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arXiv:2610.27050

Field Theory of the Gutzwiller Projection

Abhijat Sarma, Cenke Xu, Chong Wang

Published Oct 19, 2026• 0 citations
arXivHTMLPDF

Abstract

The Gutzwiller projection produces a well-known ansatz for spin-liquid and quantum Hall ground-state wavefunctions that is amenable to numerical simulation, with the Laughlin wavefunction being the most celebrated example. A long-standing issue is the precise relation between the Gutzwiller projection and the closely related parton gauge theory. Here, we clarify this issue by providing a field-theoretic description of the Gutzwiller projection. In a path integral description, the universal physics of the projected state is described in terms of a gauge field supported on a temporal slab of vanishing width $ε\rightarrow 0$ coupled to the bulk matter fields. We analytically derive the universal physics of various Gutzwiller projected states. Our construction is also applicable to discrete symmetries, and we use it to analyze various projected SPT states. Among our results, we show that the Gutzwiller projection is not a good ansatz for the long-distance properties of $U(1)$ Dirac spin liquids, spinon Fermi surface spin liquids or composite Fermi liquids, and that the Gutzwiller projection of a quantum spin Hall state has only quasi-long range order. In contrast, in examples where the gauge field is gapped, including Laughlin states, $\mathbb{Z}_2$ spin liquids and Luttinger liquids, we show that the slab gauge theory from Gutzwiller projection reproduces the correct universal physics of the full parton gauge theory. We also present a generalization of our field theory that describes $G$-intertwining defects, with application to self-duality defects of recent theoretical interest.

DOI
10.48550/arxiv.2610.27050

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The Gutzwiller projection produces a well-known ansatz for spin-liquid and quantum Hall ground-state wavefunctions that is amenable to numerical simulation, with the Laughlin wavefunction being the most celebrated example. A long-standing issue is the precise relation between the Gutzwiller projection and the closely related parton gauge theory. Here, we clarify this issue by providing a field-theoretic description of the Gutzwiller projection. In a path integral description, the universal physics of the projected state is described in terms of a gauge field supported on a temporal slab of vanishing width ε→0ε\rightarrow 0ε→0 coupled to the bulk matter fields. We analytically derive the universal physics of various Gutzwiller projected states. Our construction is also applicable to discrete symmetries, and we use it to analyze various projected SPT states. Among our results, we show that the Gutzwiller projection is not a good ansatz for the long-distance properties of U(1)U(1)U(1) Dirac spin liquids, spinon Fermi surface spin liquids or composite Fermi liquids, and that the Gutzwiller projection of a quantum spin Hall state has only quasi-long range order. In contrast, in examples where the gauge field is gapped, including Laughlin states, Z2\mathbb{Z}_2Z2​ spin liquids and Luttinger liquids, we show that the slab gauge theory from Gutzwiller projection reproduces the correct universal physics of the full parton gauge theory. We also present a generalization of our field theory that describes GGG-intertwining defects, with application to self-duality defects of recent theoretical interest.
We introduce a method to design a local spin Hamiltonian to realize a Gutzwiller-projected parton wave functions (GPWF) as its ground state. For example, the Dirac spin liquid (DSL) state is quite close to the true ground state of the spin-1/2 Heisenberg model on kagome lattice. We examine what kind of perturbations we should add in order to drive the DSL to more stable chiral spin liquid (CSL), valence bond solid (VBS) or Gutzwiller-projected spin Hall (GSH) states. We compute the two-body reduced-density-matrices (2-RDM) of GPWFs for those target states, and compare them to the 2-RDM of the DSL. This allows us to design local spin models with only two-body interactions that may realize those interesting target states. Our results agree very well with recent numerical calculations for CSL on kagome lattice. We also study spin-1 systems on kagome lattice, and design local spin models that may realize CSL, VBS and symmetry-protected topological (SPT) states. Our work establishes a directional guide for further numerical simulations.
The parton approach for quantum spin liquids gives a transparent description of low-energy elementary excitations, e.g., spinons and emergent gauge-field fluctuations. The latter ones are directly coupled to the hopping/pairing of spinons. By using the fermionic representation of the U(1)U(1)U(1) Dirac state on the kagome lattice and variational Monte Carlo techniques to include the Gutzwiller projection, we analyze the effect of modifying the gauge fields in the spinon kinematics. In particular, we construct low-energy monopole excitations, which are shown to be gapless in the thermodynamic limit. States with a finite number of monopoles or with a finite density of them are also considered, with different patterns of the gauge fluxes. We show that these chiral states are not stabilized in the Heisenberg model with nearest-neighbor superexchange couplings, and the Dirac state corresponds to the lowest-energy ansatz within this family of variational wave functions. Our results support the idea that spinons with a gapless conical spectrum coexist with gapless monopole excitations, even for the spin-1/2 case.
The diamond lattice has been a promising playground for both classical and quantum spin liquids. Here, the authors conduct a comprehensive symmetry classification of SSS=\textonehalf{} quantum spin liquids with SU(2), U(1), and \ensuremath{\mathbb{Z}}2{}_{2}2​ gauge fields, and present dynamical spin structure factors within a self-consistent simulation. The authors highlight three discoveries: spinon Ans\"atze with robust gapless nodal loops, topological bands realizing topological insulators, and that Gutzwiller projection of the 0- and $\ensuremath{\pi}$-flux SU(2) spin liquids generates long-range N\'eel order.
We study Gutzwiller-projected wave functions for triangular-lattice U(1) Dirac spin liquids in a Zeeman field, where we allow the U(1) gauge field to develop a gauge flux, resulting in (spin-split) spinon Landau levels. We find that at a given magnetization, the optimal candidate state has a finite flux chosen such that the spinon filling lies in a |C|=1 Landau-level gap: it gives the lowest variational energy and the smallest energy variance within our correlation-matrix reconstruction for local Heisenberg-type models. By symmetry, we argue that the finite gauge flux results in a nonzero (staggered) scalar spin chirality, as also numerically observed, and further find that the |C|=1 state exhibits dominant quasi-long-ranged 120° magnetic correlations. Studying the next-to-optimal wave function with a |C|=2 Landau-level gap, we observe unusual spin-nematic correlations. Our results may provide guidance for analyzing the magnetic-field response of Dirac spin liquid candidate materials and offer numerical diagnostics that can connect to the underlying theory of spinons coupled to an emergent U(1) gauge field.
While most of the features of the Kitaev honeycomb model are already clarified from the combined Z2{\mathbb Z}_2Z2​ plaquette and Majorana fermion picture, we show that retrieving the exact quantum many-body form gives a clue to fix the remaining issues on its spin liquid property. We provide an exact two-dimensional tensor network description of a full set of solutions from a random product state, by sequential quantum-number projection and Gutzwiller projection, both expressed as local matrix-product operators of dimension D=2D=2D=2. The ground and excited states share the same bond dimension yet exhibit distinct Schmidt spectra, distinguishing their area-law and volume-law entanglement. We demonstrate that the topological entanglement entropy emerges {\it for all eigenstates only after} the Gutzwiller projection onto the physical spin Hilbert space, where the gauge constraint removes exactly half of the Schmidt states. This establishes that excited states are likewise topologically ordered. We further show that there exists a projection-induced collapse of the wave function into a single gauge sector in the Parton mean-field states that is known to give a good description of the spin liquids. The demonstrations on the S=1/2S=1/2S=1/2 kagome and triangular Heisenberg models suggest that the present perspective is applied to a wider class of models hosting Z2{\mathbb Z}_2Z2​ spin liquids. Our finding visualizes a bulk-boundary correspondence with the Wilson-loop characterization of topological order.
Slave particle approaches are widely used in studies of exotic quantum phases. A complete description beyond mean field also contains dynamical gauge fields, while a simplified procedure considers Gutzwiller-projected trial states. We apply this in the context of bosonic models with ring exchanges realizing so-called exciton Bose liquid (EBL) phase and compare a Gutzwiller wave function against an accurate EBL wave function. We solve the parton-gauge theory and show that dynamical fluctuations of the spatial gauge fields are necessary for obtaining qualitatively accurate EBL description. On the contrary, just the Gutzwiller projection leads to a state with subtle differences in the long-wavelength properties, thus suggesting that Gutzwiller wave functions may generally fail to capture long-wavelength physics.