A mononuclear Ru(II) complex with meloxicam: DNA- and BSA-binding, molecular modeling and anticancer activity against human carcinoma cell lines

Physics Letters B 754 (2016) 264–269 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb From domain wall to overlap in 2 + 1d Simon Hands Department of Physics, College of Science, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom a r t i c l e i n f o Article history: Received 27 December 2015 Received in revised form 12 January 2016 Accepted 20 January 2016 Available online 22 January 2016 Editor: J.-P. Blaizot Keywords: Lattice gauge ﬁeld theories Field theories in lower dimensions Global symmetries a b s t r a c t The equivalence of domain wall and overlap fermion formulations is demonstrated for lattice gauge theories in 2 + 1 spacetime dimensions with parity-invariant mass terms. Even though the domain wall approach distinguishes propagation along a third direction with projectors 12 (1 ± γ3 ), the truncated overlap operator obtained for ﬁnite wall separation L s is invariant under interchange of γ3 and γ5 . In the limit L s → ∞ the resulting Ginsparg–Wilson relations recover the expected U(2N f ) global symmetry up to O(a) corrections. Finally it is shown that ﬁnite-L s corrections to bilinear condensates associated with dynamical mass generation are characterised by whether even powers of the symmetry-breaking mass are present; such terms are absent for antihermitian bilinears such as i ψ̄ γ3 ψ , markedly improving the approach to the large-L s limit. © 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 . 1. Introduction Relativistic fermions moving in 2 spatial dimensions are the focus of much attention, in part due to the stability of Dirac points in graphene and surface states of topological band insulators when the underlying Hamiltonian is symmetric under time reversal and spatial inversion (see, e.g. [1]). Even in this case a gap may develop at the Dirac points in the presence of interactions. The corresponding issue in quantum ﬁeld theory is the stability of the vacuum with respect to spontaneous generation of a parity-invariant bilinear condensate of the form ψ̄i ψ = 0. Since the transition to a gapped phase generically occurs for strong interactions, it deﬁnes a quantum critical point (QCP) [2]; the phase diagram for planar fermionic systems with various interactions and characterisation of possible QCPs as a function of the number of fermion species N f remain open questions [3]. To date there have been many lattice ﬁeld theory simulations probing QCPs using the staggered fermion formulation [4] (a notable recent exception employs the SLAC derivative [5]); N staggered fermions describe N f = 2N continuum ﬂavors each having 4 spinor components [6], with global symmetry group U( N ) ⊗ U( N ) spontaneously broken by a parity-invariant mass to U( N ). However, because there are two matrices γ3 and γ5 which anticommute with the kinetic operator, the correct continuum symmetry breaking is U(2N f ) → U( N f ) ⊗ U( N f ). For the strongly-interacting E-mail address: s.hands@swan.ac.uk. continuum limit at a QCP, there is no reason a priori to expect the correct symmetry-breaking pattern to be recovered. For this reason the properties of domain wall fermions, which purportedly more faithfully reproduce continuum symmetries, were explored for 2 + 1 + 1d in Ref. [7]. In particular bilinear condensates and meson correlators constructed from distinct spinor combinations, but which should yield identical results in a U(2)-invariant theory, were investigated as a function of the extent L s of the “third” direction separating the domain walls. Numerical results obtained in the context of quenched non-compact QED3 with variable coupling strength support U(2) symmetry being restored as L s → ∞. In 2 + 1d the Ginsparg–Wilson relation specifying the optimal requirements for lattice fermions to avoid species doubling while retaining as much of the continuum global symmetry as possible [8] generalises to a set of three relations (since chiral rotations are now speciﬁed by an element of U(2) rather than U(1)). These were set out in [7], along with the speciﬁcation of an overlap Dirac operator D ov [9] deﬁned in 2 + 1d in which realises them. As it must, D ov has equivalent properties under the U(2) rotations generated by γ3 and γ5 . Overlap formulations of massless fermions in 2 + 1d and the relation with the parity anomaly occurring for an odd number of two-component spinor ﬂavors have previously been discussed in [10]. In the domain wall approach, the 2 + 1d ﬁelds ψ, ψ̄ are deﬁned ¯ ± which are approxiin terms of surface states of ﬁelds ± , mately localised on the walls and are ± eigenstates of γ3 [11]. Some questions which remain unanswered in [7] are: the extent to which the domain wall formulation, in which propagation along http://dx.doi.org/10.1016/j.physletb.2016.01.037 0370-2693/© 2016 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 . S. Hands / Physics Letters B 754 (2016) 264–269 265 ⎡ DW − M + 1 0 ⎢ −1 DW − M + 1 ··· −1 .. . 0 ⎤ the direction separating the walls is governed by γ3 , can maintain the equivalence between γ3 and γ5 rotations for ﬁnite L s ; the reason for O (a) violations of U(2) symmetries even in the overlap limit L s → ∞; and a better understanding of why ﬁnite-L s corrections are minimised by choosing i ψ̄ γ3 ψ, rather than ψ̄ψ, as the bilinear condensate to focus on. In this brief technical Letter I outline how the overlap operator is recovered in the L s → ∞ limit of the domain wall formulation using a by now familiar sequence of matrix algebra operations. In particular, it will prove possible to extend the key results on the equivalence of γ3 and γ5 to a truncated overlap operator deﬁned by domain wall fermions with ﬁnite L s . As well as providing a ﬁrm conceptual foundation for domain wall fermions and their symmetry properties in 2 + 1d, the proof sheds light on each of these outstanding issues. δs,1 δs , L s ] P − + δs,s P + so that 2. From domain wall to overlap with First we review the passage from the domain wall formulation of lattice fermions to the overlap operator. The corresponding treatment for 4d gauge theories is well-known [12]: here we follow closely the treatment of [13]. We begin from the 2 + 1d domain wall operator deﬁned in [7], correcting an overall (unphysical) sign: Q ± = ( D W − M + 1) P ± − P ∓ ; (8) 1 1 C ± (mh ) = (1 − mh ) ± (1 + mh )γ3 = P ± − mh P ∓ . (9) 2 2 Now deﬁne the block diagonal matrix Q = Q + 14V N c ×4V N c ; it is important to note that Q ± = Q ± (mh ), Q = Q(mh ), P = P (mh ). With D̃ ≡ Q−1 D P , we deduce S dw = ¯ x, s) D (x, s| y , r )( y , r ). ( (1) x, y s,r ¯ are four-component spinors deﬁned in 2 + 1 + 1 The ﬁelds , dimensions, and D (x, s| y , s ) = δs,s ( D W (x| y ) − M ) + δx, y D 3 (s|s ), (2) where the ﬁrst term is the 2 + 1d Wilson operator deﬁned on spacetime volume V ( D W − M )x, y 1 † =− (1 − γμ )U μ (x)δx+μ̂, y + (1 + γμ )U μ ( y )δx−μ̂, y 2 μ=0,1,2 + (3 − M )δx, y , (3) and D 3 controls hopping along the dimension separating the domain walls at s = 1 and s = L s , which we will refer to as the third direction: D 3 s,s = − P − δs+1,s (1 − δs , L s ) + P + δs−1,s (1 − δs ,1 ) + δs,s , (4) where the projectors P ± ≡ 12 (1 ± γ3 ). Following convention, in (3) we include interaction with a SU( N c ) valued gauge connection ﬁeld U μ (x) located on the lattice links, noting in passing that some models relevant for 2 + 1d QCPs share the global U(2N f ) symmetries of gauge theories. Initially we supplement (1) with a hermitian mass term coupling ﬁelds on opposite walls: mh S h = mh ¯ x, 1) P + (x, L s ). ¯ x, L s ) P − (x, 1) + ( ( (5) x The operator D W − M + D 3 + mh S h can be represented as a L s × L s matrix consisting of 4V N c × 4V N c blocks: ⎡D ⎢ ⎢ D (mh ) = ⎢ ⎢ ⎣ W −M +1 −1 0 .. . 0 0 DW − M + 1 −1 ··· +mh 0 .. ⎤ ⎥ ⎥ ⎥P+ ⎥ ⎦ . −1 D W − M + 1 ⎢ +⎢ ⎣ . .. 0 −1 DW − M + 1 +mh ⎥ ⎥ ⎥ P −. ⎦ (6) Now deﬁne the cyclical shift operator Ps,s ≡ [δs−1,s (1 − δs,1 ) + ⎡Q ⎢ Q− ⎢ ⎢ DP = ⎢ 0 ⎢ . ⎣ . . 0 Q− .. Q −C− ⎤ 0 ⎥ ··· 0 Q+ + . .. . .. . . .. . Q− 0 Q +C+ .. 0 ⎥ ⎥ ⎥ ⎥ ⎦ (7) det[ D̃ (1)−1 D̃ (mh )] ≡ det[ D (1)−1 D (mh )], where ⎡ 1 ⎢ − T −1 0 1 0 − T −1 ⎢ ⎢ D̃ = ⎢ ⎢ ⎣ .. . ··· − T −1 C − 0 0 1 .. . .. 0 . .. . − T −1 0 C+ (10) ⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦ (11) −1 with T = − Q − Q +. In more detail, T = −[( D W − M + 1) P − − P + ]−1 [( D W − M + 1) P + − P − ] = 1 − γ3 = − 1 (D W − M) (D W − M) 1 + γ3 2 + (D W − M) 2 + (D W − M) 1− H (12) 1+ H where the hermitian 4V N c × 4V N c matrix H is deﬁned H = −γ3 [2 + ( D W − M )]−1 [ D W − M ] ≡ −γ3 A . (13) Hermiticity of H requires γ3 A γ3 = A , which is the case for A deﬁned by (3). Up to an unphysical sign and with γ3 assuming the role played by γ5 in 4d gauge theories, H is identical with the Shamir kernel [14]. Next observe that in the form (11), D̃ = L D U with † ⎡ 1 ⎢ ⎢ −T 1 ⎢ L=⎢ ⎢ 0 ⎢ . ⎣ .. 0 ··· 1 0 − T −1 .. . .. . 0 ⎡1 0 ··· ⎢0 1 0 ⎢. ⎢ ⎢ ⎣ U = ⎢ .. 0 1 0 ⎤ .. ⎥ .⎥ ⎥ ⎥; ⎥ ⎥ ⎦ − T −1 1 − T −1 C − ⎤ −( T −1 )2 C − ⎥ ⎥ .. . −( T −1 )3 C − ⎥ ⎥ ⎥ .. .. ⎦ . . 1 (14) 266 and S. Hands / Physics Letters B 754 (2016) 264–269 ⎡ ··· 1 0 ⎢ ⎢0 1 ⎢. . D=⎢ ⎢. ⎢ ⎣ 0 ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦ .. . 0 1 .. . 0 Taking into account a benign wavefunction renormalisation, this is the propagator for a continuum species with mass proportional to mh . By contrast near a doubler pole p̃ μ = p μ − (i , j , k)π ≈ 0, i , j , k ∈ {+1, −1}, ⎤ (15) sgn( H ) ≈ −γ3 C + − ( T −1 ) L s C − det[ D (1) D ov ≈ 1 + D (mh )] = det[ D̃ (1)−1 D̃ (mh )] = det[D L s , L s (1)−1 D L s , L s (mh )], (16) where the 4V N c × 4V N c matrix D L s , L s is the Schur complement of D̃: D L s , L s (mh ) = C + − ( T −1 ) L s C − = (1 + T −1 )γ3 = D L s , L s (1) 1 2 1 2 (1 + mh ) − (1 − mh )γ3 (1 + mh ) − (1 − mh )γ3 1−T 1+T 1−T , 1+T (17) 1 with T ≡ T L s . We now multiply both sides of (17) by D − L s , L s (1) to ﬁnd that the combination of domain wall fermion determinants det[ D (1)−1 D (mh )] is the same as the determinant of the truncated overlap operator ⎡ D Ls [ H ] = ≡ 1⎢ 2 ⎣(1 + mh ) − (1 − mh )γ3 1 2 1− 1+ 1− H 1+ H 1− H 1+ H L s ⎤ ⎥ L s ⎦ (18) (1 + mh ) − (1 − mh )γ3 tanh( L s tanh−1 H ) . (19) In order for the tanh function to be deﬁned by a power series the second equality (19) requires H to be a bounded operator, namely | H | < 1. The factor D (1)−1 can be thought of as modelling Pauli– Villars boson ﬁelds which cancel the contributions of the fermions from the 4d bulk. Now, tanh( L s tanh−1 (x)) is an analytic approximation to the signum function sgn(x) which becomes exact in the limit L s → ∞. So long as H is hermitian and bounded, we therefore recover the overlap operator [9]: lim D L s = D ov L s →∞ 1 DW − M 2 2 where the unphysical nature of the sign of γ3 is manifest. For mh → 0 (20) coincides with the 2 + 1d overlap operator given in [7]. Next let’s check the overlap operator (20) has the expected weak-coupling limit. For link ﬁelds U μ = 1, and with lattice spacing set to unity, in momentum space D W = i μ γμ sin p μ + μ (1 − cos p μ ), implying propagator poles at p μ ≈ 0 and near the Brillouin Zone corners p μ ≈ π . At the origin D W ≈ i γμ p μ so ip / M −1 (21) so that the overlap operator D ov ≈ ip / (1 − mh ) 2M + mh . (2n − M ) +1 (23) (1 − mh ) ip /̃ . 2(2n − M ) (24) So long as (2n − M ) is not too small, the species has a mass of O(1) in cutoff units, and decouples from low-energy physics. Since mh and M have opposite signs, for strong enough coupling there is the possibility of the system entering a paritybreaking Aoki phase signalled by a bilinear condensate with the quantum numbers of an isotriplet pion. This was investigated in the context of a 3d Gross–Neveu model in [15], where it was found that the Aoki phase was manifest for mh < 0 with the width of the parity-broken region vanishing exponentially as L s → ∞. 3. Equivalence of γ3 and γ5 Despite the manifest independence of the overlap operator D ov (20) of which matrix γ3 or γ5 is used to deﬁne the hermitian argument H of the signum function, for ﬁnite L s it remains unclear whether the distinction is important or not [7], since clearly the deﬁnition (4) of the domain wall operator D 3 distinguishes them. We can address this using the analytic approximation for signum (19). First, the series expansion for tanh−1 H is well-deﬁned since H = γ3 A is a bounded operator, i.e. | H | = M /(2 − M ) < 1 for 0 < M < 11 : tanh−1 H = H + H3 3 + H5 5 + ··· (25) Each term is an odd power, so can be reexpressed using γ3 A γ3 = A † : H 2n+1 = γ3 A ( A † A )n . (26) The signum approximation is then bn ( A † A )n ) n tanh( L s γ3 A bn ( A A ) ) = cosh( L s γ3 A n bn ( A † A )n ) n ⎛ (20) A† A H (ip/ − M ) (2 − M ) sgn( H ) = √ ≈ −γ3 = −γ3 (2 − M ) M H2 = −γ3 ip /̃ † n sinh( L s γ3 A (27) with bn = (2n + 1)−1 . In the McLaurin series expansions of the hyperbolic functions on the RHS of (27), expansion of the argument yields a general term of the form (1 + mh ) − (1 − mh )γ3 sgn −γ3 2 + (D W − M) 1 A = (1 + mh ) + (1 − mh ) √ , = (2n − M ) with n = |i | + | j | + |k|, so the overlap is Again, note L = L (mh ), and detL = detU = 1. We conclude −1 ip /̃ + (2n − M ) (22) ⎝ Lm s ∞ ∞ n 1 =0 n 2 =0 ⎞ ∞ m ⎠ [bni (γ3 A )( A † A )ni ] ··· nm (28) i =1 For the sinh series, m is an odd integer so that the term in square brackets reads ( bni )(γ3 A )( A † )n1 (γ3 A )( A † A )n2 . . . (γ3 A )( A † A )nm = ( bni )(γ3 A )( A † )n1 ( A † A )n2 +1 ( A † A )n3 . . . ( A † A )nm−1 +1 ( A † A )nm = ( bni )(γ3 A )( A † A ) i ni +(m−1)/2 . (29) 1 For free fermions the most stringent limit on M comes from the origin of momentum space. In practice on any ﬁnite lattice with antiperiodic temporal boundary conditions M = 1 is safe since | H | = 1/ 5 − 4 cos Lπ < 1 for L t < ∞. t S. Hands / Physics Letters B 754 (2016) 264–269 For the cosh series m is even and a similar argument gives the general term ( bni )( A † A ) i ni +m/2 . The ﬁnal step is to observe that [(γ3 A )−1 , ( A † A )n ] = 0 for any n; the RHS of (27) can therefore be manipulated to bring γ3 A to the left of all terms in the expansion, whereupon the γ3 cancels in the expression (19) for the truncated overlap. Now using the fact that γ5 has identical properties with respect to commutation with A, we can reverse all the steps to rewrite the truncated overlap operator D Ls [ H ] = 1 2 (1 + mh ) + (1 − mh )γ5 tanh( L s tanh −1 γ5 A ) . (30) 267 Next consider the mass term m5 S 5 . Even though this term differs from the other masses by coupling ﬁelds on the same domain wall, rather than on opposite ones, the matrix manipulations of Sec. 2 still arrive at (7), with this time C 5± = P ± − im5 γ5 P ± = P ± − im5 P ∓ γ5 , (38) where the second step is crucial. The truncated overlap in this case is D Ls [ H ] = (1 + im5 γ5 ) − γ3 tanh( L s tanh−1 H )(1 − im5 γ5 ) ; 1 2 (39) This establishes that the truncated overlap operator is equally blind to the distinction between γ3 and γ5 as the overlap (20). however the considerations of Sec. 3 permit this to be rewritten 4. Introducing m3 , m5 = 0 D Ls = In [7] we exploited the possibility of U(2)-rotating the ﬁelds leaving the kinetic term unaltered while changing the form of the mass term. In terms of continuum ﬁelds deﬁned in 2 + 1d the alternative but physically equivalent, antihermitian but parity-invariant mass terms are im3 ψ̄ γ3 ψ , im5 ψ̄ γ5 ψ . In the domain wall approach (5) is replaced by one of m3 S 3 = im3 ¯ x, L s )γ3 P − (x, 1) ( x ¯ x, 1)γ3 P + (x, L s ); + ( ¯ x, L s )γ5 P + (x, L s ) m5 S 5 = im5 ( (31) x ¯ x, 1)γ5 P − (x, 1). + ( (32) 1 2 (1 + im5 γ5 ) − γ5 tanh( L s tanh−1 (γ5 A ))(1 − im5 γ5 ) . (40) The complete equivalence between (40) and (35) is manifest. 5. Ginsparg–Wilson relations Whilst the previous two sections have established the equivalence of the domain wall formulation with respect to a discrete interchange of the matrices γ3 and γ5 , in order to study restoration of the full U(2N f ) symmetry it is more convenient to examine the overlap operator. Following (20), (36), in the large-L s limit we can write Lagrangian densities in terms of “Ginsparg–Wilson” ¯: ﬁelds , ¯ D 0ov + mh (1 − D 0ov )]; Lh = [ (41) First consider a mass term m3 S 3 . The matrix manipulations outlined in Sec. 2 leading to eqn. (7) go through as before, but with (9) replaced by ¯ D 0ov + im3 (1 − D 0ov ) 3 ], L3 = [ (42) C 3± = P ± ± im3 P ∓ . 1 D 0ov = (33) D L s , L s (mh = 1) 2 (1 + im3 γ3 ) − γ3 1−T 1+T (1 − im3 γ3 ) , (34) implying a truncated overlap D Ls = 1 2 (1 + im3 γ3 ) − γ3 tanh( L s tanh−1 (γ3 A ))(1 − im3 γ3 ) , (35) with A still given by (13). An important technical point is that the passage from domain wall to overlap requires the Pauli–Villars matrix D L s , L s (1) = (1 + T −1 )γ3 to continue to be deﬁned with the hermitian mass term 1 × S h . The overlap operator found in the limit L s → ∞ is thus D ov = 1 2 (1 + im3 γ3 ) + √ A A† A (1 − im3 γ3 ) (36) with A deﬁned in (13). In the weak coupling long wavelength limit D ov ≈ ip / (1 − im3 γ3 ) 2M + im3 γ3 . where D 0ov is the overlap operator for massless fermions 2 The Schur complement of D̃ = Q−1 D P is then 1 γ (37) This time there is an O (a) term proportional to p / γ3 not present in the continuum action, which cannot be absorbed by wavefunction rescaling. It seems highly plausible that this lies at the heart of the O (a) departures from U(2) symmetry observed when rotating fermion bilinears according to the remnant symmetries derived from the 3d Ginsparg–Wilson (GW) relations in Sec. 3 of [7]. 1+ √ A A† A (43) . In both cases there is an O (a) correction to the expected continuum form, but as noted above for the hermitian mass case (41) the correction can be absorbed into a harmless rescaling of the kinetic term. For the antihermitian case (42) by contrast the correction is not of the same form as a term in the continuum Lagrangian, as ﬁrst noted in [7] (although (42) differs in detail from eq. (34) of that paper). The reconciliation is made by ﬁrst observing that the GW relation appropriate for the domain wall operator (2) is [8,13] γ3 D 0ov + D 0ov γ3 = 2D 0ov γ3 D 0ov . (44) As expected, there are further GW relations, ﬁrst with γ5 replacing γ3 in (44), and also a rotation generated by i γ3 γ5 which along with a simple global phase rotation completely speciﬁes the U(2) [7]: γ5 D 0ov + D 0ov γ5 = 2D 0ov γ5 D 0ov ; γ3 γ5 D 0ov − D 0ov γ3 γ5 = 0. (45) 0† We can further exploit 3 D 0ov 3 = 5 D 0ov 5 = D ov to express both 0† 0† non-trivial GW relations as D 0ov + D ov = 2D 0ov D ov [10]. The associ- γ γ γ γ ated symmetry in the massless limit is then [16,7] 0 ¯ → ¯ e (i α (1− D 0ov )γ3 ) → e (i αγ3 (1− D ov )) ; 0 ¯ → ¯ e (i α (1− D 0ov )γ5 ) → e (i αγ5 (1− D ov )) ; ¯ → ¯ e αγ3 γ5 . → e −αγ3 γ5 ; (46) 268 S. Hands / Physics Letters B 754 (2016) 264–269 Strictly speaking, therefore, symmetry under global U(2) rotations of local ﬁelds is only recovered as a → 0, under the assumption that the overlap operator D 0ov is suﬃciently localised in this limit. Next, deﬁne projection operators as follows: P± = 1 2 (1 ± γ3 ); P̃ ± = P ± ∓ D 0ov γ3 (47) where we assume mh is small enough to justify the binomial expansion. Since the trace over an odd number of gamma matrices is zero, all even powers of mh vanish on taking the trace, which makes sense since ψ̄ψ should be an odd function of mh . The mass term m3 S 3 yields the same series: i ψ̄ γ3 ψ = tr(D / + im3 γ3 )−1 i γ3 with the property P̃ ± D 0ov = D 0ov P ∓ following from (44). With pro- = tr 1 − ¯± = ¯ P̃ ∓ , we can write jected ﬁelds ± = P ± , ¯ + D 0ov + + ¯ − D 0ov − = ¯ D 0ov ; L0 = (48) ¯ − + + ¯ + − ) = mh ( ¯ 1 − D 0ov ); mh S hGW = mh ( (49) ¯ − 3 + + ¯ + 3 − ) = im3 ( ¯ 1 − D 0ov ) 3 m3 S 3GW = im3 ( γ γ γ (50) consistent with (41), (42). The extension to the terms involving γ5 is trivial [7].2 = −4 ∂ ln Z ∂M , = trM −1 ∂ mi ∂ mi 2 i 2 i ψ̄ψ L s = ψ̄ γ3 ψ L S →∞ + ψ̄ γ3 ψ L s = ψ̄ γ3 ψ L S →∞ + 3 ( L s ); / γ3 + · · · 1 D / ( i γ3 ) + ··· D4 (55) i ψ̄ γ5 ψ L s = ψ̄ γ3 ψ L S →∞ + 5 ( L s ). (52) 2 The numerically dominant residual is h , deﬁned to be the imaginary component of i ψ̄ γ3 ψ evaluated on just the + component of : ψ̄ γ3 ψ L s + i h ( L s ). (53) The imaginary contribution from the − component has opposite sign and hence cancels even for ﬁnite L s . In order to understand why h only contributes for the hermitian condensate, ﬁrst consider the continuum case with M =D / + mh : ψ̄ψ = tr(D / + mh )−1 = tr 1 1− D / = −4 mh D2 [1 + γ3 ε L s ] [1 + γ3 ε L s ]2 + · · · . + mh2 [1 − γ3 ε L s ] [1 − γ3 ε L s ]2 mh D / + + mh3 D4 Here ε L s [ H ] ≡ tanh( L s tanh the signum function. Now, 1 + γ3 ε L s 1 − γ3 ε L s mh2 D /2 − mh3 D /3 + ··· + ··· (54) 2 Note that in order to recover the expressions for the antihermitian mass terms derived in [7] we should have chosen a matrix decomposition of D (m3,5 ) with the projectors P ± multiplying to the left rather than to the right as in (6). −1 H ) is the ﬁnite-L s approximation to = [1 − γ3 ε L s ]−1 [1 + ε L s γ3 ]−1 [1 + ε L s γ3 ][1 + γ3 ε L s ] = (1 − ε L2s − [γ3 , ε L s ])−1 (1 + ε L2s + {γ3 , ε L s }). (57) In the limit L s → ∞, ε L2s = 1, and the long-wavelength weak coupling limit (21) gives L s →∞ 2 i D3 [1 + γ3 ε L s ] [1 − γ3 ε L s ] lim {γ3 , ε L s } = 2; i 2 / − (56) h ( L s ) + h ( L s ); 2 ¯ 1)γ3 P + ( L s ) = i ( + × 1 − mh (51) where detM is the part of the functional measure coming from the fermions. For a U(2)-invariant theory the condensates generated by the masses mh , m3 , m5 should all coincide, and indeed numerical evidence for this as L s → ∞ was presented for quenched noncompact QED3 [7]. A particular useful result was that ﬁnite-L s corrections are minimised by choosing the mass term antihermitian. We parametrise these in terms of residuals h , h , 3 , 5 which vanish exponentially as L s → ∞ by writing: i = tr 2 D2 m33 D2 imh3 where we have used γ3 D / γ3 = −D/ . This time the even powers vanish because they consist of products of an odd number of matrices γμ (μ = 0, 1, 2) with γ3 , so are proportional to either trγμ γ3 or trγ5 . Now, for a theory with functional weight detD Ls [ H ] the corresponding expression for ψ̄ψ is The freedom to specify variants of the parity-invariant mass term can be exploited in the study of the corresponding bilinear condensates deﬁned via 1 m3 D / γ3 + m23 trM −1 M = tr[1 − γ3 ε L s + mh (1 + γ3 ε L s )]−1 [1 + γ3 ε L s ] 6. Bilinear condensates ψ̄i ψ = im3 lim [γ3 , ε L s ] = − L s →∞ 2ip / M , (58) so (57) ≈ 2M /i p / and we are on the right track. However, for ﬁnite L s 1 − ε L2s is a real quantity, and now there is no reason for the terms in (56) corresponding to even powers of mh necessarily to vanish. Another way of saying this is that the form of A deﬁning εL s dictates that it is no longer the case that even powers of mh are proportional to the trace over an odd number of gamma matrices. We conclude that the function ψ̄ψ(mh ) in general contains an even component, labelled h in (52), weakly dependent on mh as mh → 0 and only vanishing as L s → ∞. Now repeat the exercise for the mass term m3 S 3 : trM −1 M = tr[1 − γ3 ε L s + im3 γ3 (1 + ε L s γ3 )]−1 i γ3 [1 + ε L s γ3 ] = tr 1 + im3 − 1 [1 + γ3 ε L s ] [1 + γ3 ε L s ] γ3 ( i γ3 ) [1 − γ3 ε L s ] [1 − γ3 ε L s ] (59) Now, from (57) and the considerations of Sec. 3, all the terms in the binomial expansion of the ﬁrst factor in (59) can only contain 2q L s dependence in terms of the form (γ3 ε L s ) p , ε L s with p , q inte- ger, which have the property that trγ3 (γ3 ε L s ) p = trγ3 (ε L s )2q = 0. This implies that only odd powers of m3 survive the trace. Hence i ψ̄ γ3 ψ(m3 ) is an odd function of m3 , and the dominant residual h is necessarily absent. For ﬁnite L s when the limiting forms (58) do not hold, we cannot exclude corrections which are odd functions of m3 , corresponding to the residual 3 in (52). S. Hands / Physics Letters B 754 (2016) 264–269 Finally, the arguments of Sec. 3 then imply the identical property for the condensate i ψ̄ γ5 ψ, consistent with the numerical results of [7]. 269 1 the Pauli–Villars bulk correction detD − L s , L s (1) requires the hermitian mass 1 × S h . Acknowledgements 7. Summary In Sec. 2 we showed that the 2 + 1d domain wall fermion formulation introduced in [7] coincides with the overlap operator in the limit L s → ∞, and, importantly not simply in the continuum limit as suggested in the abstract of that paper. Whilst the Dirac matrices γ3 and γ5 enter the domain wall formulation (1) in very different ways, it was shown in Sec. 3 that the resulting 2 + 1d truncated overlap operator (19), (30) is blind to the distinction between them even for L s ﬁnite. There seems to be no obstruction to modelling U(2N f ) → U( N f ) ⊗ U( N f ) symmetry breaking in lattice simulations of 2 + 1d fermions, so long as it is understood that the nature of the a > 0 corrections to continuum symmetry operations, encapsulated in the GW relations (44), (45), and needed, say for identifying interpolating operators for Goldstone modes [7], is more complicated than for 4d gauge theories, as discussed in Sec. 5. In particular the antihermitian mass term (50) consistent with the GW relations contains an O(a) correction of a form not present in the continuum action. Ultimately, successful control of these corrections will depend on the locality properties of the overlap operator D ov [17], which is a dynamical question. 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