PHYSICAL REVIEW LETTERS 121, 041602 (2018)

Parity Cancellation in Three-Dimensional QED with a Single Massless Dirac Fermion

† Nikhil Karthik1,* and Rajamani Narayanan2, 1Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA 2Department of Physics, Florida International University, Miami, Florida 33199, USA (Received 23 March 2018; revised manuscript received 19 May 2018; published 26 July 2018)

We study three-dimensional noncompact QED with a single two-component massless fermion and two infinitely massive regulator fermions of half the charge using the lattice overlap formalism. The parity anomaly is expected to cancel exactly between the massless and regulator fermions in the continuum, but this cancellation is inexact on a lattice akin to lattice chiral gauge theories. We show nonperturbatively that parity-breaking terms vanish in the continuum limit at any finite volume. We present numerical evidence that the resulting parity-invariant theory spontaneously breaks parity in the infinite volume limit.

DOI: 10.1103/PhysRevLett.121.041602

Introduction.—The standard model of particle physics The odd-dimensional analog to chiral anomalies is parity is anomaly free due to an exact nontrivial cancellation of anomaly [9–12] and this also can be discussed geometri- gauge anomalies [1] from different representations to all cally [13]. In this Letter, we consider a three-dimensional orders of perturbation theory. Chiral anomalies outside analog to the chiral gauge theories, where there is a nontrivial perturbation theory can be discussed geometrically [2] and cancellation of parity anomaly between massless fermions the relation between consistent and covariant currents [3] and infinitely massive fermions, which is a property unique plays a central role. Such fundamental issues should be to three dimensions. The theory we consider is an Abelian addressed in any nonperturbative formalism of chiral gauge Uð1Þ with one massless Dirac fermion of theories. The overlap formalism of chiral gauge theories on charge q and two infinitely massive fermions of charges the lattice [4] was motivated [5] by an attempt to regularize q=2 in a three-torus with physical size, l3. This corresponds a specific chiral gauge theory using an infinite number of to the Euclidean continuum theory, with an implicit Pauli-Villars fields [6] and the ability to use domain walls , to create a chiral zero mode [7]. In order to discuss the problem of chiral anomalies in a gauge covariant and 2 1 L ¼ ψ¯ ð∂ þ Þψ −q iϵ ∂ þ μν ð Þ geometric manner, a two-form in the space of gauge fields = iq=A 8π μνρAμ νAρ 4FμνF ; 1 defined through the curl of the difference between the covariant and consistent currents was introduced within written in standard notation in units where the coupling the overlap formalism in Ref. [8], and it was identified g2 ¼ 1 to be Berry’s curvature. Two sources contribute to this constant . This theory has phenomenological rel- ’ evance to the low-energy physics of fractional quantum Hall Berry s curvature for a chiral fermion in an anomalous – representation—the first is due to the genuine continuum effect at half-filled Landau level [14 16].Likeineven gauge anomaly that cannot be removed, and the second is dimensions, lattice regularization of this theory within the overlap formalism [4,17,18] does not succeed in an exact due to the spatial smearing of the anomalous contribution cancellation of the parity anomaly. A salient result in this due to finite lattice spacing. There is just the contribution due to smearing in an anomaly free chiral theory which Letter is the numerical evidence for the restoration of parity can only be removed by fine-tuning the irrelevant terms in invariance in the continuum at any finite physical volume without the need for fine-tuning the fermion action, which fermion action on the lattice [8]. The exceptions to the fine- tuning are QCD-like vector theories where the anomaly suggests a similar situation to hold in even dimensional cancellation is trivial. chiral gauge theories as anticipated in Ref. [8]. This will also establish the existence of such three-dimensional theories outside perturbation theory. We will then present a numerical study of this theory in the infinite volume limit and provide Published by the American Physical Society under the terms of evidence for spontaneous breaking of parity. the Creative Commons Attribution 4.0 International license. Modus operandi.—As is standard in lattice field theory, Further distribution of this work must maintain attribution to 3 3 the author(s) and the published article’s title, journal citation, we discretize the physical volume l using L lattice points and DOI. Funded by SCOAP3. with the lattice spacing being l=L [19]. The continuum

0031-9007=18=121(4)=041602(5) 041602-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 121, 041602 (2018) limit is achieved by taking the L → ∞ limit at fixed value 1 þ Vθ − ðθÞ¼ Sg ð Þ of l. For the Abelian theory, the dynamical real lattice pþ det 2 e ; 6 variables are θμðnÞ at the link connecting the lattice point at n to n þ μˆ. The lattice regularized partition function of the and consider the phase eiA to be part of the observables. model in Eq. (1) using the overlap formalism [17,18,20] is Our first aim is to study the distribution of A generated at a Z given l and L and show that the distribution has a tendency − ðθÞ 1 þ Vθ 2 † to approach a delta function for all l as we take L → ∞.As ðl Þ¼ ½ θ Sg ð Þ Z ;L d e det det V1θ; 2 2 2 the lattice spacing increases with l in a range of numeri- cally feasible values of L, we can only provide reasonable numerical evidence for parity anomaly cancellation over a where SgðθÞ is the noncompact gauge action on the lattice 2 limited but wide range of l. Our second aim in this Letter (obtained by discretizing the Fμν term). The unitary is to assume that parity anomaly cancellation holds for all operator Vqθ depends on the compact link variables l q θ ðnÞ values of and study the infrared physics of the model in UμðnÞ¼eiq μ q where is the charge of the fermion Eq. (1) using pþðθÞ as the measure. ¼ 1 coupled to the gauge field. We have set q in Eq. (1) and Anomaly cancellation.—Using the LAPACK subroutines the first determinant factor realizes the effective action [26], we determined the phase AðθÞ. Figure 1 shows the obtained by integrating out the massless fermion in Eq. (1) distribution PðAÞ of AðθÞ as sampled using pþðθÞ in three and the second determinant factor realizes the Chern- panels, top to bottom, for l ¼ 4, 32, 200, respectively. Simons term in Eq. (1) as induced by an infinitely massive Within each panel for a fixed l, the different symbols fermion. correspond to different lattice spacings. Due to the parity- 2A If we define the induced action q from the infinite invariant measure pþðθÞ, the distributions are almost 2 mass fermion via, det Vqθ ≡ exp ð2iq AqÞ, then we expect symmetric with small deviations resulting from finite AqðθÞ to be independent of q for smooth gauge fields statistics. We notice from the l ¼ 4 and 32 panels that [11,12,21,22], and be the same as the level-one Chern- PðAÞ gets sharper as one approaches the continuum limit Simons action. If we perform the Euclidean parity trans- L → ∞. However, this approach of the width of the → † distribution to zero is hard to see in the l ¼ 200 panel, formation, under which Vqθ Vqθ, the path integral in Eq. (2) transforms to and it is understandable since the finest lattice spacing (L ¼ 16)atl ¼ 200, where we were able to compute A is Z 5.4 times larger than the one at l ¼ 32 (L ¼ 14). By − ðθÞ 1 þ Vθ 2 † −2 AðθÞ ðl Þ¼ ½ θ Sg i ð Þ A l Z ;L d e det det V1θe ; 3 putting together the data for from all and L,wenow 2 2 justify that the distributions at larger l will indeed get sharper at prohibitively large values of L. Since one expects where the remnant phase A to be a volume integral of local irrelevant terms, we show the variance per unit physical AðθÞ¼A1ðθÞ − A1ðθÞ: ð4Þ −3 2 volume, l VarðAÞ, as a function of lattice spacing, l=L,

Parity anomaly cancellation in the continuum means that 2A ¼ 0 or equivalently, A ¼ nπ for n ¼ 0, 1 as L → ∞. On the lattice, however, the nontrivial anomaly cancellation between two different charges will result in 2AðθÞ being zero only on classically smooth backgrounds. An ensemble of gauge field configurations on the lattice away from the continuum limit will not be smooth and we do not expect 2AðθÞ¼0ðmod2πÞ, leading to 1þVθ 1þVθ 2 † ¼ iAðθÞ; A ∈ ð−π π ð Þ det det V1θ det e ; ; 5 2 2 2 which forms the core of the problem addressed in this Letter. Our strategy can be summarized as follows. Using the rational hybrid Monte Carlo (RHMC) [23–25] method, an algorithm based on molecular dynamics evolution, we FIG. 1. The distributions of AðθÞ at different physical volumes numerically simulate the theory on the lattice using the l3 are shown in the three panels. The different symbols positive definite measure correspond to different L.

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which is a fermion-induced pseudovector current in lattice units, and the expectation value of its magnitude is EqðnÞ¼ hJqðnÞ JqðnÞi qðnÞ · þ. One expects Ji to depend locally on the flux ∼ϵijkΔjθk, but need not be ultralocal and get smeared around n as discussed in Ref. [8]. In the absence of such an ultralocality, EðnÞ¼E1ðnÞ − E1=2ðnÞ will not vanish at finite lattice spacing but it must vanish faster than E1ðnÞ and E1=2ðnÞ as one approaches the continuum. In Fig. 3, we put together the data from all l and L for E1 and n FIG. 2. The variance (left panel) and the height of the E1=2 at an arbitrarily chosen , and show it as a function of distribution at A ¼ 0 (right panel) for PðAÞ, both scaled by lattice spacing l=L. The data from different values of l fall appropriate powers of l, are shown as functions of lattice on the same curve due to the local nature of this observable. spacing. The lattice spacing scaling of E1 and E1=2 is l=L, the same as the average local energy density. With this combined data, we see that E falls off with the lattice spacing like 3 in the left panel of Fig. 2. The data points of the same ðl=LÞ , faster than E1 or E1=2 by two powers of lattice colored symbol belong to a fixed value of l but differ in L, spacing, ensuring again that the theory will be parity- while different colored symbols correspond to different l invariant at all values of l studied here. as specified near them. The data approximately falls on a Having demonstrated the path integral measure is −1 universal curve, with VarðAÞ ∼ L at smaller l=L. On the anomaly free in the continuum limit, it is also imperative right panel of Fig. 2, we show the scaled peak height of that we show the vacuum expectation values of parity-odd 3=2 the distribution, l PðA ¼ 0Þ, as a functionpffiffiffiffi of l=L. The observables vanish in the continuum limit. Decomposing approximate data collapse suggests a L increase in the any observable O into its parity-even and odd components O O peak-height at smaller l=L. As expected, higher order e and o, respectively, its expectation value can be effects in lattice spacing come into play in both figures for written as larger l=L. Based on these empirical observations, we find hO ðθÞ AðθÞi hO ðθÞ AðθÞi reasonable evidence for PðAÞ to approach a delta function e cos þ o sin þ hOðθÞi ¼ þ i : ð8Þ in the continuum limit at a fixed l and it is important that hcos AðθÞiþ hcos AðθÞiþ one takes the continuum limit before taking the infinite volume limit. We want to show that in the continuum limit, the parity- We now discuss the sign of the fermion determinant. The even first term on the right-hand side becomes hOeiþ and distribution PðAÞ on the coarser lattices, such as the one at the parity-odd second term vanishes. We consider the l ¼ 200 ð−π π , covers the entire range ; , but still remains correction Ce þ iCo ¼hOi − hOeiþ as a function of L. peaked at zero. Based on the arguments above, this implies For O, we used the dimensionless lowest positive eigen- þ that the distribution in the continuum limit will be peaked value λ1 ðθÞl of the inverse of massless Hermitian overlap A π −1 around zero, in spite of values of close to being allowed Dirac propagator, iG ðθÞL ¼ ið1 þ Vθ=1 − VθÞL, at dif- in the essentially continuous molecular dynamics evolution ferent L. In Fig. 4, we show the decreasing behavior of of gauge fields used by the RHMC algorithm on coarser both Co and Ce at different fixed l,asL is increased. The lattices. In principle, we could have found a separation of different colored symbols in the plot belong to different l. our ensemble into two sectors on coarser lattice spacings At any finite L, C is significantly nonzero and indeed Að ¼ ∞Þ¼0 π o (corresponding to L and ) easily identi- decreases when the lattice spacing is made smaller. On finer ðAÞ fied by a doubly peaked P . In this case, it would have lattices, a distinct L−Δ behavior with an empirical value been necessary to have a zero of the fermion determinant along the RHMC’s canonical evolution as the continuum limit is approached. Since we did not find this to be the case, our result is consistent with the absence of topological zero modes in odd-dimensional space without a boundary [27,28]. In this manner, we have succeeded in demonstrat- ing that Eq. (2) has a parity invariant as well as an effectively positive measure in the continuum. Another quantity relevant to the anomaly cancellation is

q δ ðnÞ¼ A ðθÞ ð Þ FIG. 3. The dependence of E1, E1 2 and their difference, E,on Ji δθ ðnÞ q ; 7 = i lattice spacing l=L.

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FIG. 4. The approach of C (left) and C (right) to 0 in the o e λ Σ ðlÞl3 continuum limit. FIG. 5. Comparison of the distributions of i i (symbols connected by dotted lines), and the RMT eigenvalues zi (solid curves). The red, green, blue, and purple curves correspond to i ¼ 1, 2, 3, 4, respectively. Top: Volume dependence at fixed Δ ≈ 1.5 is seen. For the data at larger l=L, a downward L ¼ 20. Bottom: Lattice spacing dependence at l ¼ 250. curvature is seen implying the asymptotic values of Δ will be greater than what can be extracted from the data (which then ΣiðlÞ for different i should approach the same nonzero is about 1.2). On the other hand, the ultraviolet physics of value Σ (the value of the condensate) as l → ∞. anomaly cancellation seems to decouple from the infrared Figure 5 shows a comparison of the distributions of parity-even expectation values, as seen from the fact that Ce l3Σ ðl Þ 1 þ þ the scaled, four low-lying Dirac eigenvalues, i ;L is much less than 0.1% of h2 ½λ1 ðθÞþλ1 ðθpÞliþ (and λiðθ; l;LÞ, to the distributions from the RMT, which are about 2 to 3 orders of magnitude lesser than the corre- shown as solid curves in the plots. The top panel shows the l sponding Co) in the range of we studied. In fact, for volume dependence of the distributions at a fixed number 10 ≈ 0 1 5–σ L> , Ce within a . error range. of lattice points L ¼ 20. One can see the distributions Spontaneous symmetry breaking of parity.—Having approaching the RMT as l is increased from l ¼ 64 to numerically established a parity-invariant theory with a l ¼ 250. The bottom panel shows this agreement between positive measure in a certain range of l that was numeri- the Dirac and RMT eigenvalue distributions at l ¼ 250 cally accessible, we will assume this to be the case for is robust as the number of lattice points L is made larger higher values of l and study the infrared behavior of from L ¼ 14 to 20. A quantitative estimate shows that the the theory by taking the l → ∞ limit using the pþðθÞ deviation of the data from the RMT distributions becomes measure. A possibility is the spontaneous symmetry break- smaller with increasing l and approaches zero in the ing (SSB) of parity leading to a nonzero bilinear condensate infinite volume limit. This agreement with RMT shows Σ, i.e., at finite fermion mass m and infinite volume, the presence of SSB. hψψ¯ ið Þ¼Σð j jÞ þ ð Þ m m= m O m . To study this, we focus on Figure 6 shows ΣiðlÞ as extracted from the matching the discrete dimensionless Dirac operator eigenvalues with RMT using Eq. (9), as a function of l. The different ordered by magnitude, 0 < lλ1ðθ; lÞ < lλ2ðθ; lÞ < …, symbols are the values of Σ ðlÞ; i ¼ 1, 2, 3, 4 in the −1 i (which are technically obtained from LjG ðθÞj), at finite continuum at different fixed l. At any finite l the values l → ∞ hλ ðlÞli . We first take the L continuum limit of i of ΣiðlÞ from different i do not agree, as expected. (using L from 12 to 24) at different fixed l ranging from 4 Assuming the existence of the finite nonzero value of to 250 for this study before considering the l → ∞ limit. the condensate Σ in the infinite volume limit, we used λ ðθ; lÞ −1 −2 The probability distribution of i , as sampled in ΣiðlÞ¼Σi þ k1l þ k2l , to fit the entire range of finite the Monte Carlo calculation, will exhibit several well separated peaks consistent with a spectrum that is discrete. Perturbation theory will hold as l → 0 and hλiðlÞi will be proportional to l−1. If the theory spontaneously breaks −3 parity as l → ∞, then hλiðlÞi ∼ l (due to a finite eigenvalue density near zero [29]) and in addition, the distributions of the individual eigenvalues should also match with those from an appropriate random matrix theory (RMT) ensemble [30–32]. If we define ΣiðlÞ through the means hλiðlÞi and zi of the two respective distributions,

FIG. 6. The infinite volume extrapolation of ΣiðlÞ. The red, ¼ 1 hλ ðlÞiΣ ðlÞl3 ¼ ð Þ green, blue, and purple points and curves correspond to i ,2, i i zi; 9 3, 4, respectively.

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