Deconfined criticality and a gapless $\mathbb{Z}_2$ spin liquid in the square lattice antiferromagnet

The theory for the vanishing of N\'eel order in the spin $S=1/2$ square lattice antiferromagnet has been the focus of attention for many decades. A consensus appears to have emerged in recent numerical studies on the antiferromagnet with first and second neighbor exchange interactions (the $J_1$-$J_2$ model): a gapless spin liquid is present for a narrow window of parameters between the vanishing of the N\'eel order and the onset of a gapped valence bond solid state. We propose a deconfined critical SU(2) gauge theory for a transition into a stable $\mathbb{Z}_2$ spin liquid with massless Dirac spinon excitations; on the other side the critical point, the SU(2) spin liquid (the `$\pi$-flux' phase) is presumed to be unstable to confinement to the N\'eel phase. We identify a dangerously irrelevant coupling in the critical SU(2) gauge theory, which contributes a logarithm-squared renormalization. This critical theory is also not Lorentz invariant, and weakly breaks the SO(5) symmetry which rotates between the N\'eel and valence bond solid order parameters. We also propose a distinct deconfined critical U(1) gauge theory for a transition into the same gapless $\mathbb{Z}_2$ spin liquid; on the other side of the critical point, the U(1) spin liquid (the `staggered flux' phase) is presumed to be unstable to confinement to the valence bond solid. This critical theory has no dangerously irrelevant coupling, dynamic critical exponent $z \neq 1$, and no SO(5) symmetry. All of these phases and critical points are unified in a SU(2) gauge theory with Higgs fields and fermionic spinons which can naturally realize the observed sequence of phases with increasing $J_2/J_1$: N\'eel, gapless $\mathbb{Z}_2$ spin liquid, and valence bond solid.

IV. Renormalized perturbation expansion for the critical SU (2)  Antiferromagnetism on the square lattice became a topic of intense study soon after the discovery of high temperature superconductivity in the cuprates, and it continues to be a wellspring of interesting experimental and theoretical physics. It was established early on that the insulating antiferromagnet with S = 1/2 spins on each site, and only nearest neighbor antiferromagnetic exchange interactions (J 1 ) has long-range Néel order in its ground state i.e. global SU(2) spin rotation symmetry was broken with the spin expectation value S i = η i N 0 where S i is the spin operator on site i, η i = ±1 on the two checkerboard sublattices, and N 0 is the antiferromagnetic moment. Much attention has since been lavished on the insulating J 1 -J 2 antiferromagnet [1][2][3][4][5][6], which also has a second-neighbor antiferromagnetic exchange interaction J 2 . The key questions Towards a final phase diagram  of Refs. [7][8][9][10], all of which agree that the spin liquid is gapless. Each ellipse in the valence bond solid (VBS) represents a singlet pair of electrons. Lower part of figure adapted from Ref. [11].
are the nature of the quantum phases of the model, and of the quantum phase transitions between them, as a function of increasing J 2 /J 1 after the Néel order vanishes at a critical value of J 2 /J 1 .
These questions are also the focus of our attention here.
An early proposal [5,6,12,13] was that there was a direct transition from the Néel state to a valence bond solid (VBS) (see Fig. 1) which restores spin rotation symmetry but breaks lattice symmetries (followed by a first order transition at larger J 2 /J 1 to a 'columnar' state which breaks spin rotation symmetry, and which we do not address in the present paper). A theory of 'deconfined criticality' was developed [14][15][16] showing that a continuous Néel-VBS transition was possible, even though it was not allowed in the Landau-Ginzburg-Wilson framework because distinct symmetries were broken in the two phases. Evidence has since accumulated for the presence of a VBS phase in the J 1 -J 2 model, but the nature of the Néel-VBS transition in this model has remained a question of significant debate. However, in the past year, a consensus appears to have emerged [11] among the groups investigating this question by different numerical methods [7][8][9][10], and is summarized in Fig. 1: there is a narrow window with a gapless spin liquid phase between the Néel and VBS states. This gapless phase has been identified [8,[17][18][19][20] as a Z 2 spin liquid [5,6,21,22] with gapless, fermionic, S = 1/2 spinon excitations with a Dirac-like dispersion [18,[23][24][25][26], labeled Z2Azz13 in Wen's classification [24].
The starting point of our analysis is the fermionic spinon dual [27][28][29][30] of the bosonic spinon CP 1 model used earlier [12,13,15,16] [31][32][33][34] have indicated that the 2 fermion flavor SU(2) gauge theory does not ultimately describe a conformal field theory needed for Néel-VBS criticality, but exhibits a 'pseudocriticality' associated with a proximate fixed point at complex coupling [27,[35][36][37]. Ref. [28] used connections to bosonic spinon theories to argue that the 2 fermion flavor SU(2) gauge theory was ultimately unstable to confinement and symmetry breaking leading to the appearance of Néel order. We assume this is the case, and we can then describe the transition to the Z 2 spin liquid by the condensation of Higgs fields which break the SU(2) gauge symmetry down to Z 2 : see Fig. 2. The Néel-Z 2 spin liquid transition is a confinement-Higgs transition, and the critical theory is proposed to be a 2-flavor SU(2) gauge theory with critical Higgs fields [28]. We note that a similar critical theory was proposed in Ref. [38] for a continuous transition from the Néel state to a different gapless state with a Z 2 gauge field (the 'orthogonal semi-metal'), and this scenario was supported there by quantum Monte Carlo simulations. Evidently, it is possible that critical Higgs fields can stabilize a scale-invariant critical point of the 2-flavor SU(2) gauge theory at the boundary of a Higgs phase where the SU(2) gauge symmetry is broken down to Z 2 .
As we will see below, an important difference between our critical Higgs SU(2) gauge theory and that of Ref. [38] is that our theory does not preserve Lorentz invariance. The Lorentz symmetry is broken by the Yukawa couplings between the Higgs fields and fermions. The Yukawa couplings also do not preserve the SO(5) flavor symmetry of the SU(2) gauge theory with only fermionic matter; this symmetry rotates between the Néel and VBS states. Both these features have important consequences for the Néel-Z 2 spin liquid critical point, and lead to predictions described below which can tested by numerical studies.
In earlier work, Ran and Wen [39,40] had considered the 2-flavor SU(2) gauge theory as the description of an extended gapless phase on the square lattice-also called the π-flux phase [41].
They proposed a theory for a transition from the π-flux phase to the Z2Azz13 spin liquid by the condensation of a pair of adjoint Higgs fields, which we denote Φ 1,2 (the vector symbol implies with increasing J 2 /J 1 . However, as we discuss in Section VI, we cannot rule out interchanging the assignments of the confining states between the SU(2) and U(1) spin liquids, in which case the orientation of the blue arrow would be reversed. The critical SU(2) gauge theory has a dangerously irrelevant coupling, but the critical U(1) gauge theory does not. The mean-field analysis was performed with w = u = 1, v = −1,ũ = 0.75, and v 4 = 0.5 in Eq. (3.26). We use the ansatz Φ a 1 = c 1 δ ax , Φ a 2 = c 1 δ ay , and Φ a 3 = c 2 δ az , so the terms in V (Φ) proportional to v 1 , v 3 are automatically zero.
gauge SU(2) adjoint index). In light of our arguments above on the confining instability of the π-flux phase to the Néel state, the critical Higgs theory of Ran and Wen [39,40] can serve as the deconfined critical theory for the Néel to Z2Azz13 spin liquid transition. However, as we shall see in Section IV, additional 'dangerously irrelevant' terms are needed to fully define the critical theory in a 1/N f expansion, and these contribute a logarithm-squared renormalization.
The mean-field phase diagram of the SU(2) gauge theory with adjoint Higgs fields describing the π-flux to Z2Azz13 transition turns out to naturally acquire an additional phase, as explained in Section III D-this is the U(1) staggered flux spin liquid [41]. We show that the adjoint Higgs (3.30)) is precisely that required to go from the SU(2) π-flux phase to the U(1) staggered flux phase. Specifically, starting from the π-flux phase, if both Φ 1,2 condense with Φ 1 × Φ 2 = 0, we obtain the gapless Z 2 spin liquid (the simultaneous condensation of Φ 1 and Φ 2 does not require fine tuning because of symmetry constraints that we will describe). On the other hand, Eq. (1.1) implies that if only the composite field Φ 1 × Φ 2 condenses, but the individual fields Φ 1,2 do not, then the π-flux phase turns into the U(1) staggered flux phase. Speaking imprecisely, starting from the parent π-flux phase, the Higgs condensate for the gapless Z 2 spin liquid is the 'square root' of the Higgs condensate for the staggered flux phase. (Let us also note that Song et al. [29] proposed that a trivial monopole would drive the the staggered flux state into the π-flux state: so the Higgs field Φ 3 can be viewed as a 'dual' description of the trivial monopole, and induces a transition in the opposite direction. Four-fermion terms have also been proposed as a route to reducing the emergent symmetry of the staggered flux state to that of the π-flux state [42].) The phase diagram of the Higgs fields Φ 1,2,3 is computed in Section III D and shown in Fig. 2. We propose here that the transition from the gapless Z 2 spin liquid to the VBS state is described by the deconfined critical theory appearing at the onset of the U(1) spin liquid. Other works [25,29,30] have discussed the possible instability of this U(1) spin liquid to either Néel or VBS order via monopole proliferation.
The critical U(1) gauge theory is described briefly in Section V, where we show that it does not contain the dangerously irrelevant terms found in the critical SU(2) theory.
We will review the derivation of the Ran-Wen theory, and discuss its symmetry properties in some detail in Section II and Appendix A. A continuum SU(2) gauge theory coupled to 3 adjoint Higgs fields and gapless Majorana fermions will be obtained in Section III. The critical SU(2) gauge theory for the onset of the gapless Z 2 spin liquid phase from the π-flux phase will be presented in Section IV, along with an analysis of its properties in a 1/N f expansion. The critical U(1) gauge theory for the onset of the same gapless Z 2 spin liquid from the staggered flux phase appears in Section V.

II. GAPLESS Z 2 SPIN LIQUID
The fermionic spinon theory of Z 2 spin liquids proceeds by re-expressing the spin operators in terms of spinons f iα , α =↑, ↓ at site i = (i x , i y ) of the square lattice using     We write down a Bogoliubov Hamiltonian for the f iα to obtain a Z 2 spin liquid. Following Wen's notation [24], we introduce the Nambu spinor resulting in the Bogoliubov Hamiltonian Here, with τ a Pauli matrices acting on the Nambu indices of ψ i . Invariance under global SU(2) s spin rotation requires that the u µ ij are all real numbers obeying This fermionic spinon representation has a SU(2) g gauge symmetry, under which and a corresponding transformation for u ij .
We will provide 3 different ansatzes for the u ij in the Z2Azz13 spin liquid, each suited for different purposes. The 3 ansatzes are, of course, related to each other by SU(2) g gauge transformations. Wen's ansatz for the Z2Azz13 spin liquid is given in Appendix A, where the continuum Lagrangian describing the different spin liquid phases is deduced from symmetry fractionalization considerations. In the main text, we obtain the continuum theory directly from the lattice model, for which the ansatz given in Eq. (2.11) will be most useful. To derive this ansatz, we first describe the Z2Azz13 spin liquid by starting from the familiar staggered flux phase with U(1) gauge symmetry [41], and perturbing it with d xy pairing. Explicitly, the ansatz is The first four terms in (2.7) represent the fermion hopping, which is sketched in Fig. 3, and the last 2 terms are the d xy pairing. With this ansatz, three distinct spin liquids may be described depending on the choice of parameters. These spin liquids are shown in Fig. 2, and we list them below: • The π-flux phase with SU(2) gauge symmetry corresponds to φ = π/4, and no fermion pairing γ 1,2 = 0.
The U(1) gauge field corresponds to a nearly spatially uniform modulation in the phases of the fermion hopping terms.
• The Z2Azz13 spin liquid is obtained when the d xy pairing γ 1 + iγ 2 is present, and breaks the Note that we have d xy pairing in the Z 2 spin liquid only, with opposite signs on the two sublattices.
In momentum space, we choose the A and B checkerboard sublattices as the basis sites (shown in Fig. 3), and the Hamiltonian acting on h z X q y X q x 3 6 2 P R W r K K m S r 8 g f X 5 A y y Q l i g = < / l a t e x i t > The eigenvalues of (2.8) are and these co-incide with those obtained in Wen's gauge in (A3). Note that the dispersion depends only on |γ 1 + iγ 2 |, and not on γ 1,2 separately. This is natural in the staggered flux gauge, where U(1) the gauge transformation acts simply as f iα → f iα e iφ i , and so the d xy pairing acts like a charge 2 Higgs field: a simple identification of the charge 2 Higgs field is the advantage of the present gauge. This dispersion is plotted in Fig. (4). The staggered flux phase has Dirac nodal points at (±π/2, ±π/2). Introducing d xy pairing does not gap these nodal points, but moves them away from these high symmetry points. Although the dispersion does not have full square lattice symmetry, all gauge-invariant observables do, and this is verified by the analysis in Appendix A.

A. Majorana gauge
For the remainder of the analysis in the body of the paper we map (2.7) onto the gauge used by Wang et al. [27] for the π-flux phase, which is convenient for eventual representation in Majorana fermions and making the gauge and spin rotation symmetries manifest. In this gauge, the ansatz of the Z2Azz13 spin liquid (which is gauge equivalent to Eq. (2.7)) is As in the previous gauge, the π-flux phase is obtained when φ = π/4 while the staggered-flux phase corresponds to general φ.
A. π-flux state with SO(5) symmetry We begin by working out the continuum SU(2) gauge theory with the 2-flavor massless Dirac fermion from the mean-field ansatz for the π-flux phase, using the Majorana gauge given in Eq. (2.11).
In this gauge, we replace the Nambu spinor in Eq. (2.2) by the matrix operator The spinon SU(2) gauge symmetry of Eq. (2.6) now acts on X i as The physical spin symmetry acts on X i on the left: We write the Bogoliubov Hamiltonian Eq. (2.3) as The correspondence with the notation in Eq. (2.3) is The additional γ ij hoppings involve projective realizations of the spin rotation symmetry, and will not be relevant. The degrees of freedom in this Hamiltonian can be represented by four Majorana fermions, The SU(2)-invariant π-flux state comes from the hoppings β a = 0 and The low-energy behavior of this mean-field ansatz is described by an SU(2) gauge theory with an emergent SO(5) symmetry. To work out the dispersion relation of this Hamiltonian, we increase our unit cell by one lattice site in the x direction and so χ acquires an additional sublattice index m = A, B. Note that this unit cell differs slightly from the one used in the staggered flux gauge.
In momentum space, we then have Eq. (3.8) has Dirac points at k y = 0, π, k x = 0. Labelling these Dirac points by another index v = 1, 2, and expanding around these two points, we decompose our Majorana operator as With this, the Hamiltonian reduces to with the sublattice and 0, a indices implicit. This gives the continuum Lagrangian Here we have chosen to express L M F in the Minkowski metric (+, −, −); we ultimately move to the Euclidean metric below to perform calculations.
We now define the 4 × 2 matrix operator andX = X † γ 0 , where the sublattice/Dirac index m is left implicit. This lets us write our Lagrangian as as the x ↔ y hoppings will be identical. We start with the hoppings in the x-direction, expand our Majorana operators in terms of low-energy modes, and keep only the lowest-order gradient terms.
As in Eq. (3.12), the two indices on χ correspond to (0, x, y, z) and valley, respectively, with the sublattice index implicit.
In our final term, we have reintroduced the x ↔ y hoppings. For the hoppings in the y-direction, Note that here and in Eq. (3.15) the Pauli matrix σ z is acted on by the SU(2) gauge symmetry of the π-flux phase. Gauge invariance requires there exist nearly identical continuum model bilinears containing instead σ x and σ y Pauli matrices. It is therefore useful to express the perturbation in a gauge independent fashion using an adjoint Higgs field Φ a 3 , where a = x, y, z is a SU(2) gauge index: Condensing the Higgs field with Φ a 3 leads to a transition from the SU(2) π-flux state to the U(1) staggered flux state. For concreteness, we continue to work in the gauge where Φ a 3 condenses in the z direction, as implied by Eqs. (3.15) and (3.16).

C. From π-flux to Z2Azz13
We now evaluate the effects of a non-zero γ 1,2 in the π-flux phase, using the Majorana gauge as given in Eq. (2.11). We first consider turning on the perturbation Recall that in the Majorana basis, terms proportional to τ x (τ y ) correspond to hoppings between the 0 ↔ x(y) and z ↔ y(x) Majorana fermions. Focusing on the γ 1 term, we expand in low-energy The perturbation is identical for the γ 2 term, but with σ y ⇒ −σ x .
As in the previous section, the addition of the hopping parameters of Eq. (3.19) can be formulated in a gauge-invariant fashion by coupling the bilinear above to an adjoint Higgs field Φ ā 1 , a = x, y, z (the bar on the "1" will be apparent below). In particular, when a term Φ ā 1 Tr σ aX (µ z γ x + µ x γ y ) X is added to the Lagrangian, we reproduce the continuum version of Eq. (3.19) we just derived when Φ ā 1 condenses as We perform the same analysis for the second term proportional to γ 1,2 : The continuum derivation of this is essentially identical to as before, yielding prompting use to introduce Φ ā 2 Tr σ aX (µ z γ x − µ x γ y ) X . The continuum version of Eq. (3.19) is obtained through the condensation Φ ā 2 such that Φ ā 2 = γ 1 δ ax + γ 2 δ ay .

D. Majorana-Higgs Lagrangian
We now combine the results of Sections III A, III B, and III C to obtain the low energy Lagrangian for the Majorana field X, and 3 real, adjoint Higgs scalars, which we now identify as Φ a 1 , Φ a 2 , Φ a 3 (Φ a 1,2 are rotations of Φ ā 1,2 in the 1, 2 plane). We do not explicitly write out the coupling to the SU(2) gauge field in this subsection, which can be included by the usual requirements of minimal coupling.
The Lagrangian is (3.24) The staggered flux state is obtained when Φ 3 ∝ (0, 0, δφ). The Z2Azz13 state follows from The Higgs potential V (Φ) arises from integrating out the high energy spinon degrees of freedom.
We deduce its form by carefully considering the symmetry properties of the theory, which are described in some detail in Appendix A. Here, we note that the theory should respect time reversal and the lattice symmetries, and we summarize the transformations of the Higgs fieds here: From this, we can deduce that the following gauge-invariant terms are allowed to quartic order in the Higgs potential where abc is the antisymmetric unit tensor.
An important feature of V (Φ) is the cubic term proportional to w. This term implies that if any two of the Higgs fields are condensed, then so must the third. It also shows that even if we were only considering the transition from the SU(2) π-flux phase to the gapless Z 2 spin liquid by the condensation of Φ a 1,2 , we would be forced to include Φ a 3 in our theory, and hence the additional possibility of a U(1) staggered flux phase. The symmetry transformations show that Φ a 3 is the unique adjoint Higgs field that can be made from the tensor product of the Higgs fields needed to describe the gapless Z 2 spin liquid, Φ a 1 and Φ a 2 : so the staggered flux phase is a natural partner of this gapless Z 2 spin liquid and the π-flux phase.
We can perform a mean-field minimization of Eq. (3.26), and typical results are shown in Fig. 2.
There are 3 phases as a function of the tuning parameters s and s, which correspond to exactly those obtained in the lattice mean-field theory described in Section II. The presence of the w term implies that there is a first order transition line near the point where the 3 phases meet [43], as shown in Fig. 2. We summarize and re-express the lattice theory results in terms of the continuum model parameters below.

SU(2) π flux phase
Here, there is no Higgs condensate Φ a 1,2 = 0, Φ a 3 = 0, and the system lies in the red region on the top right of Fig. 2: the SU(2) π-flux phase. The continuum model possesses an SU(2) gauge symmetry, along with the corresponding gauge bosons. The theory is believed to confine to the Néel or VBS phase-as discussed in Section I and VI, we view the Néel phase to be more likely.

U(1) Staggered flux phase
This state as Φ a 3 non-zero, while Φ a 1,2 = 0, resulting in the U(1) staggered flux phase represented on the top left of Fig. 2. Making contact with the lattice ansatz, we have Again, the theory has a continuous unbroken gauge degree of freedom, now with only a U (1) symmetry. There is a single gauge boson, which we nevertheless assume triggers confinement. As argued, the most likely fate of the theory is the VBS state, but we cannot preclude the Néel phase.

Z 2 spin liquid Z2Azz13
The Z 2 spin liquid Z2Azz13 corresponds to a Higgs condensate satisfying Φ a 1,2 = 0; it is shown in the lower half of the phase diagram of Fig. 2. The symmetry transformations imply that Φ a 1 and Φ a 2 have the same mass, so only a single tuning parameter, s, is required to make them condense from the SU(2) π-flux phase. From the symmetry transformations, we also see that the absence of a broken symmetry requires that the gauge-invariant bilinears obey Such saddle points are obtained from the Higgs potential for a range of v 1 positive and v 2 negative.
Moreover, such a saddle point is indeed present in the lattice ansatz of the previous section where We note that this implies Φ 1 ⊥ Φ 2 and | Φ 1 | = | Φ 2 |, where we use a vector shorthand for the indices a = x, y, z of the Higgs fields. By minimizing the potential V (Φ) in Eq. (3.26), we see that this Z 2 spin liquid also implies the condensation of the remaining Higgs field: It follows that s can change sign within this phase without any phase transition.

E. Visons
The Z 2 spin liquid is obtained from the theory in Eq. (3.24) + SU(2) gauge fields (which is Eq. (4.2) below) by condensing Φ a 1,2 . This spin liquid has gapless fermionic spinon excitations, whose low energy dispersion can also be determined from the continuum theory. However, as in all Z 2 spin liquids, there must also be vison excitations, which are mutual semions with respect to the spinons. In the theory in Eq. (4.2), the vison is a finite energy excitation associated with vortex-like saddle point in which the Higgs fields Φ a 1,2 undergo a topologically non-trivial SO(3) rotation, associated with π 1 (SO(3))= Z 2 , around the core of the vortex: see Ref. [44] for an explicit solution in a theory without the fermionic spinons. Given that the vison appears in a lattice model with a background spinon density of one spinon per site, we expect the vison transforms projectively under translational symmetries with T x T y = −T y T x , where T α is translation by one lattice spacing in the α direction [23,[45][46][47][48]. For the case of gapped spinons, this fact now has a modern interpretation in the theory of symmetry fractionalization in topological phases [49][50][51][52][53].
We expect that a similar result applies in the present gapless spinon case, but this has not been explicitly established. For the case of gapped spinons, the vison projective transformation can be derived from a parent U(1) gauge theory (which is Higgsed down to Z 2 ) in which the monopoles carry Berry phases [45,46,48]. Such monopole Berry phases are in-turn related to a SO(5) Wess-Zumino-Witten term in an effective theory the Néel and VBS order parameters [54,55]. Notably, this SO(5) WZW term is also linked to an anomaly of the Majorana theory in Eq. (4.2) [27]. It would therefore be interesting to establish T x T y = −T y T x for gapped visons in the presence of gapless spinons starting directly from Eq. (4.2) and condensing the Higgs fields: we leave such an analysis for future work.

GAUGE THEORY
This section will present an analysis of the transition obtained by tuning the Higgs 'mass' s in Eq. (3.26) across a quantum critical point at s = s c , for s > 0 in Fig. 2, between the SU(2) and Z 2 spin liquids. We have Φ a 1,2,3 = 0 for s > s c , yielding the π-flux spin liquid. For s < s c , we have Φ a 1,2 = 0 yielding the Z 2 spin liquid Z2Azz13. As we noted below Eq. (3.29), Φ a 3 will also be non-zero once both Φ a 1,2 are non-zero. However, as Φ a 3 is quadratic in Φ a 1,2 (see Eq. (3.30)), it is not a primary order parameter for the transition. So we can entirely neglect Φ a 3 in the analysis of the criticality in the present section.
It is also convenient to write the theory in terms of 2 flavors of complex Dirac fermions which also carry a fundamental SU(2) gauge charge, ψ α,v ; Here α is the SU(2) gauge index, v = 1, 2 is the valley index, and the Dirac/sublattice index is suppressed. The global SU(2) spin symmetry is not manifest in this formalism, unlike in the earlier Majorana formalism. Since the Lagrangian in Eq. 3.24 does not contain terms that act on the physical SU(2) spin, our Lagrangian nevertheless has a simple form in terms of these Dirac fermions, although a more careful analysis will be required to calculate the behavior of the Néel order parameter, which does involve the physical SU(2) spin.

Explicitly, the relationship between the Dirac and Majorana fermions is
Applying this change of variables to Eq. (3.24), and including the SU(2) gauge field A a µ , we obtain the Lagrangian for ψ and the Φ a 1,2 Higgs fields We will henceforth work in Euclidean signature, with (γ µ ) 2 = 1 for all µ. This Lagrangian includes an important new term not present in Eq. (3.24): a bare spatial gradient term for the Higgs field proportional to the coupling K (we will define N f shortly). This coupling is allowed by symmetry, and will turn out to be 'dangerously irrelevant' i.e. under renormalization, K flows to zero, but it cannot be set to zero at the outset because of some singular effects that we will describe below.
In contrast, the quartic couplings u, v 1,2 are geniunely irrelevant at the critical point, and will not be considered further.
The theory L is invariant under SU(2) gauge, SU(2) spin rotation, time-reversal, and space group transformations, as it must be, because these are symmetries of the underlying Hamiltonian and its parton representation. However, the Yukawa coupling λ breaks both the emergent Lorentz and SO(5) symmetries of the fermion kinetic term. As we will show below, λ is not an irrelevant perturbation, and so the absence of these emergent symmetries will be apparent in the critical correlation functions.
We will analyze the critical properties of Eq. exponent z: we will compute z to order 1/N f and find it to be a universal number at this order.
Next, we shall examine the renormalization of the field scales. As in the Ref. [56], a convenient choice, as we explain below, is to renormalize the boson field scale Φ so that the Yukawa coupling λ = 1; we will assume λ = 1 below. As usual, the renormalization of the fermion field, Z ψ , is determined from the fermion self energy, which then determines a fermion anomalous dimension η ψ . Here we will find an unusual phenomenon, which is one of our main results: the value of η ψ is not universal at order 1/N f , but has a logarithmic dependence upon the irrelevant coupling K. Finally, we will also compute the renormalization of the fermion bilinears associated with the Néel and VBS order parameters: these are not equal to each other because the SO(5) symmetry is explicitly broken.

A. Boson propagators
The first step in the large N f expansion is to integrate out the large number of fermions ψ, which allows us to determine the propagators of the bosons: the Higgs fields and the gauge fields.
To leading order in 1/N f , we have to evaluate the diagrams in Fig. 6, and this leads to an effective quadratic action of the following form We work in the Euclidean time signature, and k is a 3-momentum.
We first calculate the one-loop corrections to the Higgs propagators. The correction to the Φ 1 propagator is shown in the first diagram in Fig. 6, and is We have omitted a constant term, which will be tuned to zero at the critical point.
The correction to the Φ 2 propagator is identical to the Φ 1 correction, with k x ↔ k y .
The reader should now notice some key features. As in Ref. [56], the overall scaling in momentum is Γ 1,2 ∼ |k|. So, this fermion-induced contribution to the Φ propagators is more important at low momenta than the k 2 terms which would be present in the bare theory. In general, the bare boson k 2 terms are irrelevant, and this is why we choose to set the field scale of Φ with the renormalization condition λ = 1. However, unlike Ref. [56], we will see below in some detail that we cannot entirely ignore the bare k 2 term. The expression for Γ 1 (Γ 2 ) is not an increasing function of k x (k y ) when it is larger than the other momentum components, and this will lead to infrared singularities at first order in 1/N f . Specifically, the integral over the propagator 1/Γ 1 (1/Γ 2 ) has an infrared divergence in the k 0 ,k y (k 0 ,k x ) plane. Consequently, we do need to include the dangerously irrelevant Kk 2 x (Kk 2 y ) term in the bare action for Φ a 1 (Φ a 2 ), as we have anticipated in Eqs. (4.2) and (4.3). The O(N f ) propagator for the gauge field is obtained from

B. Fermion self-energy
We first calculate the one-loop corrections to the fermion self-energy, which will determine the anomalous dimension of the fermion operators as well as the dynamical critical exponent z.
Although the anomalous dimension of the fermion is not a gauge-invariant observable, it will be needed to calculate the critical behavior of the gauge-invariant SO (5)  contributions to the fermion self-energy, as shown in Fig. 7, come from the two Higgs bosons and the gauge field, Σ = Σ 1 + Σ 2 + Σ A .
We have introduced ξ as a gauge-fixing parameter to obtain the gauge boson propagator.
Focusing on the Higgs corrections (Eqs. (4.6) and (4.7)), we analyze the behavior at small external momenta k i . Note that the self-energy integrals are fully regulated by the presence of K and a non-zero external momenta. Since Σ 1 (Σ 2 ) is invariant under k y ↔ k 0 (k x ↔ k 0 ), and the two transform into each other under a 90 degree spatial rotation, there are two distinct types of contributions for small external momenta. The first is proportional to k x γ x for Σ 1 , and k y γ y for Σ 2 . The second type includes all other possible choices of momenta, such as k 0 γ 0 .
As we shall justify below and in Appendix B, we can focus on the regime |p x | |p 0 |, |p y | for graphs with a Φ 1 propagator. In this limit, we can approximate the Φ 1 propagator as (4.9) At K = 0, this propagator has an infrared divergence when integrated over the p 0,y plane -so K is needed an infrared regulator. With this, we extract the γ x correction to the self-energy from the Φ 1 propagator by considering the k 0 = k y = 0 limit: We have indicated a cutoff Λ to regulate the theory at large momenta, and this is needed in conformal gauge theories in 2+1 dimensions. However, with our inclusion of the irrelevant K to control the infrared singularity, we find that the integrand vanishes faster at large momenta. It is not difficult to see that for K = 0 Eq. (4.10) is finite as Λ → ∞, and we will take this limit in the present section. The theory with a finite Λ will be examined in Appendix B in a renormalization group computation.
We will now show that Eq. (4.10) has a leading k x ln 2 (k x ) contribution. One factor of ln(k x ) is the usual one: it follows from the fact that at K = 0 the integrand divided by k x is a homogeneous function of momenta of dimension −3. The other comes from the infrared divergence regulated by K noted below Eq. (4.9).
Extracting the coefficient of the k x ln 2 (k x ) contribution requires a number of approximations.
To understand the values of p that dominate the integral in Eq. (4.10), it is useful to perform the integral over p 0 and p y : By examining the form of the integrals in Eqs. (4.10) and (4.11), one can verify that the dominant term at small k x and K is proportional to k x ln 2 (Kk x ), and arises from the integration regime The scale K appears both as an ultraviolet cutoff and in defining the infrared bound. For future calculations, this integration regime will prove to be the relevant one in isolating similar log 2 contributions in other diagrams, although in principle one must still carry out an explicit calculation like in Eq. (4.11) to verify that no other integration regimes give comparable contributions. We provide these calculations in Appendix C in addition to numerical evaluations of the one-loop integrals which confirm the validity of our approximations, and simply evaluate the one-loop integrals in the Eq. (4.12) limit in the main text.
We can extract the coefficient of this log 2 term by performing the integral in this regime, (4.13) Another discussion of the origin of the k x ln 2 (k x ) is presented in Appendix B using a renormalization group analysis.
We now calculate the form of the second type of corrections using the limits in Eq. (4.12), evaluating the Φ 1 contribution to the self-energy with external momentum k 0 for concreteness. (4.14) Combining the corrections from both Higgs propagators, we obtain the full expression for the self-energy for small external momenta at log 2 order, In principle, the dependence on external momenta inside the logarithms could be more complicated for general k, i.e. Σ(k)γ 0 ∼ k 0 ln 2 (Kf (k 0 , k x , k y )), but since we have verified that f (k 0 , 0, 0) = k 0 , then corrections to this are subleading.
These divergent corrections are absorbed into the renormalization of the fermion field, ψ = where we have renormalized the theory at some momentum scale µ. This counterterm only cures the divergence at log 2 order, since the renormalized self-energy at some other momentum scale k will scale as ln 2 (Kµ) − ln 2 (Kk) = ln(µ/k) ln K 2 kµ . (4.17) This, along with the RG analysis in Appendix B, indicates that the subleading single-logarithm corrections will generically give non-universal behavior. However, these log 2 corrections to the self-energy are Lorentz invariant, and do not affect the renormalization of the dynamical critical exponent, z. Therefore, the subleading single-logarithm correction to the velocity anistropy will lead to a universal correction to the dynamical critical exponent. To extract the subleading correction to z using K and the external momenta as a regulator, we start with the expression To leading order in k, we set k = 0 inside the integrand and simply use it as an IR cutoff, which This result can be obtained analytically by approximating the integration region k ≤ |p x | ≤ 1/K, and can be verified by a numerical evaluation of Eq. 4.18. This implies a renormalization of the The logarithmic derivative with respect to 1/K determines the renormalization of the dynamical critical exponent, The one-loop calculation defined in Eq. (4.18) is actually well-defined when K = 0 and can be regulated via more standard approaches, such as dimensional regularization, as shown in Appendix D. The same value of z is also obtained in a renormalization group computation in Appendix B.

C. SO(5) order parameter
In the absence of the Higgs fields, our theory possesses an emergent SO(5) symmetry corresponding to rotations between Néel and VBS order parameters. This SO(5) symmetry is broken by the critical Higgs fields, and as a result, the scaling behavior of Néel and VBS order parameters will differ. In terms of Dirac fermions, the fermion bilinears corresponding to the two-component VBS order parameter -determined by the action of the square lattice symmetries on the bilinears -may be written as The three-component Néel order parameter has a less concise expression in terms of Dirac fermions -this is due to the fact that the Dirac fermion representation obfuscates the action of the physical SU(2) spin rotation symmetry. In terms of the Majorana field X, the order parameter is Tr X µ y σ a X , a = x , y , z. In order to calculate corrections to the Néel order parameter, we focus on the σ z component, which happens to be simply expressible in terms of a Dirac fermion bilinear:  where the first and second terms arise from interactions with Φ a 1 and Φ a 2 , respectively. The gauge field correction does not break SO(5) symmetry and does not contribute to the renormalization at log 2 order, so we will focus on the Higgs corrections. Additionally, there is a possible two-loop diagram shown in Fig. 8 that contributes at O N −1 f , but we show explicitly in Appendix F that these corrections also do not contribute to the renormalization at log 2 order. At zero external momenta, the log 2 Higgs corrections to the VBS order parameter (µ i = µ x , µ z ) drops out entirely, leaving only Higgs corrections coming from the fermion renormalization.
We focus on vertex corrections to the Néel order parameter (µ i = µ y ). As is the case in the fermion self-energy, the spatial anisotropy in the Higgs propagators gives rise to log 2 divergences in their corrections to the SO(5) vertex. We isolate log 2 divergences in the Higgs correction to the SO(5) order parameter by including an external momenta 2k x to the order parameter, which is distributed symmetrically between the two fermion fields. We calculate this for the Φ 1 propagator-approximating the Higgs propagator as 4|p x |/(p 2 0 + p 2 y + 4K|p x | 3 ) as in the previous section and taking the limit in Eq. (4.12), the one-loop correction is The Φ 2 propagator gives an identical correction. Since the external momenta only play the role of an IR cutoff to leading order, we generalize this result to an arbitrary external momentum and obtain the composite operator renormalizations [57] Z VBS = 1 Z Néel = 1 + 3 N f π 2 ln 2 (Kµ) .

(4.26)
We can state these results in terms of the perturbative corrections to the two-point correlator of the order parameters, ψ Γ i ψ(k)ψΓ j ψ(−k) , i.e. the corresponding susceptibilities χ VBS and χ Néel ; these combine the consequences of the composite operator renormalizations in Eq. (4.26), and Z ψ in Eq. (4.16), to yield (4.27) After a Fourier transform to real space, these correlators are The renormalization group analysis in Appendix B shows how the above results may be renormalized to large r; we find where the exponents of the prefactors, a and b, are non-universal numbers.
Leading logarithm-squared corrections have appeared earlier in a few other problems in quantum many-body theory. They appear in the theory of weakly disordered two-dimensional metals with Coulomb interactions [58][59][60]. More recently, log 2 terms have also been found in computations of the density of states of clean bilayer graphene with Coulomb interactions [61,62]. Renormalization group analyses of these cases [59,62] also yield an exponentiation similar to that in Eq. (4.29).
As an aside, we note that the one-loop vertex corrections to the bilinearψψ, whose symmetry properties identify it as the scalar spin chirality [63], have the same magnitude and opposite sign as the Néel order parameter. Because of this, the log 2 divergence is in fact cancelled by the fermion self-energy. As shown in Appendix F, the two-loop corrections coming from the Higgs fields vanish, meaning that correlations of the scalar spin chirality should have power law decay at O N −1 f . Since this power law decay is slower than the Néel and VBS correlations, this may indicate proximity to a chiral spin liquid.

V. TRANSITION FROM U(1) STAGGERED FLUX TO GAPLESS Z 2 SPIN LIQUID
This section discusses the critical U(1) gauge theory for the transition between the U(1) staggered flux spin liquid and the gapless Z 2 spin liquid Z2Azz13 in Fig. 2. A similar theory has been considered earlier [25] for the Néel-Z 2 spin liquid transition.
Both phases have the Higgs field Φ a 3 = 0. So let us fix Φ a 3 = δ az Φ, with Φ a non-zero constant, which will turn into a coupling constant in the low energy theory below. In this situation, the SU(2) gauge symmetry is broken down to U(1), and we need only consider a U(1) gauge theory with the U(1) gauge field A µ ≡ A z µ . Also important is the consequence of the w term in the Higgs potential Eq. (3.26): Choosing a gauge with w Φ < 0, and diagonalizing the quadratic form of the Higgs potential for Φ x,y 1,2 , we deduce that we need only focus on a single low energy complex Higgs field near the critical point It can now be checked that H transforms as a charge 2 Higgs field under the unbroken U(1) gauge symmetry. Other linear combinations of Φ x,y 1,2 can be ignored for the critical theory. We can now obtain the critical theory for the fermions ψ, the complex Higgs field H, and the U(1) gauge field A µ from Eq. (3.24): We define the covariant derivative D µ = ∂ µ − iA µ σ z and operators σ ± = (σ x ± iσ y )/2. Note that Φ is a marginal coupling constant here, not a fluctuating field. A crucial feature of L sf is that it does not contain the K gradient terms: these terms are now truly irrelevant. This can be seen in the large N f expansion: upon integrating the fermions, we obtain, in place of Eq. (4.3), where Γ 1,2 (k) are specified in Eqs. under the symmetries of the underlying square lattice antiferromagnet, and an analysis of these symmetries occupy a significant portion of this paper.
Our main results can be summarized in the context of the mean-field phase diagram in Fig. 2 obtained from the SU(2) gauge theory with 3 adjoint Higgs field Φ a 1,2,3 in Eq. (3.24). This mean field theory yields 3 spin liquids, with deconfined SU(2), U(1), and Z 2 gauge fields. We assume that the spin liquids with continuous gauge symmetries confine, except at possible deconfined critical transitions to the Z 2 spin liquid. This phase diagram maps onto the J 1 -J 2 model along the trajectory of the dotted blue line, and our proposed deconfined critical theories are at the boundaries between the mean field SU(2) and Z 2 spin liquids, and the U(1) and Z 2 spin liquids.
The numerical evidence for the confinement of the SU(2) π-flux spin liquid was reviewed in Section I. This confining state should have either Néel or VBS order [27], and Ref. [28] argued by comparing to bosonic spinon theories that it should be the Néel state. The structure of the critical theory from such a confining state to the gapless Z 2 spin liquid was presented in Section IV, and we found some unusual log 2 corrections to both the Néel and VBS critical correlators. From the geometry of the mean field phase diagrams in Fig. 2, and the numerical studies on the square lattice antiferromagnet noted in Fig. 1, it is then natural to propose that the U(1) staggered flux spin liquid confines to the VBS state. The critical U(1) gauge theory for the boundary between the U(1) and Z 2 spin liquid was presented in Section V, and this has no log 2 terms. We also note that the log 2 correlators in Eqs. (4.28) and (4.29) show a faster decay of the Néel order than the VBS order, which might be evidence that the SU(2) critical theory is proximate to the VBS state rather than the Néel state, which would reverse the direction of the arrow in Fig. 2.
Irrespective of the assignment of the Néel or VBS confining states to the SU(2) or U(1) spin liquids in Fig. 2, we expect any direct phase boundary between the Néel and VBS states to be a first order transition. This follows from the numerical studies [31,32,34] noted in Section I.
Our critical SU(2) gauge theory for the S = 1/2 square lattice antiferromagnet has massless 2-component Dirac fermions with 2 flavors and 2 colors, and real critical Higgs fields with 2 flavors and 3 colors, and is shown in Eq. (4.2). This derives from a theory for the π-flux to gapless Z 2 spin liquid transition proposed by Ran and Wen [39,40], and includes an additional 'dangerously irrelevant' coupling K, which is the coefficient of a spatial gradient term in the Higgs fields. We analyzed this theory along the lines of the 1/N f expansion of Ref. [56] (the case of interest to us here is N f = 1). We found that the theory with K = 0 has infrared divergencies that arise from the highly anisotropic spatial structure of the Higgs correlations, which is in turn a consequence of the non-Lorentz invariant Yukawa couplings between the Higgs fields and the fermions. So even though the coupling K is formally irrelevant, it must be included to understand the long-distance and long-time behavior of the theory i.e. the coupling K is dangerously irrelevant. We found that the coupling K leads to leading logarithm-squared corrections to various correlators, such as those in Eqs. It would be useful to examine numerical studies of the square lattice antiferromagnet for logarithmic violations of scaling, Lorentz invariance, and SO(5) symmetry, and compare to our predictions. In particular, we note the violations of scaling observed in Ref. [65], although for a different square lattice antiferromagnet.
Finally, we note that gapless Z 2 spin liquid studied is an attractive candidate for the ancilla model of doped antiferromagnets [66][67][68], as it can realize a stable state in the second ancilla layer for the pseudogap state.
As we were completing this paper, we became aware of some related work: (i ) Superconductivity has been observed [69,70] in the doped J 1 -J 2 model; doping the gapless Z 2 spin liquid is a known to be a natural route to d-wave superconductivity [18,25].
(ii ) Yang et al. [71] have detected a gapless spin liquid phase next to the Néel phase on the Shastry-Sutherland model, which is obtained from the J 1 -J 2 model by removing 3/4 of the J 2 bonds. [24]. Wen In terms of the spinons f iα , this can be written as So in this gauge, the Z2Azz13 spin liquid has both d x 2 −y 2 + is and d xy pairing and no hopping, and the fermion dispersion relation is In the ansatz in Eq. (A1), the 3 spin liquids are • The π-flux phase with SU(2) gauge symmetry corresponds to χ = η = 0, γ 1 = 0.
• The Z2Azz13 spin liquid is obtained when the d xy pairing γ 1 breaks the U(1) down to Z 2 .
For our purposes, and in general, a complex Higgs field is needed to break U(1) down to Z 2 . We have characterized the d xy pairing above by a real parameter γ 1 , and we need to generalize this to a complex parameter. From the analysis in Section II, we deduce that this is obtained by taking a complex d xy order parameter which has opposite phases on the two sublattices i.e.

Lattice PSGs
We first recall the spin liquid classification scheme of Ref. 24. If u ij is the mean field ansatz for a spin liquid symmetric under the group action G, it transforms as where W G (i) is a gauge transform. In addition to the symmetries, these gauge transformations characterize the spin liquid, yielding the projective symmetry group (PSG) [24].
Using the notation from Ref. 24, the spin liquid Z2Azz13 is defined by the PSG while the PSG of U1Cn01n (the staggered flux phase) is where g 3 (θ) = e iθτ z . From these PSGs we can extract the symmetry fractionalization through the group relations given in the appendix of Ref. 28 (Eq. (B8)). These are provided in Table I. Note that instead of P xy : (i x , i y ) → (i y , i x ), we consider the 90 • rotation R π/2 = P xy P y . Similarly, P x is related to the other symmetries through R π/2 P y R −1 π/2 .

Identification of staggered flux in continuum model
The staggered flux state (U1Cn01n) can be obtained by coupling a Higgs field to the bilinear O a 3 = tr σ aX µ y (γ x i∂ y + γ y i∂ x )X (A8) giving something like The U(1) spin liquid U1Cn01n is then obtained upon condensing one component of Φ 3 . This was determined by considering the symmetry fractionalization of the U(1) spin liquid obtained by condensing the z-component of Φ 3 : Φ z 3 = 0. Based on the symmetry transformations outlined in Table II, this condensate has a corresponding continuum PSG Group relations Z2Azz13 U1Cn01n lattice U1Cn01n cont where g 3 (φ) = e iφσ z is an arbitrary gauge transformation. Importantly, in the U(1) spin liquid, the phases φ can take any value. When these phases are rewritten in terms of the U(1) phases from Eq. (A7), θ G , according to , −θ y , θ t , θ pxy + θ py (A11) the symmetry fractionalizations given in columns 4 and 5 of Table I are identical. It is possible that two distinct spin liquids (as defined by having distinct PSGs) could nevertheless have identical symmetry fractionalization. This seems unlikely in this situation and is, moreover, proven false by the explicit derivation of the continuum action from the lattice model.

Operators
T

Identification of Z2Azz13 in continuum model
The spin liquid Z2Azz13 is proximate to U1Cn01n in that the PSG of Eq. (A6) may be obtained through gauge transformations and judicious choices of the angles θ G in Eq. (A7). It is, however, simpler to determine the U(1) transformations (i.e. the angles φ µ ) that map the symmetry fractionalization of U1Cn01n to the symmetry fractionalization of Z2Azz13. That is, we find that the assignment (φ x , φ y , φ py , φ t , φ r ) = θ + π 4 , θ − π 4 + n y π, (2n py + 1) π 2 , (2n t + 1) π 2 , θ + (2n r + n y + 1) transforms the 5th column of Table I into a set of ±1s that match the third column. Inserting these φ µ s into the PSG defined in Eq. (A10) and selecting n µ = 0, µ = y, py, t, r and θ = π/4, we obtain the Z 2 continuum PSG[72] We can now ask what form of operator needs to couple to a new Higgs field in order to realize this PSG and hence the Z 2 spin liquid Z2Azz13. Firstly, it's clear that the σ x or σ y components of the Higgs field must condense-condensing in the σ z channel, ∼ Φ z tr(σ zXM X), would not break the U(1) symmetry. However, in considering condensates in x or y, we see that the gauge transformations corresponding to the translations T x and T y are different and, further, the rotation R π/2 exchanges σ x and σ y , meaning that both must be present in a symmetric spin liquid.
Appendix B: Renormalization group analysis of the SU(2) gauge theory In this appendix, we describe the origin of the log 2 terms in the critical SU(2) gauge theory in a renormalization group (RG) framework. Integrating the RG equations will lead to an exponentiated prediction for the correlators.
We start with the expression in Eq. (4.10), keep the full Higgs propagator as in Eq. (4.6), and perform a standard momentum shell RG in the window Λ − dΛ < (p 2 0 + p 2 Expanding to linear order in k x , using spherical co-ordinates with and setting µ = cos θ, we obtain Under normal circumstances, the µ integral would be a finite numerical constant, and the coefficient of dΛ/Λ would the usual RG log which would then contribute (in this case) to the exponent η ψ . However, that is not the case here, because of the logarithmic divergence of the µ integral near µ = 1. Evaluating the µ integral , we obtain In a similar manner, we obtain for the frequency dependence of the self energy In spherical co-ordinates this simplifies to The expression for δγ y Σ 1 (k y ) is the same as δγ 0 Σ 1 (k 0 ), after mapping k 0 ⇒ k y .
We can also examine the vertex correction for the SO(5) order parameter in a similar manner.
From Eq. (4.25) at zero external momentum, we note that the vertex correction needs the integral We now proceed as usual to obtain the RG equations from the momentum shell results under the rescaling Importantly, we note the flow of the irrelevant coupling K under this transformation For the fermion field we define The field ψ is not gauge-invariant, and neither is its anomalous dimension η ψ . However, the leading log 2 term we shall find shortly is gauge invariant. In the presence of the log 2 term, we will also see that the usual logarithm terms have a non-universal co-efficient. So we ignore the gauge field contributions here (the gauge field induced renormalizations have been computed in Refs. [39,40]), because they only contribute logarithm terms which become part of overall terms which are non-universal.
We can now obtain the momentum dependence of physical observables by evaluating them at a scale = * = ln(Λ/|p|). Note that the co-efficient of * involves the bare value of K 0 , and hence the co-efficient of the logarithm term is non-universal, as claimed earlier. The leading term is log 2 , and its co-efficient is universal and agrees with that in Eq. We state in the main text that logarithm-squared divergences in the critical SU(2) gauge theory arise in the one-loop diagrams in a certain parameter regime, given by Eq. (4.12). This is shown in the main text for the simplest one-loop calculation, which is the Φ 1 (Φ 2 ) contribution to the fermion self-energy with external momenta k x (k y ). Here, we provide more general calculations for other cases.
We first analyze the Φ 1 contribution to the fermion self-energy with external momenta k 0 . This is equivalent to the k y external momenta, as well as the Φ 2 contribution with external momenta We see that the dominant term is proportional to k 0 log 2 (Kk 0 ), arising in the same limit as in Eq. 4.12. This log 2 dependence comes from the inverse tangents, since 2 tan −1 (x) = i log 1+ix 1−ix . Assuming K|p x | 3 p 2 x , the expression in the denominator of the inverse tangent argument is ≈ ip 2 x , so our integrand ≈ i tan −1 (i(1 + 4K|p x |)). If we further assume K|p x | 1, we get an integrand that scales like k 0 /p x ln(Kp x ), and hence the full expression scales as k 0 log 2 (Kk 0 ).
These log 2 contributions are verified by numerically evaluating Eq. 4.10 and Eq. C1 and analyzing the behavior at small k, as shown in Fig. 9. with K = 1 as f A (k x ) and f B (k 0 ), we plot a numerical evaluation of 8π 2 f i (k)/(k ln 3 (k)) vs 1/(ln(k)). The form of this expression is designed to isolate the log 2 contribution at small momenta, and agreement with our analytic predictions should give a straight line with a slope of 1. The approximation of the Higgs propagator as Eq. 4.9 allows for greater numerical precision, as the dimensionality of the integral can be reduced by performing portions of the integral analytically. These numerical evaluations give good agreement with analytic predictions as well as calculations using the full form of the Higgs propagator.
For the Higgs vertex correction to the SO(5) order parameter, we can also isolate a log 2 divergence. Calculating the Φ 1 correction to the vertex, we regulate the integral by including an external momenta 2k x evenly distributed between the two outgoing fermions. The vertex correction is We can obtain a log 2 from the second logarithm in the brackets in the limit given by Eq. (4.12).
Appendix D: Alternate computation of z in the SU(2) gauge theory In the main text we emphasize that although irrelevant terms in the Higgs propagator turn out to strongly influence the renormalization of the critical theory, these effects cancel in the dynamical critical exponent, and its value can be computed through more standard methods. In this section, we compute z via dimensional regularization after explicitly setting the irrelevant Higgs terms to zero, and show that is gives the same answer for z as in the main text. For the calculation of z, we are interested in the counterterms generated by This integrand is well-behaved for p 2 = 0, ∞, and one can see that the second term in brackets vanishes, since the Γ 2 propagator is invariant under p x ↔ p 0 .
The integrals over p 0 , p y can be performed exactly in radial coordinates, which gives Continuing the p x integral to 1 − dimensions and imposing a UV cutoff Λ, yields Which gives the same answer for z as when the irrelevant Higgs terms were used to regulate the divergences in the self-energy. fact that these two-loop diagrams require both types of Higgs Φ 1,2 , as they vanish trivially when both Higgs propagators are of the same type. As the log 2 divergences are connected to the rotational symmetry breaking in the O(N f ) effective action for the Higgs propagators, it is naturalalthough still a non-trivial fact -that these two-loop diagrams which respect rotational symmetry only contribute single logarithm divergences.
We combine this with the remaining loop integral, setting the coefficient K of the irrelevant operators to zero, to give .

(F5)
Hence, this two-loop contribution only contributes a standard logarithm divergence, and is subleading in comparison to the one-loop Higgs corrections.