A real-space renormalization-group calculation for the quantum Z_2 gauge theory on a square lattice
AA real-space renormalization-group calculation for thequantum Z gauge theory on a square lattice Steve T. Paik ∗ Santa Monica College,Santa Monica, CA 90405 (Dated: September 16, 2020)We revisit Fradkin and Raby’s real-space renormalization-group method to study the quantum Z gauge theory defined on links forming a two-dimensional square lattice. Following an old suggestionof theirs, a systematic perturbation expansion developed by Hirsch and Mazenko is used to improvethe algorithm to second order in an intercell coupling, thereby incorporating the effects of discardedhigher energy states. A careful derivation of gauge-invariant effective operators is presented in theHamiltonian formalism. Renormalization group equations are analyzed near the nontrivial fixedpoint, reaffirming old work by Hirsch on the dual transverse field Ising model. In addition torecovering Hirsch’s previous findings, critical exponents for the scaling of the spatial correlationlength and energy gap in the electric free phase are compared. Unfortunately, their agreement ispoor. The leading singular behavior of the ground state energy density is examined near the criticalpoint: we compute both a critical exponent and estimate a critical amplitude ratio. I. INTRODUCTION
We study the quantum Hamiltonian for a two-dimensional Z gauge theory on a square lattice us-ing a real-space renormalization-group method. Themethod, due to Fradkin and Raby, is a gauge-invariance-preserving block-spin algorithm with length rescaling fac-tor two. A variational approximation is made for theground state of the theory and the Hilbert space isthinned so that low-energy states and long-distance cor-relations are preserved. Despite the crudeness of thetruncation, we demonstrate, without recourse to dual-ity, that spatial correlations decay exponentially in theelectric free phase.The quantum Hamiltonian may be obtained from clas-sical statistical mechanics by starting with a euclideanthree-dimensional Z gauge theory on a lattice withanisotropic couplings β t , along a particular direction cho-sen as “time,” and β s in the orthogonal directions. Whenthe partition function is expressed in terms of a trans-fer operator, a special limit exists in which the Trotterproduct formula allows for the transfer operator to be ex-pressed as the exponential of some Hamiltonian H . Thisis an infinite-volume limit that is also highly anisotropicand requires β t → ∞ and β s → β s e β t re-mains a fixed and arbitrary dimensionless coupling. At each link l there exists spin- operators (cid:126)σ l obeyingthe Pauli algebra. Operators belonging to different linkscommute. The Hamiltonian is H = − h (cid:88) l σ zl − J (cid:88) p Φ p , (1)where σ zl measures the discrete electric flux running alonglink l , and Φ p = (cid:81) l ∈ ∂p σ xl measures the discrete magneticflux through plaquette p .Local gauge transformations are defined by operatorsassociated to the sites or vertices between links. At such order in Hirsch–Mazenkoperturbation theory ν t ν s α first (Refs. 3 and 4) 0.62 1.20 − . ν t ), spatial correlation length ( ν s ), and singularity in theground state energy ( α ) in the Hirsch–Mazenko perturbationexpansion. a site (cid:126)r in the lattice the generator is G (cid:126)r = (cid:89) links l emergingfrom (cid:126)r σ zl . (2) G (cid:126)r commutes with H .The renormalization-group transformation developedby Fradkin and Raby fixed an important shortcomingof previous real-space schemes. Although blocking spinoperators into cells makes a variational approximation tothe lattice ground state analytically tractable, such ap-proximations are engineered to preserve the low-energyspectrum without regard for spatial correlations. How-ever, in the Hamiltonian formalism, time and space aretreated on very different footings, so one generally needsto ensure that both dimensions scale equally under therenormalization transformation. Consequently, while thegap is well-approximated, (equal-time) correlations ex-hibit qualitatively incorrect behavior such as power-lawdecay away from criticality. Fradkin and Raby foundthat such long-range correlations may be suppressed inthe ground state by designing the neighboring block spinsto share boundary conditions. This prevents cells frombeing disconnected because the magnetic flux of one cellis not entirely independent of the flux of its nearest neigh-bor. They proved that correlation functions decay expo-nentially in the disordered phase of the one-dimensional a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p transverse field Ising chain. Since the same conclusionholds for the two-dimensional Ising model in a transversefield, Fradkin and Raby invoke duality to argue that itmust be true for the Z gauge theory in the electric freephase. We shall, in fact, demonstrate this directly usingthe transformation relations in the gauge theory.Despite the qualitative success of the real-spacerenormalization-group transformation, quantitative suc-cess eludes this method. In their analysis, Fradkin andRaby used a square block of linear size 2 (in units of thelattice spacing). Thus, length scales double after eachiteration of the blocking transformation. Unfortunately,time does not scale the same way. They find that, evenat the fixed point, where one expects the rotational in-variance of the classical three-dimensional gauge theoryto be fully restored, time increases only by a factor of1 . In studies of the one-dimensionaltransverse field Ising model enlarging the cell size doesimprove the computation of critical exponents. However,the convergence to exact results are slow and the com-putation ceases to be analytically tractable beyond cellscontaining more than a few spins. Certainly, real-spacerenormalization has not been as successful in producingprecise output in the critical regime as other techniques(e.g., epsilon expansion, high-temperature series, MonteCarlo, conformal bootstrap). The difficulty of systemat-ically correcting the variational approximation also castssome doubt as to whether the procedure is physicallyreasonable—it is hard to judge the efficacy of the ap-proximation based solely on the numerical proximity ofexponents as this might be accidental.It was suggested by Fradkin and Raby that the asym-metric scaling of space and time in their transformationcould be remedied by applying a perturbative formalismdeveloped by Hirsch and Mazenko in Ref. 8. In this ap-proach virtual effects arising from decimated degrees offreedom generate effective operators connecting nearbyplaquettes and links. The expansion is organized arounda parameter, g , such that the original results of Fradkinand Raby are obtained at order g , and quantum fluctua-tions from truncated cell spectra influence the renormal-ized Hamiltonian at order g and beyond by generatingnew effective interactions. The order g calculation hasbeen studied by Hirsch in the context of the Ising model. In the present work we show how the same formalismcan be applied in the Z gauge theory. We find the sameresults for critical exponents as Hirsch because our effec-tive operators in the lattice gauge theory map into thoseof the Ising model according to the well-known dualitytransformation. The results confirm that the scaling ofspatial correlations improves significantly in going fromorder g to g . We understand this to be due to the inclu-sion of more delocalized operators in the effective Hamil-tonian. Unfortunately, the gap critical exponent worsens I FIG. 1. The cell I consists of four plaquettes. significantly. We understand this to mean that the pa-rameter g does not encode small corrections to the vari-ational ground state. This is unsurprising since g is not related a priori to the dimensionless ratio J/h . Rather, g arises due to an artificial separation of the Hamilto-nian into intra- and intercell terms. Therefore, includinghigher-order-in- g corrections will not necessarily improvethe estimate for the gap.The main objective of this article is to exploreHirsch and Mazenko’s renormalization-group perturba-tion method at second order in the real-space frame-work of Hamiltonian lattice gauge theory. To the best ofour knowledge, a calculation based on such an approachhas not been presented in the literature. The quanti-tative improvements for critical exponents garnered inthis approach are modest compared to prior real-spacefindings—they are not, by any means, state-of-the-art.We are able to cross-check our calculations with Hirsch’sfor the transverse field Ising model. The existence of adual model without local symmetry is obviously helpful,but not requisite.This article is organized as follows. In Section II wereview the basic formalism of Fradkin and Raby’s real-space renormalization-group approach and how it fitsinto the perturbation theory of Hirsch and Mazenko. InSection III we pause to prove the exponential decay ofspatial correlations between Z magnetic monopoles inthe ground state of the electric free phase. Our resultsnear and at criticality are presented in Section IV. A briefdiscussion is given in Section V. The technical aspects ofthe renormalization calculation are explained in detail inthe appendix. II. METHODOLOGY
Following Fradkin and Raby, partition the lattice intoregular, repeating square cells I each comprised of fourplaquettes p . See Fig. 1. A given link l belonging to acell is classified into one of two groups: internal (denotedby a dedicated index i ) corresponding to the four centrallinks situated inside the cell, and external (denoted by adedicated index b ) corresponding to the eight boundarylinks around the edge of the cell. Let each cell have itsown cellular Hamiltonian given by H I = − h (cid:88) i ∈ I σ zi − J (cid:88) p ∈ I Φ p . (3)Since only the transverse field operators σ zi of the fourinternal links are included, only these degrees of free-dom act quantumly. The operators σ xb of the eight ex-ternal links behave classically and their eigenvalues serveas boundary conditions on the cell spectrum. We denoteexternal link eigenstates and eigenvalues as σ xb | x b (cid:105) = x b | x b (cid:105) , x b = ± . (4)Since cell I contains four qubits and eight bits, and thecell Hamiltonian commutes with a generator of Z gaugetransformations located at its center site, the cell Hilbertspace has dimensionality 2 − = 8 per boundary con-figuration. The spectrum is easily worked out analyti-cally. Cell eigenstates depend parametrically on the x b around the boundary of the cell. Let us denote the cellground state as | { x b } ) (cid:105) I . Gauge symmetry constrainsits energy eigenvalue (cid:15) I to depend only on the productΦ I = (cid:81) b ∈ I x b .Define interactions by V = − h (cid:88) I (cid:88) b ∈ I σ zb , (5)where it is understood that links are not repeated in thesum. The original lattice Hamiltonian is then H = (cid:88) I H I + gV, (6)where the intercell coupling g has been introduced toaid in organizing a perturbation expansion—its value isultimately set to 1.The goal is to construct a renormalized Hamiltonian H ren governing a new set of spin- operators { X B , Z B } again obeying the Pauli algebra and defined on links B corresponding to the sides of the cells I . The renor-malized electric and magnetic flux operators Z B and (cid:81) B ∈ I X B will then come with renormalized couplings h (cid:48) and J (cid:48) , respectively. But we also expect that more com-plicated gauge-invariant operators are generated. Thiswill proliferate more couplings. H ren is constructedsuch that, for an arbitrary configuration of the exter-nal link eigenvalues, its lowest eigenvalue is identical tothat of H . At g = 0 this is done by projecting H onto a subspace spanned by states | n (cid:105) which are formedfrom tensor products over all cells I of | { x b } ) (cid:105) I andthe | x b (cid:105) (without repetition). Such states have energy (cid:15) n = (cid:80) I (cid:15) I (Φ I ). The truncated Hilbert space is spannedby states | µ n (cid:105) which are simply products of the | x b (cid:105) . Inessence, the internal links are decimated by the trun-cation. At the surviving links we define new operators µ xb = ( | + (cid:105)(cid:104) + | − |−(cid:105)(cid:104)−| ) b and µ zb = ( | + (cid:105)(cid:104)−| + |−(cid:105)(cid:104) + | ) b .Then, for each pair of contiguous links b and b (cid:48) in the lat-tice, we define renormalized operators X B = µ xb µ xb (cid:48) and Z B = ( µ zb + µ zb (cid:48) ) / × I × p qR FIG. 2. A correlation function of disorder operators at pla-quettes at p and q separated by a line γ of R + 1 vertical links(bold). Attached to each one of these links l is a transversefield operator σ zl . One iteration of the decimation procedure(cells are shown dashed) reduces the separation to R/
2. Linkson the edges of the cells are boundary links and operators liv-ing on these links do not act directly on the Hilbert space ofthe cell.
There is a set of vectors {| α (cid:105)} much larger than and or-thogonal to the set {| n (cid:105)} that, when combined with {| n (cid:105)} ,span the original Hilbert space. We construct | α (cid:105) simi-larly to | n (cid:105) except that one or more cell eigenstates mustbe chosen in an excited cell state. For g > H and (cid:80) I H I are not the same. Correct-ing this order by order in g constrains the renormalizedHamiltonian to have the form H ren = H (0) + gH (1) + g H (2) + · · · , (7a) H (0) = (cid:88) n (cid:15) n | µ n (cid:105)(cid:104) µ n | (7b) H (1) = (cid:88) n,n (cid:48) (cid:104) n (cid:48) | V σ | n (cid:105)| µ n (cid:105)(cid:104) µ n (cid:48) | (7c) H (2) = (cid:88) n,n (cid:48) (cid:88) α (cid:104) n (cid:48) | V σ | α (cid:105)(cid:104) α | V σ | n (cid:105)× (cid:16) (cid:15) n − (cid:15) α + 1 (cid:15) n (cid:48) − (cid:15) α (cid:17) | µ n (cid:105)(cid:104) µ n (cid:48) | . (7d)The detailed computation of these terms may be foundin the appendix. III. PROOF OF EXPONENTIAL DECAY
Let λ = J/h . The electric free phase—the ground statein which lines of electric flux can meander throughout thelattice without energy cost—corresponds to λ (cid:29)
1. De-note the ground state of the lattice Hamiltonian by | gs λ (cid:105) .’t Hooft disorder operators—which are string-like yet stilllocal—create and annihilate magnetic monopoles. Theircorrelation function, when separated by a row of R = 2 n plaquettes, is given by C λ ( R ) = (cid:104) gs λ | (cid:89) l ∈ γ σ zl | gs λ (cid:105) , (8)where γ is the set of R +1 vertical links separating the twoplaquettes. See Fig. 2. Fradkin and Raby’s decimationprocedure amounts to the approximate replacement | gs λ (cid:105) (cid:39) (cid:89) I | { x b } ) (cid:105) I | gs λ (cid:48) (cid:105) . (9)Here | gs λ (cid:48) (cid:105) is the ground state of H ren with λ (cid:48) = J (cid:48) /h (cid:48) ,which now has a fourth as many plaquettes. This is atruncation of the the original Hilbert space to a sub-space of states that may be expressed solely in terms ofboundary link eigenstates | x b (cid:105) . Substitution gives C λ ( R ) = (cid:104) gs λ (cid:48) | (cid:89) b σ zb R/ (cid:89) I =1 (cid:104) | σ zi | (cid:105) I | gs λ (cid:48) (cid:105) . (10)The appendix contains the explicit cell ground statewavefunction, Eq. (A.17) or Eq. (A.19), and represen-tation of the internal σ zi matrices, Eq. (A.15), neededto compute the cell matrix element. Irrespective of thechoice for i , the matrix element turns out to depend onlyon the sign of the magnetic flux, (cid:104) | σ zi | (cid:105) I = ( A − λ + A + λ )11 I − ( A − λ − A + λ )Φ I , (11a)where A + λ = (1 − λ + (1 + λ ) / )(1 + λ + (1 + λ ) / ) / √ λ ) / , (11b) A − λ = 12 (cid:16) λ ) / (cid:17) . (11c)To O ( g ), the renormalized couplings J (cid:48) and h (cid:48) may beread off from Eqs. (A.29c) and (A.30e), respectively. Foran initial choice of λ (cid:29)
1, there is the asymptotic equiv-alence λ (cid:48) ≈ λ. (12)Since the renormalized coupling increases with iteration,the ground state has no magnetic energy and the oper-ator Φ I evaluates to +1. In terms of renormalized linkoperators, this yields the multiplicative recursion relation C λ ( R ) ≈ ( A + λ ) R/ C λ (cid:48) ( R/ . (13)Iterating n times starting from λ , C λ (2 n ) = ( A + λ ) n − ( A + λ ) n − · · · ( A + λ n − ) n − n C λ n (1) . (14)Substituting A + λ ≈ / λ and λ n ≈ n λ yields C λ (2 n ) ≈ n (4 λ ) n − C n λ (1) . (15)But C λ (1) ∼ λ − up to some constant factor, so C λ ( R ) ∼ R (4 λ ) R ∝ exp( − R log(4 λ )) . (16) 𝒪 = Z B Z B′ 𝒪 = Z B Z B′ 𝛷 I elbow 𝒪 = Z B Z B′ 𝒪 = Z B Z B′ 𝛷 I sandwiched Z B 𝛷 I 𝒪 = 𝛷 I 𝛷 J I elbow I sandwiched B B ′ B B ′ BIB B ′ B B ′ I J
FIG. 3. Effective operators present in renormalized Hamil-tonian H ren to second order in the intercell coupling g . Theseare dual to the operators found by Hirsch in his study of thetwo-dimensional Ising model in a transverse field. IV. RESULTS
Our main technical achievement is the explicit expres-sion for the renormalized Hamiltonian calculated fromEq. (7). The details of the calculation are given in theappendix. The reader interested only in the final form of H ren can see the precise operators in Eq. (A.85), althoughseveral definitions needed to understand the coefficientsof these operators are scattered throughout the appendix.It turns out that five new effective operators are createdin addition to the effective electric flux hZ B on each link B , and effective magnetic flux J Φ I on each cell I . Wedenote these new operators as K α O α for α = 1 , . . . , F I on each cell I . In the appendix we study numerically the recursion re-lations for the operator coefficients h , J , K α , and F . Fol-lowing Fradkin and Raby, we adopt h as an energy scaleand consider the dimensionless couplings ( J/h, K α /h )packaged into a six-dimensional vector. Iterations of therecursion relations produce a sequence of points in thisvector space (an “RG flow”) that describes Hamiltoniansdefined over successively coarser lattices. We are inter-ested in flows that begin on the axis ( J/h,(cid:126)
J/h = (
J/h ) c for which the flowconverges onto a nontrivial fixed point ( J/h, (cid:126)K/h ) ∗ . Butthis requires fine tuning. Generically, for J/h (cid:54) = (
J/h ) c ,flows veer away from this fixed point and tend towardeither the origin or infinity. Thus, the nontrivial fixedpoint is unstable and infrared-repulsive, whilst the triv-ial fixed points at the origin and infinity are stable andinfrared-attractive. In the neighborhood of the nontrivialfixed point there is a linear space with a single relevantscaling variable that we call u . Iterations of the recur-sion relations renormalize u to Λ u , where Λ > A. Critical point
Numerical analysis of flows was performed with
Math-ematica 10 . A critical coupling was found at (
J/h ) c =3 . B. Energy gap
Consider the electric free phase in which
J/h is onlyslightly greater than (
J/h ) c . The flow is observed to be-have as follows: a small and finite number of steps N brings the flow from ( J/h,(cid:126)
0) into the neighborhood of thenontrivial fixed point; for some large, but finite, numberof steps N , the flow dawdles and remains quite close tothe fixed point; further steps finally allow the flow to es-cape this region and head off to infinity. Numerical anal-ysis shows that whilst the five couplings K α /h remain oforder one, the coupling J/h grows without bound. Thisis expected since the trivial fixed point at infinity shoulddescribe the deconfined phase of the lattice gauge the-ory with infinitely heavy magnetic monopoles. So nowconsider the recursion relations just for the coefficients h and J , which may be expressed in the form h (cid:48) = hζ ( J/h, (cid:126)K/h ) , (17a) J (cid:48) = Jη ( J/h, (cid:126)K/h ) . (17b)Specifically, the function ζ is given by dividing the right-hand side of Eq. (A.85b) by h , and the function η isgiven by dividing the right-hand side of Eq. (A.85c)by J . Numerical evidence suggests that repeated iter-ations cause ζ to approach some number less than 1,and η to approach 1. Therefore, in the limit of infinitelymany renormalization-group iterations the coefficient h will vanish and only J will remain. Thus, the originalHamiltonian defined on the fine lattice will exhibit thesame energy gap as a Hamiltonian defined on the coarselattice containing only the operators (cid:80) I Φ I . Since thelatter theory is weakly coupled, it may be analyzed per-turbatively.The ground state is characterized by the absence ofmagnetic flux for all plaquettes (i.e., Φ I = +1 for all I ).The lowest energy excitation creates a unit of magneticflux on a single plaquette. The energy of the first excitedstate (relative to the ground state) is G (cid:39) J N , where N indicates the total number of iterations of the recur-sion relations. Since J N = J (cid:81) N − n =0 η ( J n /h n , (cid:126)K n /h n ), we need only keep a record of the point sequence alongthe flow in order to calculate the gap. However, close tocriticality the product may be decomposed as G (cid:39) J N − (cid:89) n =0 η N + N (cid:89) n = N η N (cid:89) n = N + N η. (18)The first product in Eq. (18) corresponds to the inflow.It will be analytic in the difference J/h − ( J/h ) c . Thelast product corresponds to the outflow and therefore weexpect it asymptotes to the value 1. However, the middleproduct corresponds to the dawdle near the fixed point.We can evaluate η at the fixed point—this approximationgets better the closer the flow starts to criticality. Thenumber N may be estimated by asking how many stepsneed to be taken to multiplicatively renormalize the rele-vant scaling variable u —which we assume is exceedinglysmall at step N —into an arbitrary, but fixed and smallnumber U for which the linearized approximation to therecursion relations is still valid. This condition is U = Λ N u = ⇒ N = log( U /u ) / log Λ . (19)Hence,nonanalytic part of G ∼ η (( J/h, (cid:126)K/h ) ∗ ) N . (20)Since u arises from the inflow and N is finite, it followsthat u itself is some analytic function of J/h − ( J/h ) c .Finally,nonanalytic part of G ∼ ( J/h − ( J/h ) c ) ν t , (21)where ν t = − log η (( J/h, (cid:126)K/h ) ∗ )log Λ (cid:39) . . (22)Obtaining this critical exponent, which was not com-puted in Ref. 5, was one of the original motivations forthis work. C. Spatial correlation length
In the free electric phase, the correlation function ofdisorder operators given by Eq. (8) ought to have a cor-relation length ξ that diverges as J/h approaches (
J/h ) c from above. After some large number N of decimations,the dimensionless correlation length will be an order onenumber because the flow will be far from the neighbor-hood of the fixed point where the linearized recursionrelations hold. Therefore, the part of N that depends onthe reduced coupling J/h − ( J/h ) c must be the same as N as estimated in Eq. (19). Since our renormalizationscale factor is 2, ξ ∼ N ∼ ( J/h − ( J/h ) c ) ν s , (23)where ν s = log 2log Λ (cid:39) . . (24) D. Ground state energy
From the coefficient of the identity operator on eachcell it is possible to represent the ground state energy bythe expression E gs ( J/h, (cid:126)K/h ) = lim n →∞ F ( n ) N plaq n , (25)where the superscript ( n ) denotes the n th iteration ofthe recursion relation given by Eq. (A.85i). The case n = 0 indicates the original bare coupling. For instance, h (0) = h . Define the energy density by ε gs = E gs /N plaq .Rather than a limit, let us express the ground state en-ergy density as an infinite sum. For clarity, we writethe six-dimensional vector of dimensionless couplings as κ ( n ) = ( J ( n ) /h ( n ) , (cid:126)K ( n ) /h ( n ) ). Knowing that the recur-sion relation for F , Eq. (A.85i), takes the form F ( n +1) = 4 F ( n ) + h ( n ) ∆( κ ( n ) ) , (26) where ∆ is an analytic function of its argument, we ob-tain ε gs ( κ (0) ) = ∞ (cid:88) n =0 n +1 h ( n ) ∆( κ ( n ) ) . (27)We have assumed that F (0) = 0. Since the recursionrelation for h , Eq. (A.85b), takes the form h ( n +1) = h ( n ) ζ ( κ ( n ) ) , (28)it follows that h ( n ) = h (0) ζ ( κ (0) ) ζ ( κ (1) ) · · · ζ ( κ ( n − ) . (29)Therefore, in Eq. (27), specification of κ (0) completely de-termines the right-hand side since the recursion relationsmay be applied to to obtain κ (1) , κ (2) , etc. FollowingRef. 4, Eq. (27) may be written ε gs ( κ (0) ) = h (0) ∆( κ (0) ) + ζ ( κ (0) ) (cid:20) h (0) ∆( κ (1) ) + h (0) ζ ( κ (1) )∆( κ (2) ) + h (0) ζ ( κ (1) ) ζ ( κ (2) )∆( κ (3) ) + · · · (cid:21) . (30)Notice that the bracketed term in Eq. (30) is just theright-hand side of Eq. (27) but started at the point κ (1) .Thus, we arrive at a recursion relation satisfied by theground state energy density, ε gs ( κ (0) ) = h (0) ∆( κ (0) ) + ζ ( κ (0) ) ε gs ( κ (1) ) . (31)We are interested in extracting the leading singularbehavior of the ground state energy density as the criticalpoint is approached. Since ∆ is differentiable, even at thefixed point, Eq. (31) implies the following homogeneoustransformation law for the singular part of ε gs , ε singgs ( κ ) = 14 ζ ( κ ) ε singgs ( κ (cid:48) ) . (32)Close to the fixed point, we can write this using scalingvariables. Ignoring irrelevant variables and iterating n times, ε singgs ( u ) = 4 − n n − (cid:89) r =0 ζ (Λ r u ) ε singgs (Λ n u ) . (33)Since u grows under iteration we need to apply a stop-ping condition. We take n = N as specified by Eq. (19).If Λ r u remains small, then we may approximate each ζ by its value at the fixed point u = 0. Then ε singgs ( u ) ≈ (4 /ζ (0)) − n ε singgs ( U ) ∼ u log(4 /ζ (0))log Λ1 . (34)Finally,singular part of ε gs ∼ | J/h − ( J/h ) c | − α , (35) where α = 2 − log(4 /ζ (0))log Λ (cid:39) . . (36)A direct numerical calculation of Eq. (27) at the criticalcoupling ( J/h ) c produces ε gs (cid:39) − . h (0) . E. A critical amplitude ratio
We also studied the divergence of the second deriva-tive of the ground state energy density with respect to
J/h , denoted ε (cid:48)(cid:48) gs , in a small region around the criticalcoupling. A plot of this divergence is visible in Fig. 4as the prominent spike. The main difficulty in obtain-ing these values, besides discretization error associatedwith differentiating, is the inability to evaluate Eq. (27)to arbitrarily high n . Our renormalization group trans-formation κ ( n ) → κ ( n +1) cannot be iterated indefinitelywithout running into a nonsensical value for J/h (e.g.,negative values). In practice, we could iterate the flowbetween 10 and 20 steps, more steps being possible thecloser
J/h begins to the true critical coupling.The leading singular behavior in ε gs comes fromthat part of the sum in Eq. (27) corresponding to therenormalization flow along the outflow trajectory (seeFig. 10) . Since the flow away from the fixed point isdifferent for the two phases of the gauge theory, we ex-pect the amplitude A to be different for J/h > ( J/h ) c − 𝜀 ′′ gs J/h
FIG. 4. A plot of a numerical calculation of − ∂ ε gs /∂ ( J/h ) .It diverges at the critical coupling ( J/h ) c = 3 . J/h ) c E gs /hN plaq first (Refs. 3 and 4) 3.28 − . − . J/h ) c and critical ground stateenergy per plaquette E gs /hN plaq in the Hirsch–Mazenko per-turbation expansion. and J/h < ( J/h ) c . We may writeleading singular part of ε gs ∼ A >,< | J/h − ( J/h ) c | − α . (37)The amplitude ratio A > /A < is a universal quantity, butunlike critical exponents, it depends on the entire flow,not just the linearized flow in the vicinity of the fixedpoint.We applied a naive procedure to obtain a cursory es-timate for the amplitudes. After transforming data tothe form (log | J/h − ( J/h ) c | , log ε (cid:48)(cid:48) gs ), we made a least-squares fit to the line y = b − αx , where b is the singlefree parameter and α is constrained to be the value inEq. (36). We remark that, even with a two-parameter fitlike y = b − b x , the slope parameter b comes within10% of α . Notice that we completely ignore correction-to-scaling terms in these fits. We obtain b > = − . b < = − . A > /A < = exp( b > − b < ) (cid:39) . . (38)For comparison, the ratio of specific heat amplitudes inthe three-dimensional Ising universality class is known tobe about 0.52. V. DISCUSSION
Using Hirsch–Mazenko perturbation theory we havecalculated some critical properties properties of the quan-tum Z gauge theory on a square lattice. Universal crit-ical exponents are given in Table I while nonuniversaldata are collected in Table II. Most indications are thatthe second-order theory is an improvement over the first-order theory.It is known from Monte Carlo simulations of the simplecubic Ising model that the critical inverse temperatureis K c (cid:39) .
22 and the critical exponent for the corre-lation length is ν Ising (cid:39) . . Hyperscaling then im-plies that the critical exponent for the specific heat is α Ising = 2 − ν Ising (cid:39) .
11. Using duality we are ableto transfer these values over to the gauge theory: thecritical coupling ought to be (
J/h ) c = K − (cid:39) .
51, thecritical exponents for the energy gap and spatial corre-lation length—equal due to rotational invariance at thecritical point—ought to be ν s = ν t = ν Ising (cid:39) .
63, andthe critical exponent for the leading singular behavior ofthe ground state energy ought to be α = α Ising (cid:39) . ν s and α , which we stress were computedindependently, both improved dramatically at second or-der. To wit, α made a qualitative switch from negativeto positive!Furthermore, at the fixed point, we find that the recip-rocal of the gap energy scales, under a renormalizationtransformation, by a factor of 1 /η (( J/h, (cid:126)K/h ) ∗ ) (cid:39) . O ( g ) result of 1 .
43. This supports Fradkinand Raby’s suggestion that systematic improvement ispossible using a perturbative framework like Hirsch andMazenko’s.In order to evaluate the accuracy of the critical groundstate energy density, we may use a duality relation be-tween quantum Hamiltonians for the two-dimensional Z lattice gauge theory (“LGT”) and the two-dimensionaltransverse field Ising model (“TFIM”) . If λ = J/h and E is any eigenvalue, then E LGT ( λ ) = λE TFIM ( λ − ). Us-ing the critical data in Table II, the critical ground stateenergy per spin in the TFIM is approximately − .
03 (firstorder) and − .
04 (second order). A numerical calcula-tion of the lowest eigenvalue of the TFIM Hamiltonian ona 4 × K c yields a groundstate energy per spin of − .
02. The agreement is decent.Unfortunately, not all Hirsch–Mazenko perturbativecorrections are improvements. There is a clear worseningof the gap critical exponent: the second-order value for ν t is much worse than its first-order value. This suggeststhat the artificial separation of the Hamiltonian given byEq. (6) is not a small correction to the variational groundstate energy.One may straightforwardly improve ν t by enlarging thecell size. For instance, we have done an O ( g ) analysisusing 3 × × cell size ( J/h ) c ν t ν s × × × O ( g ) inthe Hirsch–Mazenko perturbation expansion for different cellsizes. These are most easily computed from the transversefield Ising model via duality. Table III. There is modest improvement after increasingthe cell Hilbert space dimensionality from 2 to 2 andthen to 2 , indicating that the variational approximationis a little better. I I+x ^ I−x ^ I+y ^ I−y ^ ( I ,0,2)( I ,0,3) ( I ,0,7)( I ,0,6) ( I ,1,0)( I ,3,0) ( I , , )( I , , )( I , , )( I , , )( I , , )( I , , ) ( I , i , j )cell internal link external link FIG. 5. Our notation for the links.
Appendix: Details
In this appendix we use a different and more thoroughnotation than in the main body of the article.
1. Notation
Let I denote a cell of four plaquettes. See Fig. 5. Thefour neighboring cells are called I + ˆ x , I − ˆ x , I + ˆ y , and I − ˆ y . Each cell has four internal links with spin- operators (cid:126)σ I,i, , i = 1 , . . . ,
4. Also, each cell is surrounded by eightexternal links with spin- operators (cid:126)σ I, ,j , j = 1 , . . . , I, , j ) is a given link in a given cell, then the neigh-boring cell that shares that link will be denoted I [ j ] , andthe same link will, from this cell’s perspective, be called[ j ]. Explicitly, ( I, , j ) = ( I [ j ] , , [ j ]) (A.1)( I, ,
8) = ( I + ˆ x, , I, ,
1) = ( I + ˆ x, , I, ,
2) = ( I + ˆ y, , I, ,
3) = ( I + ˆ y, , I, ,
4) = ( I − ˆ x, , I, ,
5) = ( I − ˆ x, , I, ,
6) = ( I − ˆ y, , I, ,
7) = ( I − ˆ y, , . Separate the Hamiltonian into an intracell part andintercell part, H σ = H σ + V σ . (A.2)The intracell part is a sum over all cells of the internallinks, including both σ x and σ z operators, and the ex-ternal links, but including only the σ x operators. Let H σ = (cid:88) I H I , (A.3a)where H I = − h (cid:88) i =1 σ zI,i, − J (cid:88) i =1 σ xI,i, σ xI,i +1 , σ xI, , i − σ xI, , i . (A.3b)The intercell part is a collection of all transverse fieldoperators acting on the external links of each cell, V σ = − h (cid:88) I (cid:88) j =1 σ zI, ,j . (A.4)It is understood that all external links are to be summedover just once. As a shorthand we will write (cid:80) I,j .It is important to notate eigenvalues of σ x operatorson external links. For any I and j , denote σ xI, ,j | x I,j (cid:105) I, ,j = x I,j | x I,j (cid:105) I, ,j , x I,j = ± . (A.5)Since [ H I , σ xI, ,j ] = 0 for each j , each external linkoperator σ xI, ,j may be replaced by its eigenvalue x I,j .Thus, the eight bits x I,j behave as classical bound-ary conditions. For the cell Hamiltonian we may write H I ( x I, , . . . , x I, ).Each cell, because it has four qubits and eight bits,would seem to have a 2 -dimensional Hilbert space for ev-ery classical configuration of its external links. However,gauge invariance reduces this large space of possibilitiesso that, ultimately, it matches the information encodedin the quantum Ising model with a block of four sites.First, [ H I , (cid:81) i =1 σ zI,i, ] = 0 and we are interested only inthe gauge-invariant sector (cid:81) i σ zI,i, = +1. This halvesthe dimensionality from 2 to 2 . And each eigenstate of H I has an eigenvalue that depends only on the sign ofthe gauge-invariant flux operator,Φ I = (cid:89) j =1 σ xI, ,j = (cid:89) j =1 x I,j . (A.6)That is, if we denote an eigenstate of H I by | i I ( x I, , . . . , x I, ) (cid:105) I , i I = 0 , . . . , , (A.7)then its corresponding eigenvalue is some (cid:15) c i I (Φ I ) . (A.8)The superscript “c” stands for “cell.” Thus, we are reallydealing with a Hilbert space containing 16 eigenstates:8 in the sector with Φ I = +1, and 8 in the sector withΦ I = −
1. We reserve i I = 0 to indicate the lowest-energystate in either sector.
2. Cell spectrum
The explicit wavefunctions and energies for H I may beworked out by following the procedure in Ref. 3. We usea basis for the internal links in which σ zI,i, is diagonal(i.e., σ zI,i, ↑ i = ↑ i and σ zI,i, ↓ i = − ↓ i , and σ xI,i, ↑ i = ↓ i and σ xI,i, ↓ i = ↑ i ). The basis ordering is {↑ ↑ ↑ ↑ , ↑↓↑↓ , ↓↑↓↑ , ↓↓↓↓ , ↑↑↓↓ , ↓↓↑↑ , ↑↓↓↑ , ↓↑↑↓} . Since all eigen-states are going to be expressed in this basis, the con-dition (cid:81) i σ zI,i, = +1 is automatically satisfied. Define A k = x I, k − x I, k , k = 1 , . . . , . (A.9)Then H I = (cid:32) P QQ T (cid:33) , (A.10a)where P = − h − , (A.10b) Q = − J A A A A A A A A A A A A A A A A . (A.10c)Clearly, the wavefunctions are not functions of the eight x I,j , but rather the four combinations A k , | i I ( x I, , . . . , x I, ) (cid:105) I = | i I ( A , . . . , A ) (cid:105) I . (A.11)We wish to solve the eigenvalue problem H I | i I (cid:105) I = (cid:15) c i I | i I (cid:105) I subject to additional constraints inherited fromgauge invariance. In the full Hamiltonian H , gauge trans-formations are possible at each of the nine sites in cell I .However, because of the artificial nature of the blockingscheme the eight transformations around the cell perime-ter no longer manifest as symmetries from the point ofview of the cell Hamiltonian H I . Instead, they mani-fest as the following identities (for the sake of brevitywe only write parameters that are being affected in someway, e.g., by being negated): H I ( x I, , x I, ) = H I ( − x I, , − x I, ) , (A.12a) H I ( x I, , x I, ) = H I ( − x I, , − x I, ) , (A.12b) H I ( x I, , x I, ) = H I ( − x I, , − x I, ) , (A.12c) H I ( x I, , x I, ) = H I ( − x I, , − x I, ) , (A.12d)and σ zI, , H I ( x I, , x I, ) σ zI, , = H I ( − x I, , − x I, ) , (A.13a) σ zI, , H I ( x I, , x I, ) σ zI, , = H I ( − x I, , − x I, ) , (A.13b) σ zI, , H I ( x I, , x I, ) σ zI, , = H I ( − x I, , − x I, ) , (A.13c) σ zI, , H I ( x I, , x I, ) σ zI, , = H I ( − x I, , − x I, ) . (A.13d)0Eqs. (A.12) correspond to the four corners of the cell,while Eqs. (A.13) correspond to the midpoints of eachside. The former are trivially satisfied if we express thewavefunctions in terms of the A k . However, the latterare nontrivial and require that σ zI, , | i I ( A , A ) (cid:105) I = | i I ( − A , − A ) (cid:105) I , (A.14a) σ zI, , | i I ( A , A ) (cid:105) I = | i I ( − A , − A ) (cid:105) I , (A.14b) σ zI, , | i I ( A , A ) (cid:105) I = | i I ( − A , − A ) (cid:105) I , (A.14c) σ zI, , | i I ( A , A ) (cid:105) I = | i I ( − A , − A ) (cid:105) I . (A.14d)Note that, in our chosen basis, σ zI, , = diag(+ , + , − , − , + , − , + , − ) , (A.15a) σ zI, , = diag(+ , − , + , − , + , − , − , +) , (A.15b) σ zI, , = diag(+ , + , − , − , − , + , − , +) , (A.15c) σ zI, , = diag(+ , − , + , − , − , + , + , − ) . (A.15d) a. Φ I = + sector Cell energies arranged in increasing order are (cid:15) c0 (+) = − / [ h + J + ( h + J ) / ] / , (A.16a) (cid:15) c1 (+) = − / [ h + J − ( h + J ) / ] / , (A.16b) (cid:15) c2 (+) = 0 , (A.16c) (cid:15) c3 (+) = 0 , (A.16d) (cid:15) c4 (+) = 0 , (A.16e) (cid:15) c5 (+) = 0 , (A.16f) (cid:15) c6 (+) = 2 / [ h + J − ( h + J ) / ] / , (A.16g) (cid:15) c7 (+) = 2 / [ h + J + ( h + J ) / ] / . (A.16h)There is no level crossing. For i I = 0 , , ,
7, let E = (cid:15) c i I (+) for brevity. The unnormalized wavefunction is | , , , (cid:105) I = J ( E + 4 h ) − Γ( E − h ) A A Γ( E − h ) A A Γ EA A − A − A − A − A , (A.17a)where Γ = E + 4 hE − J J ( E + 4 h )(3 E − h ) . (A.17b) For i I = 2 , , ,
5, the normalized wavefunctions are | (cid:105) I = 1 √ A A − A A , | (cid:105) I = 1 √ A − A , | (cid:105) I = 1 √ A − A , | (cid:105) I = 12 A − A − A A . (A.17c)It is important to note that {| (cid:105) I , | (cid:105) I , | (cid:105) I , | (cid:105) I } forman orthonormal basis in the zero-energy subspace. Ob-taining these particular wavefunctions required judicioususe of identities like A k = 1 /A k , and A = A A A , A A = A A , etc. which follows from the fact that A A A A = 1. b. Φ I = − sector Cell energies arranged in increasing order are, for h > (cid:15) c0 ( − ) = − h + J ) / − h (A.18a) (cid:15) c1 ( − ) = − J (A.18b) (cid:15) c2 ( − ) = − J (A.18c) (cid:15) c3 ( − ) = − h + J ) / + 2 h (A.18d) (cid:15) c4 ( − ) = 2( h + J ) / − h (A.18e) (cid:15) c5 ( − ) = 2 J (A.18f) (cid:15) c6 ( − ) = 2 J (A.18g) (cid:15) c7 ( − ) = 2( h + J ) / + 2 h. (A.18h)1For i I = 0 , , ,
7, let E = (cid:15) c i I ( − ) for brevity. The nor-malized wavefunctions are | , (cid:105) I = 1 (cid:112) E /J − EJ − A A A A , (A.19) | , (cid:105) I = 1 (cid:112) E /J − EJ − A A A A − A − A , | (cid:105) I = 1 √ A A A − A A − A , | (cid:105) I = 1 √ − A A A − A A − A , | (cid:105) I = 1 √ A A − A A A − A , | (cid:105) I = 1 √ − A A − A A A − A .
3. Lattice eigenstates as products of cell andexternal link eigenstates
For any given configuration on the external links, thelowest energy eigenstate of H σ is obtained from a productover all cells with i I = 0, | i (cid:105) = (cid:89) I,j | x I, , . . . , x I, ) (cid:105) I | x I,j (cid:105) I, ,j , (A.20)where it is understood that each external link contributesjust once. The H σ -eigenvalue is (cid:15) i = (cid:88) I (cid:15) c0 (Φ I ) . (A.21) When summing over all possible | i (cid:105) we shall use the short-hand (cid:88) i = (cid:89) I,j (cid:88) x I,j = ± . (A.22)Corresponding to each state | i (cid:105) in the Hilbert space ofthe original lattice Hamiltonian is a state | µ i (cid:105) belongingto the smaller Hilbert space of the renormalized Hamil-tonian. Quite simply, it is everything in | i (cid:105) but the wave-function of the internal links. | µ i (cid:105) = (cid:89) I,j | x I,j (cid:105) I, ,j . (A.23)It is in this sense that the internal links have been“decimated.” On the thinner lattice of external links {| x I,j (cid:105) I, ,j } we define new Pauli operators11 I,j = ( | + (cid:105)(cid:104) + | + |−(cid:105)(cid:104)−| ) I, ,j , (A.24a) µ xI,j = ( | + (cid:105)(cid:104) + | − |−(cid:105)(cid:104)−| ) I, ,j , (A.24b) µ zI,j = ( | + (cid:105)(cid:104)−| + |−(cid:105)(cid:104) + | ) I, ,j . (A.24c)Since {| i (cid:105)} is merely a small subset of the energy basisof H σ , the remaining higher-energy lattice eigenstates areconstructed from cells with any value of i I , | α (cid:105) = (cid:89) I,j | i I ( x I, , . . . , x I, ) (cid:105) I | x I,j (cid:105) I, ,j , i I = 0 , . . . , , (A.25)with the caveat that at least one i I >
0. Its H σ -eigenvalue is (cid:15) α = (cid:88) I (cid:15) c i I (Φ I ) . (A.26)When summing over all possible | α (cid:105) we shall use theshorthand (cid:88) α = (cid:89) I,j (cid:88) i I =0 (cid:12)(cid:12)(cid:12)(cid:12) some i I (cid:54) = 0 (cid:88) x I,j = ± . (A.27)
4. Hirsch–Mazenko perturbation expansion
To second order in the intercell coupling, H ren µ = H (0) µ + H (1) µ + H (2) µ + · · · , (A.28a) H (0) µ = (cid:88) i (cid:15) i | µ i (cid:105)(cid:104) µ i | , (A.28b) H (1) µ = (cid:88) i,i (cid:48) (cid:104) i (cid:48) | V σ | i (cid:105)| µ i (cid:105)(cid:104) µ i (cid:48) | , (A.28c) H (2) µ = (cid:88) i,i (cid:48) (cid:88) α (cid:104) i (cid:48) | V σ | α (cid:105)(cid:104) α | V σ | i (cid:105)× (cid:16) (cid:15) i − (cid:15) α + 1 (cid:15) i (cid:48) − (cid:15) α (cid:17) | µ i (cid:105)(cid:104) µ i (cid:48) | . (A.28d)Refer to Ref. 8 for a derivation of these expressions. Theyhave been written in a simplified form following Eq. (3)of Ref. 5.When there is no chance of confusion, we will abbrevi-ate the state | x I,j (cid:105) I, ,j as | x I,j (cid:105) .2 a. Computation of H (0) µ Consider Eq. (A.28b), H (0) µ = (cid:16)(cid:89) I,j (cid:88) x I,j (cid:17)(cid:16)(cid:88) I (cid:15) c0 (Φ I ) (cid:17) (cid:89) I,j | x I,j (cid:105)(cid:104) x I,j | . (A.29a)Bring the sum over cells I out so that only the linksbelonging to a given Φ I will be non-identity operators. Call the eight links of cell I , b , . . . , b . Then H (0) µ = (cid:88) I (cid:88) x b · · · (cid:88) x b (cid:15) c0 ( x b · · · x b ) ×| x b (cid:105)(cid:104) x b | · · · | x b (cid:105)(cid:104) x b | (A.29b)= (cid:88) I (cid:16) (cid:15) c0 (+) + (cid:15) c0 ( − )2 11 b · · · b − (cid:15) c0 ( − ) − (cid:15) c0 (+)2 µ xb · · · µ xb (cid:17) . (A.29c) b. Computation of H (1) µ Consider Eq. (A.28c), H (1) µ = (cid:16)(cid:89) I,j (cid:88) x I,j (cid:17)(cid:16)(cid:89)
I,j (cid:88) x (cid:48) I,j (cid:17) (cid:89)
I,j (cid:104) { x (cid:48) I,j } ) | I (cid:104) x (cid:48) I,j | · − h (cid:88) I,j σ zI, ,j · (cid:89) I,j | { x I,j } ) (cid:105) I | x I,j (cid:105) · (cid:89)
I,j | x I,j (cid:105)(cid:104) x (cid:48) I,j | . (A.30a)By an abuse of notation, each instance of “ (cid:81) I,j ” serves toremind us that there are as many copies of the expressionimmediately to the right of this symbol but left of “ · ” or“)” as external links in the lattice. Pull out (cid:80) I,j . Fora single external link at ( I, , j ), we get the constraint (cid:104) x (cid:48) I,j | σ zI, ,j | x I,j (cid:105) = δ x (cid:48) I,j , − x I,j . At all other external linksthe eigenvalues x (cid:48) I,j and x I,j are equal. Therefore, allcells ˜ I not containing link ( I, , j ) on their border yield (cid:104) | (cid:105) ˜ I = 1. Only cells I and I [ j ] share this link. So H (1) µ = − h (cid:88) I,j (cid:88) x I,j (cid:88) x b · · · (cid:88) x b ×(cid:104) − x I,j ) | x I,j ) (cid:105) I (cid:104) − x I [ j ] , [ j ] ) | x I [ j ] , [ j ] ) (cid:105) I [ j ] ×| x I,j (cid:105)(cid:104)− x I,j | (cid:89) b =1 | x b (cid:105)(cid:104) x b | , (A.30b)where b , . . . , b denote all links in cells I and I [ j ] be-sides ( I, , j ) = ( I [ j ] , , [ j ]). Consider the matrix element (cid:104) − x I,j ) | x I,j ) (cid:105) I . Besides the choice of j and the valueof x I,j it could also depend on the seven additional ex-ternal link eigenvalues forming the rest of the boundaryof I . Let us call them x b , . . . , x b (we have suppressedwriting these as they are not negated in the inner prod-uct). However, it turns out that this matrix element iscompletely independent of boundary conditions. That is, the matrix element evaluates to the same quantity forany choice of j and the values x I,j , x b , . . . , x b . For con-venience let us select j = 1, x I,j = x b = · · · = x b = +.So the matrix element is an inner product between thetwo lowest energy states from the Φ I = + and Φ I = − sectors, respectively. For future convenience define thesestates to be (with specific boundary conditions) | + (cid:105) I = | A = + , A = + , A = + , A = +) (cid:105) I , (A.30c) | − (cid:105) I = | A = + , A = − , A = + , A = +) (cid:105) I . (A.30d)Although it seems peculiar to make A negative inEq. (A.30d) rather than, say, A , in hindsight this choiceallows us to write all matrix elements using only thesetwo states. We will return to this point later. Thus, H (1) µ = − h |(cid:104) − | + (cid:105) I | (cid:88) I,j µ zI,j . (A.30e)The factor of 2 arises from the fact that each side of acell contributes two links. c. Computation of H (2) µ When fully written out Eq. (A.28d) is3 H (2) µ = (cid:16)(cid:89) I,j (cid:88) x (cid:48)(cid:48) I,j (cid:17)(cid:16)(cid:89)
I,j (cid:88) x (cid:48) I,j (cid:17)(cid:16)(cid:89)
I,j (cid:88) i I =0 (cid:12)(cid:12)(cid:12) some i I (cid:54) = 0 (cid:88) x I,j (cid:17)(cid:89)
I,j (cid:104) { x (cid:48) I,j } ) | I (cid:104) x (cid:48) I,j | · − h (cid:88) I,j σ zI, ,j · (cid:89) I,j | i I ( { x I,j } ) (cid:105) I | x I,j (cid:105)× (cid:89)
I,j (cid:104) i I ( { x I,j } ) | I (cid:104) x I,j | · − h (cid:88) I (cid:48) ,j (cid:48) σ zI (cid:48) , ,j (cid:48) · (cid:89) I,j | { x (cid:48)(cid:48) I,j } ) (cid:105) I | x (cid:48)(cid:48) I,j (cid:105)× (cid:16) (cid:80) I [ (cid:15) c0 (Φ (cid:48)(cid:48) I ) − (cid:15) c i I (Φ I )] + 1 (cid:80) I [ (cid:15) c0 (Φ (cid:48) I ) − (cid:15) c i I (Φ I )] (cid:17) · (cid:89) I,j | x (cid:48)(cid:48) I,j (cid:105)(cid:104) x (cid:48) I,j | , (A.31)where Φ I = (cid:81) j =1 x I,j , Φ (cid:48) I = (cid:81) j =1 x (cid:48) I,j , and Φ (cid:48)(cid:48) I = (cid:81) j =1 x (cid:48)(cid:48) I,j , Pull out (cid:80)
I,j and (cid:80) I (cid:48) ,j (cid:48) ; eventually, wewill want to keep just one of these sums unevaluated.It is possible to completely evaluate (cid:81) I,j (cid:80) x (cid:48)(cid:48) I,j and (cid:81)
I,j (cid:80) x (cid:48) I,j by collapsing Kronecker deltas for externallinks. For links b (cid:54) = ( I, , j ), x (cid:48) b = x b , but for thethe special link b = ( I, , j ), x (cid:48) b = − x b . Similarly, forlinks b (cid:54) = ( I (cid:48) , , j (cid:48) ), x (cid:48)(cid:48) b = x b , but for the special link b = ( I (cid:48) , , j (cid:48) ), x (cid:48)(cid:48) b = − x b .Consequently, two kinds of inner product between cellwavefunctions may result. If a given cell ˜ I does notcontain the special link ( I, , j ), then orthonormality re-quires that i ˜ I = 0. Likewise, if a given cell I does notcontain the special link ( I (cid:48) , , j (cid:48) ), then i I = 0. This re-sults in a drastic simplification of the energy denomina-tors. For convenience define R i I ,i I [ j ] (Φ I , Φ I [ j ] ) = (cid:104) (cid:15) c0 ( − Φ I ) + (cid:15) c0 ( − Φ I [ j ] ) − (cid:15) c i I (Φ I ) − (cid:15) c i I [ j ] (Φ I [ j ] ) (cid:105) − . (A.32) However, if a cell does have one of these special linkssitting on its boundary—there will be two in the caseof ( I, , j ): I and I [ j ] , and two in the case of ( I (cid:48) , , j (cid:48) ): I (cid:48) and I (cid:48) [ j (cid:48) ] —then an extra minus sign appears in oneof the eight parameters in one of the two wavefunctionsparticipating in the inner product. It shall be convenientto define ζ ji I (Φ I ) = (cid:104) − x I,j , { x I,k } k (cid:54) = j ) | i I ( x I,j , { x I,k } k (cid:54) = j ) (cid:105) I . (A.33)Eq. (A.33) has an important property: although it de-pends on the choice of i I and j , it does not depend onthe precise choice of the x I,j except for the overall signof Φ I .At this step, H (2) µ = h (cid:88) I,j (cid:88) I (cid:48) ,j (cid:48) (cid:16)(cid:89) I,j (cid:88) i I (cid:12)(cid:12)(cid:12) some i I (cid:54) = 0 (cid:88) x I,j (cid:17) (cid:89) ˜ I (cid:54)(cid:51) ( I, ,j ) δ ,i ˜ I (cid:89) I (cid:54)(cid:51) ( I (cid:48) , ,j (cid:48) ) δ ,i I × ζ ji I (Φ I ) ζ [ j ] i I [ j ] (Φ I [ j ] ) ζ j (cid:48) i I (cid:48) (Φ I (cid:48) ) ζ [ j (cid:48) ] i I (cid:48) [ j (cid:48) ] (Φ I (cid:48) [ j (cid:48) ] ) (cid:16) R i I ,i I [ j ] (Φ I , Φ I [ j ] ) + R i I (cid:48) ,i I (cid:48) [ j (cid:48) ] (Φ I (cid:48) , Φ I (cid:48) [ j (cid:48) ] ) (cid:17) × (cid:89) b (cid:54) =( I, ,j ) , ( I (cid:48) , ,j (cid:48) ) ( | x b (cid:105)(cid:104) x b | ) b × (cid:40) ( | − x I,j (cid:105)(cid:104)− x I,j | ) I, ,j ( I (cid:48) , , j (cid:48) ) = ( I, , j )( | − x I (cid:48) ,j (cid:48) (cid:105)(cid:104) x I (cid:48) ,j (cid:48) | ) I (cid:48) , ,j (cid:48) ( | x I,j (cid:105)(cid:104)− x I,j | ) I, ,j ( I (cid:48) , , j (cid:48) ) (cid:54) = ( I, , j ) . (A.34)If we use up the remaining Kronecker deltas over cells,then the decimation of internal links will be complete.However, there is not necessarily one Kronecker delta percell since the number of such constraints that get enforceddepends on the relative location of link ( I (cid:48) , , j (cid:48) ) to link( I, , j ). For imagine that ( I, , j ) is fixed. If ( I (cid:48) , , j (cid:48) ) islocated at...1. Any of the two links on the shared boundary of I and I [ j ] , then I (cid:48) and I (cid:48) [ j (cid:48) ] coincide with I and I [ j ] precisely. Therefore, Kronecker deltas will not exist for these two cells. And (cid:80) i I and (cid:80) i I [ j ] will be leftundone;2. Any of the twelve links on the outer perimeter ofthe union of I and I [ j ] , then either I (cid:48) or I (cid:48) [ j (cid:48) ] willcoincide with one of I and I [ j ] . Therefore, a Kro-necker delta will not exist for that doubly-coveredcell. Say this is I . Then (cid:80) i I will be left undone;3. Any other link on the lattice, then I (cid:48) and I (cid:48) [ j (cid:48) ] willnot overlap I and I [ j ] at all. Therefore, a Kronecker4 I I [ j ] I I [ j ] FIG. 6. Subcases 1a and 1b. Links ( I, , j ) and ( I (cid:48) , , j (cid:48) )are represented by the bold and dashed links, respectively. InSubcase 1a they are the same link. delta exists for all cells.In terms of these three general cases, let us write H (2) µ = h (cid:88) I,j (Case 1 + Case 2 + Case 3) . (A.35) a. Case 3 If every cell is constrained to be in itsground state, then there cannot be an intermediate ex-cited state (i.e., (cid:80) i I | some i I (cid:54) = 0 is null). Thus,Case 3 = 0 . (A.36) b. Case 1 There are two subcases: (a) ( I (cid:48) , , j (cid:48) ) =( I, , j ), and (b) ( I (cid:48) , , j (cid:48) ) (cid:54) = ( I, , j ), such thatCase 1 = Subcase 1a + Subcase 1b . (A.37)See Fig. 6. For all external links that are not onthe two neighboring cells that share the links ( I, , j )and ( I (cid:48) , , j (cid:48) ), we obtain the identity operator since (cid:80) x b ( | x b (cid:105)(cid:104) x b | ) b = 11 b . There are fourteen link sums leftto do.Consider Subcase 1a. Define C j,i I ,i I [ j ] (Φ I , Φ I [ j ] ) = ζ ji I (Φ I ) ζ [ j ] i I [ j ] (Φ I [ j ] ) × R i I ,i I [ j ] (Φ I , Φ I [ j ] ) . (A.38)This is, essentially, the second line in Eq. (A.34). Sum-ming over the fourteen external links and using the iden- tity µ xb µ xb = 11 b givesSubcase 1a = (cid:32) (cid:88) i I [ j ] =1 (cid:12)(cid:12)(cid:12) i I =0 + (cid:88) i I =1 (cid:12)(cid:12)(cid:12) i I [ j ] =0 + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 (cid:33)(cid:34) S j,i I ,i I [ j ] ++++ (cid:89) b ∈ I b · (cid:89) b ∈ I [ j ] b + S j,i I ,i I [ j ] + −− + (cid:89) b ∈ I µ xb · (cid:89) b ∈ I [ j ] µ xb − S j,i I ,i I [ j ] ++ −− (cid:89) b ∈ I µ xb · (cid:89) b ∈ I [ j ] b − S j,i I ,i I [ j ] + − + − (cid:89) b ∈ I b · (cid:89) b ∈ I [ j ] µ xb (cid:35) , (A.39)where S j,i I ,i I [ j ] σ σ σ σ = 14 (cid:104) σ C j,i I ,i I [ j ] (+ , +)+ σ C j,i I ,i I [ j ] (+ , − )+ σ C j,i I ,i I [ j ] ( − , +)+ σ C j,i I ,i I [ j ] ( − , − ) (cid:105) . (A.40)Further consolidation is achieved by defining S = (cid:88) i I [ j ] =1 S j, ,i I [ j ] ++++ + (cid:88) i I =1 S j,i I , + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 S j,i I ,i I [ j ] ++++ , (A.41a) S = (cid:88) i I [ j ] =1 S j, ,i I [ j ] + −− + + (cid:88) i I =1 S j,i I , −− + + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 S j,i I ,i I [ j ] + −− + , (A.41b) S = (cid:88) i I [ j ] =1 S j, ,i I [ j ] ++ −− + (cid:88) i I =1 S j,i I , −− + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 S j,i I ,i I [ j ] ++ −− (A.41c)= (cid:88) i I [ j ] =1 S j, ,i I [ j ] + − + − + (cid:88) i I =1 S j,i I , − + − + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 S j,i I ,i I [ j ] + − + − . (A.41d)We have checked that expressions (A.41c) and (A.41d)are equivalent. Furthermore, none of the expressions for S , S , and S depend on the choice of parameter j .This is expected since the lattice remains unchanged by590 ◦ rotations, or reflections about a horizontal or verticalline. Letting Φ I = (cid:81) b ∈ I µ xb , we getSubcase 1a = S + S Φ I Φ I [ j ] − S (Φ I + Φ I [ j ] ) . (A.42)Next consider Subcase 1b. Define D j,i I ,i I [ j ] (Φ I , Φ I [ j ] ) = ζ ji I (Φ I ) ζ [ j ] i I [ j ] (Φ I [ j ] ) ζ j (cid:48) i I (Φ I ) ζ [ j (cid:48) ] i I [ j ] (Φ I [ j ] ) × R i I ,i I [ j ] (Φ I , Φ I [ j ] ) . (A.43)Summing over the fourteen external links and using theidentity ( |−(cid:105)(cid:104) + | − | + (cid:105)(cid:104)−| ) b = i µ yb = µ zb µ xb givesSubcase 1b = (cid:32) (cid:88) i I [ j ] =1 (cid:12)(cid:12)(cid:12) i I =0 + (cid:88) i I =1 (cid:12)(cid:12)(cid:12) i I [ j ] =0 + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 (cid:33) µ zI,j µ zI,j (cid:48) (cid:34) T j,i I ,i I [ j ] ++++ (cid:89) b ∈ I b · (cid:89) b ∈ I [ j ] b + T j,i I ,i I [ j ] + −− + (cid:89) b ∈ I µ xb · (cid:89) b ∈ I [ j ] µ xb − T j,i I ,i I [ j ] ++ −− (cid:89) b ∈ I µ xb · (cid:89) b ∈ I [ j ] b − T j,i I ,i I [ j ] + − + − (cid:89) b ∈ I b · (cid:89) b ∈ I [ j ] µ xb (cid:35) , (A.44)where T j,i I ,i I [ j ] σ σ σ σ = 14 (cid:104) σ D j,i I ,i I [ j ] (+ , +)+ σ D j,i I ,i I [ j ] (+ , − )+ σ D j,i I ,i I [ j ] ( − , +)+ σ D j,i I ,i I [ j ] ( − , − ) (cid:105) . (A.45)Once again, further consolidation is achieved by defining T = (cid:88) i I [ j ] =1 T j, ,i I [ j ] ++++ + (cid:88) i I =1 T j,i I , + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 T j,i I ,i I [ j ] ++++ , (A.46a) T = (cid:88) i I [ j ] =1 T j, ,i I [ j ] + −− + + (cid:88) i I =1 T j,i I , −− + + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 T j,i I ,i I [ j ] + −− + , (A.46b) T = (cid:88) i I [ j ] =1 T j, ,i I [ j ] ++ −− + (cid:88) i I =1 T j,i I , −− + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 T j,i I ,i I [ j ] ++ −− (A.46c)= (cid:88) i I [ j ] =1 T j, ,i I [ j ] + − + − + (cid:88) i I =1 T j,i I , − + − + (cid:88) i I =1 7 (cid:88) i I [ j ] =1 T j,i I ,i I [ j ] + − + − . (A.46d)We have checked that expressions (A.46c) and (A.46d)are equivalent, and that none of the expressions for T , T , and T depend on the choice of parameter j . Thus,Subcase 1b = µ zI,j µ zI,j (cid:48) [ T + T Φ I Φ I [ j ] − T (Φ I + Φ I [ j ] )] . (A.47) c. Case 2 There are twelve subcases such thatCase 2 = Subcase 2a + · · · + Subcase 2l . (A.48)Each subcase falls naturally into one of two groups basedon the location of ( I (cid:48) , , j (cid:48) ) on the perimeter of the rectan-gular region formed by I and I [ j ] : (a)–(d) have ( I (cid:48) , , j (cid:48) )as one of the four links on the short sides of the rectangle;(e)–(l) have ( I (cid:48) , , j (cid:48) ) as one of the eight links on the longsides of the rectangle. See Fig. 7.Consider Subcases 2a, b, e, f, g, h. It is possible toregard the link ( I (cid:48) , , j (cid:48) ) as ( I, , j (cid:48) ) if we identify cell I (cid:48) with I . Define E j,j (cid:48) ,i I (Φ I , Φ I [ j ] , Φ I [ j (cid:48) ] )= ζ ji I (Φ I ) ζ [ j ]0 (Φ I [ j ] ) ζ j (cid:48) i I (Φ I ) ζ [ j (cid:48) ]0 (Φ I [ j (cid:48) ] ) × (cid:104) R i I , (Φ I , Φ I [ j ] ) + R i I , (Φ I , Φ I [ j (cid:48) ] ) (cid:105) . (A.49)Summing over the twenty external links gives6 I I [ j ] I I [ j ] I I [ j ] I I [ j ] I [ j′ ] I′I′
I I [ j ] I I [ j ] I [ j ] I I [ j ] I I I [ j ] I′ I I [ j ] I′ I I [ j ] I′ I I [ j ] I′I [ j′ ] I [ j′ ] I [ j′ ] I [ j′ ] I [ j′ ] FIG. 7. Subcases 2a through 2l. Links ( I, , j ) and ( I (cid:48) , , j (cid:48) ) are represented by the bold and dashed links, respectively. (cid:88) i I =1 µ zI,j µ zI,j (cid:48) (cid:34) U j,j (cid:48) ,i I ++++ | ++++ (cid:89) b ∈ I [ j (cid:48) ] b · (cid:89) b ∈ I b · (cid:89) b ∈ I [ j ] b + U j,j (cid:48) ,i I + − + −| + − + − (cid:89) b ∈ I [ j (cid:48) ] µ xb · (cid:89) b ∈ I b · (cid:89) b ∈ I [ j ] b − U j,j (cid:48) ,i I ++++ |−−−− (cid:89) b ∈ I [ j (cid:48) ] b · (cid:89) b ∈ I µ xb · (cid:89) b ∈ I [ j ] b − U j,j (cid:48) ,i I + − + −|− + − + (cid:89) b ∈ I [ j (cid:48) ] µ xb · (cid:89) b ∈ I µ xb · (cid:89) b ∈ I [ j ] b − U j,j (cid:48) ,i I ++ −−| ++ −− (cid:89) b ∈ I [ j (cid:48) ] b · (cid:89) b ∈ I b · (cid:89) b ∈ I [ j ] µ xb − U j,j (cid:48) ,i I + −− + | + −− + (cid:89) b ∈ I [ j (cid:48) ] µ xb · (cid:89) b ∈ I b · (cid:89) b ∈ I [ j ] µ xb + U j,j (cid:48) ,i I ++ −−|−− ++ (cid:89) b ∈ I [ j (cid:48) ] b · (cid:89) b ∈ I µ xb · (cid:89) b ∈ I [ j ] µ xb + U j,j (cid:48) ,i I + −− + |− ++ − (cid:89) b ∈ I [ j (cid:48) ] µ xb · (cid:89) b ∈ I µ xb · (cid:89) b ∈ I [ j ] µ xb (cid:35) , (A.50)where U j,j (cid:48) ,i I σ σ σ σ | σ σ σ σ = 18 (cid:104) σ E j,j (cid:48) ,i I (+ , + , +)+ σ E j,j (cid:48) ,i I (+ , + , − )+ σ E j,j (cid:48) ,i I (+ , − , +)+ σ E j,j (cid:48) ,i I (+ , − , − )+ σ E j,j (cid:48) ,i I ( − , + , +)+ σ E j,j (cid:48) ,i I ( − , + , − )+ σ E j,j (cid:48) ,i I ( − , − , +)+ σ E j,j (cid:48) ,i I ( − , − , − ) (cid:105) . (A.51)Define U j,j (cid:48) = (cid:88) i I =1 U j,j (cid:48) ,i I ++++ | ++++ (A.52a) U j,j (cid:48) = (cid:88) i I =1 U j,j (cid:48) ,i I + − + −| + − + − (A.52b)= (cid:88) i I =1 U j,j (cid:48) ,i I ++ −−| ++ −− (A.52c) U j,j (cid:48) = (cid:88) i I =1 U j,j (cid:48) ,i I ++++ |−−−− (A.52d) U j,j (cid:48) = (cid:88) i I =1 U j,j (cid:48) ,i I + − + −|− + − + (A.52e)= (cid:88) i I =1 U j,j (cid:48) ,i I ++ −−|−− ++ . (A.52f)We have checked that expressions (A.52b) and (A.52c),and expressions (A.52e) and (A.52f) are equivalent. Also, (cid:88) i I =1 U j,j (cid:48) ,i I + −− + | + −− + = 0 (A.53a) (cid:88) i I =1 U j,j (cid:48) ,i I + −− + |− ++ − = 0 . (A.53b) Thus, Subcase 2a, b, e, f, g, h= µ zI,j µ zI,j (cid:48) [ U j,j (cid:48) + U j,j (cid:48) Φ I [ j (cid:48) ] − U j,j (cid:48) Φ I [ j ] − U j,j (cid:48) Φ I − U j,j (cid:48) Φ I [ j (cid:48) ] Φ I + U j,j (cid:48) Φ I Φ I [ j ] ] . (A.54)Notice that the mapping j → [ j ] j (cid:48) → [ j (cid:48) ] I → I [ j ] I [ j ] → II [ j (cid:48) ] → I (cid:48) (A.55)converts the diagrams for Subcases 2a, b, e, f, g, andh into the diagrams for Subcases 2c, d, i, j, k, and l,respectively, up to a reflection in the plane. However,this reflection does not affect the mathematical expres-sion since it does not alter the relative locations of links.Applying rules (A.55) to Eq. (A.54) and rememberingthat now we cannot identify cells I (cid:48) and I , yieldsSubcase 2c, d, i, j, k, l= µ zI,j µ zI (cid:48) ,j (cid:48) [ U [ j ] , [ j (cid:48) ]1 + U [ j ] , [ j (cid:48) ]2 Φ I (cid:48) − U [ j ] , [ j (cid:48) ]2 Φ I − U [ j ] , [ j (cid:48) ]3 Φ I [ j ] − U [ j ] , [ j (cid:48) ]4 Φ I (cid:48) Φ I [ j ] + U [ j ] , [ j (cid:48) ]4 Φ I [ j ] Φ I ] . (A.56)
5. Renormalized spin- operators Since each link B of the new lattice corresponds to twolinks of a cell, say b and b (cid:48) , a prescription is needed todefine new link operators { X B , Z B } from the old ones { µ xb , µ zb } and { µ xb (cid:48) , µ zb (cid:48) } . Although it appears that ex-ternal links b and b (cid:48) have two qubits worth of freedom,Fradkin and Raby showed that gauge invariance restrictsthis freedom to just one. Here we translate their argu-ment into a format that comports with the perturbativeframework of Hirsch and Mazenko.8Consider the site at the midpoint of the bound-ary between cells I and I + ˆ x in Fig. 5. The gen-erator of gauge transformations at this site is G σ = σ zI, , σ zI, , σ zI, , σ zI +ˆ x, , . Physical states in Hilbert spacemust satisfy G σ = +1. Since [ G σ , H σ ] = 0, the bi-vector-valued quantity T = (cid:80) i | i (cid:105) ⊗ | µ i (cid:105) must satisfy T = G σ T . (A.57)Whence does T come? Briefly, according to Ref. 8, T is the lowest-order-in- g approximation to T [ µ | σ ], whichis a vector-to-vector projection operator that allows therenormalized Hamiltonian to be computed by a trace. In Eq. (2.5) T is seen to satisfy a normalization con-straint, Tr σ ( T T † ) = 11 µ . This is equivalent to11 µ = Tr σ ( G σ T T † ) = (cid:88) i,i (cid:48) (cid:104) i (cid:48) | G σ | i (cid:105)| µ i (cid:105)(cid:104) µ i | , (A.58)with the second equality following from (cid:104) i | α (cid:105) = 0. Simplyput, the identity operator in the reduced Hilbert spacecan be computed by restricting G σ to the subspace {| i (cid:105)} of lattice eigenstates formed by cell ground states; theorthogonal subspace {| α (cid:105)} is completely overlooked bythe projector T . Using Eq. (A.14), G σ | i (cid:105) = | − x I, , − x I, ) (cid:105) I | − x I +ˆ x, , − x I +ˆ x, ) (cid:105) I +ˆ x | − x I, (cid:105) I, , | − x I, (cid:105) I, , · · · , (A.59)where ellipses represent cell and external link states thatare not acted upon by G σ . Note that x I +ˆ x, = x I, and x I +ˆ x, = x I, . When (cid:104) i (cid:48) | is applied to Eq. (A.59), the re-sulting matrix element is nonzero only if x (cid:48) I, , = − x I, , and x (cid:48) I, , = − x I, , , with all other x (cid:48) b = x b . If thisholds, then the matrix element is unity. This constraintcollapses the sum over i (cid:48) . Performing the remaining sumover i then leads to (cid:88) x I, = ± | x I, (cid:105)(cid:104)− x I, | (cid:88) x I, = ± | x I, (cid:105)(cid:104)− x I, | = µ zI, µ zI, , (A.60)with implicit identities on all other external links. Thus,we have proven that µ zI, µ zI, = 11 µ . (A.61)Therefore, µ zI, = µ zI, on the reduced Hilbert space ofthe thinned lattice.Define renormalized link operators X B = µ xb µ xb (cid:48) , Z B = ( µ zb + µ zb (cid:48) ) / , (A.62)where b and b (cid:48) are the two contiguous links from the sameedge B of a cell. See Fig. 8. It is easily checked that theyreproduce the Pauli algebra X B Z B = − Z B X B , X B =11 B , and Z B = 11 B . Note that the identity µ zb µ zb (cid:48) = 11 B implies that Z B = µ zb = µ zb (cid:48) .Cells become plaquettes in the thinned lattice. Forthese we define the magnetic flux in the obvious way,Φ I = (cid:89) B ∈ ∂I X B . (A.63) IMP JNQK L bb′B B B B B B B FIG. 8. Cells involved in effective operators gen-erated at second order in the intercell coupling. Cells
I, J, K, L, M, N, P, Q become the plaquettes of the thinnedlattice once internal links (dashed) are decimated. Externallinks (solid) b and b (cid:48) comprise the link B in the thinned lat-tice.
6. Renormalized Hamiltonian
In terms of the renormalized spin- operators,Eq. (A.29c) reads H (0) µ = (cid:15) c0 (+) + (cid:15) c0 ( − )2 N plaq − (cid:15) c0 ( − ) − (cid:15) c0 (+)2 (cid:88) I Φ I , (A.64)where N plaq is the number of plaquettes in the originallattice. And for Eq. (A.30e) we make a minor notationalchange: instead of referring to links as “ I, j ” we use “ B ,” H (1) µ = − h |(cid:104) − | + (cid:105) I | (cid:88) B Z B . (A.65)We are not quite ready to write a gauge-invariant ex-pression for H (2) µ . First, Eq. (A.35) needs to be assem-bled by combining all subcases from Cases 1 and 2. Sincepairs of contiguous links that form the edge of a cell willeventually become a single link B in the thinned lattice,it is convenient to reduce the scope of the sum as H (2) µ = h (cid:88) B (Case 1+Case 2) j =1 +(Case 1+Case 2) j =8 . (A.66)Note that we could have also picked the pair j = 2 , ,
5, or 6 ,
7. However, this choice is immaterial due tothe 90 ◦ rotation symmetry of the lattice. Also, the choice j = 1 , b, b (cid:48) in Fig. 8 whose sim-plified notation we henceforth employ. Accordingly, forCase 1 we have Subcase 1a = S + S Φ I Φ J − S (Φ I +Φ J )and Subcase 1b = T + T Φ I Φ J − T (Φ I + Φ J ) because µ zb µ zb (cid:48) = 11 B . For Case 2, we must be careful to properlyoverlay Fig. 8 onto each diagram shown in Fig. 7 so that9the cells labeled “ I ” coincide. For instance, consider Sub-case 2c with j = 1. Then I [ j ] = J , I (cid:48) = L , and j (cid:48) = 4.Therefore,(Subcase 2c) j =1 = Z B Z B [ U [1] , [4]1 + U [1] , [4]2 Φ L − U [1] , [4]2 Φ I − U [1] , [4]3 Φ J − U [1] , [4]4 Φ L Φ J + U [1] , [4]4 Φ J Φ I ] . (A.67)By using relations (A.1) we can set [1] = 4 and [4] = 1.For another example, consider Subcase 2f with j = 8.The diagram for this may be obtained by reflecting di-agram 2f in Fig. 7 about a horizontal line. OverlayingFig. 8 then gives I [ j ] = J , I [ j (cid:48) ] = P , and j (cid:48) = 6. There- fore,(Subcase 2f) j =8 = Z B Z B [ U , + U , Φ P − U , Φ J − U , Φ I − U , Φ P Φ I + U , Φ I Φ J ] . (A.68)In order to combine all subcases under Case 2 define V k = U , k + U , k + U , k + U , k (A.69)= U , k + U , k + U , k + U , k = U , k + U , k + U , k + U , k = U , k + U , k + U , k + U , k ,W k = U , k + U , k + U , k + U , k (A.70)= U , k + U , k + U , k + U , k , for k = 1 , . . . ,
4. Notice that V k are coefficients fromterms in which Z ’s reside on nearest-neighbor (i.e., diag-onally adjacent) links, whereas W k are coefficients fromterms in which Z ’s reside on next-nearest-neighbor (i.e.,directly opposite) links. The contribution from link B is(Case 1 + Case 2) B = 2( S + T ) + 2( S + T )Φ I Φ J − S + T )(Φ I + Φ J )+ Z B Z B [ V + V Φ M − V Φ J − V Φ I − V Φ M Φ I + V Φ I Φ J ]+ Z B Z B [ V + V Φ P − V Φ J − V Φ I − V Φ P Φ I + V Φ I Φ J ]+ Z B Z B [ V + V Φ Q − V Φ I − V Φ J − V Φ Q Φ J + V Φ J Φ I ]+ Z B Z B [ V + V Φ N − V Φ I − V Φ J − V Φ N Φ J + V Φ J Φ I ]+ Z B Z B [ W + W Φ K − W Φ J − W Φ I − W Φ K Φ I + W Φ I Φ J ]+ Z B Z B [ W + W Φ L − W Φ I − W Φ J − W Φ L Φ J + W Φ J Φ I ] . (A.71)Observe that not all of these terms are hermitian. For ex-ample, ( Z B Z B Φ K ) † = Φ K Z B Z B = − Z B Z B Φ K be-cause X B anticommutes with Z B . Such terms must dis-appear when we add similar contributions from the otherlinks. For instance, if we look at (Case 1+Case 2) B thenwe also get a term prefaced with Z B Z B . Combining liketerms eliminates non-hermitian operators. In particular,operators with coefficients V k and W k for k = 2 , B and B (cid:48) atright angles (denoted B ⊥ B (cid:48) ), H (2) µ contains operators of the form h Z B Z B (cid:48) ( V − V Φ I ) , (A.72)where plaquette I is bounded by links B and B (cid:48) . Wemight say that I sits in the “elbow” of the hook made by B and B (cid:48) .For any pair of next-nearest-neighbor links B and B (cid:48) directly opposite from each other (denoted B (cid:107) B (cid:48) ), H (2) µ contains operators of the form h Z B Z B (cid:48) ( W − W Φ I ) , (A.73)where plaquette I is bounded by links B and B (cid:48) . Wemight say that I is “sandwiched” in-between B and B (cid:48) .The full second-order correction is H (2) µ = 2 h ( S + T ) N plaq h ( S + T ) (cid:88) (cid:104) I,J (cid:105) Φ I Φ J − h ( S + T ) (cid:88) I Φ I + h V (cid:88) B ⊥ B (cid:48) Z B Z B (cid:48) − h V (cid:88) B ⊥ B (cid:48) Z B Z B (cid:48) Φ I elbow + h W (cid:88) B (cid:107) B (cid:48) Z B Z B (cid:48) − h W (cid:88) B (cid:107) B (cid:48) Z B Z B (cid:48) Φ I sandwiched , (A.74)0where (cid:104) I, J (cid:105) denotes nearest-neighbor plaquettes I and J . It is important to remark that all operators Z B Z B (cid:48) associated to hooks B ⊥ B (cid:48) are being summed over, eventhose that may be gauge-equivalent to others. The effec-tive operators generated by renormalization are depictedin Fig. 3.Inspection of Eq. (A.74) reveals five new gauge-invariant effective operators that have been generated bythe renormalization transformation. The coefficients ofthese new operators will influence those obtained by suc-cessive iterations of the decimation procedure. There-fore, we must go back and include these operators in H σ .We choose to append them to V σ in Eq. (A.2) so thatthe eigenvalue problem on each cell remains unchanged.Using the notation of Fig. 3,add to V σ = (cid:88) α =1 K α O α + F N plaq . (A.75)As in Refs. 8 and 5 we treat the coefficients K α as being O ( h ) since they are generated at second order in Hirsch–Mazenko perturbation theory. This means that we needonly compute the analogue of Eq. (A.28c),add to H ren µ = (cid:88) i,i (cid:48) (cid:104) i (cid:48) | (add to V σ ) | i (cid:105)| µ i (cid:105)(cid:104) µ i (cid:48) | . (A.76)For the identity operator, a single renormalization stepreduces the number of plaquettes by a factor of 4. There-fore, F (cid:48) = 4 F .Consider (cid:80) i,i (cid:48) (cid:104) i (cid:48) |O α | i (cid:105)| µ i (cid:105)(cid:104) µ i (cid:48) | . Generically, O α con-sists of a product of Z ’s and X ’s (or σ z ’s and σ x ’s in theoriginal Hilbert space). Depending on how this operatoris situated on the lattice some of the spin operators willact on internal links of a cell and some will act on ex-ternal links. Since operators on different links commute,we can always write O α as a product of operators—onereferring to internal links only and the other referringto external links only. Let O α = O int α O ext α . Since inter-nal links always belong to a specific cell, we may furtherdecompose O int α = (cid:81) I O int Iα , where I is the cell index.Then (cid:104) i (cid:48) |O α | i (cid:105) = (cid:89) I (cid:104) { x (cid:48) } ) |O int Iα | { x } ) (cid:105) I ×(cid:104) x (cid:48) | · · · (cid:104) x (cid:48) |O ext α | x (cid:105) · · · | x (cid:105) , (A.77)where all external links x and x (cid:48) are involved in the ma-trix element. If we now perform (cid:80) i (cid:48) , which amounts tosumming over all possible configurations of the externallinks x (cid:48) , then any external link not directly touched by O ext α will receive a Kronecker delta setting x (cid:48) = x . Since | µ i (cid:105)(cid:104) µ i (cid:48) | = (cid:81) | x (cid:105)(cid:104) x (cid:48) | , | x (cid:105)(cid:104) x (cid:48) | corresponding to untouchedlinks become | x (cid:105)(cid:104) x | . Next, when (cid:80) i is performed wemight expect to get (cid:80) x = ± | x (cid:105)(cid:104) x | = 11 for those untouchedexternal links. However, this is only true if another con-dition is met: those untouched external links must notlie on a cell that contains a touched link somewhere elseon its border or has O int α acting on it. We shall refer to B ′ I BB i)ii)iii) FIG. 9. How O generates effective operators. such cells as “touched” cells. Otherwise, there will bea non-unit matrix element factor sitting inside the sum (cid:80) x = ± . Our calculational algorithm can be stated as (cid:88) i,i (cid:48) (cid:104) i (cid:48) |O α | i (cid:105)| µ i (cid:105)(cid:104) µ i (cid:48) | = (cid:88) x . . . (cid:88) x (cid:124) (cid:123)(cid:122) (cid:125) directly touchedexternal links (cid:88) x . . . (cid:88) x (cid:124) (cid:123)(cid:122) (cid:125) untouchedexternal linkson a touched cell (cid:88) x (cid:48) . . . (cid:88) x (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) directly touchedexternal links (cid:89) touched cells I (cid:104) { x (cid:48) } ) |O int Iα | { x } ) (cid:105) I (cid:12)(cid:12)(cid:12) x (cid:48) = x for untouched links × (cid:104) x (cid:48) | · · · (cid:104) x (cid:48) | (cid:124) (cid:123)(cid:122) (cid:125) directly touchedexternal links O ext α | x (cid:105) · · · | x (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) directly touchedexternal links × | x (cid:105)(cid:104) x (cid:48) | · · · | x (cid:105)(cid:104) x (cid:48) | (cid:124) (cid:123)(cid:122) (cid:125) directly touchedexternal links · | x (cid:105)(cid:104) x | · · · | x (cid:105)(cid:104) x | (cid:124) (cid:123)(cid:122) (cid:125) untouchedexternal linkson a touched cell . (A.78)Direct computation of expression (A.76) is straight-forward using Eq. (A.78) but tedious. As an example,consider O . There are three qualitatively distinct waysit can cover the original lattice. See Fig. 9. In Case ( i )the hook lies entirely on internal links. Define χ = (cid:104) { A } ) | σ zI, , σ zI, , | { A }(cid:105) I . (A.79)It turns out that χ is a function of the flux Φ I = (cid:81) k =1 A k χ (+) = (cid:104) + | σ zI, , σ zI, , | + (cid:105) I , (A.80a) χ ( − ) = (cid:104) − | σ zI, , σ zI, , | − (cid:105) I . (A.80b)Thus, Case ( i ) = 4 K χ (+) + χ ( − )2 N plaq K χ (+) − χ ( − )2 (cid:88) I Φ I . (A.81)There is a factor of 4 because there are four differentorientations in which the hook can lie entirely inside theplaquette. In Case ( ii ) the hook can lie in one of fourways on the shared boundary between two plaquettes.So Case ( ii ) = 4 K |(cid:104) − | + (cid:105) I | (cid:88) B Z B . (A.82) In Case ( iii ) there is only a single orientation of the hookthat straddles all three plaquettes. SoCase ( iii ) = K |(cid:104) − | + (cid:105) I | (cid:88) B ⊥ B (cid:48) Z B Z B (cid:48) . (A.83)The contributions from O α for α = 2 , . . . , (cid:126)σ I,i, ,will be shortened to (cid:126)σ i . External-link spin operators willnot appear explicitly in the matrix elements so there is nochance of confusion. And we shall suppress the subscript I on matrix elements. We findadd to H ren µ = N plaq (cid:104) F + 2 K ( (cid:104) + | σ z σ z | + (cid:105) + (cid:104) − | σ z σ z | − (cid:105) ) + 2 K ( (cid:104) + | σ z σ z σ x σ x | + (cid:105) + (cid:104) − | σ z σ z σ x σ x | − (cid:105) )+2 K (cid:0) (cid:104) + | σ x σ x | + (cid:105) + ( (cid:104) + | σ x σ x | + (cid:105) + 2 (cid:104) + | σ x σ x | + (cid:105)(cid:104) − | σ x σ x | − (cid:105) + (cid:104) − | σ x σ x | − (cid:105) ) (cid:1)(cid:105) + (cid:88) B Z B (cid:104) K |(cid:104) − | + (cid:105)| + 4 K (cid:104) − | + (cid:105)(cid:104) − | σ x σ x | + (cid:105) + 4 K |(cid:104) − | + (cid:105)| + 4 K (cid:104) − | + (cid:105)(cid:104) − | σ x σ x | + (cid:105) (cid:105) + (cid:88) I Φ I (cid:104) K ( (cid:104) + | σ z σ z | + (cid:105) − (cid:104) − | σ z σ z | − (cid:105) ) + 2 K ( (cid:104) + | σ z σ z σ x σ x | + (cid:105) − (cid:104) − | σ z σ z σ x σ x | − (cid:105) )+2 K ( (cid:104) + | σ x σ x | + (cid:105) + (cid:104) + | σ x σ x | + (cid:105) − (cid:104) − | σ x σ x | − (cid:105) ) (cid:105) + (cid:88) B ⊥ B (cid:48) Z B Z B (cid:48) (cid:104) K |(cid:104) − | + (cid:105)| + K |(cid:104) − | + (cid:105)| ( (cid:104) + | σ x σ x | + (cid:105) + (cid:104) − | σ x σ x | − (cid:105) ) (cid:105) + (cid:88) B ⊥ B (cid:48) Z B Z B (cid:48) Φ I elbow (cid:104) K |(cid:104) − | + (cid:105)| ( (cid:104) + | σ x σ x | + (cid:105) − (cid:104) − | σ x σ x | − (cid:105) ) (cid:105) + (cid:88) (cid:104) I,J (cid:105) Φ I Φ J (cid:104) K ( (cid:104) + | σ x σ x | + (cid:105) − (cid:104) + | σ x σ x | + (cid:105)(cid:104) − | σ x σ x | − (cid:105) + (cid:104) − | σ x σ x | − (cid:105) ) (cid:105) (A.84)All matrix elements appearing above have been written interms of | ± (cid:105) as given by Eqs. (A.30c) and (A.30d). It isworth noting that expression (A.84) with the alternativedefinition | − (cid:105) I = | A = − , A = + , A = + , A = +) (cid:105) I would not be correct.The renormalized Hamiltonian to second order in thecoupling h is given by adding expressions (A.64), (A.65),(A.74), and (A.84). This yields H ren µ = − h (cid:48) (cid:88) B Z B − J (cid:48) (cid:88) I Φ I + K (cid:48) (cid:88) B ⊥ B (cid:48) Z B Z B (cid:48) + K (cid:48) (cid:88) B ⊥ B (cid:48) Z B Z B (cid:48) Φ I elbow + K (cid:48) (cid:88) B (cid:107) B (cid:48) Z B Z B (cid:48) + K (cid:48) (cid:88) B (cid:107) B (cid:48) Z B Z B (cid:48) Φ I sandwiched + K (cid:48) (cid:88) (cid:104) I,J (cid:105) Φ I Φ J + F (cid:48) N plaq , (A.85a)2where h (cid:48) = 2 |(cid:104) − | + (cid:105)| ( h − K − K ) − (cid:104) − | + (cid:105)(cid:104) − | σ x σ x | + (cid:105) ( K + K ) (A.85b) J (cid:48) = (cid:15) c0 ( − ) − (cid:15) c0 (+)2 + 4 h ( S + T ) + 2 K ( (cid:104) − | σ z σ z | − (cid:105) − (cid:104) + | σ z σ z | + (cid:105) )+2 K ( (cid:104) − | σ z σ z σ x σ x | − (cid:105) − (cid:104) + | σ z σ z σ x σ x | + (cid:105) )+2 K ( (cid:104) − | σ x σ x | − (cid:105) − (cid:104) + | σ x σ x | + (cid:105) − (cid:104) + | σ x σ x | + (cid:105) ) (A.85c) K (cid:48) = h V + |(cid:104) − | + (cid:105)| (cid:0) K + K ( (cid:104) + | σ x σ x | + (cid:105) + (cid:104) − | σ x σ x | − (cid:105) ) (cid:1) (A.85d) K (cid:48) = − h V + K |(cid:104) − | + (cid:105)| ( (cid:104) + | σ x σ x | + (cid:105) − (cid:104) − | σ x σ x | − (cid:105) ) (A.85e) K (cid:48) = h W (A.85f) K (cid:48) = − h W (A.85g) K (cid:48) = h ( S + T ) + K ( (cid:104) + | σ x σ x | + (cid:105) − (cid:104) − | σ x σ x | − (cid:105) ) (A.85h) F (cid:48) = (cid:15) c0 (+) + (cid:15) c0 ( − )2 + 2 h ( S + T ) + 4 F + 2 K ( (cid:104) + | σ z σ z | + (cid:105) + (cid:104) − | σ z σ z | − (cid:105) )+2 K ( (cid:104) + | σ z σ z σ x σ x | + (cid:105) + (cid:104) − | σ z σ z σ x σ x | − (cid:105) )+2 K (cid:0) (cid:104) + | σ x σ x | + (cid:105) + ( (cid:104) + | σ x σ x | + (cid:105) + (cid:104) − | σ x σ x | − (cid:105) ) (cid:1) . (A.85i)The matrix elements appearing above may be calculatedusing the cell-basis representations for internal σ z matri-ces given in Eqs. (A.15), and σ x σ x = , (A.86a) σ x σ x = . (A.86b)Eqs. (A.85) are recursion relations for the operator co-efficients in the Hamiltonian. We have checked that theyare equivalent to the recursion relations derived by Hirschin Ref. 5 for the quantum Hamiltonian of the square-lattice Ising model in a transverse magnetic field. Thischeck is accomplished using a duality transformation. The ’t Hooft disorder operator, given by a string of σ z ’sin the lattice gauge theory, corresponds to the order pa-rameter, σ x , in the Ising model. And the magnetic fluxoperator, given by a product of σ x ’s around a plaquette,maps to the transverse field operator, σ z , living at thesite dual to the plaquette.It is certainly more challenging to obtain the recursionrelations in the lattice gauge theory than in the Ising model—gauge invariance enlarges the number of possiblestates, and additional formalism is necessary to restrictto the gauge-invariant sector. For instance, in the formerone is forced to consider the cellular magnetic flux Φ I as potentially dependent on eight boundary conditions,whereas in the Ising model no such boundary conditionsare required. The fact that our results agree with thoseobtained from the simpler (non-gauged) formulation ofthe theory is a reassuring check that we correctly imple-mented the Hirsch–Mazenko procedure.The renormalized Hamiltonian given by expression(A.85a) maintains gauge invariance on the thinnedsquare lattice which has a quarter as many plaquettesas the original lattice. Local gauge transformations aredefined by operators associated to the sites or verticesbetween links. At any site (cid:126)r in the thinned lattice thegenerator is G (cid:126)r = (cid:89) links B emergingfrom (cid:126)r Z B . (A.87) G (cid:126)r commutes with H ren µ .
7. Critical point, fixed points, and eigenvalues
Let us analyze the recursion relations given byEqs. (A.85). Treating h as an energy scale, we shallwork with dimensionless couplings grouped into the tu-ple ( J/h, K /h, . . . , K /h ). This quantity, which we mayabbreviate as ( J/h, (cid:126)K/h ) is a six-dimensional real-valuedvector. By starting at an arbitrary vector, iterations ofthe recursion relations in the form(
J/h, (cid:126)K/h ) (cid:55)→ ( J (cid:48) /h (cid:48) , (cid:126)K (cid:48) /h (cid:48) ) (A.88)yield a sequence of points which describe a “flow” of theHamiltonian defined over successively thinner lattices.3 non tri v i a l fi xed po i n t o u t f l o w c r iti ca l s u rf ace K/h → i n f l o w J/h ( , ) → ou t f l o w con fi n i ng phase c riti ca l po i n t e l ec tri c fr ee phase FIG. 10. A heuristic picture of the sequence of itera-tions of the recursion relations visualized as a flow in thesix-dimensional space of dimensionless couplings. (cid:126)K/h repre-sents a vector of couplings K α /h for α = 1 , . . . ,
5. The criticalsurface has codimension 1. The outflow trajectory from thenontrivial fixed point to the origin is a line.
This flow should preserve the low-energy spectrum ofthe original lattice Hamiltonian. Note that Eq. (A.88)is obtained by dividing Eqs. (A.85c) through (A.85h) byEq. (A.85b).We study numerically flows that begin on the Ising axis(
J/h,(cid:126)
J/h ) c = 3 . . (A.89)For J/h > ( J/h ) c , flows have the property that | J/h | grows without bound. And for J/h < ( J/h ) c , flows ap-proach the origin (0 ,(cid:126) J/h ) c ,(cid:126)
0) is the criti-cal point, i.e., the intersection of the critical surface withthe Ising axis. A flow near this point will initially ap-proach (along an “inflow”) a nontrivial fixed point beforeveering away (hugging the “outflow”) toward the stablefixed points at infinity and the origin. See Fig. 10. The nontrivial and unstable fixed point is foundby applying Newton’s method to the beta function( J (cid:48) /h (cid:48) , (cid:126)K (cid:48) /h (cid:48) ) − ( J/h, (cid:126)K/h ). It is located at(
J/h, (cid:126)K/h ) ∗ = (2 . , − . , . , − . , . , − . . (A.90)Our results corroborate those obtained by Hirsch for thequantum Ising model in a transverse field. Renormalization for flows in the vicinity of the nontriv-ial fixed point are particularly simple since the recursionrelations may be linearized. Let us denote the vector(
J/h, (cid:126)K/h ) more succintly by κ . Then κ (cid:48) (cid:39) κ ∗ + R ( κ − κ ∗ ) , (A.91)where R = ∂κ (cid:48) /∂κ | κ = κ ∗ is the Jacobian matrix of par-tial derivatives evaluated at the fixed point. Our matrix R turns out not to be symmetric and therefore, not alleigenvalues are guaranteed to be real. The eigenvaluesare (ordered from largest to smallest magnitude):Λ = { . , . , . . , . − . , . , } . (A.92)Note that there is a complex conjugate pair of eigen-values. This strange feature was noted by Hirschand it seems to indicate an inconsistency in therenormalization-group equations . Notwithstanding thisblemish, if the left eigenvector corresponding to Λ k is de-noted e k , then we may form scaling variables by taking adot product: u k = e k · κ . These scaling variables renor-malize multiplicatively, i.e., u (cid:48) k = Λ k u k . Since Λ , . . . , Λ all have absolute value less than 1, the scaling variables u , . . . , u renormalize to zero. These five coordinatescorrespond to the irrelevant directions along the criticalsurface that guide flows into the nontrivial fixed point.However, since Λ >
1, the scaling variable u is relevantand iterations of the recursion relations will tend to makethis coordinate grow. Thus, the eigenvector e must de-fine the outflow trajectory in the linear space around thenontrivial fixed point. ∗ paik [email protected] T. D. Schultz, D. C. Mattis, and E. H. Lieb, Rev. Mod.Phys. , 856 (1964). E. H. Fradkin and L. Susskind, Phys. Rev.
D17 , 2637(1978). E. H. Fradkin and S. Raby, Phys. Rev.
D20 , 2566 (1979). D. C. Mattis and J. Gallardo, Journal of Physics C: SolidState Physics , 2519 (1980). J. E. Hirsch, Phys. Rev. B , 3907 (1979). V. Privman, P. C. Hohenberg, and A. Aharony,
UniversalCritical-Point Amplitude Relations, Phase Transitions andCritical Phenomena (edited by Domb, C. and Lebowitz, J.L.) , Vol. 14 (Academic Press Limited, 1991). R. Jullien, P. Pfeuty, J. N. Fields, and S. Doniach, Phys.Rev. B , 3568 (1978). J. E. Hirsch and G. F. Mazenko, Phys. Rev. B , 2656(1979). At order g , it is easy to show analytically thatlim J/h →∞ ζ = and lim J/h →∞ η = 1. Specifically, ∆ is everything on the right-hand side ofEq. (A.85i) with 4 F set to 0 and h set to 1. J. Cardy,
Scaling and Renormalization in StatisticalPhysics (Cambridge University Press, 1996). A. M. Ferrenberg, J. Xu, and D. P. Landau, Phys. Rev. E , 043301 (2018). J. B. Kogut, Rev. Mod. Phys. , 659 (1979). S. D. Drell, M. Weinstein, and S. Yankielowicz, Phys. Rev.D , 1769 (1977). J. S´olyom, Phys. Rev. B , 230 (1981). Take, for instance, σ zI, , H I ( x I, , x I, ) σ zI, , = H I ( − x I, , − x I, ) and apply it to the state σ zI, , | ψ ( x I, , x I, ) (cid:105) , where | ψ ( x I, , x I, ) (cid:105) is an eigen-state of H I ( x I, , x I, ) with eigenvalue (cid:15) c ( x I, , x I, ). Sincethis becomes H I ( − x I, , − x I, )( σ zI, , | ψ ( x I, , x I, ) (cid:105) ) = (cid:15) c ( x I, , x I, ) σ zI, , | ψ ( x I, , x I, ) (cid:105) , but (cid:15) c ( x I, , x I, ) = (cid:15) c ( − x I, , − x I, ), we must have σ zI, , | ψ ( x I, , x I, ) (cid:105) = | ψ ( − x I, , − x I, ) (cid:105) . In Eq. (2.3) of Ref. 8, the projection is written as H ren µ =Tr σ ( H σ T [ µ | σ ] T † [ µ | σ ]). Our couplings h , J , K , K , K , K , K , and F are dual toHirsch’s couplings ∆, (cid:15) , − µ/ α , − δ , λ , − β , and d , respec-tively. See Ref. 5. Minus signs account for different conven-tions used to define renormalized operators. The reason fora factor of a half in K = − µ/ Z B Z B (where B ⊥ B ) is gauge-equivalent to Z B Z B if B , B , B ,and B are four links meeting at the same site. It shouldbe also noted that errant signs and minor typos exist inHirsch’s recursion relations. See Eqs. (A1) and (A2) in theappendix of Ref. 5. A recomputation of the renormalizedHamiltonian in the tranverse field Ising model reveals thefollowing corrections. In the equation for (cid:15) n +1 , all instancesof β n should have the opposite sign as the one written. Inthe equation for β n +1 , there should be a minus sign insteadof a plus sign in front of the . In the expression for B ,1 / ( E − E n ) ought to be 1 / ( E − E n ). In the expressionfor C , ( n, n (cid:48) ) (cid:54) = (0 ,
1) ought to read ( n, n (cid:48) ) (cid:54) = (1 , Hirsch’s fixed point is given by Eq. (9) in Ref. 5, but webelieve there is a typo: the values for λ/ ∆ and δ/ ∆ shouldbe swapped. His critical coupling is given by Eq. (10) andit is 3 ..