Implementation of General Background Electromagnetic Fields on a Periodic Hypercubic Lattice
Implementation of general background electromagnetic fields on a periodic hypercubic lattice
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Citation Davoudi, Zohreh, and William Detmold. "Implementation of general background electromagnetic fields on a periodic hypercubic lattice." Phys. Rev. D 92, 074506 (October 2015). © 2015 American Physical Society
As Published http://dx.doi.org/10.1103/PhysRevD.92.074506
Publisher American Physical Society
Version Final published version
Citable link http://hdl.handle.net/1721.1/99362
Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 92, 074506 (2015) Implementation of general background electromagnetic fields on a periodic hypercubic lattice
† Zohreh Davoudi* and William Detmold Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 12 August 2015; published 19 October 2015) Nonuniform background electromagnetic fields, once implemented in lattice quantum chromodynamics calculations of hadronic systems, provide a means to constrain a large class of electromagnetic properties of hadrons and nuclei, from their higher electromagnetic moments and charge radii to their electromagnetic form factors. We show how nonuniform fields can be constructed on a periodic hypercubic lattice under certain conditions and determine the precise form of the background Uð1Þ gauge links that must be imposed on the quantum chromodynamics gauge-field configurations to maintain periodicity. Once supplemented by a set of quantization conditions on the background-field parameters, this construction guarantees that no nonuniformity occurs in the hadronic correlation functions across the boundary of the lattice. The special cases of uniform electric and magnetic fields, a nonuniform electric field that varies linearly in one spatial coordinate (relevant to the determination of quadruple moment and charge radii), nonuniform electric and magnetic fields with given temporal and spatial dependences (relevant to the determination of nucleon spin polarizabilities) and plane-wave electromagnetic fields (relevant to the determination of electromagnetic form factors) are discussed explicitly.
DOI: 10.1103/PhysRevD.92.074506 PACS numbers: 12.38.Gc, 12.38.−t, 13.40.Gp
I. INTRODUCTION met with success in several cases, but is generally chal- lenging (see Ref. [12] for a review of lattice QCD The electromagnetic (EM) properties of hadrons and calculations of nucleon structure). It is therefore useful nuclei arise from the underlying interactions among their to consider further alternatives. strongly interacting quark constituents and the photons. When interested in the response of hadronic systems to Consequently, any reliable theoretical approach that aims to weak EM fields, which determines properties such as EM elucidate the electromagnetic structure of hadronic systems moments and polarizabilities, a powerful method is to must necessarily account for the nonperturbative nature of introduce electromagnetism through a classical background quantum chromodynamics (QCD). Lattice QCD, which is a field. In this approach, fixed U(1) gauge links are simply Monte Carlo evaluation of the QCD path integral regulated imposed on the QCD gauge links in the lattice formulation. through a finite discrete spacetime, is the only such method Depending on the computational resources available and by which to perform first-principles calculations of had- the type of quantities that are being considered, this ronic systems. It relies on the fundamental degrees of imposition may be performed solely on the valence quark freedom of QCD, i.e., quarks and gluons, and as input, only sector of QCD, where only the computation of quark takes the QCD parameters, the mass of quarks and the propagators is influenced by the additional Uð1Þ links, strength of the coupling constant. To be compared with or on both the sea and valence quark sectors, where the nature, the results of lattice QCD calculations must be U(1) gauge links are also incorporated in the generation of systematically extrapolated to the continuum and infinite- the QCD gauge configurations. The former is not a reliable volume limits. Quantum electrodynamics (QED) can be approximation when there are sea-quark (disconnected) treated along with QCD in a similar fashion, although contributions to hadronic correlation functions in back- the computational cost associated with a Monte Carlo ground fields. A low-energy (multi-)hadronic theory that evaluation of the QCD and QED path integral is consid- describes the interaction of the hadron (or nucleus) with – erably higher; see Refs. [1 11] for recent progress in this weak external EM fields can be matched onto appropriate direction. Alternatively, given that EM interactions are lattice QCD correlation functions. This matching constrains perturbatively small, one can constrain the EM structure those parameters of the hadronic theory that characterize of hadrons and nuclei, for example their EM form factors, the response of the hadron (nucleus) to the applied external by evaluating the matrix elements of current operators in fields [13–25]. While uniform background fields already the presence of solely QCD interactions. This approach has provide a means to constrain a wealth of EM quantities such as the magnetic moment and the electric and magnetic *[email protected] polarizabilities (see Refs. [23,24,26–38] for recent progress † [email protected] in discerning EM properties of hadrons and light nuclei
1550-7998=2015=92(7)=074506(22) 074506-1 © 2015 American Physical Society ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) using lattice QCD with the background field method), more A condition that supports periodic background Uð1Þ general background fields potentially provide sensitivity to gauge fields on a hypercubic lattice1 is well known for the many additional parameters of the low-energy theory. In case of uniform EM fields, namely the ’t Hooft quantization particular, spin-dependent structure parameters beyond the condition (QC) [13,14,46–48], and has been commonly magnetic moment can show up at low orders in the weak- implemented in lattice QCD calculations with the use of field strength if one allows the spatial and/or temporal background fields [23,24,32–38,49]. This condition derivatives of the background fields to be nonvanishing. requires the magnitude of the electric, E, or magnetic, B, Among such quantities are spin polarizabilities of nucle- field on a torus to be quantized, and follows from a simple ons [23,39], and the quadrupole moment of hadrons and argument: for a closed surface geometry, the net flux of the nuclei with spin, s ≥ 1 [15]. Moreover, nonuniform back- EM field through that surface is required to be quantized. ground fields do not constrain one to the static limit of EM The same condition can be obtained by imposing more form factors, and by injecting energy and momentum into general boundary conditions, namely electro/magneto- the system, provide a means to directly evaluate the PBCs. These boundary conditions require the gauge and corresponding off-forward hadronic matrix elements of matter fields to be periodic up to a gauge transformation, the EM current from a response to the background fields and have been studied by ’t Hooft for the case of Abelian [40], with different systematic uncertainties than other and non-Abelian gauge theories, to explore the properties of methods. Another application of such nonuniform back- the flux of the corresponding field strength tensors in the ground fields, as is recently proposed and implemented confinement regime [46,50,51]. When spacetime is discre- in Ref. [41], is to evaluate the hadronic vacuum polari- tized on a hypertorus, special care must be given to the zation function (as the leading hadronic contribution to implementation of the Uð1Þ gauge links at each spacetime the muon anomalous magnetic moment) through evalu- point (see, e.g., Refs. [49,52,53]) to guarantee that the values ating the magnetic susceptibilities with a plane-wave of the Uð1Þ elementary plaquettes remain constant in uni- background field. form background fields. In order to explore such possibilities, one first needs to To implement background EM fields with arbitrary properly implement the desired EM background fields in spacetime dependences, a similar procedure must be lattice QCD calculations. A class of nonuniform EM fields, undertaken to ensure that, despite the gauge links having for example, have been implemented in Refs. [42,43] to been set to satisfy the PBCs at the boundary, the expected obtain some preliminary results for spin polarizabilities of values of the Uð1Þ elementary plaquettes are correctly the nucleon [43]. These studies do not consider background produced adjacent to the boundaries. This requirement gauge potentials that are periodic at the boundary. enables one to determine the modified links near the Retaining the periodic boundary conditions (PBCs) that boundary, as well as the conditions that the parameters are imposed on the gauge fields in the majority of lattice of the chosen background fields must satisfy. We first QCD calculations requires a nontrivial implementation of demonstrate this procedure for a rather special case of an the Uð1Þ gauge links. However, such an implementation electric field generated from a scalar gauge potential with enforces smooth behavior of correlation functions across an arbitrary dependence on only one spatial coordinate. the boundary of the lattice and is a desired feature. This allows one to generalize, rather straightforwardly, to Moreover, quantifying the behavior of the system in the the case of background fields generated from more general hadronic theory is generally more straightforward in gauge potentials, but also makes one appreciate the subtle- periodic EM potentials. Although nonperiodic implemen- ties and limitations encountered in the general case. We use tations of uniform background fields have been pursued in this special case to study in detail the examples of a uniform some earlier lattice QCD studies (by placing hadronic electric field and an electric field that varies linearly in one sources away from the boundary effects) [26–31], quanti- spatial coordinate. We verify the periodicity of the setup by fying uncertainties associated with these nonuniformities is numerically evaluating the correlation functions of neutral difficult [24,44,45]. These issues can be prevented by pions in these background fields. This special case, along explicitly modifying the naive Uð1Þ gauge links, and with the examples, is presented in Sec. II. The general imposing conditions on the background field parameters, considerations are presented in Sec. III, where it is shown such that when setting the value of the gauge links at that a periodic implementation of gauge links in our one boundary of the lattice to its value on the opposite framework is possible if the flux of the electric and boundary, no nonuniformities occur in the value of the Uð1Þ plaquettes. This guarantees smooth behavior of hadronic 1Although the words “hypercubic” and “hypertorus” are used correlation functions across the boundaries. This paper throughout to refer to the lattice geometry, this paper considers presents in detail a procedure for the periodic implementa- the more general case of an anisotropic geometry where both the lattice spacing and the extent of the volume in temporal and tion of background gauge fields, under certain conditions spatial directions are different. In particular, the general result in that are enumerated, along with several examples that follow Sec. III accounts for distinct lattice spacings and volume extents from these general considerations. in all directions.
074506-2 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) magnetic fields through each plane on the lattice is L and temporal extent T, the value of the plaquette with independent of the coordinates transverse to that plane, 0 ≤ x3 074506-3 ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) Since the additional link introduced in Eq. (7) is t with constant E and R, gives rise to a uniform electric field dependent, and given that the gauge link is also required in the x3 direction, to be periodic with respect to the time variable, one must study the value of the plaquette located at x3 ¼ L − as and E ¼ Exˆ 3: ð12Þ t ¼ T − at more closely. For this plaquette, which is located at the far corner of the lattice,2 This electric field can be achieved by implementing the link, Pð0;3ÞðL − as;T− atÞ ðQCDÞ ðQCDÞ ð Þ ˆ ð Þ U ðxÞ → U ðxÞ QCD ieQA0ðL−asÞat QCD μ μ ¼ U0 ðL − as;T− atÞe U3 ðL − as; 0Þ −ieQEaˆ ðx −R−½x3 LÞ δ ieQELˆ ðt−½ t TÞ δ δ ð Þ† ˆ e t 3 L × μ;0 e T × μ;3 x3;L−as ; QCD −ieQA0ð0Þat × × U0 ð0;T− atÞe ð Þ ð Þ† ˆ e 13 QCD −ieQ½A0ð0Þ−A0ðLÞ ðT−atÞ × U3 ðL − as;T− atÞe ˆ e ðQCDÞ which gives rise to the desired value of the plaquette in the ¼ eieQ½A0ðL−asÞ−A0ðLÞ at P ðL − a ;T− a Þ ð0;3Þ s t 0–3 plane −ieQˆ ½A ð0Þ−Ae ðLÞ T × e 0 0 : ð8Þ ˆ ð Þ ieQEatas QCD Pð0;3Þðx3;tÞ¼e Pð0;3Þ ðx3;tÞ; ð14Þ Although the first phase factor correctly accounts for the 0 ≤ x ≤ L − a 0 ≤ t ≤ T − a total flux through a plaquette located at x3 ¼ L − as, the for 3 s and t. In addition to the last factor modifies the value of this plaquette away from modification to the naive Uð1Þ links as presented in the desired value. By demanding Eq. (13), the plaquettes at the boundary are correctly valued if the magnitude of the electric field is quantized, ieQˆ ½A ð0Þ−AeðLÞ T e 0 0 ¼ 1; ð9Þ 2πn E ¼ ; ð15Þ eQTLˆ the value of this plaquette is fixed to its desired value. Equation (9) places a QC on the net flux of the electric field with n ∈ Z, in agreement with previous works, e.g., through the 0–3 plane on the lattice, Refs. [13,14,47,48,52,53]. ðEÞ;net f We have numerically verified that, in contrast with the Φð0;3Þ ≡ ½A0ð0Þ − A0ðLÞ T x Z Z nonperiodic case where the additional link in the 3 ˆ T L direction is not implemented, the choice of the background ¼ eQ dt E3ðx3Þdx3 ¼ 2πn; ð10Þ gauge field in Eq. (13), supplemented by the QC in 0 0 Eq. (15), results in the periodicity of the correlation where n ∈ Z. This QC could indeed be deduced a priori by functions of neutral pions. The upper panel of Fig. 1 shows recalling that the 0–3 plane represents a closed surface area the pion correlation function (projected to zero transverse ðsrcÞ due to PBCs (the surface area of a torus), and the net flux momentum), Cðx3; τÞ, as a function of x3 − x3 at a fixed through this surface must necessarily be quantized. Euclidean time, τ=at ¼ 18 (τ ¼ it). Explicitly, this corre- Equations (7) and (10) are the main results of this section. lation function is defined as Two particular cases of a uniform electric field and a ð Þ linearly varying electric field along the x3 direction can be src Cðx3; τÞ ≡ Cðx3; τ; x3 ; 0Þjp ¼p ¼0 readily worked out from this general result, and are 1 2 L−a L−a presented in the following. Xs Xs ¼ Cðx; τ; xðsrcÞ; 0Þ; ð16Þ x1¼0 x2¼0 A. Example I: A constant electric field x in the 3 direction where An external periodic Uð1Þ gauge field ðsrcÞ † ðsrcÞ Cðx; τ; x ; 0Þ¼h0jOπðx; τÞOπðx ; 0Þj0iE: ð17Þ x3 Aμ ¼ −E × x3 − R − L ; 0 ; ð11Þ † L OπðOπÞ is a lattice interpolating operator that creates (annihilates) any hadronic states with the quantum numbers of the neutral pion. Subscript E refers to the 2Strictly speaking, there is no notion of a corner in a toroidal geometry. However, once an origin is specified for the coordinate fact that the expectation value is evaluated in the back- system, we can choose to define the corner as the plaquette ground of an electric field, E. The calculation only located at x3 ¼ L − as and t ¼ T − at. involves imposing the Uð1Þ gauge links on the QCD 074506-4 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) FIG. 1 (color online). A comparison of choices of background gauge fields that result in a uniform electric field in the x3 direction. “Modified links—Quantized” denotes the choice of the gauge links in Eq. (13) with R ¼ 0, supplemented by the QC in Eq. (15), with 1 n ¼ 3. We have quantized the field for the d quark, corresponding to jQj¼3, which guarantees the quantization of the field for the u quark as well. “Modified links—Nonquantized” corresponds to the same choice of the gauge links but with a nonquantized value of the field, n ¼ e. “No modified links—Quantized” and “No modified links—Nonquantized” denote the naive choice of the gauge links without including the additional links in the x3 direction, with quantized and nonquantized values, n ¼ 3 and n ¼ e, respectively. The ðsrcÞ upper panel shows the correlation function (projected to zero transverse momentum), Cðx3; τÞ, as a function of x3 − x3 at a fixed Cðx3;τÞ ðsrcÞ Euclidean time, τ=at ¼ 18, while the lower panel depicts the dependence of the quantity Mðx3; τÞ ≡ log on x3 − x at Cðx3;τþ1Þ 3 3 τ=at ¼ 18. In this demonstrative study, the lattice volume is taken to be V ¼ð12asÞ × ð24atÞ with as ¼ at ≈ 0.145 ½fm , the QCD gauge configurations are quenched, the Uð1Þ gauge links are solely imposed on the QCD gauge links in the valence quark sector, and ðsrcÞ only the connected pieces of the correlation functions are evaluated for the neutral pion. The values of x3 and x3 in the figure are in units of as. The dashed lines denote the boundary of the lattice. gauge links in the valence sector. xðsrcÞ denotes the uniform magnetic field). Such breakdown is most evident location of the source, which for the upper panel is in the quantity taken to be xðsrcÞ ¼ð0; 0; 0Þ. Since for a neutral pion in a uniform electric field, the finite-volume correlation Cðx3; τÞ Mðx3; τÞ ≡ log ; ð18Þ function with PBC must be symmetric about the point Cðx3; τ þ 1Þ L ðsrcÞ 2 þ x3 , the deviation of the correlation function from symmetricity for nonperiodic gauge-link choices, includ- ðsrcÞ as is plotted as a function of x3 − x for τ=at ¼ 18 in ing those with the correct link structure but with non- 3 quantized values of electric field, signals the breakdown the lower panel of Fig. 1. Here, the source is located xðsrcÞ ¼ð0; 0; 9a Þ of translational invariance in units of L in the x3 direction s and therefore the boundary point ðsrcÞ (this translational invariance is the analogue of the x3 ¼ L ≡ 0 corresponds to x3 − x3 ¼ 3as in these magnetic translation group discussed in Ref. [48] for a plots. Nonuniformities in Mðx3; τÞ when crossing this 074506-5 ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) FIG. 2 (color online). Correlation function of neutral pions (projected to zero transverse momentum), Cðx3; τÞ, as a function of ðsrcÞ ðsrcÞ x3 − x3 at a fixed Euclidean time, τ=at ¼ 18, for two different choices of the x3 , and with different implementations of background gauge fields that result in a uniform electric field, as explained in the caption of Fig. 1. The lack of translational invariance in correlation ðsrcÞ functions is signaled by the dependence of the correlation functions on the value of x3 , as is observed in all the nonperiodic implementations (the upper right and the lower panels). boundary (denoted by the dashed line) are observed uniform background fields are expected to depend in all the cases considered, except for the “Modified upon a gauge-dependent, nontranslationally invariant links—Quantized” case.3 This is again a signature of phase (see, e.g., Refs. [15,38]). This phase must be losing translational invariance in units of L in the x3 factored out before comparing properties of correlation direction. In Refs. [24,44], a similar kinked feature functions under translation with different implementa- was observed in the correlation function of neutral tions of background fields. For nonuniform background pions with nonperiodic implementations of a uniform fields, the translational invariance is fully broken in the electric field with the choice of a time-dependent gauge correlation functions [15], and the dependence on the potential. source location does not provide much insight into Alternatively for the neutral pions, the breakdown of the lack of periodicity of different implementations of periodicity can be examined by studying the dependence background fields. of the correlation functions on the location of the source. Since the correlation functions of neutral par- B. Example II: A linearly varying electric ticles in uniform background fields are translationally field in the x direction invariant, as long as the implementation of background 3 fields does not break translational invariance, they must An external periodic Uð1Þ gauge field, ðsrcÞ be only a function of x3 − x3 . As is plotted in Fig. 2, all the nonperiodic implementations indeed give rise to 2 E0 x3 correlation functions that are not solely a function of the Aμ ¼ − x3 − R − L ; 0 ; ð19Þ ðsrcÞ 2 L distance to the source, and vary as a function of x3 . For the charged particles, correlation functions in gives rise to a linearly varying electric field in the x3 3These nonuniformities may be quantified more precisely by direction, evaluating (the finite-difference approximation to) the derivative x of the functions with respect to 3. As the continuum limit is x3 approached, this (numerical) derivative diverges near the boun- E ¼ E0 × x3 − R − L xˆ 3; ð20Þ dary as a result of nonperiodic implementations. L 074506-6 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) L FIG. 3 (color online). The scalar potential in Eq. (19) (the left panel) with E0 < 0 and R ¼ 2 is a finite harmonic oscillator potential between 0 ≤ x3 ≤ L. It produces a linearly varying electric field, Eq. (20), as depicted in the right panel. The periodic images of the potential and the electric field are also shown in the figures. FIG. 4 (color online). A comparison of choices of background gauge fields that result in a linearly varying electric field in the x3 direction. “Modified links—Quantized” denotes the choice of the gauge links in Eq. (21) with R ¼ 0, supplemented by the QC in 1 Eq. (23), with n ¼ 3 and jQj¼3. “Modified links—Nonquantized” corresponds to the same choice of the gauge links but with a nonquantized value of the field slope, n ¼ e. “No modified links—Quantized” and “No modified links—Nonquantized” denote the naive choice of the gauge links without including the additional links in the x3 direction, with quantized and nonquantized values of the field slope, n ¼ 3 and n ¼ e, respectively. The left panel shows the correlation function (projected to zero transverse momentum), ðsrcÞ Cðx3; τÞ, as a function of x3 − x3 at a fixed Euclidean time, τ=at ¼ 18, while the left panel depicts the dependence of the quantity Cðx3;τÞ ðsrcÞ Mðx3; τÞ ≡ log on x3 − x at τ=at ¼ 18. The details of this numerical study are the same as in Fig. 1. The values of x3 and Cðx3;τþ1Þ 3 ðsrcÞ x3 in the figure are in units of as. The dashed lines denote the boundary of the lattice. asplottedinFig.3. This electric field can be imple- which gives rise to the following value for a plaquette in mented in a lattice QCD calculation through the follow- the 0–3plane, ing links, ˆ as ðQCDÞ ieQE0atasðx3−Rþ 2 Þ Pð0;3Þðx3;tÞ¼e Pð0;3Þ ðx3;tÞ; ð22Þ ðQCDÞ ðQCDÞ −ieQEˆ a ðx −R−½x3 LÞ2 δ 0 ≤ x ≤ L − a 0 ≤ t ≤ T − a Uμ ðxÞ → Uμ ðxÞ × e 2 0 t 3 L × μ;0 for 3 s and t. There exists a R ≠ L ieQEˆ Lð−RþLÞðt−½ t TÞ δ δ QC when 2 which constrains the value of the slope e 0 2 T × μ;3 x3;L−as ; ð Þ × 21 of the electric field to be quantized, 074506-7 ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) FIG. 5 (color online). A comparison of choices of background gauge fields that result in a linearly varying electric field in L the x3 direction, as described in the caption of Fig. 4, but with the offset value being set to R ¼ 2. The left panel shows the ðsrcÞ correlation function (projected to zero transverse momentum), Cðx3; τÞ, as a function of x3 − x3 at a fixed Euclidean time, τ=at ¼ 18, Cðx3;τÞ ðsrcÞ while the left panel depicts the dependence of the quantity Mðx3; τÞ ≡ log on x3 − x at τ=at ¼ 18. The details of Cðx3;τþ1Þ 3 ðsrcÞ this numerical study are the same as in Fig. 1. The values of x3 and x3 in the figure are in units of as. The dashed lines denote the boundary of the lattice. 2πn E ¼ ; ð Þ For a neutral pion in an electric field that varies as a 0 ˆ L 23 x eQTLð−R þ 2Þ function of 3, the finite-volume correlation function with PBCs will no longer be symmetric about the point L þ xðsrcÞ with n ∈ Z. When the offset value R is chosen to 2 3 (see Ref. [15] and discussions associated with L Fig. 5), a feature that is indeed observed in the left be 2,Eq.(9) trivially holds and no quantization 4 panel of the figure. However, by displacing the source constraint is placed on E0. Moreover, the additional xðsrcÞ ≠ 0 link at point x3 ¼ L − as in Eq. (21) will be equal to 3 , the breakdown of translational invariance to unity. in units of L is again manifest through nonuniformities We have implemented a linearly varying background when crossing the boundary of the lattice. This feature, electric field, with and without the additional links in as is plotted in the right panel of the figure, is the x3 direction [see Eq. (21)], for quantized and observed in quantity Mðx3; τÞ at a fixed time when ðsrcÞ ðsrcÞ nonquantized values of the electric field slope [see crossing x3 − x3 ¼ 3as, with x3 ¼ 9as,andis Eq. (23)]. Figure 4 corresponds to the choices of electric more prominent for implementations that do not use field with the offset value R ¼ 0. Consequently, the modified values of the gauge links near the accordingtoEq.(23),aQCisnecessarytoguarantee boundary. the full periodicity. In the left panel of the figure, the According to Eq. (21), when the field offset value is L neutral pion correlation function (projected to zero set to R ¼ 2, no additional link is needed to guarantee transverse momentum), Cðx3; τÞ, is plotted as a function periodicity. As a result, our implementations of both ðsrcÞ of x3 − x3 at a fixed Euclidean time, τ=at ¼ 18. cases are trivially identical as is shown in Fig. 5. However, a nontrivial check is to show that the perio- 4 dicity of correlation functions is retained with arbitrary When there is no QC placed on the slope of the field, E0, the field can become arbitrarily strong at the boundaries of the (nonquantized) values of the electric field slope. lattice in the x3 direction in the large-volume limit. When Although there exists no QC for this case, we have interested in the response of the system to the external field at continued to label the choices of the electric field slope leading orders in the field strength, one needs to make sure as in the case of R ¼ 0. Then, as is evident from the right that the field remains weak at any spacetime point on the panel of Fig. 5, the function Mðx3; τÞ smoothly crosses lattice. This can be achieved by tuning E0 to have appropri- ately small values, e.g., values that are power-law suppressed over the boundary of the lattice (up to the uncertainty of in volume. the data points), and correlation functions respect the 074506-8 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) periodicity of the fields.5 Additionally, the left panel of In order to generate electric or magnetic fields with the figure suggests that the correlation function is arbitrary dependences on space and time coordinates, one L ðsrcÞ needs to consider a generic gauge potential with all four approximately symmetric about the point 2 þ x3 when L components being nonvanishing and having arbitrary R ¼ 2, in contrast with the case of R ¼ 0. This feature is understood by recalling that for a neutral pion, the spacetime dependences, leading dependence of the correlation function on a A ðx;tÞ¼ðA ðx;tÞ; −Aðx;tÞÞ: ð Þ nonuniform external field arises from its charge radius. μ 0 24 The corresponding contribution then scales as ∼E0 for the case of a linearly varying field in Eq. (12),whichis All the functions are to be understood to be periodified constant. The next contribution results from the non- according to the floor-function prescription introduced 2 vanishing polarizability of the pion and scales as E . in the previous section. Explicitly, all the xi depend- Although this contribution is x3 dependent, it only ences of the functions must be understood as a depend- xi depends on the square of the field, a quantity which ence on ðxi − ½L LÞ. Similarly, all the time dependences L ðt − ½ t TÞ is symmetric about x3 ¼ 2 if the field offset value is set aretobereplacedwith T dependences. As L ðsrcÞ introduced above, once we deperiodify the functions to R ¼ 2 [with x3 ¼ 0]; see Fig. 3. This also explains in either space or time coordinates, we place a tilde over the asymmetry in the neutral pion correlation function in 0 ≤ x 074506-9 ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) ð Þ ˆ ð Þ ˆ QCD ieQA0ðxi;tÞat QCD ieQAiðxi;tþatÞas Pð0;iÞðxi;tÞ¼U0 ðxi;tÞe Ui ðxi;tþ atÞe ð Þ† ˆ ð Þ† ˆ QCD −ieQA0ðxiþas;tÞat QCD −ieQAiðxi;tÞas × U0 ðxi þ as;tÞe Ui ðxi;tÞe ˆ ˆ ð Þ −ieQ½Aiðxi;tÞ−Aiðxi;tþatÞ asþieQ½A0ðxi;tÞ−A0ðxiþas;tÞ at QCD ¼ e × Pð0;iÞ ðxi;tÞ; ð27Þ where we have dropped the xj and xk dependences of the functions for brevity as they are fixed in this expression. Here, and in the following, i; j and k assume distinct values. The Uð1Þ phase appearing in the value of the plaquette is clearly the expected value by noting that the electric flux through the corresponding surface area in the 0 − i plane is Z Z x þa tþa ðEÞ i s 0 t 0 0 0 Φð0;iÞðxj; xkÞ¼ dxi dt Eiðxi; xj; xk;tÞ xi t ¼ −½Aiðxi; xj; xk;tÞ − Aiðxi; xj; xk;tþ atÞ as þ½A0ðxi; xj; xk;tÞ − A0ðxi þ as; xj; xk;tÞ at 2 2 þ Oðasat;at asÞ: ð28Þ ðEÞ In the continuum limit, the flux per unit area, Φð0;iÞ=asat, is finite and nonvanishing for any nonvanishing value of the electric field. As a result, the exponent of the Uð1Þ phase factor in Eq. (27) correctly accounts for the flux of electric field through the corresponding surface area in the continuum limit. Since the value of the gauge links at the boundary will be set to their value at the origin due to the use of periodic functions in Eq. (24), the value of the plaquettes adjacent to the boundary of the lattice will not necessarily produce their value as expected for the electric-field flux through these plaquettes. This can be easily seen by evaluating the value of plaquettes at xi ¼ L − as and 0 ≤ t ð Þ ˆ ð Þ ˆ QCD ieQA0ðL−as;tÞat QCD ieQAiðL−as;tþatÞas Pð0;iÞðxi ¼ L − as;tÞ¼U0 ðL − as;tÞe Ui ðL − as;tþ atÞe ð Þ† ˆ ð Þ† ˆ QCD −ieQA0ð0;tÞat QCD −ieQAiðL−as;tÞas × U0 ð0;tÞe Ui ðL − as;tÞe ˆ ˆ ~ ð Þ ˆ ~ −ieQ½AiðL−as;tÞ−AiðL−as;tþatÞ asþieQ½A0ðL−as;tÞ−A0ðL;tÞ at QCD ieQ½A0ðL;tÞ−A0ð0;tÞ at ¼ e × Pð0;iÞ ðL − as;tÞ × e ; ð29Þ where again the xj and xk dependences of the functions are direction without changing the value of adjacent plaquettes. suppressed. One may now choose to modify the value of This is exactly the case we considered in the previous the link along either the 0 or the i directions in such a way section, Eq. (6) (upon setting Ai ¼ 0 in the current case). as to cancel out the extra phase in Eq. (29).However,itis However, in contrast to the case considered previously, the easy to show that by modifying the link along the 0 scalar potential that appears in Eq. (29) carries an arbitrary t direction, the values of all the adjacent plaquettes are dependence. As a result, the prescription of Eq. (7) will not affected. In particular, one cannot consistently make this work and we need to find a more general modified link in modification throughout the lattice without shifting the the i direction to cancel the last phase factor in Eq. (29).Itis value of another set of plaquettes from their desired values. easy to see that the following modification to the Ui link at However, one can make the modification along the i xi ¼ L − as, ð Þ ð Þ ˆ ˆ ~ QCD QCD ieQAiðx;tÞas ieQ½A0ðxi¼0;xj;xk;tÞ−A0ðxi¼L;xj;xk;tÞ fi;0ðxj;xk;tÞ×δx ;L−a Ui ðx;tÞ → Ui ðx;tÞ × e × e i s ; ð30Þ achieves the desired result provided that the boundary function fi;0ðxj; xk;tÞ introduced above satisfies the following relation, ~ ½A0ðxi ¼ 0; xj; xk;tþ atÞ − A0ðxi ¼ L; xj; xk;tþ atÞ fi;0ðxj; xk;tþ atÞ ~ ¼½A0ðxi ¼ 0; xj; xk;tÞ − A0ðxi ¼ L; xj; xk;tÞ ðfi;0ðxj; xk;tÞþatÞ: ð31Þ 074506-10 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) Note that xj and xk denote coordinates that are trans- prescribed in Eq. (7). In general, if the function fi;0 verse to the 0 − i plane. In order for the function fi;0 to depends on xj and xk for the chosen gauge field, not spoil the periodicity of gauge links in the temporal additional conditions must be placed on fi;0.Wewill and spatial directions, its dependences upon space and deduce these relations once we extend the above time coordinates must be understood through the floor- considerations to the plaquettes in other planes. function prescription. Equation (31) is a recursive The fi;0 function, as well as other gauge fields, have relation, and once the initial value of the function, explicit time dependence and are periodic in this variable. corresponding to fi;0ðxj; xk;t¼ 0Þ, is input, its value at As a result, fixing the value of plaquettes at xi ¼ L − as every other point t, fi;0ðxj; xk;tÞ, can be obtained. For may not necessarily guarantee that its desired value is the special case of the previous section, fi;0 is only a produced at xi ¼ L − as and t ¼ T − at. However, before function of t and satisfies a simple recursive relation, studying the value of this last plaquette, one needs to fix the fi;0ðt þ atÞ¼fi;0ðtÞþat.Oncefi;0ð0Þ is set to zero, the value of all plaquettes located at 0 ≤ xi ð Þ ˆ ð Þ ˆ QCD ieQA0ðxi;T−atÞat QCD ieQAiðxi;0Þas Pð0;iÞðxi;t¼ T − atÞ¼U0 ðxi;T− atÞe Ui ðxi; 0Þe ð Þ† ˆ ð Þ† ˆ QCD −ieQA0ðxiþas;T−atÞat QCD −ieQAiðxi;T−atÞas × U0 ðxi þ as;T− atÞe Ui ðxi;T− atÞe ˆ ~ ˆ ð Þ ˆ ~ −ieQ½Aiðxi;T−atÞ−Aiðxi;TÞ as ieQ½A0ðxi;T−atÞ−A0ðxiþas;T−atÞ at QCD −ieQ½Aiðxi;TÞ−Aiðxi;0Þ as ¼ e e Pð0;iÞ ðxi;T− atÞ × e ; ð32Þ where we have suppressed the xj and xk dependences of the functions. To eliminate the unwanted phase factor, one can modify the value of the link along the 0 direction as follows: ð Þ ð Þ ˆ ˆ ~ QCD QCD ieQA0ðx;tÞat ieQ½Aiðx;t¼0Þ−Aiðx;t¼TÞ f0;iðxÞ×δt;T−a U0 ðx;tÞ → U0 ðx;tÞ × e × e t ; ð33Þ where the boundary function f0;iðxiÞ satisfies ~ ½Aiðxi þ as; xj; xk;t¼ 0Þ − Aiðxi þ as; xj; xk;t¼ TÞ f0;iðxi þ as; xj; xkÞ ~ ¼½Aiðx;t¼ 0Þ − Aiðx;t¼ TÞ ðf0;iðxÞþasÞ: ð34Þ The dependence of the new functions f0;i on any coordinate also produced in the 0 − j plane with j ≠ i when xi t ¼ T − a variable xi must be realized through ðxi − ½L LÞ as before. t. We will return to this point below. As will be shown shortly, although this condition on f0;i With the modifications of the U0 and Ui links according guarantees that the expected values of plaquettes at 0 ≤ to Eqs. (30) and (33), we are ready to inspect the value of xi ð Þ ˆ ˆ ~ QCD ieQA0ðL−as;T−atÞat ieQ½AiðL−as;0Þ−AiðL−as;TÞ f0;iðL−asÞ Pð0;iÞðL − as;T− atÞ¼U0 ðL − as;T− atÞe e ð Þ ˆ ˆ ~ QCD ieQAiðL−as;0Þas ieQ½A0ð0;0Þ−A0ðL;0Þ fi;0ð0Þ × Ui ðL − as; 0Þe e ð Þ† ˆ ˆ ~ QCD −ieQA0ð0;T−atÞat −ieQ½Aið0;0Þ−Aið0;TÞ f0;ið0Þ × U0 ð0;T− atÞe e ð Þ† ˆ ˆ ~ QCD −ieQAiðL−as;T−atÞas −ieQ½A0ð0;T−atÞ−A0ðL;T−atÞ fi;0ðT−atÞ × Ui ðL − as;T− atÞe e ˆ ~ ˆ ~ ð Þ −ieQ½AiðL−as;T−atÞ−AiðL−as;TÞ asþieQ½A0ðL−as;T−atÞ−A0ðL;T−atÞ at QCD ¼ e × Pð0;iÞ ðxi;tÞ LY−as TY−at ˆ ~ ˆ ~ × eieQ½Aiðxi;0Þ−Aiðxi;TÞ as e−ieQ½A0ð0;tÞ−A0ðL;tÞ at ; ð35Þ xi¼0 t¼0 where we have used Eqs. (31) and (34) to arrive at the final expression. As is required, the value of the plaquette is independent of the fi;0 and f0;i functions. In order for this plaquette to have the desired value, one can impose the condition 074506-11 ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) LY−as TY−at ˆ ~ ˆ ~ e−ieQ½Aiðxi;;xj;xk;t¼0Þ−Aiðxi;xj;xk;t¼TÞ as eieQ½A0ðxi¼0;xj;xk;tÞ−A0ðxi¼L;xj;xk;tÞ at ¼ 1; ð36Þ xi¼0 t¼0 to set the extra phase factors in Eq. (35) to unity. This condition and its implication for the allowed gauge field choices require further discussion. Let us first point out that Eq. (36) arises from adding up the value of the Uð1Þ plaquettes that are corrected by introducing modified links throughout the lattice in the 0 − i plane, LY−as TY−at ˆ ˆ ½e−ieQ½Aiðxi;xj;xk;tÞ−Aiðxi;xj;xk;tþatÞ asþieQ½A0ðxi;xj;xk;tÞ−A0ðxiþas;xj;xk;tÞ at xi¼0 t¼0 LY−as TY−at ˆ ~ ˆ ~ ¼ e−ieQ½Aiðxi;xj;xk;0Þ−Aiðxi;xj;xk;TÞ as eieQ½A0ð0;xj;xk;tÞ−A0ðL;xj;xk;tÞ at xi¼0 t¼0 h P P i ˆ L−as ~ T−at ~ ieQ − ½Aiðxi;xj;xk;0Þ−Aiðxi;xj;xk;TÞ asþ ½A0ð0;xj;xk;tÞ−A0ðL;xj;xk;tÞ at ¼ e xi¼0 t¼0 : ð37Þ This latter exponent is ieQˆ times the net flux of electric field through the 0 − i plane. Explicitly in the continuum limit, Z Z T L ðEÞ;net Φð0;iÞ ðxj; xkÞ¼ dt dxiEiðxi; xj; xk;tÞ 0Z 0 Z L ~ T ~ ¼ − dxi½Aiðxi; xj; xk; 0Þ − Aiðxi; xj; xk;TÞ þ dt½A0ð0; xj; xk;tÞ − A0ðL; xj; xk;tÞ : ð38Þ 0 0 So the condition in Eq. (36) states that the net flux of the Eq. (33) must be generalized as follows to incorporate electric field through the 0 − i plane must be quantized, additional links in the ðj ≠ iÞ directions: ΦðEÞ;netðx ; x Þ¼2πn n ∈ Z ð0;iÞ j k eQˆ with . This condition can only ð Þ ð Þ ˆ QCD QCD ieQA0ðx;tÞat hold if the left-hand side is independent of xj and xk U0 ðx;tÞ → U0 ðx;tÞ × e Y coordinates. It therefore constrains the class of electric fields ieQˆ ½A ðx;t¼0Þ−A~ ðx;t¼TÞ f ðxÞ δ × e i i 0;i × t;T−at : that can be implemented in this relatively general framework i¼1;2;3 on a periodic lattice.6 As we will see below, the same ð Þ conclusion can be drawn by inspecting the consistency of the 39 conditions on the fμ;ν functions (μ; ν ¼ 0; 1; 2; 3). Let us not constrain our discussion to the classes of At first glance, it might appear that a similar equation to gauge fields that can be quantized through Eq. (36) for the that in Eq. (30) must represent the value of the link in the i moment, and continue to allow a general coordinate direction with i ¼ 1; 2; 3. However, one should bear in dependence for the gauge fields. We will obtain all such mind that in this general scenario there exist a nonzero constraints that arise on the spacetime dependence of gauge magnetic field and the value of the links in the i direction fields shortly. Having studied the case of electric field in the can only be fully fixed after inspecting the value of i direction in great detail, it is easy to deduce the modified plaquettes in all Cartesian planes. links as well as the QC for the scenario where electric field The value of a plaquette in the i − j plane with 0 ≤ xi < is nonvanishing along other spatial directions. Firstly, L − as and 0 ≤ xj ð Þ ˆ ð Þ ˆ QCD ieQAiðxi;xjÞas QCD ieQAjðxiþas;xjÞas Pði;jÞðxi; xjÞ¼Ui ðxi; xjÞe Uj ðxi þ as; xjÞe ð Þ† ˆ ð Þ† ˆ QCD −ieQAiðxi;xjþasÞas QCD −ieQAjðxi;xjÞas × Ui ðxi; xj þ asÞe Uj ðxi; xjÞe ˆ ˆ ð Þ ieQ½Aiðxi;xjÞ−Aiðxi;xjþasÞ as−ieQ½Ajðxi;xjÞ−Ajðxiþas;xjÞ as QCD ¼ e Pði;jÞ ðxi; xjÞ; ð40Þ 2 which correctly accounts for the magnetic field flux through the surface area as in the i − j plane in the continuum limit, 6More general frameworks than what we have presented here may potentially exist that could allow more complicated spacetime dependences for the fields to be implemented periodically on the lattice. Finding such frameworks requires further investigations. 074506-12 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) Z Z 1 xiþas xjþas ΦðBÞ ðx ;tÞ¼− ϵ dx0 dx0 B ðx0;x0 ;x ;tÞ Equation (40) suggests that the results for the modified ði;jÞ k 2 ijk i j k i j k xi xj links, and the QC that was obtained from the consideration 0 − i ¼½Aiðxi;xj;xk;tÞ − Aiðxi;xj þ as;xk;tÞ as of the plaquettes in the plane, can be simply carried over to the case of the i − j plane by substituting 0 → i, − ½Ajðxi;xj;xk;tÞ − Ajðxi þ as;xj;xk;tÞ as i → j, at → as and T → L in those relations. In particular, þ Oða3Þ: ð Þ s 41 one finds that ð Þ ð Þ ˆ ˆ ~ QCD QCD ieQAiðx;tÞas ieQ½A0ðxi¼0;xj;xk;tÞ−A0ðxi¼L;xj;xk;tÞ fi;0ðxj;xk;tÞ×δx ;L−a Ui ðxÞ → Ui ðxÞ × e × e i s Y ieQˆ ½A ðx ¼0;x ;x ;tÞ−A~ ðx ¼L;x ;x ;tÞ f ðx ;x ;tÞ δ × e j i j k j i j k i;j j k × xi;L−as ; ð42Þ j≠i where the boundary function fi;jðxjÞ must satisfy ~ ½Ajðxi ¼ 0; xj þ as; xk;tÞ − Ajðxi ¼ L; xj þ as; xk;tÞ fi;jðxj þ as; xk;tÞ ~ ¼½Ajðxi ¼ 0; xj; xk;tÞ − Ajðxi ¼ L; xj; xk;tÞ ðfi;jðxj; xk;tÞþasÞ; ð43Þ ~ ½Ajðxi ¼ 0; xj; xk þ as;tÞ − Ajðxi ¼ L; xj; xk þ as;tÞ fi;jðxj; xk þ as;tÞ ~ ¼½Ajðxi ¼ 0; xj; xk;tÞ − Ajðxi ¼ L; xj; xk;tÞ fi;jðxj; xk;tÞ; ð44Þ ~ ½Ajðxi ¼ 0; xj; xk;tþ atÞ − Ajðxi ¼ L; xj; xk;tþ atÞ fi;jðxj; xk;tþ atÞ ~ ¼½Ajðxi ¼ 0; xj; xk;tÞ − Ajðxi ¼ L; xj; xk;tÞ fi;jðxj; xk;tÞ: ð45Þ While the first condition arises from requiring the plaquettes in the i − j plane to have their desired value when xi ¼ L − as, the last two conditions arise from setting the value of plaquettes in the i − k and i − 0 plane to their desired values. Furthermore, by studying carefully the value of the plaquettes in the i − j and i − k planes when xi ¼ L − as, and given the modified links in Eq. (42), one arrives at the following conditions on fi;0: ~ ½A0ðxi ¼ 0; xj þ as; xk;tÞ − A0ðxi ¼ L; xj þ as; xk;tÞ fi;0ðxj þ as; xk;tÞ ~ ¼½A0ðxi ¼ 0; xj; xk;tÞ − A0ðxi ¼ L; xj; xk;tÞ fi;0ðxj; xk;tÞ; ð46Þ ~ ½A0ðxi ¼ 0; xj; xk þ as;tÞ − A0ðxi ¼ L; xj; xk þ as;tÞ fi;0ðxj; xk þ as;tÞ ~ ¼½A0ðxi ¼ 0; xj; xk;tÞ − A0ðxi ¼ L; xj; xk;tÞ fi;0ðxj; xk;tÞ: ð47Þ These conditions add to the condition in Eq. (31) and must be satisfied simultaneously to obtain a consistent solution for fi;0. Similarly, if the values of the plaquette in the 0 − j and 0 − k planes are considered when t ¼ T − at, one arrives at ~ ½Aiðxi; xj þ as; xk;t¼ 0Þ − Aiðxi; xj þ as; xk;t¼ TÞ f0;iðxi; xj þ as; xkÞ ~ ¼½Aiðx;t¼ 0Þ − Aiðx;t¼ TÞ f0;iðxÞ; ð48Þ ~ ½Aiðxi; xj; xk þ as;t¼ 0Þ − Aiðxi; xj; xk þ as;t¼ TÞ f0;iðxi; xj; xkÞ ~ ¼½Aiðx;t¼ 0Þ − Aiðx;t¼ TÞ f0;iðxÞ; ð49Þ which supplement the previous condition on f0;i in Eq. (34). With these modified links, one can see that, in addition to the three QCs in Eq. (36) for i ¼ 1; 2; 3, three more QCs arise, LY−as LY−as ˆ ~ ˆ ~ e−ieQ½Aiðxi;xj¼0;xk;tÞ−Aiðxi;xj¼L;xk;tÞ as eieQ½Ajðxi¼0;xj;xk;tÞ−Ajðxi¼L;xj;xk;tÞ as ¼ 1; ð50Þ xi¼0 xj¼0 074506-13 ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) by requiring that the desired value of the plaquette located extra factor in Eq. (35), which is a product of all the at xi ¼ L − as and xj ¼ L − as is generated. Equation (50) coordinate-dependent phase factors above, is automatically is the statement that the net flux of the magnetic field equal to 1 because of the micro-QCs. However, it is evident ðBÞ;net that due to the coordinate dependence of the new con- through the i − j plane must be quantized, Φ ðxk;tÞ¼ ði;jÞ ditions, this procedure will not always work (note that even 2πn n ∈ Z x eQˆ with . Therefore, if the flux is dependent on the k if the gauge fields are chosen to be independent of the (k ≠ i; j) and t coordinates, this condition cannot be transverse coordinate, the QCs still carry a longitudinal satisfied in general. coordinate dependence). On the lattice, where space and Equations above for the fμ;ν functions must be satisfied time coordinates are discretized, gauge fields with simple simultaneously to ensure that the proper values of the rational dependences on the coordinates could allow such elementary plaquettes are produced throughout the lattice. micro-QCs to be satisfied. However, such conditions are However, it is not guaranteed that, for any given A field, more restrictive on the background field parameters than these equations possess consistent solutions for each fμ;ν. the QC derived in this section on the total flux of the field, To clarify this point, consider the fi;0 function which must leading to large quanta of background fields that are not be obtained recursively from Eqs. (31), (46) and (47).Itis desired for most applications. As an example, consider the straightforward to see that these equations are consistent case of a uniform electric field generated by the gauge with one another only when A0 depends solely on the xi potential in Eq. (11). The only micro-QC is and t coordinates. In general, there exists a valid fμ;ν only if 2πn Aν solely depends on xμ and xν coordinates. Note that if Aν E ¼ ; ð53Þ eQLaˆ is independent of the xμ coordinate, no discontinuity occurs t μ − ν x ¼ in the value of plaquette in the plane when μ n L − a L ¼ T a ¼ a μ ¼ 0 L ¼ L with being an integer, which arises from setting the μ μ ( μ and μ t for while μ and following phase factor equal to unity, aμ ¼ as for μ ¼ i). As a result no fμ;ν needs to be ˆ ~ introduced to guarantee periodicity. Interestingly, all such eieQ½A0ðL;xj;xk;tÞ−A0ð0;xj;xk;tÞ at conditions on the spacetime dependence of Aμ can be ieQˆ ½A~ ðLÞ−A ð0Þ a −ieQELaˆ shown to be consistent with the statement that the net ¼ e 0 0 t ¼ e t ¼ 1: ð54Þ electric or magnetic flux through any plane on the four- dimensional lattice (a closed surface in the toroidal geom- This shows that once the micro-QC is satisfied, one does etry) must be spacetime independent. This is exactly the not need to introduce any additional link near the boundary condition we deduced by examining the QCs in Eqs. (36) as is done in Eq. (13). However, such QC requires large and (50). In the next section, we present the setup for the electric-field strengths which are usually not desired. implementation of several chosen background fields on a periodic lattice and will specify the corresponding QCs. IV. EXAMPLES: PERIODIC IMPLEMENTATION Before proceeding to the next section, it is worth OF SELECTED BACKGROUND FIELDS mentioning that an alternative way to ensure that the ON A HYPERCUBIC LATTICE DEDUCED desired values of the plaquettes are produced adjacent to FROM THE GENERAL CASE the boundaries of the lattice is to enforce a set of micro- The examples that follow provide a setup for the QCs. These QCs can be deduced by setting the extra factor implementation of selected background electric and/or in the value of plaquettes near the boundaries equal to 1 magnetic fields that preserves the periodicity of the lattice (without requiring any gauge link to be modified). For calculation. These are deduced from the general case of the example, one can set the coordinate-dependent factors previous section, the results of which will be summarized here for convenience. In order for the background Uð1Þ ieQˆ ½A~ ðL;x ;x ;tÞ−A ð0;x ;x ;tÞ a e 0 j k 0 j k t ð51Þ gauge links to be implemented periodically, they must be introduced as in Eq. (29) and ðQCDÞ ðQCDÞ ieQAˆ ðxÞa Uμ ðxÞ → Uμ ðxÞ × e μ μ ˆ ~ −ieQ½Aiðxi;xj;xk;TÞ−Aiðxi;xj;xk;0Þ as Y e ð52Þ ieQˆ ½A ðx ¼0;x Þ−A~ ðx ¼L ;x Þ f ðx Þ δ × e ν μ ν ν μ μ ν μ;ν ν × xμ;Lμ−aμ ; ν≠μ in Eq. (32) equal to 1, such that not only the correct values of plaquettes in the 0 − i plane at 0 ≤ xi 074506-14 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) ~ 2πn ½Aνðxμ ¼ 0;xν þ aνÞ − Aνðxμ ¼ Lμ;xν þ aνÞ fμ;νðxν þ aνÞ a ¼ ; ð Þ 1 ˆ 62 ~ eQTL ¼½Aνðxμ ¼ 0;xνÞ − Aνðxμ ¼ Lμ;xνÞ ðfμ;νðxνÞþaνÞ: ð56Þ with n ∈ Z. 7 fμ;ν is vanishing if Aν is independent of xμ. Under the B. Example II: A constant magnetic field conditions specified, the net electric or magnetic flux in the x3 direction through any plane in the continuum limit is constant and One can pick the following gauge potential, must be quantized. On the lattice these QCs read b b 1 1 LYμ−aμ Aμ ¼ðA0; −AÞ¼ 0; x2; − x1; 0 ; ð63Þ ˆ ~ 2 2 e−ieQ½Aμðxμ;xν¼0Þ−Aμðxμ;xν¼LνÞ aμ xμ¼0 to generate a uniform magnetic field in the x3 direction, L −a Yν ν B ¼ b1xˆ 3 ieQˆ ½A ðx ¼0;x Þ−A~ ðx ¼L ;x Þ a . The nontrivial gauge links that are required to × e ν μ ν ν μ μ ν ν ¼ 1: ð57Þ implement this background field are xν¼0 ðQCDÞ ðQCDÞ ieQbˆ x a ieQbˆ Lf ðx Þ δ U → U × e2 1 2 s × e2 1 1;2 2 × x1;L−as ; All the spacetime dependences in the following examples 1 1 must be understood through the floor-function prescription ð64Þ of Sec. II. ð Þ ð Þ i ˆ i ˆ QCD QCD − eQb1x1as − eQb1Lf2;1ðx1Þ×δx ;L−a U2 → U2 × e 2 × e 2 2 s : A. Example I: A constant electric field ð65Þ in the x3 direction We have already discussed this case in Sec. II. Here we The conditions on f1;2 and f2;1 are, respectively, choose a different gauge than that taken in Sec. II, f1;2ðx2 þasÞ¼f1;2ðx2Þþas and f2;1ðx1 þasÞ¼f2;1ðx1Þþ as, with the solutions f1;2 ¼ x2 and f2;1 ¼ x1, once the Aμ ¼ðA0; −AÞ¼ð0; 0; 0;a1tÞ; ð58Þ initial values of the functions are set to zero. There is one QC for this choice of the field, which produces an electric field in the x3 direction, E ¼ a xˆ 1 3. The nontrivial gauge links that produce this LY−as LY−as ieQbˆ La ieQbˆ La ieQbˆ La L background field are e2 1 s e2 1 s ¼½e 1 s as x ¼0 x ¼0 ð Þ ð Þ ˆ 1 2 QCD QCD −ieQa1Tf0;3ðx3Þ×δt;T−a U → U × e t ; ð59Þ ˆ 2 0 0 ¼ eieQb1L ¼ 1; ð66Þ ðQCDÞ ðQCDÞ ˆ U → U eieQa1tas ; ð Þ 3 3 × 60 which constrains the strength of the magnetic field parameter, where f0;3 satisfies f0;3ðx3 þ asÞ¼f0;3ðx3Þþas, with the 8 solution f0;3ðx3Þ¼x3 once one sets f0;3ð0Þ¼0. There is 2πn b1 ¼ ; ð67Þ only one QC in this case, eQLˆ 2 L−as Y L n ∈ Z −ieQaˆ Ta −ieQaˆ Ta −ieQaˆ TL with . e 1 s ¼½e 1 s as ¼ e 1 ¼ 1; ð61Þ x ¼0 3 C. Example III: A space-dependent magnetic which constrains the value of the field strength, a1, field and a constant electric field When interested in extracting the spin polarizabilities of 7 In cases where Aν does not depend on the xν coordinate, as is nucleons, one can choose the following gauge potential, the case in most of the examples in this section, the fμ;ν satisfies the relation fμ;νðxν þ aνÞ¼fμ;νðxνÞþaν with the solution Aμ ¼ðA0; −AÞ¼ð0;b2x1x2; 0;a2tÞ; ð68Þ fμ;νðxνÞ¼xν, once one sets fμ;νð0Þ¼0.IfAν depends on both x x μ and ν coordinates, it is possible to transform to a gauge where to produce a space-dependent magnetic field, B ¼ b2x1xˆ 3, A x ∂ ∂ Fμν ¼ 0 ν does not depend on ν, as long as the condition μ ν is as well as a constant electric field, E ¼ a2xˆ 3. These Fμν satisfied, where is the EM field strength tensor and no background fields generate a nonvanishing interaction summation over μ and ν is assumed. 1 8 σ ð∇ B þ ∇ B ÞE σ Note that the initial value of the function is arbitrary as it proportional to 2 i i j j i j (with i denoting drops out of the value of plaquettes. Pauli matrices) in the nonrelativistic Hamiltonian of the 074506-15 ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) 1 _ spin-2 hadron in external fields, which gives access to the nonvanishing interaction proportional to σ · E × E in the γ 1 E1M2 spin polarizability of the hadron; see Ref. [23].To nonrelativistic Hamiltonian of the spin-2 hadron in external γ produce this electric field on a periodic lattice, the non- fields, which gives access to the E1E1 spin polarizability of trivial gauge links to be implemented are the hadron; see Ref. [23]. To produce this electric field on a periodic lattice, the nontrivial gauge links to be imple- ðQCDÞ ðQCDÞ −ieQaˆ Tf ðx Þ δ U → U e 2 0;3 3 × t;T−at ; ð Þ 0 0 × 69 mented are ðQCDÞ ðQCDÞ ieQbˆ x x a U → U e 2 1 2 s ; ð Þ ð Þ ð Þ i ˆ 2 1 1 × 70 QCD QCD − eQa3T f0;1ðx1Þ×δt;T−a U0 → U0 × e 2 t ð Þ ð Þ ˆ −ieQaˆ Tf ðx Þ δ QCD QCD −ieQb2x1Lf2;1ðx1Þ×δx ;L−a e 4 0;2 2 × t;T−at ; ð Þ U2 → U2 × e 2 s ; ð71Þ × 77 ðQCDÞ ðQCDÞ ieQaˆ ta U → U e 2 s : ð Þ ð Þ ð Þ i ˆ 2 3 3 × 72 QCD QCD eQa3t as U1 → U1 × e2 ; ð78Þ The f0;3 and f2;1 functions satisfy recursive relations x1 ðQCDÞ ðQCDÞ ieQaˆ ta f0;3ðx3 þasÞ¼f0;3ðx3Þþas and f2;1ðx1 þasÞ¼ × U → U e 4 s : ð Þ x1þas 2 2 × 79 ðf2;1ðx1ÞþasÞ, respectively, with solutions f0;3ðx3Þ¼x3 x1−as f f and f2;1ðx1Þ¼ 2 for x1 > 0, once the initial values of The 0;1 and 0;2 functions satisfy recursive relations the functions are set to zero. The only QCs are f0;1ðx1 þasÞ¼f0;1ðx1Þþas and f0;2ðx2 þasÞ¼f0;2ðx2Þþ as, respectively, with trivial solutions f0;1ðx1Þ¼x1 and L−a Ys f0;2ðx2Þ¼x2, once the initial values of the functions are ieQbˆ x La ieQbˆ L2ðL−a Þ ieQbˆ L3ð1−asÞ e 2 1 s ¼ e2 2 s ¼ e2 2 L ¼ 1; ð73Þ set to zero. The only QCs are x1¼0 LY−as L−a i ˆ 2 i ˆ 2 L i ˆ 2 Ys eQa3T as eQa3T as a eQa3T L −ieQaˆ Ta −ieQaˆ Ta L −ieQaˆ TL e2 ¼½e2 s ¼ e2 ¼ 1; ð80Þ e 2 s ¼½e 2 s as ¼ e 2 ¼ 1; ð74Þ x1¼0 x3¼0 LY−as and therefore ieQaˆ Ta ieQaˆ Ta L ieQaˆ TL e 4 s ¼½e 4 s as ¼ e 4 ¼ 1; ð81Þ 0 4πn 2πn x2¼0 b2 ¼ ;a2 ¼ ; ð75Þ eQLˆ 3ð1 − asÞ eQTLˆ L and therefore with n; n0 ∈ Z. Note that if we had only quantized the flux 4πn 2πn0 of the magnetic field in the continuum limit, we would have a3 ¼ ;a4 ¼ ; ð82Þ introduced a deviation from periodicity on the lattice of eQTˆ 2L eQTLˆ as Oð L Þ. To avoid such discretization errors one must quan- 0 tize the fields according to Eq. (57), thereby respecting the with n; n ∈ Z. lattice geometry. E. Example V: A space-dependent electric field D. Example IV: A time-dependent electric field and a constant magnetic field Another spin polarizability of the nucleon can be A third spin polarizability of the nucleon can be accessed accessed via the background gauge potential with the following gauge potential, 1 A ¼ðA ; −AÞ¼ 0; a t2;a t; 0 ; ð Þ a μ 0 2 3 4 76 A ¼ðA ; −AÞ¼ − 5 x2; 0; 0;b x ; ð Þ μ 0 2 2 3 1 83 which produces a time-dependent electric field, 9 E ¼ a3txˆ 1 þ a4xˆ 2. This background field generates a which produces a space-dependent electric field, E ¼ a5x2xˆ 2, and a constant magnetic field, B ¼ b3xˆ 2. 9Any time-dependent magnetic field will necessarily generate This background field generates a nonvanishing interaction 1 a time-dependent flux through the transverse plane. This means proportional to 2 σið∇iEj þ ∇jEiÞBj in the nonrelativistic that one cannot implement a periodic background field with the 1 Hamiltonian of the spin-2 hadron in external fields, and so method of this work to access the γM M spin polarizability of 1 1 γ the hadron. This is, up to a numerical factor, the coefficient of the gives access to the M1E2 spin polarizability of the hadron; _ 1 σ · B × B term in the nonrelativistic Hamiltonian of the spin-2 see Ref. [23]. To produce this electric field on a periodic hadron in an external magnetic field. lattice, the nontrivial gauge links to be implemented are 074506-16 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) ð Þ ð Þ i ˆ 2 iq x QCD QCD − eQa5x at E ¼ a e 3 3 xˆ U0 → U0 × e 2 2 ; ð84Þ electric field of the form, e.g., 6 3, from the following gauge potential, ð Þ ð Þ ˆ QCD QCD −ieQb3Lf1;3ðx3Þ×δx ;L−a U1 → U1 × e 1 s ; ð85Þ ia6 iq x Aμ ¼ðA0; −AÞ¼ e 3 3 ; 0; 0; 0 ; ð91Þ ð Þ ð Þ i ˆ 2 QCD QCD eQa5L f2;0ðtÞ×δx ;L−a q3 U2 → U2 × e2 2 s ; ð86Þ – ðQCDÞ ðQCDÞ ˆ with a constant flux through the 0 3 plane. Since the U → U eieQb3x1as : ð Þ 3 3 × 87 form factors in the continuum (infinite-volume) limit are rotationally invariant, the constraints on the components f f The 1;3 and 2;0 functions satisfy recursive relations of the momentum transfer vector in this setup will not f ðx þa Þ¼f ðx Þþa f ðtþa Þ¼f ðtÞþa 1;3 3 s 1;3 3 s and 2;0 t 2;0 t, prevent one from accessing the form factors at any q2 f ðx Þ¼x respectively, with trivial solutions 1;3 3 3 and values. The only limitation on the (magnitude) of the f ðtÞ¼t 2;0 , once the initial values of the functions are transferred momenta may arise from the implementation set to zero. The only QCs are of fields on a periodic lattice as will be deduced below. To produce the electric field chosen above on a TY−at periodic lattice, the nontrivial gauge links to be imple- −ieQaˆ L2a −ieQaˆ L2a T −ieQaˆ L2T e 2 5 t ¼½e 2 5 t at ¼ e 2 5 ¼ 1; ð88Þ mented are t¼0 eQaˆ 6 iq3x3 ðQCDÞ ðQCDÞ − q e at U0 → U0 × e 3 ; ð92Þ LY−as −ieQbˆ La −ieQbˆ La L −ieQbˆ L2 e 3 s ¼½e 3 s as ¼ e 3 ¼ 1; ð Þ 89 eQaˆ ð Þ ð Þ − 6ð1−eiq3LÞf ðtÞ δ x ¼0 QCD QCD q 3;0 × x3;L−as 3 U3 → U3 × e 3 ; ð93Þ and therefore where f3;0 satisfies the recursive relation f3;0ðt þ atÞ¼ f3;0ðtÞþat with trivial solution f3;0ðtÞ¼t, once the 4πn 2πn0 initial value of the function is set to zero. The only a5 ¼ ;b3 ¼ ; ð90Þ eQLˆ 2T eQLˆ 2 QC is T−a 0 Yt eQaˆ h eQaˆ i T 6 iq L 6 iq L a with n; n ∈ Z. − ð1−e 3 Þat − ð1−e 3 Þat t e q3 ¼ e q3 t¼0 F. Example VI: A plane-wave electric field eQaˆ − 6ð1−eiq3LÞT ¼ e q3 ¼ 1: ð94Þ As suggested in Ref. [40], background EM plane waves can be used to calculate the off-forward matrix elements of current operators between hadronic states, For any arbitrary value of the field amplitude parameter, a enabling an extraction of hadronic form factors.10 6, this QC can be satisfied with Additionally, a recent proposal in Ref. [41] demonstrates 2πn the advantage of a plane-wave background field in q ¼ ; ð Þ 3 L 95 evaluating the hadronic vacuum polarization function with lattice QCD. This approach proceeds by con- with n ∈ Z.Thisconstraintonq3 means that the straining the polarization function using the susceptibil- additional Uð1Þ phase factor in Eq. (93) is equal to ities with respect to the background magnetic field unity. It also means that with a background field amplitude at specific momenta. Due to the condition method, the EM form factors can only be accessed at on the spacetime dependence of the flux of the back- quantized values of momentum transfer, the situation ground field in each plane, our periodic implementation which is also encountered when form factors are of EM plane waves will be limited to fields with certain calculated through a direct evaluation of hadronic matrix Fourier modes. For example, an electric field of the form iq·x elements on the lattice. However, one could allow for E ¼ e xˆ 3 with qi ≠ 0 for all i ¼ 1; 2; 3 will generate a nonquantized q3 values by placing conditions on the coordinate-dependent flux. We can however generate an ðrÞ ðiÞ real part, a6 , and imaginary part, a6 ,ofa6. Indeed, by requiring 10In general, any spacetime inhomogeneity in the background field gives access to such off-forward quantities due to injecting energy and/or momentum. Plane-wave background fields have πq n0 ðq LÞ aðiÞ ¼ 3 ;aðrÞ ¼ − sin 3 aðiÞ; ð Þ the advantage of isolating contributions with a given momentum 6 ˆ 6 1 − ðq LÞ 6 96 transfer. eQT cos 3 074506-17 ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) 0 2πn with n ∈ Z and q3 ≠ L for n ∈ Z, the QC in Eq. (94) dependence of the fields, a periodic implementation of 2πn background Uð1Þ gauge fields is possible. is satisfied. As we already saw, for q3 ¼ L the QC in Eq. (94) trivially holds.11 Theresultofthislattercaseis To make the discussions more transparent before getting indeed consistent with the periodic implementation of an into formalities of the general case, we have first presented oscillatory magnetic field (through sin and cos func- the special case of an electric field generated through a tions) in Ref. [41] in which the Fourier modes of the scalar potential with an arbitrary dependence on one spatial Uð1Þ applied field are taken to be quantized. coordinate. The necessary modifications to the naive links adjacent to the boundary of the lattice are obtained by ensuring that the expected values of the elementary pla- V. CONCLUSION quettes are produced throughout the lattice. These expected The introduction of classical electromagnetic fields in values are nothing but the Uð1Þ phases corresponding to the lattice quantum chromodynamics calculations provides a flux of the electric field in the continuum limit through powerful technique to obtain a variety of electromagnetic surface areas at × as at each spacetime point on the lattice, properties of hadrons and nuclei. To extend the utility of where at and as refer to lattice spacings in temporal and this technique, the current implementations of uniform spatial directions, respectively. Additionally, a quantization background fields [23,24,26–37] have been extended in condition is obtained that ensures that the flux of the this paper to the case of background fields that have electric field is quantized. From this special case, two temporal and/or spatial nonuniformities. Such field con- examples of a uniform, and a linearly varying electric field figurations provide access to static and quasistatic proper- in space, are constructed. We have numerically confirmed ties of hadrons and nuclei, such as their higher EM that only the periodic implementation of gauge fields, moments and polarizabilities as well as their charge radii. according to the prescription proposed, gives rise to smooth They also provide means to directly extract EM form correlation functions for neutral pions across the boundary factors as energy and momentum are injected into the of the lattice. hadronic system immersed in appropriate external fields For the general case of gauge fields with arbitrary [40]. Such possibilities can be explored in upcoming lattice spacetime dependences, one can follow the same procedure QCD calculations once the corresponding background as that of the special case above, except that obtaining the fields required for each quantity are consistently imple- modified links near the boundary is more involved. This is mented on the particular lattice geometry. Periodic boun- simply because of the fact that when a component of the dary conditions are widely used in lattice QCD gauge field depends on more than one spacetime coor- calculations, and given that such boundary conditions dinate, fixing the values of plaquettes in one plane to their typically result in a simpler hadronic theory in general desired values (by modifying the links adjacent to the to be matched to lattice QCD calculations, it is important to boundary) can potentially affect the values of the plaquettes perform the background field calculation with the impo- in other planes. By carefully accounting for such possibil- sition of these boundary conditions. In this work, we have ities, we have derived a set of equations for the functions considered the most general spacetime-dependent Uð1Þ that need to be introduced in the modified links and have gauge fields imposed on QCD gauge configurations, and discussed the conditions on classical fields that guarantee have shown that under certain conditions on the spacetime the existence of solutions for these equations. We have further shown that these conditions are equivalent to the 11A plane-wave background gauge field as in Eq. (91) has a statement that the flux of the electric and magnetic fields nonvanishing imaginary piece. It therefore results in a non- through each plane of the lattice must be coordinate Hermitian fermionic determinant (after integrating out the quark independent and quantized. The latter is a condition that fields) which can hinder the probabilistic evaluation of the must be met to ensure the expected value of the plaquette at associated path integral (analogous to the effect of a nonvanishing the far corner of the lattice in each plane is produced. In a chemical potential). For isovector quantities, one can avoid this sign problem imposed by such plane-wave gauge field by only parallel approach, we have shown with details in the implementing the background fields on the valence quarks. For Appendix that these conditions arise from a more general isoscalar quantities, where there are disconnected contributions to class of boundary conditions, namely electro/magneto- the matrix elements of the EM current, contributions from a periodic boundary conditions [13,46–48,51–53,56], where charged quark sea cannot be ignored. For these quantities, where one assumes that the gauge fields are only periodic up to a a full imposition of the background fields on the valence and sea quarks is required, the computational cost associated with a non- gauge transformation. Hermitian measure in the Monte Carlo sampling of the path We have used our general construction to explicitly work integral can be controlled by tuning the amplitude of the gauge out several examples relevant to the extraction of various field to be small. Note that the amplitude of the field can be made EM moments, spin polarizabilities and form factors arbitrarily small (but nonvanishing) only if q3 is quantized according to Eq. (95). Alternatively, one can implement only from future lattice QCD calculations. Within our construc- real oscillatory functions, i.e. sinðq3x3Þ or cosðq3x3Þ, to access tion, time-dependent magnetic fields cannot be studied in γ the desired off-forward matrix elements. this framework, which limits access to the M1M1 spin 074506-18 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) polarizability of the nucleon. However, it is plausible that a periodic implementation of Uð1Þ gauge links on the lattice, more general construction will allow for this polarizability can also be deduced from imposing more general boundary to be extracted. A rather interesting case is a plane-wave conditions, namely electro/magneto-PBCs. These boun- background EM field, which can be devised to be periodic dary conditions require the gauge and matter fields to be on a lattice by properly choosing the Fourier modes of the periodic up to a gauge transformation so that all gauge- fields. Through studying the corresponding QC, it is invariant quantities will be periodic. The electro/magneto- apparent that to require periodicity, either the Fourier PBCs have been introduced, and extensively discussed, by modes in each direction must be quantized for arbitrary ’t Hooft in Refs. [46,50,51] for the case of Abelian and non- values of the field amplitude parameter (as implemented in Abelian gauge theories, and were later adopted to derive the Ref. [41]), or the Fourier modes can be chosen to be QC for the case of uniform background fields implemented arbitrary while the amplitude parameter must be quantized. on a torus [13,47–49,52,53]. Here, we aim to make explicit Our results have applications in upcoming lattice QCD the relation between ’t Hooft’s conditions and those pre- calculations that aim to extract such quantities using the sented in this work.12 In particular, we obtain those electro/ background field method. magneto-PBCs that give rise to the same conditions on the Uð1Þ gauge fields that have been obtained in this paper with ACKNOWLEDGMENTS PBCs. It is shown that the conditions that are placed on the We thank Michael G. Endres for interesting discussions gauge functions when the electro/magneto-PBCs are and for providing the gauge configurations used in some of imposed (see below) are equivalent to the conditions on the numerical investigations performed in this work. We the flux of the EM field when PBCs are imposed. The also thank Brian C. Tiburzi for fruitful correspondence. discussions presented in this section correspond to the Z. D. was supported by the U.S. Department of Energy continuum spacetime and so can only be compared with under Grant Contract No. DE-SC0011090. W. D. was the continuum limit of the QCs presented in Eq. (57) (i.e., the supported by the U.S. Department of Energy Early conditions on the flux of the field strength tensor in the Career Research Award No. DE-SC0010495. continuum limit). With a lattice geometry, one needs to use the results presented in earlier sections of this paper. APPENDIX: ELECTRO/MAGNETO-PERIODIC Consider a gauge field Aμ that depends on all spacetime BOUNDARY CONDITIONS AND THE coordinates, Aμðxμ;xν;xρ;xσÞ, with all indices being dis- ASSOCIATED CONDITIONS ON THE 13 tinct. For the moment, let us assume that only the Aμ BACKGROUND FIELDS component of the gauge field is nonzero. We demand that The QCs obtained in this paper for the parameters of the the field be periodic up to a gauge transformation at the background fields, and the conditions that allowed a boundary. Explicitly, ðμ;μÞ AμðLμ;xν;xρ;xσÞ¼Aμð0;xν;xρ;xσÞþ∂μΩ ðxμ;xν;xρ;xσÞ; ðA1Þ ðμ;νÞ Aμðxμ;Lν;xρ;xσÞ¼Aμðxμ; 0;xρ;xσÞþ∂μΩ ðxμ;xρ;xσÞ; ðA2Þ ðμ;ρÞ Aμðxμ;xν;Lρ;xσÞ¼Aμðxμ;xν; 0;xσÞþ∂μΩ ðxμ;xν;xσÞ; ðA3Þ ðμ;σÞ Aμðxμ;xν;xρ;LσÞ¼Aμðxμ;xν;xρ; 0Þþ∂μΩ ðxμ;xν;xρÞ: ðA4Þ In order for the Ω functions to represent a gauge trans- fields depends on Ω (and not only the derivative of Ω with formation, they must transform the matter field ψ at the respect to xμ), relations (A1)–(A4) do not entirely fix the boundaries as well. Since the transformation of the matter boundary conditions on the matter fields. We now show that the following choice of electro/magneto-PBCs on matter fields, 12For another approach in elucidating the connection between these boundary conditions for the case of a uniform magnetic ψðLμ;xν;xρ;xσÞ¼ψð0;xν;xρ;xσÞ; ðA5Þ field on the lattice, see Ref. [49]. 13The coordinate dependence of the functions in this appendix ieQˆ Ωðμ;νÞðx ;x ;x Þ must not be realized through the floor-function prescription. In ψðxμ;Lν;xρ;xσÞ¼e μ ρ σ ψðxμ; 0;xρ;xσÞ; ðA6Þ fact, all the functions are tilde functions as defined in the main text, but the tilde over functions is dropped to keep the notation ˆ ðμ;ρÞ ψðx ;x ;L ;x Þ¼eieQΩ ðxμ;xν;xσ Þψðx ;x ; 0;x Þ; ð Þ cleaner. μ ν ρ σ μ ν σ A7 074506-19 ZOHREH DAVOUDI AND WILLIAM DETMOLD PHYSICAL REVIEW D 92, 074506 (2015) ieQˆ Ωðμ;σÞðx ;x ;x Þ obtained in this paper by imposing PBCs. First note that the ψðxμ;xν;xρ;LσÞ¼e μ ν ρ ψðxμ;xν;xρ; 0Þ; ðA8Þ transformations in Eqs. (A1)–(A4) and (A5)–(A8) are along with the boundary conditions on the Aμ fields above, consistent with the following solutions for the gauge gives rise to the same QCs on the background fields’ flux as functions, ðμ;μÞ Ω ðxμ;xν;xρ;xσÞ¼xμ½AμðLμ;xν;xρ;xσÞ − Aμð0;xν;xρ;xσÞ ; ðA9Þ Z x ðμ;νÞ μ 0 0 0 Ω ðxμ;xρ;xσÞ¼ dxμ½Aμðxμ;Lν;xρ;xσÞ − Aμðxμ; 0;xρ;xσÞ ; ðA10Þ 0 Z x ðμ;ρÞ μ 0 0 0 Ω ðxμ;xν;xσÞ¼ dxμ½Aμðxμ;xν;Lρ;xσÞ − Aμðxμ;xν; 0;xσÞ ; ðA11Þ 0 Z x ðμ;σÞ μ 0 0 0 Ω ðxμ;xν;xρÞ¼ dxμ½Aμðxμ;xν;xρ;LσÞ − Aμðxμ;xν;xρ; 0Þ : ðA12Þ 0 Now consider the transformation of field ψ at the corner of the lattice in the μ − ν plane, i.e., at xμ ¼ Lμ and xν ¼ Lν.One may first transform the ψ field with the Ωðμ;μÞ function and then with the Ωðμ;νÞ function, ieQˆ Ωðμ;νÞð0;x ;x Þ ψðLμ;Lν;xρ;xσÞ¼e ρ σ ψð0; 0;xρ;xσÞ; ðA13Þ or the other way around, ieQˆ Ωðμ;νÞðL ;x ;x Þ ψðLμ;Lν;xρ;xσÞ¼e μ ρ σ ψð0; 0;xρ;xσÞ: ðA14Þ These two relations are compatible if ˆ ðμ;νÞ ðμ;νÞ eieQ½Ω ð0;xρ;xσ Þ−Ω ðLμ;xρ;xσ Þ ¼ 1; ðA15Þ ðμ;νÞ which can only hold in general if Ω is independent of the xρ and xσ coordinates. From Eqs. (A1) and (A2), these ðμ;νÞ conditions should hold if Aμ is independent of these coordinates. Now given that from Eq. (A10), Ω ð0;xρ;xσÞ¼0 and ðμ;νÞ Ω ðLμ;xρ;xσÞ is the total flux of the EM field through the μ − ν plane on the lattice (recalling the assumption that only Aμ is nonvanishing), the condition in Eq. (A15) is identical to the continuum QC as obtained in Sec. III, Z Lμ 2πn dxμ½Aμðxμ;LνÞ − Aμðxμ; 0Þ ¼ ;n∈ Z: ðA16Þ 0 eQˆ It might seem that by alternatively considering the gauge-transformed field at the corner of the lattice in any other plane, one could arrive at the same condition which would be eliminating any spacetime dependence of the gauge field. However, it is easy to see that this is not the case. For example, by considering the gauge-transformed matter field in the corner of the lattice in the ρ − σ plane one arrives at ieQˆ Ωðμ;σÞðx ;x ;0Þ ieQˆ Ωðμ;ρÞðx ;x ;L Þ ψðxμ;xν;Lρ;LσÞ¼e μ ν e μ ν σ ψðxμ;xν; 0; 0Þ; ðA17Þ as well as ieQˆ Ωðμ;ρÞðx ;x ;0Þ ieQˆ Ωðμ;σÞðx ;x ;L Þ ψðxμ;xν;Lρ;LσÞ¼e μ ν e μ ν ρ ψðxμ;xν; 0; 0Þ; ðA18Þ but it is not hard to see from Eqs. (A11) and (A12) that ðμ;σÞ ðμ;ρÞ ðμ;ρÞ ðμ;σÞ Ω ðxμ;xν; 0ÞþΩ ðxμ;xν;LσÞ¼Ω ðxμ;xν; 0ÞþΩ ðxμ;xν;LρÞ: ðA19Þ 074506-20 IMPLEMENTATION OF GENERAL BACKGROUND … PHYSICAL REVIEW D 92, 074506 (2015) Z ψðx ;x ;L ;L Þ Lμ As a result, the two transformations for μ ν ρ σ dx ½A ðx ;L Þ − A ðx ; 0Þ are identical and no extra conditions must be placed on the μ μ μ ν μ μ 0 Z xμ and xν dependences of the Aμ field. Lν 2πn − dxν½AνðLμ;xνÞ − Aνð0;xνÞ ¼ ;n∈ Z; Now let us assume that a second component of the 0 eQˆ gauge field, Aν, is also nonvanishing. Then the same ð Þ equations as those in (A1)–(A3) and (A9)–(A12) can be A25 A written for the gauge-transformed ν field at the boun- conditions that were all deduced in Sec. III by imposing daries, and for the corresponding solutions for the gauge PBCs. In general, if any of the Aμ component of the gauge functions, respectively. The transformations of the ψ field field with μ ¼ 0; 1; 2; 3 is nonvanishing, the compatibility turn into relations similar to the ones considered above must be ˆ ðν;μÞ studied at the corner of the lattice in all planes, and they are ψðL ;x ;x ;x Þ¼eieQΩ ðxν;xρ;xσ Þψð0;x ;x ;x Þ; μ ν ρ σ ν ρ σ all satisfied if the flux of the field strength tensor through ðA20Þ each plane is coordinate independent and quantized. We conclude this appendix with a remark. Note that the ieQˆ Ωðμ;νÞðx ;x ;x Þ ψðxμ;Lν;xρ;xσÞ¼e μ ρ σ ψðxμ; 0;xρ;xσÞ; boundary conditions in this section are defined by only L μ ð Þ identifying the point μ with the point 0 in the direction, and A21 similarly in other directions. If one imposes more restrictive conditions such that all points xμ þ Lμ are identified with ψðxμ;xν;Lρ;xσÞ points xμ, the QCs derived in this section for the flux of the ieQˆ Ωðν;ρÞðx ;x ;x Þ ieQˆ Ωðμ;ρÞðx ;x ;x Þ ¼ e μ ν σ e μ ν σ ψðxμ;xν; 0;xσÞ; background field through the μ − ν plane will depend, in general, on the xμ and xν coordinates and cannot be quantized. ðA22Þ This is in contrast with the case of uniform background fields ψðx ;x ;x ;L Þ where both the PBCs and electro/magneto-PBCs can be μ ν ρ σ satisfied with a single space/time-independent QC. Such a ieQˆ Ωðν;σÞðx ;x ;x Þ ieQˆ Ωðμ;σÞðx ;x ;x Þ ¼ e μ ν ρ e μ ν ρ ψðxμ;xν;xρ; 0Þ; system possesses a translational invariance in units of Lμ for μ ðA23Þ all (corresponding to the magnetic translational group in the case of a uniform magnetic field; see Ref. [48]). With a with our choice of boundary conditions on the matter background EM field that is spacetime dependent, no trans- field. The same argument discussed above based on the lational invariance exists prior to imposing the boundary compatibility of the gauge transformations of the matter conditions. Therefore, the lack of discrete translational L fields at the corner of the lattice in each plane now leads invariance in units of μ should not come as a surprise. us to the condition Given our choice of periodifying the functions with the use of the floor function, we have however explicitly built up such ˆ ðμ;νÞ ðμ;νÞ ˆ ðν;μÞ ðν;μÞ e−ieQ½Ω ð0;xρ;xσ Þ−Ω ðLμ;xρ;xσÞ eieQ½Ω ð0;xρ;xσ Þ−Ω ðLν;xρ;xσÞ translational invariance in the setup presented in this paper. One may however wonder whether this choice can be ¼ 1; ðA24Þ distinguished from a choice for which only the point Lμ is identified with the point 0 on the lattice. This answer is that which requires both the Aμ and Aν fields to be with a lattice action that at most depends on the first derivative independent of the xρ and xσ coordinates. Additionally, of the gauge and matter fields, these two choices are identical it is easy to see that this condition implies that the at the practical level. 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