussian representation as an expansion in Gaussian operators, an then show how the representation establishes a novel class of exact Monte Carlo type methos for simulating the real-time ynamics or finite-temperature equilibrium of a quantum system. We show how to map quantum operator evolution onto a set of stochastic real or complex ifferential equations an give the corresponences necessary to calculate physical moments. Finally, in Sec. V, we give examples of the application of the metho. These are intene to be illustrative rather than exhaustive, an further examples an applications in greater etail will be given elsewhere. In particular, we note that any nonlinear application requires a careful analysis of the issues of sampling error an bounary term behavior. For simplicity, we focus on the ieal Fermi gas, a generic open system master equation an the finite-temperature Hubbar moel, as well as showing how to apply gauges to moify the rift evolution. Appenix A summarizes the mathematical properties of the Gaussian operators prove in Ref. 33 that are essential to making use of the phase-space representation, an Appenix B gives an alternative form of the operator mappings. II. GAUSSIAN OPERATORS Before iscussing the Gaussian representation, we first introuce the fermion operators on which it is base. Fermionic Gaussian operators are efine as exponentials of quaratic forms in the Fermi annihilation or creation operators. This simple efinition encompasses a wie range of physical applicability. Obviously, it inclues the well-known thermal ensity matrices of the free fiel. Since the efinition inclues quaratic forms involving pairs of annihilation or creation operators, it also encompasses the pure-state ensity matrices that correspon to the BCS states use in superconuctivity. A more subtle issue is that the efinition is not restricte to Hermitian operators. This has the avantage of leaing to completeness properties that are much stronger than if the efinition were restricte to only Hermitian operators. Some of these issues are iscusse elsewhere, in a more formal erivation of the mathematical properties of the Gaussian operators. 33 A. Notation Before giving mathematical results, we summarize the notation that will be use. We can ecompose a given fermionic system into a set of M orthogonal single-particle moes or orbitals. With each of these moes, we associate creation an annihilation operators bˆ j an bˆ j, with anticommutation relations bˆ k,bˆ j = kj bˆ k,bˆ j =0, 2.1 where j,k=1,...,m. Thus, bˆ is a column vector of the M annihilation operators, an bˆ is a row vector of the corresponing creation operators. For proucts of operators, we make use of normal an antinormal orering concepts. Normal orering, enote by : :, is efine as in the bosonic case, with all annihilation operators to the right of the creation operators, except that each pairwise reorering involve inuces a sign change, e.g., :bˆ ibˆ j : = bˆ j bˆ i. The sign changes are necessary so that the anticommuting natures of the Fermi operators can be accommoate without ambiguity. To enable the general Gaussian operator to be written in a compact form, we use an extene-vector notation = bˆ bˆ T 2.2 bˆ is efine as an extene column vector of all 2M operators, with an ajoint row vector efine as bˆ = bˆ,bˆ T. 2.3 Throughout the paper, we print vectors of length M an M M matrices in bol type, an inex them where necessary with Latin inices: j=1,...,m. Vectors of length 2M we enote with an unerline, while 2M 2M matrices are inicate by a ouble unerline. These extene vectors an matrices are inexe where necessary with Greek inices: =1,...,2M. For further examples of this notation, see Refs. 11 an 33. More general kins of vectors are enote with an arrow notation:

3 GAUSSIAN PHASE-SPACE REPRESENTATIONS FOR FERMIONS B. Definition of the Gaussian operator We efine a Gaussian operator to be any normally orere, Gaussian form of annihilation an creation operators. Like a complex number Gaussian, the operator Gaussian is an exponential of a quaratic form, with the exponential efine by its series representation. The most general Gaussian form is a cumbersome object to manipulate, unless proucts of o numbers of operators are exclue. Fortunately, restricting the set of Gaussians to those containing only even proucts can be physically justifie on the basis of superselection rules for fermions. Because it is constructe from pairs of operators, this type of Gaussian operator contains no Grassmann variables. With the extene-vector notation, we can write any general Gaussian operator ˆ as ˆ = 1 N :exp bˆ =bˆ/2:, 2.4 where is an amplitue, N is a normalizing factor efine so that Trˆ =, an = isa2m 2M complex matrix. For later ientification with physical observables, it proves useful to write = in the form = = = 1 2I=, 2.5 where = is a generalize covariance matrix an I= is the constant matrix is efine as I= = I I. It is convenient to introuce complex M M matrices n an ñ = I n which, as we show later, correspon to normal Green s functions for particles an holes, respectively. We also introuce two inepenent antisymmetric complex M M matrices m an m that correspon to anomalous Green s functions. These are relate to the covariance matrix = by = = ñt m ñ m = nt I m m 2.7 I n. Thus the covariance matrix has a type of antisymmetry, which can be written as = = =, where enotes a generalize transpose operation efine by a b c c T. 2.8 b a In summary, the Gaussian operators are parametrize by =,n,m,m, 2.9 corresponing to 1 p=1m2m 1 complex parameters. Where necessary, we will inex over all the inepenent phase-space variables with the notation a, a=0,...p. The normalization N contains a Pfaffian, whose square is equal to a matrix eterminant. We will show that N oes not appear explicitly in later results. The aitional variable plays the role of a weighting factor in the expansion. This allows us to represent unnormalize ensity operators like exp Ĥ an to introuce stochastic gauges that change these relative weighting factors in orer to stabilize trajectories. C. Moments Just as with classical Gaussian forms, these generalize fermionic Gaussians are completely characterize by their first-orer moments to within a weight factor. From Eq. A3 of Appenix A, we have Trbˆ ibˆ jˆ = m ij, Trbˆ i bˆ jˆ = n ij, Trbˆ jbˆ i ˆ = ñ ij, Trbˆ i bˆ j ˆ = m ij If the Gaussian operator happens to be a physical ensity matrix, these quantities correspon to the first-orer correlations or Green s functions. Thus, in many-boy terminology, n an ñ are the normal Green s functions of particles an holes, respectively, an m an m are anomalous Green s functions. From this, we see that, for the subset of Gaussians that are physical ensity matrices, we must have that m =m an n =n. Furthermore, n an ñ must be positive semiefinite because 0bˆ j bˆ j1. More generally, the phase-space function O corresponing to the normally orere operator Ô is efine as a phase-space corresponence, accoring to O Ô TrÔˆ / For higher-orer moments, a form of Wick s theorem applies to any normally orere prouct. One simply writes own the sum of all istinct factorizations into pairs, with a minus sign in front of any prouct that is an o permutation of the original form. The term istinct factorization means that neither permutation of pair orering nor reorering insie a pair is regare as significant, since these o not change the result. Thus, an Nth orer correlation expectation value of a prouct of 2N operators, is the sum of 2N!/2 N N! istinct terms, as follows: :bˆ 1 bˆ 2N : = 1 P :bˆ P1 bˆ P2 : P :bˆ P2N 1 bˆ P2N : Here the sum is over all 2N!/2 N N! istinct pair permutations P1,...,P2N of 1,...,2N, an where 1 P is the parity of the permutation i.e., the number of pairwise transpositions require to perform the permutation. Thus, for example, the secon-orer number correlation moment is bˆ i bˆ j bˆ jbˆ i = n ii n jj n ij n ji m ij m ji

4 J. F. CORNEY AND P. D. DRUMMOND D. Generalize thermal states An important subset of the Gaussian operators is the set of generalize thermal operators, for which m=m =0. These inclue the canonical ensity matrices for free Fermi gases in the case that n an ñ are each Hermitian an positiveefinite. More generally, however, we o not require n to be Hermitian. In all cases, the generalize thermal operators in normally orere Gaussian form can be written most irectly in terms of the hole population, ñ=i n ˆ = etñ:expbˆ ñ 1 2I T bˆ : Of course, there is a symmetry here: in an antinormally orere Gaussian, the role of bˆ an bˆ is reverse an, consequently, so is the role of n an ñ. Our choice of normal orering is in fact arbitrary from a physical point of view, an antinormal orering woul also serve our purpose equally well, provie all the ientities were reefine. By comparison, the usual canonical form of the fermionic thermal state with a iagonal Hamiltonian H=bˆ bˆ an a chemical potential, is an unorere form, namely ˆ = expbˆ I bˆ /Z Here, Z is the partition function an =1/k B T is the inverse temperature. In this case, the mean occupation numbers are iagonal an well-known. They are given by the Fermi-Dirac istribution, bˆ bˆ i j = n ij = ij e i However, both Gaussian forms are equivalent. A normally orere thermal Gaussian can always be chosen so that n is Hermitian, an hence ˆ=ˆ, if an only if =1 an n ij =n ij. A rather trivial example is the vacuum state, in which n=0, so that ˆ 1,0,0,0 = 00 = :exp bˆ bˆ : We emphasize that since the Gaussian forms use here are not necessarily Hermitian, the generalize thermal operators are a much larger set of operators than the usual canonical thermal ensity matrices. E. Generalize BCS states A secon important subset of the Gaussian operators is the generalization of the Bareen-Cooper-Schreiffer BCS states, which are an excellent approximation to the groun state of a weakly interacting BCS superconuctor. The BCS states are the fermionic equivalent of the squeeze states foun in quantum optics an are compose only of correlate fermion pairs. In the case of fermions, these are the funamental pure states that carry phase information. In Bose gases, coherent states can also carry phase information as in a laser or Bose-Einstein conensate, but the fermionic equivalent of these is an unphysical Grassmann coherent state. An unnormalize pure BCS state is efine as 40 BCS = expbˆ gbˆ /20, so that the corresponing ensity matrix is ˆ BCS = BCS BCS = expbˆ gbˆ /200expbˆ g bˆ / = :expbˆ gbˆ /2 bˆ bˆ bˆ g bˆ /2: Apart from being unnormalize, this correspons irectly to a Gaussian in our normal form. More general non-hermitian BCS type states are obtaine by replacing g by an inepenent matrix g. This generalize BCS Gaussian has an extene covariance matrix of = = 1 I gg 1 0 I g 0 I g g g I Clearly, from this we can see that the occupation numbers an correlations for a generalize BCS state are given by n = g I gg 1 g, ñ = I g g 1 m = I gg 1 g m = g I gg 1, 2.21 which gives the expecte result that m m=ñn. In summary, the usual BCS states have a ensity matrix which is Gaussian an has g =g. These pure states exist as a subset of a more general class of BCS-like Gaussian operators. This class also inclues operators which have g g an are, therefore, not Hermitian. While these operators o not correspon to any physical state, a linear combination of them can still correspon to a possible physical fermionic many-boy state, provie the result is Hermitian an positive-efinite. III. GAUSSIAN REPRESENTATION While the Gaussian operators inclue a large an interesting set of physical ensity operators, there are many cases where the existence of interparticle interactions leas to more general fermionic states whose correlations are of more complex, non-gaussian forms. In all such cases, the overall physical ensity operator can still be expresse as a positive istribution over the Gaussian operators. Furthermore, any two-boy operator acting on a generalize Gaussian can be written as a secon-orer erivative. These important results, prove in Ref. 33, mean that probabilistic, ranom sampling methos may be use to calculate physical observables, as we show below. A. Definition The Gaussian representation is efine as an expansion of the ensity matrix for any physical state ˆ as a positive istribution over the Gaussian basis. That is

5 GAUSSIAN PHASE-SPACE REPRESENTATIONS FOR FERMIONS ˆ = P,ˆ, 3.1 where the expansion coefficients are normalize to one P, = Here, the variable can either be real time t or imaginary time inverse temperature. This expansion efines a type of phase-space representation of the state: the vector of Gaussian parameters becomes a generalize phase-space coorinate, the function P, is then a probability istribution function over the generalize phase space, an = 2p1 is the phase-space integration measure. B. Completeness an positivity The Gaussian representation as we have efine it here always exists for any physical state. In other wors, a istribution always exists an can always be chosen to be positive. This property follows from the overcompleteness of the Gaussian basis, which can be state as follows: For any physical ensity matrix ˆ, a positive set of coefficients P j exists such that ˆ = P j ˆ j, 3.3 j or, in a continuous formulation, a positive semiefinite function P exists such that ˆ = P ˆ, 3.4 where the Gaussian operators ˆ are as efine in Eq By physical ensity matrix, we mean one in which there are no coherences between states that iffer by an o number of fermions. The proof of this result, given in Ref. 33, oes not rely on the complex amplitues that are part of the most general Gaussian operator i.e., we set =1, since using these with complex values is equivalent to having a complex istribution, although this may be useful when constructing particular exact solutions, epening on the problem. The property is analogous to a similar result known for the positive-p bosonic representation. 18 Note that the expansion P j or P is not guarantee to be unique. In fact, there may be many possible positive expansions that correspon to the same physical state. This nonuniqueness is a further aspect of the overcompleteness of the Gaussian operators. It allows the resulting phase-space representation to be use for calculating time evolution an is the basis of stochastic gauges, which we iscuss further in Sec. IV D. As a simple example, consier the single-moe ensity operator. In this case, a complete operator basis for fermions woul be the number state projection operators, both of which are in fact Gaussian operators ˆ,n 00 = ˆ 1,0, 11 = ˆ 1, Because superselection rules prohibit superpositions of states iffering by o numbers of fermions, this is the most general case possible, an clearly the Pauli exclusion principle means that there can be no anomalous moments m or m in a single-moe Gaussian. Equation 3.5 implies that just two single-moe Gaussians are sufficient to form a complete basis set for all possible single-moe ensity matrices, an of course, these must have a positive expansion coefficient to ensure overall positivity of ˆ. Thus, a single-moe ensity matrix can always be expane as a iscrete sum of Gaussians with positive coefficients, as in Eq. 3.3, since for 0n1 ˆn = 1 nˆ 1,0 nˆ 1, However, there are other possible expansions as well. In the single-moe case, all physical ensity operators are also Gaussian operators: ˆ =ˆ 11,n. This means that a continuous expansion of form given in Eq. 3.4 is also possible, with P = n n. 3.7 It is clear from these examples that the Gaussian operators are overcomplete: a physical ensity matrix may be represente by more than one positive istribution over the Gaussians. So far, we have only consiere examples in which the Gaussians are themselves physical ensity operators. However, single-moe Gaussian operators as efine here can have n complex. Incluing such operators, which o not correspon irectly to any physical ensity operators, provies even more freeom of choice in constructing expansions. C. Moments Some basic properties of P, follow from those of the Gaussian operators. For example, using the normalization of the Gaussian operators, we fin that Trˆ = P,. 3.8 Thus, the normalize istribution P can represent unnormalize ensity operators by incorporating the normalization into the mean weight. More generally, the expectation value of an operator Ô evaluates to ÔTrÔˆ/Trˆ = P,TrÔˆ / O P, where the weighte average... P is efine as

6 J. F. CORNEY AND P. D. DRUMMOND O P = P,O / A. Types of evolution We consier three general time-evolution categories. The phase-space function O corresponing to the operator Ô is efine as previously in Eq an can be evaluate using the generalize Wick result of Eq Physical quantities thus correspon to weighte moments of P. For example, from traces evaluate in Sec. II C, we fin that the normal an anomalous Green s functions correspon to first-orer moments bˆ ibˆ j = m ij P, bˆ i bˆ j = n ij P, bˆ i bˆ j = m ij P Number-number correlations correspon to averages of proucts of these moments :nˆ inˆ j: = n ii n jj n ij n ji m ij m ji P, 3.12 where nˆ ibˆ i bˆ i. Similarly, higher-orer correlations correspon to higherorer moments, the form of which are also etermine by the generalize Wick result of Eq We note that the expectation value of any o prouct of operators must vanish, e.g., bˆ i=0. Thus the istribution cannot represent a superposition of states whose total number iffer by an o number. Such superposition states we exclue from our efinition of the physical state, as they are not generate by evolution uner any known physical Hamiltonian. The Gaussian istribution can, however, represent systems in which particles are coherently ae or remove in pairs, leaing to nonzero anomalous correlations m ij P. On the other han, if the total number of particles is conserve or change only via contact with a thermal reservoir, then the anomalous correlations will be ientically zero, an we can represent the system via an expansion in only the thermal subset of Gaussian operators. IV. TIME EVOLUTION Here we show how these positive representations of ensity matrices can be put to use. By use of these representations, any quantum evolution arising from one-boy an two-boy interactions can be sample by classical stochastic processes. To see this, note that the evolution of a ensity operator is etermine by a master equation, of the general form ˆ = Lˆ ˆ, 4.1 where the Lˆ is a superoperator that premultiplies an postmultiplies the ensity operator by combinations of annihilation an creation operators an where can represent either real or imaginary time. 1. Hamiltonian quantum ynamics For unitary evolution in real time t, the superoperator is a commutator with the Hamiltonian Lˆ ˆ = i Ĥ,ˆ. 2. Irreversible quantum ynamics 4.2 More generally, for an open quantum system, there will be aitional terms of Linbla form 42,43 to escribe the coupling to the environment Lˆ ˆ = i Ĥ,ˆ 2Ô K ˆÔ K Ô K Ô K,ˆ, K 4.3 where the operators Ô K epen on the correlations of the environment or reservoir, within the Markov approximation. 3. Thermal equilibrium ensemble To calculate the canonical thermal equilibrium state at temperature T=1/k B, one can solve an inverse temperature equation for the unnormalize ensity operator ˆ = 1 2 Ĥ Nˆ,ˆ, 4.4 the solution of which will generate the unnormalize ensity operator for a gran canonical istribution: ˆ =exp Ĥ Nˆ. B. Operator mappings We wish to show how to transform a general operator time-evolution equation Eq. 4.1 into a Fokker-Planck equation for the istribution an, hence, into a stochastic equation. A crucial part of this proceure is to be able to transform the operator equations into a ifferential form. The first step is to substitute for ˆ the expansion in Eq. 3.1 P, ˆ = P,Lˆ ˆ. 4.5 Secon, we use the ifferential ientities summarize in Eq. A5 of Appenix A to convert the superoperator Lˆ ˆ into an operator Lˆ that contains only erivatives of ˆ. Next, we integrate by parts to obtain, provie that no bounary terms arise, P, ˆ = LP,ˆ, 4.6 where L is a reorere form of L, with a sign change to erivatives of o orer. Finally, we see that this equation

7 GAUSSIAN PHASE-SPACE REPRESENTATIONS FOR FERMIONS hols if the istribution function satisfies the evolution equation P, = LP,. 4.7 This proceure for going from the master equation for ˆ to the evolution equation for P can be implemente using a set of operator mappings. To write these mappings in a compact form, we introuce antinormal orering as the opposite of normal orering an enote it via curly braces: bˆ j bˆ i = bˆ ibˆ j. We also use neste orerings, in which the outer orering oes not reorer the inner one. For example, :ˆbˆ j :bˆ i= bˆ ibˆ j :ˆ:, where ˆ is some ensity operator. When orering proucts that contain the ensity operator ˆ, weo not change the orering of ˆ itself; the other operators are merely reorere aroun it. Incluing all possible orerings, we obtain the following mappings: :ˆbˆ bˆ bˆ bˆ : P For a system in which the total number is conserve, one can use the simpler thermal subset of these corresponences, i.e., incluing only those that contain terms that remain when all anomalous correlations vanish bˆ i ˆbˆ j ñ ij ñ ik ñ ljp, n lk bˆ i bˆ jˆ n ij ñ ik n ljp, n lk ˆbˆ i bˆ j n ij n ik ñ ljp, n lk ˆ P, bˆ jˆbˆ i n ij n ik n ljp. n lk 4.11 I :ˆbˆbˆ : = = = =P, :ˆbˆbˆ : = = I = =P, I :bˆbˆ ˆ: = = = = P, ˆbˆbˆ = = I = =P, 4.8 J where = =I= =. The notation x inicates a ifferentiation on both left an right sies with the orering of matrix multiplication preserve, so that I = = =. 4.9 For convenience of the reaer, these ientities are summarize in a more explicit form using the M M submatrices, in Appenix B. We note here that the mixe ientities involving neste orerings are not inepenent one can always be obtaine from the other. Also, since the kernel is analytic, the istinct analytic erivatives of the kernel are all interchangeable an lea to equivalent ientities, so that generically if a = x a i y a, then / a =/ x a = i / y a. If there are higher than quaratic terms present, the ifferential mappings are applie in sequence. The operator set closest to the operator ˆ leas to the innermost ifferential operator acting on P. Thus, for example, C. Fokker-Planck equation To be able to sample the time evolution of P with stochastic phase-space equations, which is the final goal, we must have an evolution equation that is in the form of a Fokker- Planck equation, containing first-orer an secon-orer erivatives P, = A a a=0 a p 1 p D ab P,, a,b=0 a b where a=0,...,p is an inex that ranges over all the variables in the phase space. The matrix D ab must be positiveefinite when the Fokker-Planck equation is written in terms of real variables. Fortunately, the fact that the representation kernel ˆ is analytic in the phase-space variables means that the matrix D ab can always be chosen positive-efinite after it is ivie into real an imaginary parts, 18 through appropriate choices of the equivalent analytic forms / a =/ x a = i / y a. A Monte Carlo type sampling of Eq can be realize by integrating the Ito stochastic equations a = A a B ab W b, 4.13 b where W b are Weiner increments, obeying W b W b = b,b, i.e., Gaussian white noise. The noise matrix B ab is relate to the iffusion matrix by D ab = c B ac B bc. This equation is irectly equivalent to a path integral in phase space, so that the proceures outline here can be regare as a route to obtaining a path integral without Grassmann variables

8 J. F. CORNEY AND P. D. DRUMMOND Auxiliary fiel methos 39 can also be use to obtain a non-grassmann path integral, but these are generally more restrictive. D. Stochastic gauges The final phase-space equations are far from being unique. This freeom in the final form arises from ifferent choices that are mae at ifferent points in the proceure. The choices at some points are constraine by the nee to generate a genuine Fokker-Planck equation with a positiveefinite iffusion matrix an vanishing bounary terms. Other than this, the choices are in principle free; they affect the final stochastic behavior without changing observable moments. They are thus a stochastic analogue of a gauge choice in fiel theories, an a goo choice of stochastic gauge can ramatically improve the performance of the simulations. 34 Because the Gaussian basis is analytic, methos previously use for the bosonic stochastic gauge positive-p representation are, therefore, applicable. 34,44,45 In the fermionic case, there are three sources of gauge freeom. 1. Fermi gauges For fermionic systems, there is a freeom in the choice of operator corresponences, arising from vanishing operator proucts; any term involving a square of a fermion operator, like â i 2 Ô, is zero. Terms like this an proucts of such terms can be ae to the Hamiltonian or Liouville equation without moifying the ensity matrix. The corresponing aitional ifferential terms may not vanish, hence generating a ifferent but equivalent stochastic equation. Such a fermionic stochastic gauge is necessary to avoi complex weights in imaginary-time simulations of interacting systems, such as the Hubbar moel Diffusion gauges Diffusion gauges arise from the fact that the matrix square root D ab = c B ac B bc, has multiple solutions, especially if one notes that there is no restriction on the secon imension of B ab. This changes the stochastic noise term an can lea to a reuction in sampling error Drift gauge As well as the Fermi-gauge an iffusion-gauge freeoms, it is also possible to introuce a gauge freeom in the choice of rift terms. Drift gauges are obtaine by traing off trajectory weight against trajectory irection. The possibility for rift gauges arises from the weight in the ensityoperator expansion. The first of the corresponences in Eq. 4.8 can be use to convert rift terms for the phase-space variables into iffusion terms for the weight. 19 As a result, one can a an arbitrary gauge g a of the same imension as the noise vector. Assuming B 0b =0, an using Einstein summation conventions, one obtains = A 0 g b W b, a = A a B ab W b g b Previous work 34,44 has shown that rift gauges can remove bounary terms in bosonic positive-p representation by stabilizing eterministic trajectories. V. EXAMPLES The virtue of phase-space representation is that, whereas, the Hilbert space imension grows exponentially with the number of moes M, the phase-space imension only grows quaratically. Thus, for example, a problem involving M =1000 fermion moes has a Hilbert space imension of D= = imensions. This is larger than the number of particles in the observable universe which is perhaps by current astrophysical reckoning. By contrast, the fermion phase-space imension is While large, this is not astronomical. Hamiltonians an general time-evolution equations that are only quaratic in the Fermi laer operators, i.e., constructe from one-boy operators, will map to a Fokker- Planck equation that contains only first-orer erivatives. The evolving quantum state can thus be sample by a single, eterministic trajectory. More generally, quartic terms an cubic terms if bosonic operators are inclue can also be hanle, an these result in stochastic equations or their equivalent path integrals. Examples of how some typical Fermi problems are mappe into phase-space equations are given as follows. A. Free gas As an example of quaratic evolution, consier the thermal equilibrium calculation for a gas of noninteracting particles. The governing Hamiltonian incluing the chemical potential is always iagonalizable an can be written as Ĥ = bˆ bˆ, 5.1 where ij = ij j are the single-particle energies. The gran canonical istribution at temperature T=1/k B is foun from the equation ˆ = 1bˆ bˆ ˆ ˆbˆ bˆ Now this master equation can be mappe to an equivalent equation for the istribution P by use of the thermal corresponences in Eq However, because the solution is an unnormalize ensity operator, there will be zeroth-orer terms in the equation. We can convert such terms to first orer by applying the weight ientity in Eq. 4.8, thus obtaining the Fokker-Planck equation P = k k 1 n k n k n k P. 5.3 This Fokker-Planck equation with first-orer erivatives correspons to eterministic characteristic equations

9 GAUSSIAN PHASE-SPACE REPRESENTATIONS FOR FERMIONS = k n k, k ṅ k = k n k 1 n k A= = A=, B= = B=, Integrating the eterministic equation for the moe occupation n k leas to the usual Fermi-Dirac istribution 1 n k = e k From integration of the weight equation, one fins that normalization of the ensity operator is Trˆ u = = 0 k e k n k, 5.7 i.e., the weight ecays exponentially, at a rate given by the total energy. B. General quaratic evolution More generally, one can have a quaratic Liouville operator in situations involving nonthermal terms like bˆ ibˆ j. This can occur for, example, when fermion pairs are generate from molecule or exciton issociation. These are even associate with certain spin-chain problems, 46 where the Joran- Wigner theorem is use to transform spins to fermion operators. Other quaratic Liouville operators are commonly foun in cases involving coupling to reservoirs. 43 The generic phase-space equations for a general Fermi system with a quaratic Liouville operator can be easily obtaine, for evolution both through time t an through inverse temperature. The most general master equation that covers both kins of evolution can be written ˆ = Kˆ 1 A :bˆ bˆ ˆ:B bˆ bˆ ˆ C :ˆbˆ bˆ : 2 C * :ˆbˆ :bˆ, 5.8 where the elements of 2M 2M matrices A=, B=, an C= are etermine by the coefficients of the Hamiltonian or master equation. By applying the mappings of Eq. 4.8, wefin the evolution of the covariance matrix to be = = =A= A= = = B= B= = =C= C= = = C= C= =. 5.9 This equation simply correspons to the characteristic or rift equations given by the vector A in the Ito stochastic equation 4.13, an in these cases, there is no iffusion or stochastic term. Unlike a conventional path integral, we see that a quaratic Hamiltonian or Liouville equation simply results in a noise-free, eterministic trajectory on phase space. For eterministic evolution such as this, the weight oes not affect physical observables, so we o not consier it here. In the examples that follow, we assume for simplicity but without loss of generality that the constant matrices have been chosen with an antisymmetry such that C= = C= Temperature evolution For temperature evolution, the structure of the master equation Eq. 4.4 is such that A= =B= an C==C=, giving the simpler result where we have introuce = = 1 2 I= 2=T=I= 2= =0, T= = B= C=, 5.11 = 0 = 1 I=B= C=I= For the case of a number conserving Hamiltonian H =b b, we fin that B= =0 an C= = 1 2 T The phase-space equations then reuce to n = 1 nñ ñn, which reprouces the free gas example above. 2. Dynamical evolution For real-time evolution, with possible coupling to the environment, there is a ifferent symmetry to the master equation Eq. 4.3 that means that A= B= C= C= =0=. A formal solution to the phase-space equations can now be explicitly written own =t = exp U= t=0 = exp U=t =, where U= =B= C=I= an where = satisfies I=B=I= = U= = = U= To illustrate the physical meaning of these matrices, we consier the simplest moel of a small quantum ot couple to a zero-temperature reservoir ˆ = ibˆ bˆˆ iˆbˆ bˆ bˆˆbˆ 1 2 bˆ bˆˆ 1 2 ˆbˆ bˆ In terms of the general form, this correspons to A= =0=, B= =I=, an

10 J. F. CORNEY AND P. D. DRUMMOND 1 i 2 C= = 0 0 i The general solution then reuces to =t = e i /2t 0 0 e i /2t=0 I= i /2t ei /2t 0 I=, e which implies that the ensity ecays as nt=e t n0, as expecte. The solution to a multimoe quantum ot moel also follows from Eq The relevant master equation is ˆ = i ji bˆ i bˆ jˆ i ji ˆbˆ i bˆ j ijbˆ iˆbˆ j 1 bˆ j bˆ iˆ 1 ˆbˆ j bˆ i, for which the evolution matrix is U= = e i/2 0 0 e i/2, 5.21 where we have assume that T = an T =. Physically, this correspons, as expecte, to ampe oscillatory behavior taking to be positive efinite in the moments n = e i /2t n0e i /2t, m = e i /2t m0e i /2t Here, of course, there are no electron-electron interactions inclue. However, such interactions can be ealt with via a stochastic sampling methos, as we show in Sec. V C. C. Interacting gas 1. Two-boy interactions For systems of particles with two-boy interactions, the Gaussian representation gives nonlinear, stochastic phasespace equations, which must be solve numerically. Consier a two-boy interaction of the form Ĥ 2 = U ij nˆ ii nˆ jj, 5.23 ij where nˆ ij =bˆ i bˆ j. For a number-conserving system, we can use corresponences of Eq to generate a Fokker- Planck equation for the gran-canonical evolution. The iffusion matrix D u,v in this equation is D ij,kl = U pq n ip ñ pj n kq ñ ql ñ ip n pj ñ kq n ql pq Suppose that the interaction matrix U pq is negative-efinite, such that we can write it as a sum of negative squares: U pq = b p, b q,. Then the iffusion matrix is positive efinite, as it can be written in the form D ij,kl = B ij, where the noise matrices are B ij, 1 1 B kl, 2 B 2 kl,, B ij, 1 = b p, n ip ñ pj, B ij, p 2 = b p, ñ ip n pj. p Thus for an interaction of this type, the noise terms in the final stochastic equations will be real. The form of noise terms for a more general interaction is consiere in Ref Hubbar moel As an example, we show how to apply the representation to the Hubbar moel. Some simulations of the resulting equations were reporte in Refs , along with etails of the numerical implementations an comparisons with other methos. Here we focus on how these equations are erive an on the possible gauge choices available. The Hubbar moel is the simplest nontrivial moel for strongly interacting fermions on a lattice. It is an important system in conense matter physics, with relevance to the theory of high-temperature superconuctors, 26 an in ultracol atomic physics. The full phase iagram in two imensions is not fully unerstoo as yet. Due to evelopments in atomic lattices, this moel is irectly experimentally accessible. 6,47 The Hamiltonian for the moel is 32 Hnˆ 1,nˆ 1 = ij, t ij nˆ ij, U nˆ jj,1 nˆ jj, 1, j 5.27 where nˆ ij, =â i, â j, =nˆ ij. The inex enotes spin ±1, the inices i, j label lattice location. Here, t ij =t if the i, j correspon to nearest neighbor sites, t ij = if i= j an is otherwise 0. The chemical potential is inclue to control the total particle number. Because the Hubbar moel conserves total number an spin, one can map this problem to a reuce phase space of =,n ij,1,n ij, 1. Thus the simpler mappings of Eq can be use for each spin component. The one-boy terms generate rift terms only an can be ealt with as above. The two-boy terms generate both rift an iffusion terms. Applying the mappings irectly to the Hubbar moel as written above, we obtain the iffusion matrix D ij,kl = U, n ip ñ pj n kp ñ pl p ñ ip n pj ñ kp n pl, 5.28 which, because it has zeros on the iagonal, cannot be put into a positive-efinite form with real variables. However, using the anticommuting properties of the Fermi operators, we can rewrite the interaction term in the Hubbar Hamiltonian as

11 GAUSSIAN PHASE-SPACE REPRESENTATIONS FOR FERMIONS H I = U 2 j :nˆ jj,1 Snˆ jj, 1 2 :=U i,j :nˆ ii, nˆ jj, :, j 5.29 where S=U/U = ±1. Now in this form, the interaction matrix is negative-efinite U i,j = U 2 ij, S, = U i,k s j,k s, 2 k 5.30 where s=s1/2, so that s=0 for the attractive case an s =1 for the repulsive case. From Eq. 5.25, the iffusion matrix is positive-efinite, with corresponing noise matrices 1 B ij, = U/2 s n i, ñ j,, 2 B ij, = U/2 s ñ i, n j, With this choice of noise terms, the final phase-space equations are, in Itô form, n = 1 2 ñ T 1 n n T 2 ñ, 5.32 where we have introuce the stochastic propagation matrix r T ij, = t ij ij Un jj, s r j The real Gaussian noise r j is efine by the correlations FIG. 1. Secon-orer correlation function g 2 =1/N L j nˆ jj,1 nˆ jj, 1 /nˆ jj,1 nˆ jj, 1 for a N L =100 site oneimensional lattice at half filling. Lower curves are for U=2 repulsive an upper curves are for U= 2 attractive. Soli lines give the numerical results, ashe lines give the zero-temperature analytic result 49 for an infinite system, an otte lines inicate sampling error. t=1 an 1000 paths. r j r j =2U jj rr Because the iffusion can be realize in terms of real noise, the phase-space equations will not be riven off the real manifol. This has an important implication for the weight, which enters the problem because the solution will be an unnormalize ensity operator. The weights for each trajectory evolve as physically expecte for energy-weighte averages, with weights epening exponentially on the inverse temperature an the effective trajectory Hamiltonian H: = Hn 1,n Because the equations for the phase-space variables n ij, are all real, the weights will all remain positive, thereby eliminating the traitional manifestation of the sign problem. In contrast, the analogous equation in AFQMC oes not guarantee the weights to be positive, an this is where the sign problem enters in such calculations. This metho can calculate any correlation function, at any temperature, to the precision allowe by the sampling error an subject to there being no bounary terms in Eq Simulations in one, 48 two, 29,30 an three imensions are shown in Figs They emonstrate that sampling error is well-controlle, even for very low temperatures an for cases in which the sign eteriorates for projector QMC. However, more extensive simulations of the twoimensional 2D Hubbar moel have shown that, at half filling, certain correlation functions o not appear to converge to the correct zero-temperature results at these very low temperatures. 31 Because the Gaussian basis oes not possess many of the symmetries of the Hubbar moel, they must be restore in the istribution over Gaussian basis elements. For finite sampling, this restoration may be incomplete, giving the eparture from exact results at low temperatures. There may also be systematic errors if bounary terms are present. Both of these possibilities imply that further optimization via stochastic gauge choices may be require to keep the low-temperature istributions compact an free from tails an features that woul lea to biasing. Nevertheless, it has alreay been shown that the correct results can be obtaine by applying a projection onto a symmetric subspace. 31 Importantly, accurate results were then obtaine even in cases that are beyon the reach of AFQMC. 3. Drift gauges Here, we outline how the performance of the Hubbar simulations may be improve by means of rift gauges. We FIG. 2. Two-imensional Hubbar moel with N L =16 sites. Soli lines give energy E per lattice site for chemical potentials in orer of ecreasing energy =2, =1, an =0. Dashe lines give number of particles per site for =1 upper an =0 lower. Dotte lines inicate sampling error. U=4, t=1 an 5000 paths

12 J. F. CORNEY AND P. D. DRUMMOND turn requires aitional, off-iagonal noise terms in the propagation matrix. Such noises can be introuce by use of aitional Fermi gauges. For example, the vanishing term = ij 2 V ji, ij nˆ ij, nˆ ij, ˆ ˆnˆ ij, ij nˆ ij,, 5.37 where V ij, are positive numbers, gives the aitional stochastic contribution to the propagation matrix r T ij, r T ij, r ij,, where the new noises r ij, have the correlations 5.38 FIG. 3. Two-imensional Hubbar moel with N L =256 sites. Soli lines give energy E per lattice site for chemical potentials in orer of ecreasing energy =2, =1, an =0. Dashe lines give number of particles per site for =1 upper an =0 lower. Dotte lines inicate sampling error. U=4, t=1. Breeing algorithm use with approximately 100 paths. can moify the Hubbar rift accoring to Eq by aing a term G to the stochastic propagation matrices T r. Because of the iagonal nature of the noise terms, the ae term will also be iagonal: G ij = ij s G j. The aitional iffusion term in the weight equation is then = g 2U jr G j j r The choice of gauge term G is guie, on the one han, by the nee to ensure the phase-space istribution remains boune an, on the other, by the requirement of introucing only the minimum amount of iffusion into the weight. The function shoul thus act only when necessary to control large trajectories an shoul be zero otherwise. However, because of the coupling terms t ij, a iagonal rift gauge is insufficient to remove all instabilities, making it necessary to introuce off-iagonal gauge terms. This in FIG. 4. Three-imensional Hubbar moel with N L =216 sites. Soli lines give energy E per lattice site for chemical potentials in orer of ecreasing energy =2, =1, an =0. Dashe lines give number of particles per site for =1 upper an =0 lower. Dotte lines inicate sampling error. U=4, t=1 an 50 paths. r r ij, i =4V j, ij, ii jj rr We can now introuce arbitrary off-iagonal gauge terms r G ij, into the propagation matrix, with the corresponing iffusion term in the weight equation = G r ij, r ij, /4V ij, g ijr Again there is a traeoff between gauge strength an aitional iffusion. But there is also a freeom in the choice of V ij, as to whether the noise appears in the weight equation or in the propagation matrix. With such a combination of Fermi an rift gauges, it is possible to introuce terms to stabilize the rift evolution of any of the phase-space variables n ij,, an so maintain a boune phase-space istribution. VI. CONCLUSION In summary, we have introuce a phase-space representation for many-boy fermionic states, enabling new types of first-principles calculations an simulations of highly correlate systems. Systems with one-boy an two-boy interactions can be solve by the use of stochastic sampling methos, since they can be transforme into a secon-orer Fokker-Planck equation, provie a suitable stochastic gauge is chosen to ensure that the istribution remains sufficiently boune. These techniques are potentially applicable to a wie range of fermionic problems, incluing both real-time an finite-temperature calculations. Generalize master equations for nonequilibrium fermionic open systems couple to reservoirs are a particularly suitable type of application. We have given examples of the use of fermionic ifferential ientities to transform multimoe master equations into eterministic phase-space equations, although more general interactions typically lea to stochastic equations. These equations have exponentially less complexity than the full Hilbert space equations. In contrast to Grassmann-base approaches, the Gaussian representation oes not involve anticommuting variables an thus avois the associate complexity issues. In contrast to stanar QMC methos, the phase-space approach is base