Lattice simulations for few and manybody systems
Abstract
We review the recent literature on lattice simulations for few and manybody systems. We focus on methods that combine the framework of effective field theory with computational lattice methods. Lattice effective field theory is discussed for cold atoms as well as lowenergy nucleons with and without pions. A number of different lattice formulations and computational algorithms are considered, and an effort is made to show common themes in studies of cold atoms and lowenergy nuclear physics as well as common themes in work by different collaborations.
pacs:
03.75.Ss, 21.30.x, 21.45.v, 21.60.Ka, 21.65.f, 21.65.Cd, 71.10.Fd, 71.10.HfContents
 I Introduction
 II Effective field theory
 III Lattice formulations for zerorange attractive twocomponent fermions

IV Lattice formulations for lowenergy nucleons
 IV.1 Pionless effective field theory
 IV.2 Pionless effective field theory with auxiliary fields
 IV.3 Instantaneous free pion action
 IV.4 Chiral effective field theory on the lattice
 IV.5 Chiral effective field theory with auxiliary fields
 IV.6 Nexttoleadingorder interactions on the lattice
 IV.7 Model independence at fixed lattice spacing
 V Twoparticle scattering on the lattice
 VI Monte Carlo algorithms
 VII Some recent results
 VIII Summary
I Introduction
In this article we review the literature on lattice simulations for few and manybody systems. We discuss methods which combine effective field theory with lattice methods and which can be applied to both cold atomic systems and lowenergy nuclear physics. Several recent reviews have already been written describing quantum Monte Carlo methods for a range of topics. These include Monte Carlo calculations in continuous space for electronic orbitals in chemistry Hammond et al. (1994), solid state materials Foulkes et al. (2001), superfluid helium Ceperley (1995), and fewnucleon systems Carlson and Schiavilla (1998). There are also reviews of Monte Carlo lattice methods for stronglycorrelated lattice models von der Linden (1992), lattice quantum chromodynamics at nonzero density Muroya et al. (2003), and a general introduction to lattice quantum chromodynamics Davies (2002).
Lattice simulations of quantum chromodynamics (QCD) are now able to accurately describe the properties of many isolated hadrons. In addition to isolated hadrons, it is also possible to calculate lowenergy hadronic interactions such as mesonmeson scattering Kuramashi et al. (1993); Aoki et al. (2003); Lin et al. (2003); Beane et al. (2006a, b, 2008a). Other interactions such as baryonbaryon scattering are computationally more difficult, but there has been promising work in this direction as well Fukugita et al. (1995); Beane et al. (2004, 2007a, 2006c); Ishii et al. (2007); Aoki et al. (2008); Nemura et al. (2008). A recent review of hadronic interaction results computed from lattice QCD can be found in Ref. Beane et al. (2008b).
However for few and manybody systems beyond two nucleons, lattice QCD simulations are presently out of reach. Such simulations require pion masses at or near the physical mass and lattices several times longer in each dimension than used in current simulations. Another significant computational challenge is to overcome the exponentially small signaltonoise ratio for simulations at large quark number. For few and manybody systems in lowenergy nuclear physics one can make further progress by working directly with hadronic degrees of freedom.
There are several choices one can make for the nuclear forces and the calculational method used to describe interacting lowenergy protons and neutrons. For systems with four or fewer nucleons, a semianalytic approach is provided by the FaddeevYakubovsky integral equations. Using this method, one study Nogga et al. (2000) looked at three and fournucleon systems using the Nijmegen potentials Stoks et al. (1994), CDBonn potential Machleidt et al. (1996), and AV18 potential Wiringa et al. (1995), together with the TucsonMelbourne Coon and Gloeckle (1981) and UrbanaIX Pudliner et al. (1997) threenucleon forces. A different investigation considered the same observables using a twonucleon potential derived from chiral effective field theory Epelbaum et al. (2001). Another recent study Nogga et al. (2004) considered the lowmomentum interaction potential Bogner et al. (2003a, b). This method used the renormalization group to derive effective interactions equivalent to potential models but at low cutoff momentum.
For systems with more nucleons approaches such as Monte Carlo simulations or basistruncated eigenvector methods are needed. There is considerable literature describing Green’s Function Monte Carlo simulations of light nuclei and neutron matter based on AV18 as well as other phenomenological potentials Pudliner et al. (1997); Wiringa et al. (2000); Pieper and Wiringa (2001); Pieper et al. (2001, 2002); Wiringa and Pieper (2002); Pieper et al. (2004); Nollett et al. (2007); Gezerlis and Carlson (2008). There is a review article detailing this method Carlson and Schiavilla (1998) as well as a more recent set of lecture notes Pieper (2007). A related technique known as auxiliaryfield diffusion Monte Carlo simplifies the spin structure of the same calculations by introducing auxiliary fields Fantoni et al. (2001); Sarsa et al. (2003); Pederiva et al. (2004); Chang et al. (2004); Gandolfi et al. (2007, 2008). The NoCore Shell Model (NCSM) is a different approach to light nuclei which produces approximate eigenvectors in a reduced vector space. There have been several NCSM calculations using various different phenomenological potential models Navratil et al. (2000); Fayache et al. (2001); Navratil and Ormand (2003); Caurier and Navratil (2006). There are also NCSM calculations which have used nuclear forces derived from chiral effective field theory Forssen et al. (2005); Nogga et al. (2006); Navratil et al. (2007). Recently there has also been work in constructing a lowenergy effective theory within the framework of truncated basis states used in the NCSM formalism Stetcu et al. (2006). A benchmark comparison of many of the methods listed above as well as other techniques can be found in Ref. Kamada et al. (2001).
In this article we describe recent work by several different collaborations which combine the framework of effective field theory with computational lattice methods. The idea of lattice simulations using effective field theory is rather new. The first quantum lattice study of nuclear matter appears to be Ref. Brockmann and Frank (1992), which used a momentum lattice and the quantum hadrodynamics model of Walecka Walecka (1974). The first study combining lattice methods with an effective theory for lowenergy nuclear physics was Ref. Müller et al. (2000). This study looked at infinite nuclear and neutron matter at nonzero density and temperature. After this there appeared a computational study of the attractive Hubbard model in three dimensions Sewer et al. (2002), as well as a paper noting the absence of sign oscillations for nonzero chemical potential and external pairing field Chen and Kaplan (2004). Another study looked at nonlinear realizations of chiral symmetry with static nucleons on the lattice Chandrasekharan et al. (2004), and there were also a number of investigations of chiral perturbation theory with lattice regularization Shushpanov and Smilga (1999); Lewis and Ouimet (2001); Borasoy et al. (2003). This was followed by the first manybody lattice calculation using chiral effective field theory Lee et al. (2004). From about this time forward there were a number of lattice calculations for cold atoms and lowenergy nuclear physics which we discuss in this article.
The lattice effective field theory approach has some qualitative parallels with digital media. In digital media input signals are compressed into standard digital output that can be read by different devices. In our case the input is lowenergy scattering data, and the digital format is effective field theory defined with lattice regularization. The process of sampling and compression consists of matching lowenergy scattering data using effective interactions up to some chosen order in power counting. By increasing the order, the accuracy in describing lowenergy phenomena can be systematically improved.
Just as standard digital format enables communication between different devices, lattice effective field theory enables the study of many different phenomena using the same lattice action. This includes few and manybody systems as well as ground state properties and thermodynamics at nonzero temperature and density. Another attractive feature of lattice effective field theory is the direct link with analytic calculations using effective field theory. It is straightforward to derive lattice Feynman rules and calculate diagrams using the same theory used in nonperturbative simulations. At fixed lattice spacing all of the systematic error is introduced up front when defining the lowenergy lattice effective field theory and not determined by the particular computational scheme used to calculate observables. This allows for a wide degree of computational flexibility, and one can use a number of efficient lattice methods already developed for lattice QCD and condensed matter applications. This includes cluster algorithms, auxiliaryfield transformations, pseudofermion methods, and nonlocal configuration updating schemes. We discuss all of these techniques in this article. We also review the relevant principles of effective field theory as well as different formalisms and algorithms used in lattice calculations. Towards the end we discuss some recent results and compare with results obtained using other methods.
Ii Effective field theory
Effective field theory provides a systematic approach to studying lowenergy phenomena in few and manybody systems. We give a brief overview of the effective range expansion and the application of effective field theory to cold atoms and lowenergy nuclear physics. A more thorough review of effective field theory methods applied to systems at nonzero density can be found in Ref. Furnstahl et al. (2008).
ii.1 Effective range expansion
At sufficiently low momentum the crosssection for twobody scattering is dominated by the wave amplitude, and higher partial waves are suppressed by powers of the relative momentum. The wave scattering amplitude for two particles with mass and relative momentum is
(1) 
where is the wave phase shift. At low momentum the wave phase shift for twobody scattering with shortrange interactions can be written in terms of the effective range expansion Bethe (1949),
(2) 
Here is the wave scattering length, and is the wave effective range. The radius of convergence of the effective range expansion is controlled by the characteristic length scale of the interaction. For example in lowenergy nuclear physics the range of the twonucleon interaction is set by the Compton wavelength of the pion. The generalization of the effective range expansion to partial wave has the form
(3) 
The phase shift scales as in the lowmomentum limit, and higherorder terms are suppressed by further powers of . This establishes a hierarchy of lowenergy twobody scattering parameters for shortrange interactions. For particles with intrinsic spin there is also some mixing between partial waves carrying the same total angular momentum.
For many interacting systems we can characterize the lowenergy phenomenology according to exact and approximate symmetries and loworder interactions according to some hierarchy of power counting. This universality is due to a wide disparity between the longdistance scale of lowenergy phenomena and the shortdistance scale of the underlying interaction. In some cases the simple power counting of the effective range expansion must be rearranged or resummed in order to accommodate nonperturbative effects. We discuss this later in connection with singular potentials and threebody forces. A recent review of universality in fewbody systems at large scattering length can be found in Ref. Braaten and Hammer (2006).
In manybody systems a prime example of universality is the unitarity limit. The unitarity limit describes attractive twocomponent fermions in an idealized limit where the range of the interaction is zero and the scattering length is infinite. The name refers to the fact that the wave crosssection saturates the limit imposed by unitarity, , for low momenta . While the unitarity limit has a welldefined continuum limit and strong interactions, at zero temperature it has no intrinsic physical scale other than the interparticle spacing.
Phenomenological interest in the unitarity limit extends across several subfields of physics. The ground state of the unitarity limit is known to be a superfluid with properties in between a BardeenCooperSchrieffer (BCS) fermionic superfluid at weak attractive coupling and a BoseEinstein condensate (BEC) of bound dimers at strong attractive coupling Eagles (1969); Leggett (1980); Nozieres and SchmittRink (1985). It has been suggested that the crossover from fermionic to bosonic superfluid could be qualitatively similar to pseudogap behavior in hightemperature superconductors Chen et al. (2005). In nuclear physics the unitarity limit is relevant to the properties of cold dilute neutron matter. The neutron scattering length is about fm while the range of the interaction is comparable to the Compton wavelength of the pion, fm. Therefore the unitarity limit is approximately realized when the interparticle spacing is about fm. Superfluid neutrons at around this density may exist in the inner crust of neutron stars Pethick and Ravenhall (1995); Lattimer and Prakash (2004).
ii.2 Effective field theory for cold atoms
Physics near the unitarity limit has been experimentally observed in cold degenerate gases of Li and K atoms. Alkali atoms are convenient for evaporative cooling due to their predominantly elastic collisions. For sufficiently dilute gases the effective range and higher partial wave effects are negligible while the scattering length can be adjusted using a magneticallytuned Feshbach resonance Tiesinga et al. (1993); Stwalley (1976); Courteille et al. (1998); Inouye et al. (1998). Overviews of experiments using Feshbach resonances can be found in Ref. Koehler et al. (2006); Regal and Jin (2006), and there are a number of reviews covering the theory of BCSBEC crossover in cold atomic systems Chen et al. (2005); Giorgini et al. (2007); Bloch et al. (2007).
At long distances the interactions between alkali atoms are dominated by the van der Waals interaction. Powerlaw interactions complicate the effective range expansion by producing a branch cut in each partial wave at . For the van der Waals interaction the expansion in is an asymptotic expansion coinciding with the effective range expansions in Eq. (2) and (3) up through terms involving , , and Gao (1998a, b). Beyond this the asymptotic expansion involves powers of times or odd powers of . All of the work discussed in this article involves lowenergy phenomena where these nonanalytic terms can be neglected.
The lowenergy effective field theory for the unitarity limit can be derived from any theory of twocomponent fermions with infinite scattering length and negligible higherorder scattering effects at the relevant lowmomentum scale. For example the two fermion components may correspond with dressed hyperfine states and of K with interactions given either by a full multichannel Hamiltonian or a simplified twochannel model Goral et al. (2004); Szymanska et al. (2005); Nygaard et al. (2007). The starting point does not matter so long as the wave scattering length is tuned to infinity to produce a zeroenergy resonance.
In our notation is the atomic mass and and are annihilation and creation operators for two hyperfine states. We label these as up and down spins, , even though the connection with actual intrinsic spin is not necessary. We enclose operator products with the symbols to indicate normal ordering, where creation operators are on the left and annihilation operators are on the right. The effective Hamiltonian at leading order (LO) is
(4) 
where
(5) 
(6) 
and is the particle density operator,
(7) 
The coefficient depends on the cutoff scheme used to regulate ultraviolet divergences in the effective theory. Higherorder effects may be introduced systematically as higherdimensional local operators with more derivatives and/or more local fields.
ii.3 Pionless effective field theory
For nucleons at momenta much smaller than the pion mass, all interactions produced by the strong nuclear force can be treated as local interactions among nucleons. The effective Hamiltonian in Eq. (4) also describes the interactions of lowenergy neutrons at leading order. For systems with both protons and neutrons we label the nucleon annihilation operators with two subscripts,
(8) 
(9) 
The first subscript is for spin and the second subscript is for isospin . We use with to represent Pauli matrices acting in spin space and with to represent Pauli matrices acting in isospin space. The same letters and are also used to indicate total spin and total isospin quantum numbers, but the intended meaning will be clear from the context. If we neglect isospin breaking and electromagnetic effects, the effective theory has exact SU spin and SU isospin symmetries.
Let us define the total nucleon density
(10) 
The total nucleon density is invariant under Wigner’s SU(4) symmetry mixing all spin and isospin degrees of freedom Wigner (1937). Using and , we also define the local spin density,
(11) 
isospin density
(12) 
and spinisospin density,
(13) 
At leading order the effective Hamiltonian can be written as
(14) 
where
(15) 
(16) 
(17) 
(18) 
(19) 
Due to an instability in the limit of zerorange interactions Thomas (1935), the SU(4)symmetric threenucleon force is needed for consistent renormalization at leading order Bedaque et al. (1999a, b); Bedaque et al. (2000). With the constraint of antisymmetry there are two independent wave nucleonnucleon scattering channels. These correspond with spinisospin quantum numbers , and , . Some analytic methods used in pionless effective field theory are discussed in Ref. van Kolck (1999a); Chen et al. (1999). A general overview of methods in pionless effective field theory can be found in recent reviews van Kolck (1999b); Bedaque and van Kolck (2002); Epelbaum (2006).
ii.4 Chiral effective field theory
For nucleon momenta comparable to the pion mass, the contribution from pion modes must be included in the effective theory. In the following denotes the channel momentum transfer for nucleonnucleon scattering while is the channel exchanged momentum transfer. At leading order in the Weinberg powercounting scheme Weinberg (1990, 1991) the nucleonnucleon effective potential is
(20) 
(21) 
, are defined in the same manner as in Eq. (15), (17), (18). is the instantaneous onepion exchange potential,
(22) 
where the spinisospin density is defined in Eq. (13) and
(23) 
For our physical constants we take MeV as the nucleon mass, MeV as the pion mass, MeV as the pion decay constant, and as the nucleon axial charge.
The terms in can be written more compactly in terms of their matrix elements with twonucleon momentum states. The treelevel amplitude for twonucleon scattering consists of contributions from direct and exchange diagrams. However for bookkeeping purposes we label the amplitude as though the two interacting nucleons are distinguishable. We label one nucleon as type , the other nucleon as type , and the interactions include densities for both and . For example the total nucleon density becomes
(24) 
The amplitudes are then
(25) 
(26) 
(27) 
At nexttoleading order (NLO) the effective potential introduces corrections to the two LO contact terms, seven independent contact terms carrying two powers of momentum, and instantaneous twopion exchange (TPEP) Ordonez and van Kolck (1992); Ordonez et al. (1994, 1996); Epelbaum et al. (1998, 2000). We write this as
(28) 
The treelevel amplitudes for the new contact interactions are
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
(35) 
(36) 
(37) 
The amplitude for NLO twopion exchange potential is Friar and Coon (1994); Kaiser et al. (1997)
(38) 
where
(39) 
Recent reviews of chiral effective field theory can be found in Ref. van Kolck (1999b); Bedaque and van Kolck (2002); Epelbaum (2006).
ii.5 Threenucleon forces
The systematic framework provided by effective field theory becomes very useful when discussing the form of the dominant threenucleon interactions. Fewnucleon forces in chiral effective field theory beyond two nucleons were first discussed qualitatively in Ref. Weinberg (1991). In Ref. van Kolck (1994) it was shown that the threenucleon terms at NLO cancelled, and the leading threenucleon effects appeared at nexttonextto leading order (NNLO) in Weinberg power counting.
The NNLO threenucleon effective potential arises from a pure contact potential, , onepion exchange potential, , and a twopion exchange potential, . Parts of the NNLO threenucleon potential are also contained in a number of phenomenological threenucleon potentials Fujita and Miyazawa (1957); Yang (1974); Coon et al. (1979); Coon and Gloeckle (1981); Carlson et al. (1983); Pudliner et al. (1997). However there is clear value in identifying the full set of leading interactions. Similar to our description above for twonucleon scattering, we write the treelevel amplitude for threenucleon scattering where the first nucleon is of type , the second nucleon type , and the three type . The amplitudes are Friar et al. (1999); Epelbaum et al. (2002a)
(40) 
(41) 
(42) 
In our notation , , are the differences between final and initial momenta for the respective nucleons. The summations are over permutations of the bookkeeping labels .
The coefficients are interaction terms in the chiral Lagrangian and are determined from fits to lowenergy scattering data Bernard et al. (1995). The remaining unknown coefficients and are cutoff dependent. In Ref. Epelbaum et al. (2002a) these were fit to the triton binding energy and spindoublet neutrondeuteron scattering length. The resulting NNLO effective potential was shown to give a prediction for the isospinsymmetric alpha binding energy accurate to within a fraction of MeV.
ii.6 Nonperturbative physics and power counting
When nonperturbative processes are involved, reaching the continuum limit and power counting in effective field theory can sometimes become complicated. The twocomponent effective Hamiltonian for cold atoms introduced in Eq. (4) has no such complications. Ultraviolet divergences can be absorbed by renormalizing the interaction coefficient , and the cutoff momentum can be taken to infinity. Similarly the leadingorder pionless effective Hamiltonian in Eq. (14) has a welldefined continuum limit if we neglect deeplybound threebody states that decouple from the lowenergy effective theory. While these deeplybound states generate instabilities in numerical simulations they can be removed by hand in semianalytic calculations Bedaque et al. (1999a, b); Bedaque et al. (2000).
In chiral effective field theory there has been considerable study on the consistency of the Weinberg power counting scheme at high momentum cutoff. Complications arise from the singular behavior of the onepion exchange potential. In order to avoid unsubtracted ultraviolet divergences produced by infinite iteration of the onepion exchange potential, an alternative scheme was proposed where pion exchange is treated perturbatively Kaplan et al. (1996, 1998a, 1998b). This approach, KSW power counting, allows for systematic control of the ultraviolet divergence structure of the effective theory. Unfortunately the convergence at higher order is poor in some partial waves for momenta comparable to the pion mass Fleming et al. (2000).
The most divergent shortdistance part of the onepion exchange potential is a singularity arising from the tensor force in the spintriplet channel. There are also subleading divergences at which contain explicit factors of the pion mass. Based on this observation another power counting scheme was proposed in Ref. Beane et al. (2002). This new scheme coincides with KSW power counting in the spinsinglet channel. But in the spintriplet channel the most singular piece of the onepion exchange potential is iterated nonperturbatively, while the rest is incorporated as a perturbative expansion around .
More recently a different power counting modification was proposed in Ref. Nogga et al. (2005). In this approach the onepion exchange potential is treated nonperturbatively in lower angular momentum channels along with higherderivative counterterms promoted to leading order. These counterterms are used to cancel cutoff dependence in channels where the tensor force is attractive and strong enough to overcome the centrifugal barrier. Advantages over Weinberg power counting at leading order were shown for cutoff momenta much greater than the pion mass. Further investigations of this approach in higher partial waves and power counting with onepion exchange were considered in Ref. Birse (2006, 2007).
The choice of cutoff momentum and power counting scheme in lattice effective field theory is shaped to a large extent by computational constraints. For twonucleon scattering in chiral effective field theory, small lattice spacings corresponding with cutoff momenta many times greater than the pion mass are no problem. However at small lattice spacing significant numerical problems appear in simulations of few and manynucleon systems. In attractive channels one must contend with spurious deeplybound states that spoil Euclidean time projection methods (a technique described later in this review). In channels where the shortrange interactions are repulsive a different problem arises. In auxiliaryfield and diagrammatic Monte Carlo (methods we discuss later in this review), repulsive interactions produce sign or complex phase oscillations that render the method ineffective. Due to these practical computational issues one must settle for lattice simulations where the cutoff momentum is only a few times the pion mass, and the advantages of the improved scheme over Weinberg power counting are numerically small Epelbaum and Meißner (2006).
Iii Lattice formulations for zerorange attractive twocomponent fermions
In this section we introduce a number of different lattice formulations using the example of zerorange attractive twocomponent fermions described by in Eq. (4). In Fig. (1) we show a schematic diagram of the different lattice formulations. The numbered arrows indicate the discussion order in the text.
Throughout our discussion of the lattice formalism we use dimensionless parameters and operators corresponding with physical values multiplied by the appropriate power of the spatial lattice spacing . In our notation the threecomponent integer vector labels the lattice sites of a threedimensional periodic lattice with dimensions . The spatial lattice unit vectors are denoted , , . We use to label lattice steps in the temporal direction, and denotes the total number of lattice time steps. The temporal lattice spacing is given by , and is the ratio of the temporal to spatial lattice spacing. We also define , where is the fermion mass in lattice units.
iii.1 Grassmann path integral without auxiliary field
For twocomponent fermions with zerorange attractive interactions we start with the lattice Grassmann path integral action without auxiliary fields. It is the simplest formulation in which to derive the lattice Feynman rules. Hence it is useful for both analytic lattice calculations and diagrammatic lattice Monte Carlo simulations Burovski et al. (2006a, b).
We let and be anticommuting Grassmann fields for spin . The Grassmann fields are periodic with respect to the spatial lengths of the lattice,
(43) 
(44) 
and antiperiodic along the temporal direction,
(45) 
(46) 
We write as shorthand for the integral measure,
(47) 
We use the standard convention for Grassmann integration,
(48) 
(49) 
Local Grassmann densities are defined in terms of bilinear products of the Grassmann fields,
(50) 
(51) 
(52) 
We consider the Grassmann path integral
(53) 
where
(54) 
The action consists of the free nonrelativistic fermion action
(55) 
and a contact interaction between up and down spins. We consider the case where the coefficient is negative, corresponding with an attractive interaction. Since we are considering nonrelativistic lattice fermions with a quadratic dispersion relation, the lattice doubling problem associated with relativistic fermions does not occur.
In the grand canonical ensemble a common chemical potential is added for all spins. In this case the dependent path integral is
(56) 
where
(57) 
and is the same as defined in Eq. (54), but with replaced by
iii.2 Transfer matrix operator without auxiliary field
Let and denote fermion annihilation and creation operators satisfying the usual anticommutation relations
(58) 
(59) 
For any function we note the identity Creutz (2000)
(60) 
where and are Grassmann variables. As before the symbols in Eq. (60) indicate normal ordering, and the trace is evaluated over all possible fermion states. This result can be checked explicitly using the complete set of possible functions .
It is useful to write Eq. (60) in a form that resembles a path integral over a short time interval with antiperiodic boundary conditions,
(61) 
(62) 
This result can be generalized to products of normalordered functions of several creation and annihilation operators. Let and denote fermion annihilation and creation operators for spin at lattice site . We can write any Grassmann path integral with instantaneous interactions as the trace of a product of operators using the identity Creutz (1988, 2000)
(63) 
where .
Let us define the free nonrelativistic lattice Hamiltonian
(64) 
as well as the lattice density operators
(65) 
(66) 
(67) 
Using the correspondence Eq. (63), we can rewrite the path integral defined in Eq. (53) as a transfer matrix partition function,
(68) 
where is the normalordered transfer matrix operator
(69) 
Roughly speaking the transfer matrix operator is the exponential of the Hamiltonian operator over one Euclidean lattice time step, . In order to satisfy the identity Eq. (63), we work with normalordered transfer matrix operators. In the limit of zero temporal lattice spacing, , we obtain the Hamiltonian lattice formulation with Hamiltonian
(70) 
This is also the defining Hamiltonian for the attractive Hubbard model in three dimensions.
In the grand canonical ensemble the effect of the chemical potential is equivalent to replacing by
(71) 
For the Hamiltonian lattice formulation the effect of the chemical potential has the familiar form
(72) 
iii.3 Grassmann path integral with auxiliary field
We can reexpress the Grassmann path integral using an auxiliary field coupled to the particle density. This lattice formulation has been used in several lattice studies at nonzero temperature Chen and Kaplan (2004); Lee and Schäfer (2005); Lee et al. (2004); Wingate (2005); Lee and Schäfer (2006a, b); Abe and Seki (2007a, b). Due to the simple contact interaction and the anticommutation of Grassmann variables, there is a large class of auxiliaryfield transformations which reproduce the same action.
Let us write the Grassmann path integral using the auxiliary field
(73) 
where
(74) 
One possible example is a Gaussianintegral transformation similar to the original HubbardStratonovich transformation Stratonovich (1958); Hubbard (1959) where
(75) 
(76) 
Another possibility is a discrete auxiliaryfield transformation similar to that used in Ref. Hirsch (1983). In our notation this can be written as
(77) 
(78) 
where sgn equals for positive values and for negative values. In Ref. Lee (2008a) the performance of four different auxiliaryfield transformations were compared.
We intentionally leave the forms for and unspecified, except for a number of conditions needed to recover Eq. (53) upon integrating out the auxiliary field . The first two conditions we set are
(79) 
(80) 
Since all even products of Grassmann variables commute, we can factor out the term in Eq. (73) involving the auxiliary field at . To shorten the notation we temporarily omit writing explicitly. We find
(81) 
Therefore the last condition needed to recover Eq. (53) is
(82) 
In the grand canonical ensemble, the auxiliaryfield path integral at chemical potential is
(83) 
where