Unification of parton and coupled-wire approaches to quantum magnetism in two dimensions

The fractionalization of microscopic degrees of freedom is a remarkable manifestation of strong interactions in quantum many-body systems. Analytical studies of this phenomenon are primarily based on two distinct frameworks: Field theories of partons and emergent gauge fields, or coupled arrays of one-dimensional quantum wires. We unify these approaches for two-dimensional spin systems. Via exact manipulations, we demonstrate how parton gauge theories arise in microscopic wire arrays and explicitly relate spin operators to emergent quasiparticles and gauge-field monopoles. This correspondence allows us to compute physical correlation functions within both formulations and leads to a straightforward algorithm for constructing parent Hamiltonians for a wide range of exotic phases. We exemplify this technique for several chiral and non-chiral quantum spin liquids.


References 20
A. Energy cost of topological defects 24 B. Derivation of the bosonic-parton field theory 26 C. Derivation of the fermionic-parton field theory 28 D. Gauge-theory calculations 29

I. Introduction
Determining the ground state of interacting spin systems is a quintessential problem in quantum condensed matter physics. Generic lattice Hamiltonians that are solely restricted by symmetries and locality typically yield ground states that spontaneously break one or more microscopic symmetries. Moreover, these phases exhibit only short-range entanglement and are therefore considered conventional. The tendency towards triviality can be avoided when additional ingredients, such as geometric frustration, prevent the formation of classical order and instead promote so-called quantum spin liquid (QSL) ground states. 1-5 These phases of matter are not characterized by any local order parameter. Instead, their principal feature is the existence of low-energy excitations that carry fractional quantum numbers and/or exhibit fractional statistics. In the cases of gapped QSLs, this definition can be sharpened into the notion of topological order, 6 which manifests itself in a universal non-local contribution to the ground state entanglement 7-9 and a ground state degeneracy on non-trivial manifolds.
The primary workhorse for analytically describing such phases is known as parton construction (see, e.g., Refs. 47 and 48). There, parton creation operators ψ † r,σ are introduced on the lattice scale and used to express the spin operators, e.g., S + r = ψ † r,↑ ψ r,↓ . In a constrained Hilbert space with exactly one parton per site, these operators can be used to faithfully represent any microscopic spin Hamiltonian. Temporarily ignoring this constraint, and allowing an arbitrary number of partons per site, permits new mean-field Ansätze that are highly non-trivial in terms of microscopic spins. Refining the mean-field theory to include fluctuations reveals the expected gauge structure. A key feature of this approach is its versatility in capturing various gapped and gapless QSLs as well as conventional phases. Its main drawback is an intrinsic difficulty to relate a given QSL phase to a specific spin model. The most tangible connection between parton constructions and microscopic Hamiltonians is through projected wave functions. 49 However, such analyses are biased by the choice of mean-field and are computationally expensive.
Over the past years, an alternative technique that shines precisely at this Achilles' heel of parton approaches has gained in popularity. In 'coupled-wire constructions', microscopic parent Hamiltonians for strongly correlated phases are constructed explicitly.
A natural starting point for applying this technique to QSLs is given by a spin system where all couplings in theŷ direction, say, have been switched off. The resulting model may be profitably viewed as an array of onedimensional spin-chains (aka wires) along thex direction. Each spin-chain is then taken to form a gapless onedimensional QSL. The corresponding long-wavelength degrees of freedom form a coarse-grained basis for reintroducing inter-wire couplings. When these interactions are strongly relevant in the renormalization group sense, they may drive the system into a bona fide two-dimensional phase. Unfortunately, no general principle for the construction of parent Hamiltonians within this framework is known. Instead, it has to be done on a laborious caseby-case basis. Consequently, only a limited number of QSLs have been accessed in this manner. [80][81][82][83][84][85][86][87] We present a framework that unifies the two approaches based on the well-known particle-vortex duality of bosons in (2+1) dimensions. [95][96][97] Its recent implementation in the coupled-wire formalism 93 allows us to transcribe field-theoretic insights into explicit models and thereby achieve the desired connection between parton and coupled-wire methods. We find remarkably simple relationships between microscopic degrees of freedom, such as the Néel vector, N y (x), or the valence-bond operator, y (x), and parton operators in a suitable gauge. For example, we show that ỹ f † y,σ,χσ fỹ −2,σ,χ σ + H.c. = y N + y+1 N − y + H.c. , (1) where f † y,σ,χ creates a fermionic parton with chirality χ = R/L and spin σ =↑ / ↓ on the 'dual' wireỹ = y + 1/2. In the first equation, χ σ = R(L) for σ =↑ (↓) and χ σ = L(R) is the opposite chirality.
Crucially, any coupled-wire model that separately conserves the two parton species maps onto a local spin model. Parent spin Hamiltonians for a wide range of non-trivial phases can thus be generated by constructing weakly correlated two-dimensional band insulators or superconductors of partons. An example of such a model is shown in Fig. 2, which illustrates the spin Hamiltonian for a Z 2 QSL obtained from a superconductor of fermionic partons. Moreover, coupled-wire models for various strongly correlated states of bosons and fermions are also known in the literature. 51,52,[54][55][56][57][58][59][60][61][62][63][65][66][67][68][69][70] . Each of these corresponds to a local spin model as well. These models realize QSLs that are not describable by a parton mean-field ansatz but require a further fractionalization of the partons.
FIG. 2. The parent Hamiltonian of a Z2 spin liquid can be obtained by translating the coupled-wire model that realizes a BCS superconductor of fermionic partons. The gray and white diamonds represent the indicated interaction between spins at their corners. These terms are reminiscent of those realizing Kitaev's toric code. 10 However, the present model conserves S z and realizes a phase with dynamical matter fields. See Sec. V B 5 for the derivation and a detailed discussion.
In addition to constructing parent Hamiltonians for specific gapped phases, we use the exact transformation between spins and partons to derive the gauge theory for the latter. We explicitly relate monopoles in the emergent gauge field to spin operators and determine the action of microscopic symmetries. These properties demonstrate the desired unification of partons and coupled wires. They, moreover, indicate that the same approach may be extended to gapless states in the future.
The rest of this paper is organized as follows: In Sec. II, we briefly summarize some key elements of parton constructions. In Sec. III, we review some well-known properties of spin-1/2 chains and their description using bosonization. We then describe how a two-dimensional easy-plane antiferromagnet (AFM), valence bond solid (VBS), and Ising-AFM are realized upon introducing inter-wire couplings and discuss their topological defects. In Sec. IV and Sec. V, we introduce bosonic and fermionic partons, respectively, as combinations of the aforementioned defects. We derive the gauge theory that results when the spin-model is rewritten in terms of these nonlocal degrees of freedom and tabulate their symmetries. We then analyze several phases in both their spin and parton representations, with particular attention to the fate of the emergent gauge field in the latter. We conclude with a summary of our results and an outlook on possible extensions in Sec. VI. Finally, the appendices discuss non-universal terms that are required for microscopically exact mappings but do not affect long-distance properties. They also reproduce several known properties of the pertinent gauge theories within concrete wirebased calculations.

II. Parton construction
In this section, we briefly review partons in the context of interacting spin systems. Two widely-used parton constructions are based on the Schwinger-boson or Abrikosov-fermion representations of spin-1/2, i.e., Here ψ σ are either bosonic or fermionic annihilation operators subject to the constraint of a single parton per site,n ≡ σ ψ † σ ψ σ = 1. The expression of spins through partons has built-in redundancy, i.e., physical spin operators are invariant under phase rotations ψ σ → ψ σ e iφ .
tations. Specifically, it must transform like the spatial components of a gauge field, i.e., a r,r → a r,r + φ r − φ r . Allowing fluctuations of a r,r and encoding fluctuations of the chemical potential via a temporal component, µ → µ + ia 0,r , results in the (Euclidean) action +ˆτ r,r ,σ χ r,r ψ † σ,r e −ia r,r ψ σ,r + H.c. .
A crucial aspect of this theory is its periodicity in a r,r , which permits 'monopole' events where the flux ∆ × a changes by 2π. The gauge field is thus compact. 98 Equivalently, any induced Maxwell term generated by integrating out ψ inherits the periodicity of the minimal coupling and is of the form cos (∆ × a). By contrast, a bare Maxwell term ∝ (∆ × a) 2 , which arises, e.g., in the particle-vortex duality, 95-97 would exclude isolated monopoles; for a recent review including modern developments, see Ref. 99. The same conclusion may be reached after taking the continuum limit by analyzing the microscopic operator that corresponds to the emergent gauge flux. In the present case, it is given by the scalar spin chirality 100 which is not conserved in most microscopic spin models. Consequently, monopole events exist in the corresponding gauge theory. By contrast, in particle-vortex duality, the gauge flux is identified with the microscopically conserved boson density. Monopoles are thus absent in that case, i.e., the gauge field is non-compact.
To study compact gauge theories, it is often convenient to separate the gauge field into a monopole-free part, a 0 , and a singular part, a M , that contains monopoles of strength 2πq i at space-time points r i = (τ i , r i ), i.e., This separation is useful, e.g., for assessing the relevance of monopoles in the presence of matter fields ψ. The monopole-monopole correlation function is Here, the singular gauge field a M must satisfy Eq. (9) with two opposite monopoles, q 1 = −q 2 = 1, but is otherwise arbitrary. While conceptually straightforward, the evaluation of C M based on this formula is a formidable task, only achievable in specific limits. Extensive studies of compact gauge theories in 2 + 1 dimensions have found that their infrared behavior falls into one of three categories: 1. Confining: When all matter fields trivially gapped, the low-energy theory is a pure gauge theory. There, monopoles always proliferate and result in confinement. 98

2.
Gapped : Confinement is avoided when the gauge field becomes massive due to the formation of a condensate (Higgs mass) or a topologically non-trivial insulator (Chern-Simons mass).

Gapless:
The presence of gapless matter, e.g., in the form of a large number of Dirac-fermion species or a Fermi surface, can render monopoles irrelevant. Such systems behave like non-compact gauge theories. [101][102][103] III. Coupled-wire approach Consider a two-dimensional array of antiferromagnetic spin-1/2 chains (aka wires) with conserved S z . The longwavelength properties of each chain, labeled by an integer y and extending alongx, can be efficiently described using Abelian bosonization. 104,105 This framework is, moreover, convenient for including inter-wire couplings and studying their effect. We thus introduce a pair of conjugate variables Θ y (x), Φ y (x) and describe the spin-chain array by a Euclidean path integral Z =´DΦDΘe −S . The action S contains both intra-wire terms and couplings between different chains; both will be specified below. In our convention, smooth and staggered components of the microscopic spins, S r = J r + (−1) x+y N r , are encoded as The transformation of Θ and Φ under microscopic symmetries can be readily deduced from these expressions; we summarize the ones pertinent to this work in Tab. I. In both cases, the bosonic variable Θ(x) changes by π/2, from one minimum to the next, over a distance set by the correlation length d * .
Consequently, such defects carry spin-1/2 and cost a finite energy ∝ 1/d * . When the correlation length is comparable to the lattice spacing, the domain wall can be depicted in terms of microscopic spins as shown in (b) for a VBS, and (c) for an Ising-AFM. The spin-1/2 associated with this excitation becomes readily apparent in this limit.
The Luttinger-liquid Lagrangian L LL is perturbed by the non-linear L 4π , which introduces 4π phase slips into S + . Its scaling dimension at the Gaussian fixed point, ∆ 4π = 4K, determines the nature of the phase. For K > 1/2 phase slips are irrelevant, and the ground state is gapless, with power-law correlations in S x,y,z and the VBS order parameter An array of spin-chains in this phase is easily destabilized by various types of inter-wire couplings and will thus be our starting point for accessing two-dimensional phases. It is, however, useful to briefly review the opposite case of relevant L 4π . We begin by introducing a dimensionless coupling constant, whose bare value at the microscopic length d 0 isg 4π ≡ 16πKd 2 0 g 4π /v. For K < 1/2 it grows under renormalization and reaches order unity at a length d * . For small |g 4π | and K, the scaling dimension of the cosine implies d * d 0 |g 4π | 1/(4K−2) . Beyond this scale, each field Θ y become trapped around a minimum of the cosine. To describe the low-energy fluctuations we, therefore, expand the cosine to quadratic order and write Here Θ (ny) y,0 denote the minima of the cosines, labeled by the integers n y . To identify the ground state, it is sufficient to replace Θ y → Θ (ny) y,0 in all observables. For negativeg 4π , the minima are at Θ (ny) y,0 = πn y /2, and there is VBS order r ∝ (−1) ny+y . For positiveg 4π we instead have Θ (ny) y,0 = πn y /2 + π/4 reflecting Ising-Néel order, i.e., N z ∝ (−1) ny . We denote the two possible ground states, n y even and n y odd, by VBS 1 (Ising 1 ) and VBS 2 (Ising 2 ), respectively. In both cases, they are related by x-translations (cf. Tab. I) and one is selected spontaneously when that symmetry is broken. The universal properties of these gapped phases are insensitive to the value of d * , which may be viewed as a new parameter that replacesg 4π .
Consider now a domain wall where the state of the y 0 th wire is characterized by n y0 for x < x 0 and by n y0 + 1 for x > x 0 . Near the domain wall, Θ y changes smoothly by π/2 over a distance ∼ d * to avoid incurring a large elastic energy cost (see Fig. 3). The precise form of this interpolation is not essential for our purposes; a sample function is δΘ DW (x 0 ) = tan −1 e (x−x0)/d * . The total spin associated with introducing an N -fold domain wall, Crucially, this value is universal and only depends on the asymptotic behavior of Θ. Moreover, the associated energy cost takes a finite non-universal value proportional to d −1 * (see App. A for details). By contrast, a 'half domain wall', i.e., N = 1/2, which formally carries spin 1/4, costs a finite energy density for all x > x 0 . Consequently, the total energy diverges linearly with the system size, i.e., such configurations are confined.

B. Coupled spin-chain arrays
To describe two-dimensional phases, we initially neglect 4π phase slips in the action of the decoupled array Instead, we perturb S decoupled by inter-wire couplings that drive the system to a new fixed-point and analyze the effect of 4π phase slips there. The leading coupling terms between neighboring wires that are compatible with the symmetries in Tab. I are FIG. 4. The lowest-order coupling terms between neighboring spin-chains drive the two-dimensional array either into an easy-plane AFM or into a gapless 'sliding Ising/VBS' state. Phase slips are irrelevant in the former, but strongly relevant in the latter, where they lead to a fully gapped VBS phase.
A third cosine, cos (2Θ y+1 − 2Θ y ), has the same scaling dimension at the decoupled fixed point as the one in L u . However, it can be obtained by combining the latter with 4π phase slips and thus need not be treated independently. The cosines in Eq. (18) compete to drive the wire array into different symmetry-broken states. Their topological defects are crucial for relating spins to the bosonic or fermionic partons. We, therefore, briefly discuss how their key properties arise within the coupled-wire framework.

Easy-plane antiferromagnet (AFM)
Consider K 1 such that L t of Eq. (18a) is strongly relevant while L u of Eq. (18b) flows to zero. As in Sec. III A, we introduce a dimensionless coupling constant with bare valueg t ≡ πd 2 0 g t /vK < 0. The flow ofg t to strong coupling permits us to replace where ∆ỹ ,y ≡ (δ y+1,y − δ y,y ) is the discrete yderivative, naturally centered on a dual wireỹ = y + 1/2. The scaling dimension of the cosine at the decoupled fixed point implies d * d 0 |g t | 1/(K −1 −4) for smallg t , but that is not essential for our purposes.
On a finite array of N w wires, only N w − 1 of the differences ∆Φ are linearly independent, and the system remains gapless. The missing linear combination N −1 w y Φ y cannot be pinned due to the global U (1) spin-rotation symmetry (cf. Tab. I). This property reflects the presence of a Goldstone mode due to spontaneous U (1) symmetry breaking. The effective action can be brought to a more familiar form by performing the Gaussian integral over Θ y . Additionally taking the FIG. 5. Two vortex configurations, shown in terms of the Néel vector N , may have significant differences in trial energies but are topologically equivalent. For formal manipulations the precise choice is unimportant, and the more anisotropic limit ξ → 0 turns out to be the most convenient in the present case.
It is straightforward to verify that 4π phase slips are irrelevant at this new fixed point and that it exhibits Néel order, N + r = 0. For the topological defects, the periodicity of L t is paramount. The number of domain walls in a given co- where Γ encloses the plaquette containingỹ. Consequently, Nỹ is precisely the number of magnetic vortices contained within this plaquette, or equivalently on the dual wire. A sample configuration that contains an isolated vortex is δΦ vortex,ξ According to Eq. (20), its energy exhibits the familiar logarithmic divergence with system size (see App. A). For the formal manipulations below, it is convenient to use the ξ → 0 limit of the above expression, i.e., with H(x) the Heaviside step function. In this configuration, all phase-winding is concentrated along onedimensional lines rather than uniformly spread as in the isotropic one. A vortex in the form of Eq. (22) is created by the operator which satisfies V † ΦV = Φ + δΦ vortex . The same form of this vortex operator was used in Ref. 93 to implement dualities between bosons and Dirac fermions on coupledwire arrays. Vortex configurations in both limits are illustrated in Fig. 5.
2. Intra-wire valence bond solid (VBS) For K 1/4, it is L u of Eq. (18b) that is strongly relevant while L t of Eq. (18a) flows to zero. We focus on positive g u and express it in terms of a dimensionless coupling constant. To describe low-energies, we replace where Sỹ ,y ≡ (δ y+1,y + δ y,y ) is naturally centered on a dual wire. This Lagrangian is identical to Eq. (19) upon replacing Θ y ↔ (−1) y (Φ y + πy/2) and K ↔ K −1 , which does not affect the kinetic inter-wire terms. Consequently, the analysis of the easy-plane AFM carries over.
The ground state exhibits Ising-Néel and/or VBS orders according to with a spontaneously chosen Θ 0 . Around a topological defect, Θ 0 winds smoothly by π. Such defects are illustrated in Fig. (6) and will be referred to as dislocations. Around them, the state of the system changes from Ising 1 first to VBS 1 , then to Ising 2 , next to VBS 2 , and finally returns to the Ising 1 configuration. The creation operator in the extreme vertically-deformed limit is which satisfies b † Θb = Θ − (−1) y δΘ vortex with δΘ vortex as in Eq. (22).
At this fixed point (referred to as sliding Ising/VBS in Fig. 4), the previously neglected 4π phase-slip term, Eq. (13), is strongly relevant. Its flow to strong coupling eliminates the zero-energy mode Θ y (x) → Θ y (x) + (−1) y α and results in a gapped phase. As in the case of decoupled chains, a VBS is realized for negative g 4π , while positive g 4π leads to an Ising-AFM. In both phases, the energy cost of creating an isolated dislocation diverges linearly with the system size (see App. A). Tightly bound dislocation-anti-dislocation pairs are thus the fundamental low-energy excitations. When the two are located on neighboring plaquettes, they form precisely the staggered component of the microscopic spin operators Notice the similarity to the parton decomposition S + = ψ † ↑ ψ ↓ in Eq. (3). To make this connection manifest, we introduce the redundant label σ =↑, ↓ for odd and eveñ y, respectively. We then define b † y,σ = e iϕỹ,σ and the associated dislocation density ρỹ ,σ = 1 π ∂ x θỹ ,σ with ϕỹ ,σ = 1 2 y sgn (y −ỹ) (−1) The total number of b σ -bosons (dislocations centered on even or odd dual wires) is with S z total as defined in Eq. (16). Crucially, when S z total is microscopically conserved, then the number of each of these boson species is separately conserved.
To conclude the discussion of this phase, we want to point out a close connection between dislocations and magnetic vortices. Consider a vortex in b σ only, but not in bσ. To construct its creation operator, one need only replace Θ → θ σ in Eq. (23). Explicitly, we introducẽ Using Eqs. (23) and (28b) we find that V 2ỹ = e iφ ↑,2ỹ and V 2ỹ+1 = e iφ ↓,2ỹ+1 . Dislocations are thus dual to magnetic vortices in this precise sense.

IV. Bosonic partons from coupled wires
To access a wider range of phases, including QSLs, we now develop a dual description of the coupled-wire model in terms of the topological defects described above. Such a reformulation of spin-1/2 models in terms of magnetic vortices has already been used in Ref. 106 to access, e.g., a putative 'deconfined' quantum critical point between Néel and VBS orders. To connect coupled-wire and parton techniques, we instead focus on dislocations and argue that they form a bosonic-parton representation of the microscopic spins.
As discussed at the end of the previous section, b σ possess several of the relevant bosonic-parton properties. Moreover, while the combination b † ↑ b ↓ is local in terms of microscopic operators, and carries spin 1, individual b σ are non-local and do not have well-defined spin. This property closely relates to the local gauge redundancy of conventional parton decompositions discussed in Sec. II B. Even though b σ are non-local, the inter-wire couplings introduced in Eq. (18) remain local under the mapping. Translating them via Eq. (28), we find an intra-species nearest neighbor tunneling term and an umklapp term where U (x) = U (x + 1) is a weak periodic potential. By contrast, the intra-wire interactions of the microscopic spin-chains are non-local in bosonic-parton variables. They have a natural interpretation in terms of an emergent gauge field, as we will now explain.
FIG. 6. Four fundamental domain boundaries between VBS1, Ising1, VBS2, and Ising2 regions terminate in a dislocation that carries spin-1/2. This topological property holds irrespective of the detailed configuration. It becomes apparent in the strongly anisotropic limits, where either the Ising or the VBS phases extend only along a one-dimensional line. There, the properties of the dislocation can be inferred from the one-dimensional domain walls illustrated in Fig. 3. In our formulation, bosonic partons in a specific gauge are introduced as the extreme vertically-deformed dislocations; they carry spin-1/2 and live on dual wires.

A. Gauge theory
To derive the action governing the bosonic partons, we invert the mapping in Eq. (28) and insert them into the microscopic wire-array model in Eq. (17). It is convenient to include, on top of L chain , the symmetry-allowed quadratic inter-wire couplings In the limit of weakly coupled wires, these arise as the leading renormalizations of the kinetic energy due to Eq. (18), but in generic cases, u B and u V should be viewed as independent parameters. While L t , L u , and L inter [Eqs. (18) and (32)] are local in terms of partons, the intra-wire interactions L chain are highly non-local. This non-locality can be encoded exactly through an emergent gauge field a, just as in the case of the boson-vortex duality (see App. B for details). 93 We express the resulting gauge theory as The first two contributions contain parton and gauge-field kinetic terms as well as the coupling between the two, i.e., The parameters v B , u B ,ṽ, and κ are non-universal. For their expression in terms of microscopic spin-chains parameters, see App. B. There, we also specify the last term, L int , which contains exponentially decaying interwire density-density and current-current interactions. The gauge-field Lagrangian L Maxwell describes an anisotropic Maxwell term in the a 2 = 0 gauge but missing the ∝ (∂ τ a 1,ỹ − ∂ x a 0,ỹ ) 2 contribution. Such a term will be generated upon integrating out matter fields at high energies. We demonstrate this in Sec. IV B 1 for the case of trivially-gapped partons. Finally, it is instructive to express the bosonized model in terms of b σ as where ρ 0 = π is the boson density. This Lagrangian has the expected structure for bosonic partons [cf. Eq. (7)].

Monopoles
The 4π phase slips in Eq. (13) are also non-local in bosonic parton variables. To interpret them, we introduce the operator For both bosonic partons, , where α(r) winds counter-clockwise by 2π around the origin. M r can thus be viewed as the insertion of a 2π monopole in the emergent gauge field at position r.
Since it is odd under lattice translations (cf. Tab. I), 4π monopoles created by M 2 r are the minimal ones allowed by symmetries.
It is useful to disentangle the monopoles from the matter fields. We, therefore, write M † = e iφ M and replace Eq. (13) by where λ is a Lagrange multiplier, and φ M is now an independent variable in the functional integral. A simple shift a 0,ỹ → a 0,ỹ − y sgn (y −ỹ) λ y decouples the Lagrange multiplier from the matter fields and moves it into the gauge-field action. Integrating it out then yields our final form of the monopole Lagrangian with parameters κ,ṽ as in Eq. (34) and where we have relabeled g 4π → g M . For g M = 0, the Gaussian integral over φ M is a complete square and does not affect any gauge-field or matter correlation function.

Symmetries
To complete the description of the parton-gauge theory, we now specify how the microscopic symmetries of Tab. I are implemented. The straightforward application of the mapping between spins and partons leads to the symmetry properties summarized in Tab. II. Additionally, we introduce an external probing field A that minimally couples to the conserved S z of the microscopic spins. In its presence, the theory for the decoupled spinchain is augmented to L chain → L chain + L A with Re-deriving the bosonic parton action with these terms (see App. B) amounts, at lowest order in ∆, to replacing a µ,ỹ → a µ,ỹ − 1 4 (−1)ỹ [SA µ ]ỹ in Eqs. (33) and (35).

Alternative perspective
An alternative route to the parton gauge theory begins with rewriting the wire array in terms of magnetic vortices [see Eq. (23)]. On a lattice, these vortices experience an average flux of π per plaquette. 107,108 Their band structure thus exhibits two valleys, which amounts to two vortex flavors at low energies. To see how this is reflected in the wire framework, consider the inter-wire couplings of Eq. (18). The XY spin exchange L t translates into 2π phase slips for the vortex Vỹ while The wire-array model L LL + L u + L t can thus be equivalently expressed in terms of two separately conserved vortex flavors that reside on even or odd dual wires. Moreover, all vortices are coupled to the same non-compact gauge field a vortex whose flux represents the conserved S z . (The derivation of this gauge theory within the wire framework is identical to the one performed above for bosonic partons.) Performing separate dualities for the two vortex flavors results in two species of bosonic partons (cf. the final paragraph of Sec. III B 2) coupled to a single gauge field a that is, likewise, non-compact. Its flux corresponds to the difference between the numbers of V even and V odd vortices. This difference ceases to be conserved in the presence of the 4π phase slips. Indeed, allows vortex pairs to switch their flavor. Such processes change the flux of the gauge field a by 4π and, thereby, render it compact.

B. Phases of bosonic partons / spins
To demonstrate the generality of the formalism developed above, we now apply it to several concrete examples. In the spirit of most parton constructions, we primarily consider phases that are trivial at the mean-field level, i.e., in the absence of gauge fluctuations. These include Mott insulators, superfluids, and integer quantum Hall states. Coupled-wire models that realize such phases are either known in the literature or can be constructed relatively easily. 51,55,69,90 The mapping in Eq. (28) then immediately provides a corresponding coupled-wire model in terms of the microscopic spin variables. We carefully examine both representations of each phase-in terms of partons and of spins-and demonstrate their equivalence.
We analyze the parton models as follows: First, we determine the ground state and excitations of L b +δL at the mean-field level, i.e., by treating a as static. Second, we reintroduce the gauge-field dynamics and determine how they are affected by the matter fields. If the gauge field remains massless, we analyze the effect of monopoles. Third, we examine the quasiparticle content of the gauge theory. Finally, we perform a conventional analysis of the equivalent coupled-wire model in terms of the microscopic spin variables, L = L LL +δL [Φ, Θ], and verify that its properties match the ones obtained from the parton gauge theory.

Mott insulator / Intra-wire VBS
Mean field -A trivial Mott insulator of partons forms when the umklapp term of Eq. (31b) gaps the bosons on each wire separately. This interaction does not contain ϕỹ ,σ , which can, therefore, be integrated out trivially. The parton Mott insulator is thus described by At sufficiently long length scales, each field θỹ ,σ becomes trapped around the minima of the cosine, as in Sec. III A.
Integrating out the massive fluctuations around these minima yields L ind ∝ (∂ τ a 1,ỹ − ∂ x a 0,ỹ ) 2 , which modifies the dielectric constant of the (static) gauge field a.
The periodicity of the cosine implies that θỹ ,σ winds by π at a fundamental domain wall. Such a configuration describes an isolated parton, created by the operator b † σ . At the mean-field level, these spin-1/2 excitations are the elementary quasiparticles.
Gauge fluctuations-We now reinstate the dynamics of the gauge field and supplement its bare action, Eq. (34), by L ind . The effective Lagrangian is thus where the non-universal length scale d * encodes the flow of g u to strong coupling. Recall that in our formulation, a is a non-compact gauge field, and monopoles are included through L M [cf. Eq. (38)]. In their absence, bosonic partons would interact logarithmically via the gauge field.
(The interaction potential is readily obtained from a 0 a 0 at ω = 0.) However, it is well-known that compact U (1) gauge theories may be unstable to monopole proliferation, i.e., confinement. The relevance of monopoles can be assessed from their correlation function. For an isolated monopole-anti-monopole pair at locations R i = (x i , d * y i ) and equal time, it is given by For g M = 0, the theory is Gaussian and performing the integral over φ M yields Here ε 2 = ∆a 0 /d * is the y-component of the emergent electric field, . . . denotes the Gaussian average over a, according to L MW , and k = (k x , k y ). The momentum k y is expressed in units of d −1 * , i.e., the Fourier transform is defined as ,ω e ik·R+iωτF (k, ω). Inserting the ε 2 correlation function and evaluating the integral for large |R|, we find C M R ∼ exp κd * 2|R| . Since C M R approaches a non-zero constant at long distances, monopoles are strongly relevant. Beyond a length scale l * , we thus expand the cosine in L M to quadratic order. Integrating out φ M then results in a modified theory for a given by with L MW as in Eq. (43). The corresponding analytically continued gauge-field propagator has poles at real fre- , there are no gapless gauge-field modes for finite l * .
Alternatively, C M R can be obtained from L MW alone via Eq. (10), i.e., by externally imposing the desired singularities on a. For specificity, consider a M such that This configuration is depicted in Fig. 7. It describes two flux tubes emanating from x = ∞ and extending, at fixed y = 0 and y = y 0 , to x = 0 and x = x 0 , respectively. Inserting this expression into Eq. (10), with the gaugefield action of Eq. (43), we reproduce C M R in Eq. (45). Quasiparticles-In the presence of a confining gauge field, individual partons seize to be finite-energy quasiparticles. It would be tempting-but incorrect-to compute their interaction potential directly from the effective gauge theory in Eq. (46). To obtain L confining , we expanded g M cos (2φ M ) around a specific minimum. This restriction to a single topological sector is innocent in the special case of partons on the same dual wire. There, only the trivial sector contributes, and we can indeed use L confining to find In the generic case, a careful sum over different sectors, as performed in App. A, is required. The result is linear confinement in all directions: It is impossible to isolate any excitation charged under the emergent gauge field without incurring a diverging energy cost. Consequently, A singular gauge-field configuration with monopoles of opposite signs can be realized via two flux tubes that extend from infinity at fixed τ and y. Each x-τ plane represents a dual wire at coordinateỹ = y + 1/2, and monopoles are located at y, i.e., between wires.
finite-energy excitations can only be created as combinations of the gauge-neutral quasiparticles b † σ b σ , carrying spin 0, and b † σ bσ carrying spin 1. Spin model -There are two complementary routes to identifying the microscopic phase: through symmetry considerations and by direct translation to a microscopic model. For the former, recall how monopoles transform under the microscopic symmetries (see Tab. II). The nonzero expectation value acquired by M implies that xtranslation symmetry is reduced to translations by two sites. Furthermore, for g M < 0, the site-centered xinversion is also broken, while bond-centered inversion is preserved. Other symmetries, in particular time-reversal and U (1) spin-rotation, remain intact. These properties identify the microscopic phase as an intra-wire VBS. We arrive at the same conclusion by using the transformation from parton to spin variables. The gauge theory maps onto L LL + L u , which we already analyzed in Sec. III B 2. Its gapped ground state exhibits VBS order, ε r = 0, with integer-spin excitations, exactly as we found in the gauge theory above.

Superfluid / Easy-plane AFM
Mean field -Consider now a superfluid phase where both partons condense. Recall that the parton number is separately conserved for both species. The corresponding U (1) symmetries are spontaneously broken when the tunneling term of Eq. (31a) flows to strong coupling. We proceed as before and expand where (∆ 2 ) y,y = δ y+2,y − δ y,y . The field θỹ ,σ does not enter L t and can thus be integrated out trivially. Additionally taking the long-wavelength limit, d −1 * ∆ 2 → 2∂ y , we arrive at the action S SF b =´τ ,x,y σ L SF,σ , with This low-energy theory describes two fields that disperse linearly with velocity c = √ u B v B -Goldstone modes associated with the two condensed parton species. Integrating out ϕ σ results in the familiar Meissner response where p = ( ω c , k x , k y ) and ā = ( a0 c , a 1 , a 2 ). Finally, the periodicity of the original cosine in Eq. (49) permits 2π vortices in either condensate, which are logarithmically confined as in Sec. III B 1.
Gauge fluctuations-We reinstate the gauge-field dynamics, governed by L MW + L M of Eqs. (34) and (38). The induced Meissner term renders the gauge field massive and, thereby, monopoles strongly irrelevant. We verify this explicitly by integrating out a to obtain the effective monopole Lagrangian. In the limit of small frequencies and momenta, we find The corresponding monopole-monopole correlation function decays faster than exponentially, i.e., where L is the wire length, and ξ 1,2 are non-universal length scales. Quasiparticles-To determine the fate of the partons, we focus on one species and integrate out the other. Recall that the gauge field a lives on all dual wires, while each parton species resides on wires with a specific parity. We, therefore, integrate out b ↓ and the gauge field on even wires to obtain the effective gauge theory for b ↑ on odd wires. The long-wavelength expansion of the gaugefield action reproduces the form of L Meissner , in Eq. (51), with rescaled fields and momenta (see App. D for the exact wire-based calculation). Integrating out the massive gauge fluctuations does not qualitatively change the low-energy theory for b ↑ . In the present case, it is of the form L SF in Eq. (20), and vortices in the phase of b ↑ are logarithmically confined.
It is instructive to analyze the role of the external probing field as introduced in Sec. IV A. At the mean-field level, the partons couple to A with charges (e ↑ , e ↓ ) = (1/2, −1/2). Condensation of either species forces the flux of A to be quantized in units of 4π. However, in the gauge theory, a simple shift a µ,ỹ → a µ,ỹ + 1 4 [SA µ ]ỹ leads to (e ↑ , e ↓ ) = (1, 0) and, consequently, 2π quantization. This apparent ambiguity disappears when gauge fluctuations are accounted for. Indeed, integrating out a and b ↓ in the presence of A, we find an effective field theory for b ↑ with e ↑ = 1. Further integrating out the remaining parton, b ↑ , yields a Meissner response in the form of Eq. (51) for A (see App. D for details). Consequently, U (1) spin-rotation symmetry is spontaneously broken.
Spin model -The microscopic phase breaks U (1) spinrotation as well as time-reversal symmetries (see Tab. II). Moreover, the discrete translation symmetries in thex andŷ directions are both reduced to steps of two. These properties identify the microscopic phase as the easyplane AFM described in Sec. III B 1. Indeed, the parton gauge theory maps onto L LL + L t , which was studied there in detail. We found the same gapless ground state, topological excitations, and, implicitly, the same 2π quantization of flux.

Correlated Mott insul. / Inter-wire VBS
Mean field -Consider now a superfluid phase of one parton species and a Mott insulator of the other, i.e., δL = g t cos (ϕ 2ỹ+2,↓ − ϕ 2ỹ,↓ ) + g u cos (2θ 2ỹ+1,↑ ). The mean-field analysis is the same as in the two previous cases; the low energy theory contains the Goldstone mode of the condensed b ↓ and individual b ↑ as finite energy excitation. Vortices in the b ↓ -condensate, created by V odd ∼ e −iφ ↓ [cf. Eq. (30)], are logarithmically confined.
Gauge fluctuations-As in the parton superfluid phase, Sec. IV B 2, the gauge field acquires a Higgs mass through the b ↓ condensate. Consequently, monopoles can again be safely discarded.
Quasiparticles-The effect of gauge fluctuations on the mean-field excitation b ↑ can be inferred, as in the last section, by successively integrating out b ↓ and a. Exciting a single b ↑ boson above the Mott gap thus correspond microscopically to a spin 1 excitation. In addition, vortex excitations V odd turn into local spin 0 quasiparticles. Formally this follows from the qualitatively different behavior of the correlation function [∆ 2 ϕ ↓ ] 2ỹ+1 [∆ 2 ϕ ↓ ] 2ỹ +1 at zero frequency: At the mean-field level, it falls off quadratically with distance, but in the gauge theory, it decays exponentially. (For an explicit calculation of the vortex energy, see App. A.) Spin model -In this phase, y-translation symmetry is broken, butỹ−inversion is preserved (cf. Tab. II). Moreover, U (1) spin-rotation, time-reversal, and x-translation symmetries all remain intact. These symmetry properties, along with the quasiparticle content, imply an interwire VBS. To verify this explicitly, we transcribe the cosines in δL to spin variables δLinter−VBS = gt cos [∆Φ] 2ỹ+1 + gu cos 2 [SΘ] 2ỹ+1 . (54) One readily verifies that the arguments of the two cosines commute. Consequently, the two terms can simultaneously reach strong coupling. In the resulting fixed-point Hamiltonian, only pairs of wires are coupled. To characterize the ground state, it is thus sufficient to analyze a two-leg ladder.
We diagonalize the interaction by introducing new conjugate variables Φ ± = 1 2 (Φ 2 ± Φ 1 ) and Θ ± = (Θ 2 ± Θ 1 ). The fields Φ − and Θ + get trapped around the minima of their respective cosines, and small fluctuations are massive. Fundamental domain walls in the two are created by D + = e iΦ+ and D − = e iΘ− . The former is identified with the spin-raising operator, i.e., D + ∝ S + 1/2 , with a proportionality factor determined by the pinned Φ − . The latter similarly describes 2π phase slips in S + , i.e., D − ∝ e ∓2iΘ1,2 . Consequently, the two types of defects carry spin 1 and spin 0, respectively. Alternatively, the spin can be computed via the general expression In the present case, y S z y = 1 π ∂ x Θ + and we again find

Quantum Hall states / Chiral spin liquids
Mean field -As the first example of a fractionalized phase, consider a bilayer quantum Hall state of bosonic partons. At filling factor ν = 2/n, it can be realized with the inter-wire coupling This wire construction was proposed in Ref. 51 for microscopic bosons e iϕ and analyzed in detail. Adapted to the present context, the resulting phase hosts two species of spin-1/2n excitations that are self-bosons but exhibit mutual statistics π/n, i.e., the corresponding K-matrix is n times the Pauli matrix σ x . To find K, we calculate the response to a by replacing δL QH with its quadratic expansion and integrating out the matter fields (see App. D).
The leading contribution at long wavelength is where µν is the antisymmetric tensor. Endowing a with a (redundant) spin label according to the dual-wire parity, the induced action for a σ takes the expected form.
In the continuum limit, we find in the gauge a 2 = 0 and with K = nσ x . Gauge fluctuations-We restore the status of a as dynamical, with fluctuations governed by the sum of the induced Chern-Simons and bare Maxwell terms. The latter contains the contribution ∝ [∆a µ ] 2 , which translates into ∝ (a µ,↑ − a µ,↓ ) 2 . This term renders the antisymmetric combination of a µ,σ massive, while the Chern-Simons term results in a gap for the symmetric combination. Consequently, monopoles are strongly irrelevant and can be safely discarded.
The Chern-Simons action, Eq. (57), implies that both microscopic and bosonic-parton time-reversal symmetries are broken. To see the latter, it is convenient to compute the response to the external probing field A. Including it amounts to replacing a µ,ỹ → a µ,ỹ − 1 4 (−1)ỹ [SA µ ]ỹ in the induced action (but not in the bare one). Integrating out the emergent gauge field we find the response µν Aµ∂yAν . (59) Consequently, these phases are chiral and must exhibit topologically protected edge states. Quasiparticles-To identify the quasiparticles, we transform the partons into new composite bosons. Specifically, we attach to each parton n fluxes of the opposite one. On the operator level, this procedure amounts to introducing bosons βỹ ,σ = e −iηỹ,σ with ηỹ ,σ ≡ ϕỹ ,σ + nφσ ,ỹ . Such manipulations often become more transparent in a schematic description that specifies only the couplings between particle currents, j b,σ , and gauge fields. The bosonic-parton theory in Eqs. (33) and (34) is then expressed as where the ellipsis denotes kinetic terms for partons and dynamical gauge fields as well as short-range interactions ( A, as always, is an external probing field). Attaching n mutual fluxes amounts to replacing L b → L β with Finally, we shift c σ → c σ − a to decouple a from the matter fields and integrate it out to obtain In terms of composite bosons, the inter-wire coupling reads δL QH = g QH cos (ηỹ +1,σ − ηỹ −1,σ ), i.e., β σ form a superfluid. At the mean-field level, the excitations are two flavors of logarithmically confined vortices in the phases of β σ . In the presence of the dynamical gauge fields c σ , these turn into finite-energy excitations subject to the constraint that (−2π) flux of c σ must be accompanied by σ/2n charge of each boson, i.e., Since ρ β σ is related to the physical spin [charge under A, see Eq. (62)] via S z = (ρ β ↑ − ρ β ↓ )/2, the composites carry a total spin of 1/2n. Moreover, a clockwise exchange results in a statistical phase π/2n. Spin model -The response to the external probing field, Eq. (59), implies that the microscopic phase is a chiral QSL with topological edge states and fractionalized quasiparticles in the bulk. Translating the inter-wire coupling in Eq. (56) to microscopic variables we find Precisely this coupling, with n = 1, was proposed in Refs. 81 and 82, where it was shown to realize the Kalmeyer-Laughlin chiral spin liquid; the generalization to arbitrary integers n is straightforward. (The same coupling term also describes a bosonic Laughlin state at filling factor ν = 1/2n, see Ref. 51). In particular, bulk quasiparticles carry spin 1/2n and acquire phases π/2n upon (clockwise) exchange.

Pair condensate / Z2 spin liquid
As the final example with bosonic partons, we construct a time-reversal-invariant gapped QSL. Here, the emergent gauge field must acquire a Higgs mass without condensation of either of the two species (which would lead to a symmetry-broken phase as discussed in Secs. IV B 2 and IV B 3). These requirements are satisfied when composites with higher emergent gauge charges, such as parton pairs, condense.
Gauge fluctuations-We reinstate the gauge-field dynamics and integrate out the matter fields. The pair condensate leads to a Higgs mass for the emergent gauge field, a, as in Sec. IV B 2. Consequently, monopoles are again strongly irrelevant and can be safely discarded.
Quasiparticles-The analysis of quasiparticle excitations closely mirrors the one in Sec. IV B 3. Integrating out both θ + , ϕ + and the gauge field a results, to lowest order in ∆, in with renormalized parametersv andū. Consequently, ε ∼ b † ↑ ϕ + , a ∼ b ↓ ϕ + , a creates a deconfined bosonic spin-1/2 excitation, as in the mean-field discussion. In addition, the dynamical gauge field liberates vortices, m, from their logarithmic confinement as in Sec. IV B 3; they become bona fide spin-0 bosonic quasiparticles.
To infer the mutual statistics between the two quasiparticles, consider the hopping of ε. It stems from the microscopic term i.e., the hopping amplitude is set by the pair condensate.
In the absence of vortices, t is uniform. For a static vortex-anti-vortex pair, there is instead a branch cut connecting the two, across which the phase of t jumps by π (see Fig. 9). For dynamical quasiparticles m and ε, this property implies mutual semionic statistics. Spin model -The quantum numbers and braiding properties of quasiparticles are characteristic of a Z 2 QSL. Time-reversal symmetry is preserved in this phase, but translation symmetry in theŷ direction is reduced to translations by two wires. To analyze this phase in terms of microscopic spin variables, we translate the intra-wire couplings of Eqs. (65) and (66), finding The two cosines do not compete, and their arguments can thus be pinned simultaneously. For an even number of wires with periodic boundary conditions in theŷ direction, there are as many linearly independent pinned fields as there are wires. Consequently, a fully gapped phase can form.
The microscopically allowed operator e i2Θ2y increments the arguments of two adjacent g p -cosines by 2π, i.e., it creates a pair of fundamental domain walls. Similarly, e −i2Θ2y+1 creates a strength-2 domain wall in a single g p -cosine. Consequently, a domain wall and an anti-domain wall on the wires 2y 2 + 1 and 2y 1 − 1 are created by The string operator s m 2y1,2y2 is comprised solely of the pinned combinations Θ 2y+1 + 2Θ 2y + Θ 2y−1 and acquires a non-zero expectation value. Domain walls are thus deconfined in theŷ direction. Additionally, one readily verifies that O m 2y1−1,2y2+1 ∝ m † 2y2−1 m 2y1+1 in the ground state. To move the domain wall along the wire direction, one need only apply O m 2y0−1 (x 1 , x 2 ) = e −i´x 2 x 1 dx∂xΘ2y 0 −1 . These domain-wall excitations are thus precisely the m quasiparticles discussed in the gauge-theory analysis.
The spin of m and ε can be inferred from the operators that terminate the strings in O m and O ε using Eq. (55). The m quasiparticle is spinless while ε carries spin 1/2. Finally, we compute the exchange statistics of the quasiparticles. Since all terms in Eq. (IV B 5) commute, m has trivial self-statistics, i.e., it is a boson. The same holds for ε. Their mutual statistics can be read off from where U 1 = O m 2y0−1 (x 1 , x 2 ) and U 2 = O ε 2y1,2y2 (x 0 ). Braiding occurs when the paths of ε and m interlink, i.e., for y 0 ∈ [y 1 , y 2 ] and x 0 ∈ [x 1 , x 2 ]. In that case, we find e i2α = −1, which implies that the two quasiparticles are mutual semions. Consequently, they can combine to form a fermion, schematically O ψ ∼ O ε O m . We will see that these fermions exactly coincide with the partons that are the focus of the next section.

V. Fermionic partons from coupled wires
Above, we have seen that the topological defects in a VBS form a bosonic-parton representation of spins. We now show that fermionic partons can also be constructed from topological defects, specifically as composites of magnetic vortices and dislocations. This route to fermionic partons is closely related to the well-known flux attachment 109,110 (see also Refs. 99, 111-113 for recent refinements). Recall that b † ↑ is dual to one flavor of magnetic vortices, V ↑ = e iφ ↑ , while b ↓ is dual to the other, V ↓ = e iφ ↓ [see discussion near Eq. (30)]. Therefore, fermions can be constructed as f σ = b σ V σ , i.e., by attaching 2πσ flux to the bosonic partons. Schematically this transformation can be expressed as Performing the shift c µ,σ → c µ,σ − a µ and integrating out a µ yields the constraint c ↑ = c ↓ ≡ c. The resulting theory has the same structure as the bosonic one: two species of fermions that are minimally coupled to an emergent gauge field c, which obeys Maxwell dynamics.
To translate generic inter-wire couplings, the following identities are useful In particular, the inter-wire couplings in Eq. (18), which generate the AFM and VBS phases, become simple hopping terms for the fermions y L u =g u ỹ f † y,σ,R fỹ ,σ,L + H.c.
As in the case of bosonic partons, trivial umklapp processes are allowed, which implies unit filling. When it is the most relevant term, L u opens a band gap, as would be the case for weakly interacting fermions. In the present case, such a trivial phase can be avoided by several mechanisms: Firstly, it stands in competition with a quantum Hall insulator generated by L u . Secondly, interactions can render correlated processes strongly relevant and drive the partons to a new fixed point where umklapp processes are irrelevant. Lastly, in certain microscopic spin models L u , is altogether absent. This is the case, e.g., on a triangular lattice due to geometric frustration. 115 Finally, destroying a parton of spin ↓ and creating one with spin ↑ yields the smooth component of the microscopic spin-raising operators similar to the parton decomposition of Eq.
(3). The analogous expression for bosonic partons instead gives the staggered component of the spin. Of course, both contributions must be encoded in either parton representation. The respective missing ones are encoded non-locally in monopole operators, as we will see below.

A. Gauge theory
To derive the action for fermionic partons, we proceed as we did for the bosons in Sec. IV A. We find S f = x,τ The final term, L int , contains exponentially decaying inter-wire terms (see App. C for a detailed derivation and expressions for v F , κ andṽ). It is instructive to express L f in terms of the non-chiral fermions fỹ ,σ = fỹ ,σ,R e ik F x + fỹ ,σ,L e −ik F x . We find To determine the chemical potential µ, notice that the value of k F only carries significance relative to another length. In the present case, this scale is given by the lattice spacing of the underlying spin-chain, which enters the fermionic theory through Eq. (77).

Monopoles
The 4π phase-slip term of Eq. (13) is non-local in terms of fermionic partons. In the discussion of bosonic partons, we expressed phase slips through the operator M [cf. Eq. (36)]. Since θ σ = θ f,σ , it again acts as the insertion of a fundamental (2π) monopole in the emergent gauge field a. Microscopically, monopoles encode the Néel vector through N + y = M y J + y,χ , with χ = (−1) y , and where J + is expressed using fermion operators in Eq. (78). The same procedure as for bosonic partons (cf. Sec. IV A) leads to the monopole Lagrangian with parameters κ,ṽ as in Eq. (80). As before, when g M = 0, the monopole field φ M does not affect any gauge-field or matter correlation function.

Symmetries
We conclude the description of the parton gauge theory by discussing how microscopic symmetries are encoded. One significant difference from the case of bosonic partons is, that certain microscopic symmetries are realized non-locally. This property may be readily understood from the flux-attachment interpretation of fermionic partons: Time-reversal flips the winding of dislocations, cf. Fig. 6, but not of magnetic vortices. Consequently, it transforms f σ = b σ V σ onto a dual set of fermions The same dual fermions also arise under translation alongŷ, which takes b σ → σb † σ (cf. Tab. II). We summarize the actions of all previously discussed microscopic symmetries on the fermionic partons in Tab. III. As in the case of bosonic partons, it is convenient to keep track of the U (1) spin-rotation symmetry by introducing the appropriate external probing field A [see Eq. (39)]. To lowest order in ∆, it enters L f and L f by replacing a µ,ỹ → a µ,ỹ − 1 4 (−1)ỹ [SA µ ]ỹ.

B. Phases of fermionic partons / spins
We now apply the formalism developed above to several specific phases of the fermionic-parton gauge theory. Following the same steps as for bosonic partons in Sec. IV B, we first study mean-field states without gaugefield dynamics. We then include gauge fluctuations and determine the quasiparticle content. Finally, we analyze the corresponding microscopic model. Certain symmetries such as time-reversal act non-locally as dualities, i.e., transform fσ into dσ. As before, asterisks denote anti-unitary symmetries.

Trivial band insulator / Intra-wire VBS
The fermionic partons are at unit filling and can form a trivial band insulator. To generate it, we perturb the parton gauge theory with the umklapp term of Eq. (77b). The resulting theory has the same form as the one describing Mott-gapped bosonic partons (see Sec. IV B 1), and its analysis is identical. In particular, we obtain the same microscopic model.
This response reflects the well-known property of quantum spin Hall systems that a 2π flux is accompanied by a spin 1. 116 To see this explicitly, consider a configuration of a that includes a single 2π flux tube penetrating through the plaquette delimited byỹ 0 andỹ 0 + 1. In the a 2 = 0 gauge, such a configuration satisfies ffi 11. (a) Adiabatically threading magnetic flux generates an electromotive force that creates an inwards current of ↓ and an outwards current of ↑ partons. When 2π flux is introduced in this manner, a single ↓ particle and ↑ hole are pulled in. (b) Accordingly, fundamental monopoles become dressed by f † ↓ f ↑ and carry spin 1. Their proliferation thus results in a phase with (spontaneously) broken U (1) spin-rotation symmetry, i.e., magnetic order.
Gauge fluctuations-Upon reinstating the status of a as a dynamical gauge field, we find its universal longdistance behavior to be unaffected by L ind . The field a s describes modes near momentum q = π, which are massive according to the bare Maxwell term, L Maxwell , and thus do not affect long-wavelength fluctuations. To lowest order in ∆ and for frequencies and x-momenta small compared to the QSH gap, a is governed by Here, L Maxwell is the bare gauge-field action of Eq. (81) and d * is a non-universal length scale proportional to the inverse QSH gap. While this effective action describes a propagating photon, the monopole operator M is strongly irrelevant. Its correlation function, according to Eq. (44), is with ε 2 as in Sec. IV B 1 and R = (x, d * y). The Gaussian average . . . 0 , with respect to L MW , is readily evaluated; we find the asymptotic behavior of C M R is as given in Eq. (53) (see App. D for details).
The reason for the rapid decay is, that M attempts to introduce a gauge flux without the accompanying spin discussed above. Consider instead the 'dressed' monopole Its correlation function reproduces Eq. (45), i.e., approaches a non-zero constant at long distances, and M dressed spontaneously acquires an expectation value. Notice, however, that a term of the form δL ∼ (M n dressed + H.c.) would explicitly break U (1) spinrotation symmetry for any n = 0 and is therefore disallowed. The gauge field thus remains gapless in this phase, unlike in the trivial parton Mott insulator. For the details of these calculations, see App. D. In particular, the dressed monopole correlation function coincides with the one obtained by evaluating Eq. (10) using the singular configuration introduced in Sec. IV B 1.
Quasiparticles-The above analysis implies that, conversely, spin-s operators must be accompanied by 2πs gauge flux. Since 2π is the fundamental monopole, there are no low-energy excitations with half-odd integer spin. Spinless excitations, such as f ↑ f ↓ , are charged under the emergent gauge field. They are thus subject to logarithmic confinement (cf. Sec. IV B 1).
Spin model -The parton QSH breaks the microscopic time-reversal symmetry as well as translations by a single site in thex orŷ direction. It also exhibits spontaneous breaking of U (1) spin-rotation symmetry and an associated linear spectrum, as well as logarithmically confined neutral excitations. These properties exactly match the ones of an easy-plane AFM. Indeed, ỹ δL QSH maps onto y L t , which generates the easy-plane AFM described in Sec. III B 1.

Mixed insulators / Inter-wire VBS
Mean field -Consider now an integer quantum Hall state for one parton while the second forms a trivial band insulator. This is achieved, e.g., by introducing At the mean-field level, the fermionic partons constitute gapped spin 1/2 quasiparticles. The induced action for the gauge field a is as expected for a quantum Hall state at unit filling.
Gauge fluctuations-When the gauge field a is promoted to a dynamical variable, it acquires a mass through the Chern-Simons term. Monopoles are thus strongly irrelevant and can be safely discarded. The external probing field A can be included in Eq. (90) by replacing a µ,2ỹ → a µ,2ỹ + 1 4 [SA µ ] 2ỹ (without modifying the bare Maxwell term). Integrating out a µ does not result in a Chern-Simons term for A, which raises the possibility that the microscopic time-reversal symmetry is preserved. Indeed, it translates into the combination of fermionic time-reversal and y-translation symmetries (cf. Tab. III), which is preserved by the parton band structure in Eq. (89).
Quasiparticles-The Chern-Simons term in Eq. (90) attaches 2π emergent gauge flux to the fermionic meanfield excitations, converting them into bosonic quasiparticles. The way that A enters in Eq. (90) (see above) implies that the flux of a carries physical spin 1/2. There are, thus, two types of bosonic quasiparticles, one with spin 1 and one with spin 0.
Spin model -The y-translation symmetry is broken, whileỹ-inversion is preserved (cf. Tab. II). Moreover, U (1) spin-rotation, time-reversal, and x-translation symmetries all remain intact. These symmetry properties, along with the integer-spin quasiparticles, identify the phase as an inter-wire VBS. Indeed, translating δL MI to the microscopic spin variables, we find the wire construction of Eq. (54), which realizes an inter-wire VBS.

Generic K-matrix / Chiral spin liquid
Mean field -Consider now quantum Hall states characterized by a non-singular 2 × 2 K-matrix where m σ are odd integers and det [K] = 0. The corresponding wire construction was worked out in Ref. 51 and is given by For generic m σ , m 0 such a state exhibits a quantum Hall effect (associated with the total charge), a spin quantum Hall effect (associated with the relative charge), and a quantum spin Hall effect that connects the total and relative charges. In terms of q T = (1, 1) and s T = (1, −1)m, these are given by ν cc = q T K −1 q, ν ss = s T K −1 s, and ν cs = q T K −1 s, respectively. Integrating out the matter field we find, at leading order in ∆ 2 , where we have introduced 'charge' and 'spin' gauge fields a c/s 2y+1 = ( a 2ỹ+1 ± a 2ỹ ) /2. The quasiparticles, at the mean-field level, are anyons and carry fractional charges under a. They can be determined by a standard Kmatrix analysis (see, e.g., Ref. 47).
Gauge fluctuations-Upon reinstating the dynamics of a, governed by L MW + L CS−K , we find two distinct cases. For ν cc = 0, the gauge field remains gapless, and monopoles are important. The assumption of non-singular K implies a non-zero spin Hall response. Therefore, U (1) spin-rotation symmetry is spontaneously broken, as in the special case m 0 = 0 and m σ = σ (cf. Sec. V B 2). By contrast, for non-zero ν cc , the gauge field is massive, and monopoles can be safely discarded.
Recall, that the microscopic time-reversal symmetry acts as a duality transformation on the partons. Specifically, it attaches −2πσ flux to the fermions (followed by particle-hole transformation, see Sec. V). Therefore, Kmatrix states for the f σ fermions and for the dual d σ fermions are related by −K f = K d − 2σ z . For nonzero ν cc , the two K-matrices cannot coincide, and timereversal symmetry is broken explicitly. To obtain the physical response, we include the external probing field A in Eq. (93), according to a µ,ỹ → a µ,ỹ − 1 4 (−1)ỹ [SA µ ]ỹ. Integrating out the emergent gauge field we obtain µν Aµ∂yAν , (94) where n ≡ ν cc det [K] /2. Consequently, the phase is chiral with topologically protected edge states. Quasiparticles-To characterize the quasiparticles for n = 0, we adopt the strategy employed in Sec. IV B 4. To each fermion, we attach m σ fluxes of their own species and m 0 fluxes of the opposite one; on an operator level, we define ηỹ ,σ = ϕ f,ỹ,σ − mσ 2 [S 2φf,σ ]ỹ − m 0φf,σ,ỹ . The corresponding β σ = e −iησ particles are bosons; they are governed by the schematic Lagrangian . (95) Notice that this Lagrangian is the same as Eq. (62). The quasiparticles thus carry spin 1/2n and acquire statistical phases of π/2n upon clockwise exchange. The case n = 0 can be analyzed as in Sec. V B 2. In particular, these phases feature linearly dispersing Goldstone modes associated with the broken U (1) spin-rotation symmetry and logarithmically confined topological excitations. Spin model -The response to the external probing field, Eq. (94), implies that the microscopic phase is a chiral QSL. Translating the inter-wire coupling in Eq. (92) to microscopic variables, we find (96) with m 2y±1/2 = m↑ /↓ ∓1. For non-singular K = σ z , these inter-wire couplings explicitly break time-reversal symmetry. The arguments of all cosines in δL CSL commute and can flow to strong coupling simultaneously. They are, moreover, linearly independent and can thus generate a gapped phase for non-zero n, i.e., mỹ+mỹ +1 = 2m 0 . To identify its quasiparticles and edge structure, we introduce chiral modes φ χ,y = Φ y + χ2nΘ y + 2m y+1 Θ y+1 − 2m y Θ y−1 , (97) which satisfy ∂ xφχ,y ,φ χ ,y = iχ4πnδ y,y δ χ,χ δ (x − x ) .
Crucially, this change of variables preserves the locality of both the intra-and inter-wire terms; the latter take the form δL CSL = cos (φ R,y+1 −φ L,y ). Domain walls in these cosines carry spin 1/2n and acquire exchange phases of π/2n. Finally, for n = 0 the system is gapless, which can be seen by summing the arguments of all cosines in Eq. (96), i.e., y [φ R,y+1 −φ L,y ] = 4n y Θ y . This particular linear combination thus remains unpinned for n = 0. Its conjugate describes the Goldstone mode associated with the spontaneously broken U (1) spin-rotation symmetry, precisely as in Sec. III B 1.

BCS superconductor / Z2 spin liquid
Mean field -As a final example, consider now a BCS superconductor of fermionic partons. To generate pairing, we introduce inter-wire hopping for the Cooper-pair operator Ψ f -pair,2y+1 ≡ f 2ỹ+1↑,R f 2ỹ,↓,L , i.e., When g f -pair flows to strong coupling, Ψ f -pair spontaneously acquires an expectation value (cf. Sec. III B 1). Vortices in the phase of this condensate, created by m 2y = e −i(φ f,↑,2ỹ +φ f,↓,2ỹ−1 )/2 , are logarithmically confined. While δL f -pair renders the umklapp term in Eq. (77b) irrelevant, the back-scattering term δL bs = g bs f † 2ỹ+1,↑,R f 2ỹ+1,↑,L f † 2ỹ,↓,L f 2ỹ,↓,R + H.c. , (100) can flow to strong coupling. When it does, a fully gapped phase with fermionic spin-1/2 quasiparticles, f σ , obtains. Gauge fluctuations-Since Ψ f -pair carries emergent gauge charge, its condensation leads to a Higgs mass. Monopoles are, therefore, strongly suppressed and can be safely discarded. Importantly, Ψ f -pair is neutral under the external probing field A, so the microscopic U (1) spin-rotation symmetry is preserved. Indeed, the response to A is, at leading order in derivatives, described by a Maxwell action.
Quasiparticles-Gauge field fluctuations that are rendered massive by a Higgs term do not affect the status of f σ as fermionic quasiparticles. They do, however, promote vortices m to deconfined bosonic spin-0 quasiparticles. Being superconducting vortices, they are experienced as π flux by the fermions, i.e., the two are mutual semions. Consequently, the two can combine into ε = f χ,σ m † , a spin-1/2 quasiparticle with bosonic selfstatistics.
Spin model -The quasiparticle content characterizes a Z 2 spin liquid that is, moreover, non-chiral and spinrotation symmetric. Translating δL f -pair and δL bs to mi-croscopic spin variables, we find δL Z2 =g bs cos 2S T SΘ 2y−1 + g f -pair cos ∆ T ∆Φ 2y + 2S T SΘ 2y−1 . (101) The arguments of these cosines are linear combinations of those in Eq. (69). Consequently, they lead to the same Z 2 spin liquid phase (see full analysis in Sec. IV B 5).

VI. Summary and discussion
We have introduced exact, non-local mappings between arrays of spin-1/2 chains and parton gauge theories. Any parton model that separately conserves both species maps onto a local spin Hamiltonian. The challenge of deriving spin models that realize exotic ground states is thereby reduced to constructing parent Hamiltonians for simple phases of bosons or fermions. Conversely, any S z -conserving coupling between spins transforms into a distinct interaction or hopping term for partons. The latter are obtained without reference to a specific mean-field ansatz. They, therefore, retain not only information about the symmetries of the underlying spin model, but also about more subtle aspects, such as geometric frustration. In its presence, some symmetryallowed terms in the dual parton description are absent. Geometric frustration may thus take the form of an emergent symmetry and, thereby, stabilize phases that would not readily form in more generic situations.
To demonstrate the versatility of this method, we showed how to recover trivial states and access topologically ordered ones. Relatively simple phases of partons already correspond to fractionalized ground states. As examples, we derived microscopic parent Hamiltonians for Abelian chiral spin liquids and a non-chiral Z 2 QSL. The latter corresponds to an s-wave BCS superconductor of fermionic partons. If they instead form a topological superconductor, such as p x ± ip y , the resulting QSL will be non-Abelian.
When the partons themselves form non-trivial phases, an even wider range of exotic microscopic ground states is realized. The framework introduced here applies to such cases with no additional difficulties, once a (coupled-wire) parent Hamiltonian of the parton phase is known. We have illustrated this capability by the example of a general 2 × 2 K-matrix state of fermionic partons. Explicit parent Hamiltonians for even more exotic states, such as the non-Abelian Read-Rezayi sequence of fractional quantum Hall states, are also known and can likewise be used to generate concrete spin models.
We primarily focused on fully gapped states. However, the dual description of spin-chain arrays in terms of fermions may be able to capture exotic gapless phases and unconventional quantum phase transitions as well. One example of the latter arises at the transition between the easy-plane AFM and the intra-wire VBS. It maps onto a coupled-wire model of compact QED 3 with two boson or fermion species. This is precisely the effective field theory that was derived using different methods in Ref. 106. Its fate in the infrared is thought to be confining (and consequently the transition to be first-order). A stable gapless theory may instead arise in various ways: (i) At a transition between different phases. (ii) In the presence of emergent symmetries of the parton field theory that may arise due to geometric frustration. (iii) When the emergent fermions are doped to form a Fermi surface that suppresses monopole events. All three scenarios should be amenable to exploration within the formalism developed here. Finally, we mention two possible generalizations of the methods developed here. The first is to itinerant electron systems. There, decomposing microscopic electron operators as c σ = bf σ allows exploration of many exotic ground states. Extending our approach to wire arrays with both spin and charge modes may allow wellcontrolled access to those phases, and provide concrete model systems where they arise. A second interesting direction is given by spin models that do not conserve S z , such as the celebrated Kitaev honeycomb model. 11 Systems without U (1) symmetries are not readily describable within Abelian bosonization. Instead, coupledwire techniques based on non-Abelian bosonization have been used successfully in similar contexts. 70,84 Generalizing our methods to these systems could provide a muchdesired bridge between fine-tuned solvable models and mean-field studies of generic ones.

A. Energy cost of topological defects
Energy cost of domain walls in the VBS phase of a one-dimensional spin-chain To describe the VBS phase, consider the bosonized action of a U (1) spin chain given by with K such that the dimensionless coupling constantg 4π < 0 flows to strong coupling. The minima of the cosine potential, Θ min = πn/2, correspond to different topological sectors, labeled by the integer n. On length scales larger than d * , whereg 4π has become of order unity, it is appropriate to expand the cosine in a single topological sector, i.e., replace vg4π To find the energy cost of domain walls, we allow the system to transition between different topological sectors as a function of space, i.e., n → n x and The energy cost of forcing the system into different topological sectors, relative to the uniform n = 0 vacuum, is given by where the last equality uses the specific form of the correlation function Θ x Θ x according to Eq. (A2). For a generic configuration of domain walls parameterized by n(x) = i α i H(x − x i ) with α i ∈ Z, the energy is given by In particular, the energy cost of a single domain wall is given by the prefactor ∆E DW = πv 16Kd * . The trial function provided in the main text, Θ DW (x) = tan −1 e (x−x0)/ξ , produces a variational energy cost of ∆E DW [ξ] = v 4πK 1/ξ + ξ/d 2 * , for the renormalized action, i.e., Eq. (A1) withg 4π → −d 2 0 /d 2 * . Its minimal value, attained for ξ = d * , is given by ∆E DW = v 2πKd * and is parametrically the same as the result of Eq. (A4). The somewhat smaller numerical value relative to the previous calculation arises because there, the cosine was replaced by a parabolic potential centered around the nearest minimum.

Energy cost of magnetic vortices in the easy-plane AFM
We follow the same strategy as for the one-dimensional VBS domain walls. Expanding the cosine of Eq. (18a) in topological sectors, denoted by n x,y+1/2 , we obtain where d * is the length-scale at whichg t = 2πd 2 0 g t /vK reaches order unity. In this case, n can be interpreted as counting the magnetic flux tubes in an external probing field. Consider a magnetic field B(x) in the gauge A 1 = 0, i.e., A 2 =´x B(x). Incorporating the probing field via minimal coupling amounts to replacing ∆Φ → ∆Φ − A 2 above, which identifies n = 1 2π´x B(x). The energy cost for a given configuration n can be computed as in the one-dimensional case, i.e., ∆E [n] = 2v 2 K 2 d −2 * ˆx ,x y,y n x,y+ 1 2 n x ,y + 1 Parameterizing n in terms of strength α i vortices at positions (x i ,ỹ i ), i.e., n x,y+ 1 2 = i α i δ y,yi H (x − x i ), we arrive at the final expression with R i = (x i , d * ỹi ) and k = (k x , k y ). The y-momentum k y is measured in units of d −1 * and ∆ ky ≡ e id * ky − 1 /d * . If the total number of vortices N v ≡ i α i = 0, the energy diverges logarithmically with the size of the system. When N v = 0, the energy cost is finite; for a single vortex-anti-vortex pair we find Importantly, this calculation does not make any reference to a specific form of the vortex. Instead, it fixes the topological properties and lets the functional integral over Φ find the optimal configuration. The same result can be obtained by considering a 'trial' configuration of the form where ξ is a variational parameter. Computing the corresponding energy at large vortex-anti-vortex separation, one finds with R i = (x i , ξy i ). To optimize ξ within logarithmic accuracy, it is sufficient to focus on the prefactor of the logarithm; it is minimized for ξ = d * where Eq. (A10) reduces to the result provided in Eq. (A8).

Energy cost of dislocations / bosonic partons in the intra-wire VBS
To generate the two-dimensional VBS with dimers along the wire direction,x, we introduce two different cosines per wire pair. We introduce dimensionless coupling constants,g u > 0 andg 4π < 0, and write where, for convenience, we have redefined Θ y → (−1) y (Θ y + πy/2). Beyond the length scales d * and l * , where the coupling constants renormalize to order unity, we expand the action in topological sectors. Labeling said sectors by integers n and p, we obtain As before, we parameterize the integer functions n and p by the locations and strengths of topological defects, i.e., The energy cost for a general configuration naturally decomposes into a manifestly local, system-size independent contribution, and a non-local one that may be IR divergent, i.e., ∆E [n, p] = ∆E local [n, p] + ∆E non-local n − 1 2 ∆p with A priory, n and p are independent, and the lowest energy cost for a given n must be found by optimizing over all possible p. Fortunately, this minimization can be avoided when the density of defects n is low; there, the optimal p can be inferred based on general considerations. Consider a single dislocation-anti-dislocation pair defined by α 1 = 1 and α 2 = −1, i.e., n x,ỹ = H (x − x 1 ) δỹ ,ỹ1 − H (x − x 2 ) δỹ ,ỹ2 . The corresponding p can be deduced as follows: (i) To avoid energies that diverge with system size, the topological sectors must match asymptotically, i.e., lim x→∞ (n x,ỹ − [∆p] x,ỹ /2) = 0. (ii) The minimal number of p-defects within this constraint has exactly one of strength β i = 2 for each wire Y i ∈ [ỹ 1 ,ỹ 2 ]. (iii) Their optimal locations are along the line connectingr 1 = (x 1 ,ỹ 1 ) andr 2 = (x 2 ,ỹ 2 ). To determine the energy for distances much larger than l * , we approximate where R = (x, d * y). [When appearing in discrete sums as in ∆E, δ (x i − x j ) is to be understood as d −1 0 δ xi,xj , where the microscopic length-scale d 0 acts as a UV cutoff.] In this limit we find simplified expressions for the local and non-local contributions to the energy, which in case of a single dislocation-anti-dislocation pair give Topological defects in the intra-wire VBS phase are thus linearly confined. As for the case of magnetic vortices, the same conclusion can be reached by studying appropriate trial states. For the Ising-AFM phase, the sign ofg 4π in Eq. (A11) is reversed, but the analysis is otherwise identical, i.e., defects are also linearly confined.

Energy cost of bosonic parton vortices in the correlated Mott insulator
In the correlated Mott insulator (cf. Sec. IV B 3), there are two cosines per wire pair, i.e., Beyond the length scales d * and l * , where coupling constants renormalize to order unity, we expand the action in the topological sectors. Labeling said sectors by the integer functions n and p, we write where S 0 is the effective action of the trivial topological sector with n = p = 0. Vortex-anti-vortex pairs are encoded in n, so we set p to zero. To evaluate the energy cost for a given n, we need the Green function of ∆ϕ ↓ within the Gaussian theory S 0 [ϕ σ , θ σ , a]. Its leading order in k x and ∆ ky is where the dimensionless parameter α 2 = 4v B κṽ encodes the ratio of boson and gauge-field velocities. For a single vortex-anti-vortex pair at positions R i = (x i , d * ỹi ), withỹ i odd and |R 1 − R 2 | d * we find Here, the dimensionless integral f (x) = 1 2´π −π dθ 1 √ x+2−2 cos(θ) diverges logarithmically for x → 0 but is otherwise finite. The energy required for creating an isolated vortex is thus v B /d * times a number of order unity.

B. Derivation of the bosonic-parton field theory
The equivalence between the microscopic coupled-wire model described in Sec. III B and the gauge theory for bosonic partons introduced in Sec. IV A can be shown using the methods of Refs. 92 and 93. We begin with the gauge theory S =´x ,τ y [L b + L Maxwell + L b ] in the a 2 = 0 gauge, where We here omit the redundant label σ = (−1) y+1 to lighten the notation. The final term, L b , contains exponentially decaying inter-wire interactions. Specifically, density-density and current-current interactions where W i decay exponentially with |y − y |. Crucially, L b is short-ranged in both parton and spin variables and thus does not affect the structure of the gauge theory. Performing the Gaussian integrals over a µ and expressing ϕ, θ in terms of Φ, Θ using Eq. (28), we find S =´x ,τ y L spins + L spins with where and v = 2ṽK/κ. The term L spins contains exponentially decaying inter-wire interactions, i.e., where P y,y = (−1) y δ y,y . Since vK v B < 1 4 for any choice of parameters, all terms in L spins decay exponentially with |y − y |; a suitable choice of W i in Eq. (B3) can thus be used to achieve L spins = 0. Alternatively, W i can be chosen such that y , which is the result stated in the main text, i.e., Eqs. (12) and (32).

External probing field
Before concluding this appendix, we include the external probing field A that minimally couples to the conserved S z of the microscopic spins. Translating L A of Eq. (39) to bosonic parton variables we find where S −T y,y is the inverse-transpose of Sỹ ,y . To cast the coupling into a more revealing form, we shift a µ according to This shift completely cancels L A ; the external probing field instead appears in L b of Eq. (B1) through the replacement a µ,ỹ → a µ,ỹ − 1 4 (−1)ỹ [SA µ ]ỹ and in higher-order couplings to the emergent gauge field given by C. Derivation of the fermionic-parton field theory To derive the gauge theory for the fermionic partons, we follow the same steps as for the bosonic partons. Specifically, we consider the action S =´x ,τ y L f + L Maxwell + L f + L with The first 'primed' term, L f , contains inter-wire interactions between fermion densities and currents, that decay exponentially with the wire separation, |y − y |. This term is not essential for understanding the structure of the gauge theory. For completeness, we still provide an explicit treatment below. The final term is It has an alternating sign and thus does not affect the long-distance behavior of the photon propagator (see App. D below). However, as we will see, it is essential for the microscopic time-reversal symmetry of the spin system. To understand its origin, recall that the fermionic partons are obtained by attaching fluxes of opposite sign to the two bosonic-parton species. In the schematic continuum manipulations described by Eq. (73), the two Chern-Simons terms that implement this flux attachment exactly cancel. By contrast, in the wire regularization, the two species of fermionic partons reside at different locations (even vs. odd dual wires). Consequently, the cancellation is imperfect, and a residual term L CS,even + L CS,odd ∝ L + (−1)ỹO(∆ 4 ) remains.
To map the gauge theory to spin variables, we perform the Gaussian integrals over a µ and express ϕ f , θ f in terms of Φ, Θ using with ∆ −T ỹ,y the inverse-transpose of ∆ỹ ,y . We find S =´x ,τ y L spins + L spins + L spins , where , and u F = u F + 4vK. The term L spins contains exponentially decaying inter-wire terms whose explicit form is provided below. Crucially, a suitable choice of the L parton results in L spins = 0, and can additionally be used to tune the parameters u V and u F in Eq. (C6). The final term with v = 4vK 2α α 2 +κ 2 − 1 , cannot be eliminated by any choice of L parton , which contains only higher orders in ∆. Moreover, it violates the microscopic time-reversal symmetry. To preserve this symmetry, we must, therefore, choose α in the gauge theory of Eq. (C3) such that v = 0, namely, α = 1 ± √ 1 − κ 2 ≡ α 0 . Having eliminated L spins and L spins , the action matches Eqs. (12) and (32) with the parameters specified above.

Explicit form of short-range interactions
We include generic short-range density-density, current-current and density-current interactions for the fermionic parton in the form where W i decay exponentially with |y − y |. It is straightforward to express these in terms of Φ, Θ and combine them with the exponentially decaying interactions stemming from the Gaussian integrals over a µ . We find with exponentially decaying W i . We are interested in v = 0, for which these Kernels are given by . A choice of exponentially decaying W i that results in L spins = 0 is given by External probing field Before concluding the derivation, we also include an external probing field A that minimally couples to the conserved S z of the microscopic spins. We thus supplement the action of the gauge theory action by L A as given in Eq. (39). In terms of the fermionic parton variables To cast this into a more revealing form, we shift a µ according to a 0,ỹ → a 0,ỹ − 1 4 (−1)ỹ [SA 0 ]ỹ + (−1)ỹ S −T A 0 ỹ + ivK 2 − ∆ T ∆ ∆ −T A 1 ỹ (C17a) This completely cancels L A ; the external probing field instead appears in L f of Eq. (C1) through the replacement a µ,ỹ → a µ,ỹ − 1 4 (−1)ỹ [SA µ ]ỹ and in higher-order couplings to the emergent gauge field given by D. Gauge-theory calculations
Finally, consider the case where the second parton also condenses, i.e., where δL ↑ ∼ cos (ϕ 2ỹ+1,↑ − ϕ 2ỹ−1,↑ ) flows to strong coupling as well. Treating δL ↑ in the same way as δL ↓ and integrating out b ↑ , we find where Ā = A 0 1 √ u B v B , A and p = ω √ v B u B , k x , k y . This response to the probing field A implies, that the U (1) spin-rotation symmetry is spontaneously broken.

Fermionic partons QSH
To assess the relevance of monopoles in the QSH phase of fermionic partons we compute their correlation function within the monopole-free theory. The corresponding Lagrangian was introduced in Sec. V B 2; it is given by L QSH = L matter + L gauge with where, as usual, d * is the length scale beyond which the quadratic approximation of the cosine is appropriate. (For the last term in L gauge , cf. App. C.) It is straightforward to integrate out the matter fields and obtain the induced gauge-field action. Upon expanding in frequencies and x-momenta small compared to the QSH gap, E QSH ≡ 4ṽv F κd 2 * , we find (∂ x a 0,ỹ − ∂ τ a 1,ỹ ) 2 + i 4π (−1)ỹ [S 2 a 1 ]ỹ [∆ 2 a 0 ]ỹ + L ind .
Here L ind contains terms suppressed by at least two additional powers of ∆ 2 . These contributions can easily be retained but do not affect any long-wavelength properties; we, therefore, discard them. In the absence of oscillatory terms ∼ (−1)ỹ in the total gauge-field action, L gauge + L ind , we approximate ∆ 2 = 2∆ + ∆ 2 ≈ 2∆ for describing long-wavelength properties. We thus obtain the effective gauge-field action where γ 2 = 1+ 4ṽ κv F = and η 2 = 1+ 4u F κṽ . Oscillatory terms ∼ (−1)ỹ couple gauge fields at y-momenta k y and k y +π/d * . We obtain the long-wavelength action in that case, by integrating out modes at momenta |k y | > π/2d * . Returning to real space, we again find L eff gauge , but with modified parameters γ 2 = κ 2 +(2−α) 2 κ 2 + 4ṽ κv F , η 2 = κ 2 +(2+α) 2 κ 2 + 4u F κṽ . To compute the correlation function of a probing monopole in this theory, we supplement L eff gauge by the monopole action of Eq. (82) and integrate out a µ . At leading order in ∆ ky , we find L M,eff =  ṽ 2πκ +ṽ 2πκγ 2 η 2ṽ2 ∆ ky 2 + ω 2 η 2ṽ2 ∆ ky 2 + ω 2 + η 2 γ 2ṽ The monopole-monopole correlation function, evaluated within a monopole-free background g M = 0, is given by where L is the wire length, and the ω and k x integrals are cut off by E QSH and v −1 F E QSH , respectively. The function f (y) ≡ 1 − sin(πy/2) πy/2 is bounded from below by 1 − 2/π ≈ 0.36 for non-zero y. At long distances, this correlation function decays faster than exponential. Monopoles described by M are thus strongly irrelevant.
We now turn to dressed monopoles, which were introduced in Sec. V B 2 via the short-hand M dressed ∼ f † ↓ f ↑ M. Their explicit expression is