Large-scale quantum computers can simulate nonperturbative quantum field theories which are intractable classically [1]. Alas, noisy intermediate-scale quantum (NISQ) era systems will be limited both in qubits and circuit depths. Whether any gauge theory simulations in this period are possible depends upon efficient formulations. The situation is similar to the early days of lattice field theory when computer memory was limited and the cost of storing SU(3) elements was prohibitive.

For fermionic fields, relatively efficient mappings to quantum registers are known [2–5] evidenced by most existing quantum calculations being fermionic [6–9]. The bosonic nature of gauge fields preclude exact mappings, but many proposals exist with different costs [10–28]. Digitizing reduces symmetries—either explicitly or through finite-truncations [10]. These breakings mean a priori the original model may not be recovered in the continuum limit [29–34]. Further, choices of digitization may limit the use of classical resources for Euclidean simulations or state preparation [35]. In summary, the understanding of resource costs, systematic errors, and the continuum limit for these proposals is poorly known today.

In this work, we systematize the proposal of replacing continuous gauge groups G by their discrete subgroups H [11,28] by deriving lattice actions using the group space decimation procedure of [36,37]. After deriving the general third order action, we will investigate the behavior of discretizing three distinct gauge groups U(1), SU(2), and SU(3). We begin by reviewing the discrete subgroup approximation in Sec. II. In Sec. III we discuss the general aspects of the group space decimation procedure. Following that, in Sec. IV we derive the decimated action up to 3rd order. Numerical results for V using the decimated actions are presented in Sec. V. Section VI studies the continuous group limit, and we conclude in Sec. VII.

Approximating gauge theories by replacing G→H was explored in the early days of Euclidean lattice field theory. The viability of the Zn subgroups replacing U(1) were studied in [38,39]. Further studies of the crystal-like discrete subgroups of SU(N) were performed [40–42], including with fermions [43,44]. These studies met with mixed success depending on the group and action tested.

The fundamental issue of group discretization can be understood by considering the Wilson gauge action S[U]=-∑pβNReTr(Up),(1)where Up indicates a plaquette of continuous group gauge links U (for discrete groups, up denotes plaquettes and u denotes links). As β→∞, links near the group identity 1 dominate, i.e., U≈1+ϵ, where ϵ can be arbitrarily small. Therefore the gap ΔS=S[1+ϵ]-S[1] goes to zero smoothly. For discrete groups, ϵ has a minimum given by the nearest elements N to 1, and thus ΔS=S[N]-S[1]>0. This strongly suggests a phase transition at some critical βf=c/ΔS, where c≈O(1) depends on spacetime dimensionality, gauge group, and entropy. For U(1)→Zn in 4d, βf=0.781-cos(2π/n) [41]. Above βf, all field configurations but u=1 are exponentially suppressed. Thus, H fails to approximate G for β>βf. Another way to understand this behavior follows [45], where some discrete theories are shown equivalent to continuous groups coupled to a Higgs field. The Higgs mechanism introduces a new phase missing from the continuous gauge theory when β→∞.

Both arguments suggest H be viewed as an effective field theory for G with a UV-cutoff at Λf. Provided the typical separation of scales of physics mIR≪Λf, the approximation could be reliable up to O(mIR/Λf) effects.

In lattice calculations, one replaces Λf by a fixed lattice spacing a=a(β) which shrinks as β→∞ for asymptotically free theories. To control errors when extrapolating to a→0, one should simulate in the scaling regime of a≪mIR-1. We denote the onset of the scaling regime by as, and βs. For af(βf)∼Λf-1, errors from the discrete group approximation would be small if a can be reduced such that mIR-1≫a≳af i.e., βs≤βf.

In the case of U(1) with βs=1, Zn>5 satisfies βf>βs. For non-Abelian groups, only a finite set of crystal-like subgroups exist. SU(2) has three: the binary tetrahedral BT, the binary octahedral BO, and the binary icosahedral BI. While BT has βf=2.24(8), BO and BI have βf=3.26(8) and βf=5.82(8) respectively [28], above βs=2.2. Hence, BO and BI appear useful for SU(2).

For the important case of SU(3) with βs=6, there are five crystal-like subgroups with the Valentiner group V with 1080 elements.¹

This name is most common in the mathematical literature [46,47]. It has also referred to as S(1080) [28,36,37,42] or Σ3×360 [48].

To decrease af, adding additional terms to Eq. (1) was attempted [28,36,37,39,42,52–55], although only in [28,42] were Monte Carlo calculations undertaken for SU(3). Two reasons suggest this would help. First, additional terms which have a continuum limit ∝ReTrFμνFμν, but take different values on the element of H (e.g., |Tr(up)|2-1), change ΔS and thus af. Second, new terms can reduce finite-a errors as in Symanzik improvement.

The term usually added was the adjoint trace, giving S[u]=-∑p(β{1}3ReTr(up)+β{1,-1}8|Tr(up)|2),(2)where up∈V, and the first term is normalized so for β{1,-1}=0, the S[u] matches the Wilson action (with β{1}=β). In these works, no relationship was assumed between β{1} and β{1,-1}. That Eq. (2) improves the viability of V over the Eq. (1) will be shown in [56]. For a different action, S[u]=-∑p(β03ReTr(up)+β1ReTr(up2)),(3)smaller values of af were demonstrated in [28].

With these actions, the dimensionless product Tct0 of the pseudocritical temperature and the Wilson flow parameter were found to agree in the continuum with SU(3), allowing one to set the scale of those calculations. a>0.08  fm was achieved without the effects of af being seen. This suggest that V can reproduce SU(3) in the scaling region with a modified action, such that practical quantum computations of SU(3) could be performed. While promising, the choice of new terms was ad hoc and left unclear how to systematically improve or analyze effectiveness. In the next section, we systematically derive lattice actions for H, discovering that the terms added in these two actions are in fact the first terms generated.

Our ultimate goal is to approximate the path integral of group G faithfully by a discrete subgroup H by replacing the integration over G by a summation over H. Group space decimation can be understood in analogy to Wilsonian renormalization, where we integrate out continuous field fluctuations instead of UV modes. The typical method used with discrete subgroup approximations is to replace the gauge links U∈G by u∈H such that the action S[U]→S[u]. This corresponds to simply regularizing a field theory. For strong coupling, this appears sufficient. As β→∞, correlations between gauge links increase and the average field fluctuation becomes smaller. When the average field fluctuations decrease below the distance between 1 and N of the discrete group, freeze-out occurs and the approximation breaks down–similar to probing a regulated theory too close to the cutoff. Therefore, improving this approximation and understand the systematics can be done by considering these discarded continuous field fluctuations. To do this, instead of performing the replacement U→u, we will integrate out the continuous fluctuations, following the decimation formalism developed by Flyvbjerg [36,37]. He derived the second order decimated action for U(1), SU(2), and SU(3). An important general feature of the decimated action though is missing from this second order action—while new terms are generated at each order, until third order no coefficient of an existing term is modified. One expects such terms are critical to understanding deviations from the continuous group and therefore we compute them in Sec. IV.

It is natural to associate every subgroup element u∈H with an unique set, or region, Ωu containing all closest continuous group elements U∈G: Ωu≡{U∈G|d(U,u)<d(U,u′), ∀ u′∈H\{u}},(4)where the distance is defined as d2(U,u)=Tr((U-u)†⁢(U-u)). By such a definition. the continuous group is fully covered, i.e., G=∪u∈HΩu and a graphical demonstration of Ω≡Ω1 can be found in Fig. 1. Note that for any U∈G, there exist a unique u∈H and ε∈Ω such that U=uε, where we may treat ε as the error of u approximating U. In this way, without approximation, the Euclidean path integral integrating over G can be written as a summation over H and integration over ε∈Ω: Z=∫GDUe-S[U]=∑u∈H∫ΩDεe-S[u,ε],(5)where Z is a functional integral over all gauge links U on the lattice, or equivalently a functional integral over ε and a functional sum over u. In this expression, S[u,ε]=S[U] is defined by replacing each gauge link U by uε.

We then expand the exponential in the path integral and integrate over ε producing a moment expansion Z=∑u∈H∫ΩDε(1-βS[u,ε]+β22!S[u,ε]2+⋯)=∑u∈H(1-β⟨S[u,ε]⟩+β22!⟨S[u,ε]2⟩+⋯),(6)where we have introduced the notation ⟨f⟩=∫ΩDεf with normalization ∫ΩDε=1. What we are really after is an expansion for the action S[U], writing Z in terms of a cumulant expansion Z=∑u∈Hexp(-∑n=1∞βnn!𝒮n[u]),(7)allows us to match Eq. (6) with (7) to obtain an effective action. In this way, after integrating over ε, the contributions to the action depend only on the discrete group gauge link u and the effective action can be defined as S[u]≡∑nβnn!𝒮n[u].(8)Up to O(β3) one has 𝒮1[u]=⟨S[u,ε]⟩,(9)𝒮2[u]=-⟨S[u,ε]2⟩+⟨S[u,ε]⟩2,(10)𝒮3[u]=⟨S[u,ε]3⟩-3⟨S[u,ε]⟩⟨S[u,ε]2⟩+2⟨S[u,ε]⟩3.(11)One may worry about poor convergence in the region of interest β≥βs≥1. As will be discussed more thoroughly in Sec. VI, βn terms are suppressed by powers of the average field fluctuation. Thus, the size of the discrete group, which determines the size of field fluctuations integrated out, also determines the series convergence.

Starting with the second order terms computed in Refs. [36,37], the decimated action generates multiplaquette contributions. Their inclusion in quantum simulations brings substantial nonlocality which requires high qubit connectivity and increases circuit depth. Luckily these contributions will be shown to be small in Sec. IV.

In the following section, we will calculate Eq. (9) to (11) in terms of linear combination of the group characters starting from the Wilson action of Eq. (1): S[U]≡-∑pβNReTr(Up)=-∑pβNReχ{1}.(12)Here we introduced χr, the character of the group representation²

Through out this work we suppress the argument of χr, but it can only be Up or up and context makes it clear which is meant.

In deriving the decimated action, integrating out the field fluctuations require us to reduce expressions of the form ⟨εi1j1⋯εinjn⟩. To simplify these, we use an identity derived in [57] for SU(N) and U(N) groups for any integer n≤N. The necessary relations for n≤3 are found in Appendix A. From these identities, we are left with expectation values of χr over Ω Vr≡1dr⟨Reχr⟩,(13)where dr is the dimension of representation r.

For U(1)→Zn, there is only one representation at each order of the cumulant expansion, V{h}=⟨εh⟩. These terms can be computed analytically by a change of variables ε=eiϕ [37]: V{h}=1V0∫-πnπndϕeiϕh=nπhsin(πhn)(14)with h=1,2,… being integers and the normalization constant V0=∫Ωdεε0=2πn.

Extending this to non-Abelian groups, e.g., SU(N), Ω becomes a high-dimensional polytope in SU(N) space. In [36], the Vr for BI and V were computed up to second order by approximating these polytopes with hyperspheres to two significant figures. It is crucial to remove these approximations for our purpose because the uncertainty δVr∼O(1%) is magnified in the coupling constants of the decimated action. These couplings are combinations of powers of Vr with extreme cancellations making the fraction errors grow rapidly. Hence we avoid the hypersphere approximation and numerically compute all the Vr necessary for the 3rd order actions to O(0.1%). (Results found in Table I.)

In this section, we summarize the derivation of the decimated action order-by-order. Further details can be found in Appendix C. The first order is relatively straightforward, and only contains a single plaquette term. Working from Eq. (9) β𝒮1[u]=-βN⟨ReTr(u1ε1u2ε2(u3ε3)†(u4ε4)†)⟩=-βNRe(u1abu2cdu3ef†u4gh†⟨ε1bc⟩⟨ε2de⟩⁢⟨ε3fg†⟩⟨ε4ha†⟩).(15)After applying Eq. (A2), 𝒮1[u] depends only on u: β𝒮1[u]=-V{1}4βNRe(u1abu2cdu3†efu4†gh)δbcδdeδfgδha=-V{1}4βNReχ{1}≡-β{1}(1)1NReχ{1},(16)where βr(n) is the nth order term in front of 1drReχr.

It is comforting that at O(β), no new terms are generated in S[u]. This allows for rescaling β{1}(1)→β, recovering the procedure of directly replacing U→u in the Wilson action. Although this rescaling is permitted, V{1}<1 contains content about the approximation G→H. As the number of elements of H increases, Ω shrinks and V{1}→1. This means V{1} quantifies how densely H covers G and thus the minimal fluctuation size. Since β{1}(1)=V{1}4β, decreases in V{1} signals the poorness of using Eq. (16) alone. This is discussed further in Sec. VI.

We now proceed to calculate the second order decimated action while fixing a few typos in [36] along the way. The second order decimated action 𝒮2[u]=-⟨S[u,ε]2⟩+⟨S[u,ε]⟩2 depends upon two plaquettes Up=U1U2U3†U4† and Uq=U5U6U7†U8†. A natural decomposition of 𝒮2[u] can be made into three terms based on how the two plaquettes p and q are related: p=q (one-plaquette contribution), p∩q=1-link (two-plaquette contribution), and p∩q=0-links. To all orders, the p∩q=0 contributions to the decimated action vanish. For the case of p=q, we conclude that it is 12!β2𝒮2[u]1p=-β{0}(2)-β{2}(2)2N(N+1)Reχ{2}-β{1,1}(2)2N(N-1)Reχ{1,1}-β{1,-1}(2)1N2-1χ{1,-1},(17)where the βr(2) can be found in Table II.

Next, we calculate the case of p∩q=1-link for the second order decimation. Contracting the δ’s in Eqs. (A3) and (A4), where unlike Eq. (C1), we only identify one link as the same between the two plaquettes. This leads to the following expression, 12!β2𝒮2[u]2p=-β{2r}1NRe[χ{1}(up)]1NRe[χ{1}(uq)]-β{2i}1NIm[χ{1}(up)]1NIm[χ{1}(uq)]-β{2t}1NRe[χ{1}(up*q†)]-β{2u}1NRe[χ{1}(up*q)],(18)where we have used the fact that all the Vr’s are real due to our choice of the integration region. The explicit expressions for the couplings are found in Table II. Note that this expression is also applicable to U(1).

We would now like to comment on how the two-plaquette—and general multiplaquette—terms contributes to the S[u]. It would be desirable if these terms could be neglected, because they require substantial quantum resources. By inspecting Table III, one observes that the two-plaquette βr are O(0.1) or smaller than the single-plaquette terms. The largest coupling, β2i, multiples a term Imχ1Imχ1≈0. Strong cancellations are expected from correlations between the remaining terms (shown in Fig. 2) as evident by the observation β2t≈-β2u.

It is reasonable to expect these individual reasons to persist at higher orders, suggesting that at a fixed order all multiplaquette terms can be neglected compared to their 1—plaquette counterpart. But can we argue that the multi-plaquette terms generated at order O(βn-1) are still negligible when the O(βn) contribution is introduced? To do this, we look at the continuum limit of each term being introduced. In this way, we recognize that the two-plaquette terms are related to the Lüscher-Weisz action [59]. k-plaquette terms corresponds to applying 2k-2 derivatives to a4⟨TrFF⟩ and are thus O(a2k+2). Here F is the field strength tensor. Combining this with the observation that for a coupling βj generated at O(βn) has the scaling βj≈10-nβn, we estimate that ⟨𝒮mk-plaq[u]⟩⟨𝒮n1 plaq[u]⟩≈(10β)n-ma2k+2⟨D2k-2(TrFF)⟩a4⟨TrFF⟩,(19)where D is a covariant derivative projected onto the lattice directions. The combination of higher powers of a and the associated expectation values of higher-dimensional operators should be sufficient to suppress the mild β enhancement for n>m, at least for O(β3) S[u]. For these reasons, we will neglect higher order multiplaquette terms.

For the third-order terms of Eq. (11), we, therefore, only focus on the case where three plaquettes are identical. Combining Eqs. (C3), (C4), and (C5) we arrive at the third order contribution to the single-plaquette decimated action β33!𝒮3[u]=-β{3}(3)d{3}Reχ{3}-β{2,1}(3)d{2,1}Reχ{2,1}-β{1,1,1}(3)d{1,1,1}Reχ{1,1,1}-β{2,-1}(3)d{2,-1}Reχ{2,-1}-β{1,1,-1}(3)d{1,1,-1}Reχ{1,1,-1}-β{1}(3)d{1}Reχ{1},(20)where the overall factor of 1/3! has been absorbed into the definition of βr(3). Note that, unlike the second order results where only certain decimation programs generate renormalization for existing terms, the third order S3[u] introduces corrections to Reχ1 for all G→H. Additionally, a number of the specific group identities in Eqs. (B1)–(B3) also lead to renormalization.

Putting together Eqs. (16), (17), and (20), the single-plaquette decimated action of Eq. (7) to O(β3) for a general gauge group is, S[u]=∑p-(β{1}(1)+β{1}(3))1NRe(χ{1})-β{0}(2)-β{2}(2)2N(N+1)Reχ{2}-β{1,1}(2)2N(N-1)Reχ{1,1}-β{1,-1}(2)1N2-1χ{1,-1}-β{3}(3)6N(N+1)(N+2)Reχ{3}-β{2,1}(3)3N(N2-1)Reχ{2,1}-β{1,1,1}(3)6N(N-1)(N-2)Reχ{1,1,1}-β{2,-1}(3)2N(N-1)(N+2)Reχ{2,-1}-β{1,1,-1}(3)2N(N+1)(N-2)Reχ{1,1,-1},(21)where βr are in Table II. Note that this S[u] is correct for any G→H. Referring to Eqs. (B1), (B2), and (B3), for a given G simplifications occur. For SU(3), with βr≡∑n1n!βr(n),(22)this corresponds to: S[u]=∑p-(β{1}+β{1,1})13Reχ{1}-(β{0}+β{1,1,1})-(β{2}+β{1,1,-1})16Reχ{2}-(β{1,-1}+β{2,1})18χ{1,-1}-β{3}10Reχ{3}-β{2,-1}15Reχ{2,-1}.(23)For U(1) and SU(2), we refer the reader to Appendix B.

As a demonstration, we simulated Eq. (23) to each order in β for SU(3)→V. For these computations 102 configurations separated by 103 sweeps were collected on a 44 lattice and plotted in Fig. 3. In the figure, we compare the average energy per plaquette ⟨E0⟩ versus the coupling, β{1} as defined in Eq. (22), which multiplies the Wilson term Reχ1. For SU(3), this corresponds to β, and β{1}+β{1,1} for V.

Naively, ⟨E0⟩ is monotonic in a. Including O(β2) terms, we observe a clear reduction in ⟨E0⟩ and thus an improvement over the Wilson action for small a. This suggests promise in this systematic approach. As will be discussed in Sec. VI, it also supports the effectiveness of the previously studied ad hoc actions. Instead of freezing out, the theory approaches a nonzero value of ⟨E0⟩.

At O(β3), ⟨E0⟩ displays nonmonotonic behavior due to the negative coefficient of χ1 at β3. This suggests the higher order terms (4th order and beyond) required to match Eq. (6) and (7) are dominating the action.

Taken together, the O(βn) results suggests that the decimation procedure, formulated as a strong coupling expansion, converges to the continuous theory slowly. Sufficiently large order calculations would suppress the higher order contributions in the form βn/n! for a reasonable range of β. While an O(β4) calculation would undoubtedly be insightful to understanding this convergence, other approaches such as introducing “counterterms” to absorb some higher-order contributions, or instead using a character expansion may prove fruitful. It also may prove useful to take the expansion in Eq. (7) as an effective theory where each character is subject to field redefinitions. While these possibilities are interesting to investigate, they are certainly beyond the scope of this work. We therefore leave them for future studies.

With Eq. (21), it is possible for us to investigate systematically the effect of replacing the continuous group by its finite subgroup. In order to proceed, it is useful to introduce a new parameter which approximately represents the field fluctuations. To do this, consider the representation of a continuous group lattice gauge link in terms of the corresponding generators λa in the adjoint representation, U=eiλaAa, where a summation over color indices a is implied. In this form, we see that the gauge fields correspond to amplitudes in each of the generators. For ε∈Ω, inserting its small parameter expansion ε≈1+iλaAa-12(λaAa)2+⋯ into Eq. (13) gives Vr≈1-∫ΩDA(cr(2)∑aAa2+⋯),(24)where DA is a measure over all Aa which respects gauge symmetry and cr(n) are representation and group-dependent constants. From this, we see that as the subgroup H incorporates more elements, the size of Ω approaches 0 and Vr→1 from below. This means that for finite Ω the domain size of Aa that gives rise to Ω is an indicator for deviations from G of H. Flyvbjerg defines a parameter R as the radius of a hypersphere with equal volume to Ω to get a handle on the domain of Aa. This allows him to approximate Vr analytically [36,37]. Here, we can use this idea to roughly understand the scaling of Vr.

For U(1)→Zn, the hypersphere is exactly Ω and R cleanly defines ε≤R=π/n. Beyond U(1), the connection between Ω and a single value of R is complicated because the Ω of H form polytopes in the hypervolume of their continuous partner (see Fig. 1 for a clear demonstration). In this case, while one could take Ω to be contained by a hypersphere centered at 1 whose boundary incorporates elements of the nearest neighbors of 1 in H, making some element of the hypersphere not included in Ω. On the other hand, there exists a largest hypersphere centered at 1 that only contains elements in Ω. In this way, we define an upper and lower bound for R. Note, this is different from [36,37] where the polytopes of H were always approximated by hyperspheres with definite radii. For SU(2) with BI, we find 0.09≤R2≤0.15 which can be compared to Rsphere2=0.12 of [36,37]. In the case of SU(3) with V, 0.42≤R2≤0.93 compared to Rsphere2=0.62.

While superficially the cumulant expansion has appeared as a strong-coupling expansion in β, the actual behavior is controlled by both β and R with R controlling Vr. As pointed out in [36,37], the leading order behavior for small R for a given power of βα (α>0) is actually O([βR2]αR-2). Therefore one would predict that the relative smallness of R2 for BI compare to V signals that βf should be larger for BI which is indeed the case.

For subgroups of SU(3), this scaling behavior becomes unsatisfactory because R2∼1. It is possible to study this breakdown in U(1)→Zn where the systematic effect of decimation can be studied in detail both because errors can be made arbitrarily small for large n and because Vr and βr are known analytically. In terms of R, one can expand the βr for the U(1) action of Eq. (B8) to find: β{0}≈(R23-19R490+⋯)β2,(25)β{1}+β{1,1,-1}≈(1-2R23+R45+⋯)β+(-17R490+311R6945+⋯)β3,(26)β{2}≈(-R23+53R490+⋯)β2,(27)β{3}≈(17R490-1609R62835+46303R856700-77603R10103950+⋯)β3.(28)The first thing to note is that the O([βR2]αR-2) scaling found in [36,37] continues to the third order. One might be tempted to use this leading behavior to estimate the βf or the radius of convergence of this series, but this would be incorrect. Instead, it behooves one to note that for both 2nd and 3rd order contributions, the subleading terms [R2]kR-2 with k>α initially grow until a 1/k! factor dominates over all the other factors.

But what is the origin of this behavior? For simplicity, we can understand this behavior by considering the expansion of Vj4m which form βr. The specific combination of Vj4m dictated by the cumulant expansion ensures that orders lower than O([βR2]αR-2) cancel in βr. The j representation contributes to β{r1,…,rk} in the form of V{j11,…,jl1}4m1⋯V{j1k,…,jnk}4mk under the constraint |r|=mij˜i where |r|=|r1|+⋯+|rk| and j˜i=|j1i|+⋯+|jli|. One might worry that studying the expansion of Vj4m is not representative, but one can verify that the scaling behavior observed below persists in βr, although the numerical factors become cumbersome. For Vj4m≡V{j1,…,jl}4m, j˜=|j1|+⋯|jl|, we have Vj4m≈1-23m(j˜R)2+145(10m2-m)(j˜R)4+O(m3[j˜R]6)(29)from which, we see that the coefficients of the [R2]kR-2 contributions to βr are accompanied by a factor ∝1k!mk-1j˜2k-2. While the factorials ensure the series converges, βr for higher representations r have larger m, j˜, or both leading to higher order terms in the expansion being large for moderate R. This helps explaining why Z4 with the Wilson action fails to replicate U(1) substantially above β=1—while the naive scaling would suggest R≲βk1-k would be enough to suppress higher representations, in reality a stronger bound of max{1k!mk-1(j˜R)2k-2}1≤j˜≤|r|≲1 for ∀βr is required for all subleading terms to be small. Considering the range of m with fixed |r|,j˜, the bound is strictest when |r|=j˜ yielding R≲1/|r|3/2 in order for the lowest order contribution to dominate such that R≲βα1-α provides a reasonable estimate for the range of β where the decimated action provides a reasonable approximation for its continuous partner. While these conditions are satisfied for BI, they are violated for V in which case the dominant term in the R expansion isn’t clear.

Another feature observed in the R expansion of the Zn group is that because Vr∝sinrR, the sign of the O([rR]k) terms oscillate, and therefore the sign of βr(n) can depend sensitively on R. Since Reχr(1)>Reχr(N), where N is the nearest neighbors of 1 in H (see Fig. 1), the overall sign of βr(n) determines whether or not the rth term in the action enters the frozen phase in the limit of β→∞. This behavior is observed in Fig. 6 where β{1}(1,2)>0 but β{1}(3)<0.

From the behavior observed in U(1), we can improve the quantitative understanding of how well H can approximate G, even when βr are not known analytically. Clearly, Vr→1 indicates that the R→0, and in that limit the two actions would agree. Therefore, the difference between the two actions SG-SH≈βχ{1}(U)-β{1}χ{1}(u)≈(1-V{1}4)βχ{1}(u) serves as an indicator of βf.