Jastrow-Gutzwiller Wave Functions

• Jastrow-Gutzwiller wave functions are correlated many-body states created by applying density-dependent Jastrow correlators and Gutzwiller projectors to mean-field reference states.
• They provide a unified variational framework for modeling phenomena such as Mott physics, charge and orbital ordering, antiferromagnetism, and critical behavior in low-dimensional systems.
• These approaches support quantum-circuit implementations via Hubbard-Stratonovich decompositions, bridging traditional variational methods with modern digital quantum simulation techniques.

Jastrow-Gutzwiller wave functions are correlated many-body trial states obtained by multiplying an uncorrelated or mean-field reference state by density-dependent correlators, with the Gutzwiller factor representing the on-site limit of a Jastrow correlator. In fermionic lattice problems, the standard construction starts from a symmetric Slater determinant or a BCS-like singlet state and applies a density-density Jastrow factor and, when needed, a soft or full Gutzwiller projector; in one-dimensional spin-chain contexts, the term also denotes a family of half-filled wave functions whose amplitudes are explicit Jastrow products on chord distances around a circle (Kaneko et al., 2015). Across these settings, Jastrow-Gutzwiller states serve as variational descriptions of Mott physics, charge and orbital ordering, antiferromagnetism, dimerization, critical Luttinger-liquid behavior, and Haldane-Shastry-type spin physics, while more recent work has recast the on-site Gutzwiller factor into quantum-circuit primitives via discrete Hubbard-Stratonovich decompositions (Turkeshi et al., 2019).

1. Formal definitions and canonical ansätze

In the lattice fermion setting, the basic Jastrow-Slater ansatz starts from a symmetric Slater determinant

$$|\Phi_0\rangle = \prod_{\gamma=1}^{N} \phi^\dagger_{\gamma}\,|0\rangle,$$

or equivalently from a BCS-like pairing form

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$

with $f_{ij}=f_{ji}$ enforcing singlets and full SU(2) symmetry. A density-density Jastrow correlator is then applied,

$$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$

where $n_i=n_{i\uparrow}+n_{i\downarrow}$, $v_{ij}$ are variational pseudo-potentials, and $v_{ii}\equiv g$ is the Gutzwiller parameter. The momentum-space form is

$$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$

and the multi-orbital generalization reads

$$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$

The Gutzwiller factor is the on-site limit of this density correlator. In soft form,

$$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$

while the full projector enforcing no double occupancy is

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$0

The combined variational form is

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$1

and, for spin models at half filling $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$2,

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$3

optionally supplemented by a number projector $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$4 (Kaneko et al., 2015).

A closely related normalized formulation, used in quantum-computing implementations, defines

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$5

with $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$6 and $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$7 typically a normalized Slater determinant. Provided $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$8 is an eigenstate of total particle number, this exponential form is equivalent to the original projector form

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$9

so the Gutzwiller correlator is explicitly an on-site density-density Jastrow factor (Seki et al., 2022).

In one-dimensional spin-chain work, Jastrow-Gutzwiller states also denote a distinct but related family at fixed particle number $f_{ij}=f_{ji}$0 and filling $f_{ij}=f_{ji}$1, expanded as

$f_{ij}=f_{ji}$2

with amplitudes

$f_{ij}=f_{ji}$3

Here $f_{ij}=f_{ji}$4 controls the Jastrow strength, $f_{ij}=f_{ji}$5 gives a symmetrized Slater determinant, and $f_{ij}=f_{ji}$6 yields the exact ground state of the Haldane-Shastry Hamiltonian (Turkeshi et al., 2019).

2. Symmetry preservation and spontaneous symmetry breaking

A central feature of these states is that the reference state $f_{ij}=f_{ji}$7 can be chosen to preserve SU(2) spin and all lattice symmetries, including translation, inversion, and point-group operations. The density Jastrow operator $f_{ij}=f_{ji}$8 is scalar under SU(2) and can itself be taken translationally and point-group invariant, so the variational state is symmetry-invariant at the wavefunction level for finite systems: total magnetization is zero and diagonal one-site expectation values remain uniform unless ergodicity is broken in sampling. Nevertheless, sufficiently strong correlators can induce long-range order in the thermodynamic limit without any explicit symmetry-breaking fields (Kaneko et al., 2015).

For itinerant models, the mechanism is transparent through a quantum-to-classical mapping for diagonal observables in the occupation basis: $f_{ij}=f_{ji}$9 with

$$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$0

When density fluctuations are strongly suppressed, the first term dominates and acts like a classical interacting holon-doublon model. Increasing short-range pseudo-potentials such as nearest-neighbor $$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$1 is then analogous to lowering an effective temperature, and classical phase transitions in the effective model manifest as spontaneous charge-density or orbital order in the quantum variational state. A heuristic criterion given in this framework is that a long-range Jastrow with $$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$2 at small $$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$3 suppresses compressibility and promotes Mott physics; discrete ordering may then appear once short-range components exceed thresholds determined by filling and lattice geometry (Kaneko et al., 2015).

For spin systems at half filling, the relevant mechanism is different. Under full Gutzwiller projection, charge degrees of freedom are frozen, and a projected BCS/RVB state may develop magnetic or dimer long-range order even though the underlying $$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$4 carries no explicit order parameter. Two ingredients are singled out: the no-double-occupancy constraint, which enhances spin correlations, and long-range singlet amplitudes $$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$5, which generate strong long-distance entanglement. Imposing the Marshall-compatible conditions

$$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$6

with $$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$7 in one dimension or $$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$8 in two dimensions, gives

$$|\Psi\rangle = \hat{J}\,|\Phi_0\rangle,\qquad \hat{J} = \exp\Big(-\tfrac{1}{2}\sum_{i,j} v_{ij}\, n_i n_j\Big),$$9

For gapless BCS spectra and sufficiently slow decay $n_i=n_{i\uparrow}+n_{i\downarrow}$0, Néel order emerges in two dimensions, whereas in one dimension SU(2) symmetry remains unbroken, consistent with Mermin-Wagner. By contrast, short-ranged $n_i=n_{i\uparrow}+n_{i\downarrow}$1, as produced by a gapped pairing $n_i=n_{i\uparrow}+n_{i\downarrow}$2, yields spontaneous dimerization in one dimension (Kaneko et al., 2015).

Long-range order is diagnosed through finite-size scaling of structure factors. For charge and orbital order, one requires

$n_i=n_{i\uparrow}+n_{i\downarrow}$3

while antiferromagnetic order is extracted from

$n_i=n_{i\uparrow}+n_{i\downarrow}$4

Disordered phases instead show $n_i=n_{i\uparrow}+n_{i\downarrow}$5 or $n_i=n_{i\uparrow}+n_{i\downarrow}$6 vanishing as $n_i=n_{i\uparrow}+n_{i\downarrow}$7 or faster (Kaneko et al., 2015).

3. Representative phases in spin and itinerant models

The spin-sector examples establish that full Gutzwiller projection plays qualitatively different roles in one and two dimensions. In the $n_i=n_{i\uparrow}+n_{i\downarrow}$8 Heisenberg chain, fully projected gapless states reproduce the correct critical scaling but do not break SU(2), while gapped pairing generates spontaneous dimerization despite a translationally invariant $n_i=n_{i\uparrow}+n_{i\downarrow}$9. The staggered dimer response can be monitored either through

$v_{ij}$0

or through the four-spin correlator

$v_{ij}$1

In the square-lattice Heisenberg antiferromagnet, fully projected RVB/BCS states with Marshall signs and slowly decaying $v_{ij}$2 generate Néel order. Finite-size benchmarks reported for these projected states include $v_{ij}$3 for the projected Fermi sea and $v_{ij}$4 for other parameter choices $v_{ij}$5, to be compared with the exact $v_{ij}$6. Energetically, the best two-dimensional variational energies remain about $v_{ij}$7 above exact QMC, whereas in one dimension projected gapless forms reach $v_{ij}$8 against the exact $v_{ij}$9 (Kaneko et al., 2015).

The paper also gives explicit decay exponents for Marshall-compatible gapless pairing. In one dimension,

$v_{ii}\equiv g$0

implies

$v_{ii}\equiv g$1

while in two dimensions, along diagonals,

$v_{ii}\equiv g$2

Néel order is found for all $v_{ii}\equiv g$3 in two dimensions, but no magnetic order appears for any $v_{ii}\equiv g$4 in one dimension (Kaneko et al., 2015).

In itinerant systems, partial projection and short-range Jastrow terms select competing charge or orbital patterns. For the one-dimensional extended Hubbard model at quarter filling $v_{ii}\equiv g$5, with

$v_{ii}\equiv g$6

a charge-density wave with $v_{ii}\equiv g$7 appears above a threshold in $v_{ii}\equiv g$8. With $v_{ii}\equiv g$9, the metallic regime persists for $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$0, where $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$1, while for $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$2 the system enters a CDW regime with $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$3; Monte Carlo sampling shows ergodicity breaking between the “0101…” and “1010…” patterns (Kaneko et al., 2015).

On the triangular lattice at filling $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$4, the same construction stabilizes two distinct CDW patterns. For $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$5, a “2-0-0” pattern appears for $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$6, with Bragg peaks at $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$7. For $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$8, the uniform phase survives up to $$v_{\mathbf{q}}=\sum_{\mathbf{r}} v(\mathbf{r})\, e^{i\mathbf{q}\cdot \mathbf{r}},$$9, an intermediate “1-1-0” pattern occupies $$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$0, and the “2-0-0” state reappears for $$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$1. In a two-band square-lattice Hubbard model at half filling $$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$2, with on-site intra- and inter-orbital Gutzwiller parameters and nearest-neighbor intra-orbital Jastrow terms, an orbital-density wave is stabilized when $$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$3. Using

$$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$4

the ordered phase is signaled by a peak at $$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$5 with $$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$6 (Kaneko et al., 2015).

These examples support a sharp distinction between soft and full projection. In spin models, full Gutzwiller projection is essential for stabilizing two-dimensional Néel order and one-dimensional dimer order within symmetric $$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$7, whereas the soft projector changes short-range spin correlations but leaves $$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$8 as $$\hat{J} = \exp\!\Big(-\tfrac{1}{2}\sum_{i,j,\alpha,\beta} v^{\alpha,\beta}_{i,j}\, n^\alpha_i n^\beta_j\Big), \qquad n^\alpha_i=\sum_\sigma n^\alpha_{i\sigma}.$$9. In itinerant systems, by contrast, partial projection tunes charge fluctuations and cooperates with short-range Jastrow terms to select ordered charge or orbital patterns (Kaneko et al., 2015).

4. One-dimensional Jastrow-Gutzwiller states, criticality, and parent Hamiltonians

The one-dimensional Jastrow-Gutzwiller family on a periodic chain has a particularly explicit statistical interpretation. Writing

$$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$0

one obtains a classical pseudo-energy

$$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$1

with

$$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$2

Hence $$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$3 is the Boltzmann weight of a constrained Coulomb-gas problem on a circular lattice, with $$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$4 acting as an inverse temperature (Turkeshi et al., 2019).

Several special points are exact. At $$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$5, the state is the exact ground state of the XXZ chain at $$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$6. At $$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$7, the parent Hamiltonian is free fermionic, corresponding to the XX chain up to boundary terms. At $$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$8, the state is the exact ground state of the Haldane-Shastry Hamiltonian. The participation spectrum organizes configurations by their number of ferromagnetic domain walls, with the Néel and anti-Néel states the most probable configurations for $$\hat{P}_G = \prod_i \Big[1-(1-g)\, n_{i\uparrow}n_{i\downarrow}\Big] \quad\Longleftrightarrow\quad \hat{P}_G = \exp\!\Big(-g\sum_i n_{i\uparrow}n_{i\downarrow}\Big),$$9. The participation gap increases linearly with $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$00, and the two-domain-wall interaction is logarithmic in separation, consistent with a Luttinger-liquid interpretation with $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$01 (Turkeshi et al., 2019).

The entanglement structure distinguishes critical and noncritical regimes. For the half-chain von Neumann entropy, the free-fermion point obeys

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$02

while the nonrelativistic point $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$03 exhibits

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$04

indicating $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$05 rather than relativistic conformal scaling. Finite-size scaling of a rescaled entropy and its susceptibility yields

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$06

consistent with a critical regime for $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$07. In the large-$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$08 limits, the state reduces to GHZ-like cats: for $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$09, an antiferromagnetic cat of the two Néel configurations, and for $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$10, a translation-symmetrized ferromagnetic-domain cat (Turkeshi et al., 2019).

Parent Hamiltonian reconstruction for this family uses an entanglement-guided Bisognano-Wichmann ansatz. For a subsystem $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$11,

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$12

and the candidate couplings $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$13 are optimized by minimizing the relative entropy

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$14

Within short-range XXZ-like bases,

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$15

the reconstructed couplings recover $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$16 at $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$17 and approach $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$18 near $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$19. Long-range inverse-square families of Haldane-Shastry type,

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$20

are exact only at $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$21: for $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$22, even nearby, the relative entropy grows with system size, indicating that simple inverse-square two-body terms are insufficient (Turkeshi et al., 2019).

Quality diagnostics reported for the reconstructed parents include overlaps often above $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$23 across the critical regime, small relative entropies, and relative energy errors below $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$24. The results therefore suggest that the one-dimensional Jastrow-Gutzwiller family is broadly critical for $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$25, but exact parentage outside the free-fermion and Haldane-Shastry points likely requires longer-range or more complicated operator content (Turkeshi et al., 2019).

5. Quantum-circuit implementations and algorithmic realizations

The on-site Gutzwiller factor admits an exact discrete Hubbard-Stratonovich decomposition that is well suited to digital quantum simulation. For the SU(2) Hubbard case, the charge-channel identity

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$26

with

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$27

rewrites the nonunitary Gutzwiller operator as a linear combination of unitaries. Under the Jordan-Wigner mapping,

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$28

and each factor becomes a pair of commuting single-qubit $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$29 rotations. The resulting linear-combination-of-unitaries circuit uses $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$30 ancillas, two controlled-$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$31 gates per site after simplification, and $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$32 CNOTs in total. Postselecting the ancillas in $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$33 prepares the desired Gutzwiller state, with success probability

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$34

For small $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$35, $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$36, while for $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$37 the slope saturates to zero. A complementary importance-sampling formulation introduces two sets of auxiliary fields for bra and ket and evaluates observables by Metropolis-Hastings updates. Numerical tests on the half-filled Fermi-Hubbard model on $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$38, 10-site, and 12-site systems give energy curves agreeing with exact results within statistical error bars, and IBM Q Manila reproduces the exact ground-state energy of the two-site half-filled Fermi-Hubbard model within error bars (Seki et al., 2022).

The attractive SU(3) case generalizes the same logic to multicomponent onsite Jastrow factors. Here

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$39

with $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$40, and the onsite pair-density correlator is equivalent, for number-conserving trial states, to the trion-enhancing Gutzwiller form. The discrete HS identity becomes

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$41

with

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$42

so the unitary terms are on-site color-flip SU(2) rotations. Under Jordan-Wigner, these become fermionic Givens rotations. The corresponding probabilistic preparation uses $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$43 ancillas and has success probability

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$44

Its slope obeys

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$45

so the success rate decays exponentially for small $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$46 but remains finite as $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$47. Exact-sum LCU and importance sampling reproduce the total energy, kinetic energy, interaction energy, and triple occupancy of small attractive SU(3) Hubbard systems, and experiments on the Quantinuum H1-1 trapped-ion device for the two-site model show good agreement with exact values within statistical errors (Xu et al., 25 Apr 2025).

These constructions reinforce a general identification: the Gutzwiller factor is an onsite Jastrow term, and discrete HS decompositions provide a systematic route from nonunitary correlators to hardware-native unitary blocks. A plausible implication is that increasingly general Jastrow-Gutzwiller ansätze can be compiled by enlarging the auxiliary-field space, although the cited works also emphasize the resulting ancilla, sampling, and postselection overheads (Seki et al., 2022).

6. Broader context, limitations, and failure modes

Jastrow-Gutzwiller wave functions are not uniformly reliable across all correlated phases. A direct counterexample arises in the Exciton Bose Liquid context, where a bosonic Gutzwiller-type state is built by projecting two parton liquids whose amplitudes are products of one-dimensional Jastrow factors. In coordinate representation,

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$48

which in second-quantized form corresponds to a separable pseudo-potential

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$49

This produces a harmonic structure factor

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$50

whereas the accurate EBL wave function requires a non-separable long-range Jastrow kernel

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$51

leading to

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$52

The corresponding box correlator exponents differ qualitatively: $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$53 grows logarithmically with transverse size, whereas

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$54

has a $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$55-independent exponent. The conclusion drawn there is that dynamical spatial gauge-field fluctuations are necessary for a qualitatively accurate EBL description, and that mere Gutzwiller projection may fail to capture long-wavelength physics (Tay et al., 2010).

Even in the fermionic settings where these states are successful, the limitations are explicit. Noncollinear magnetic orders, especially on frustrated lattices lacking Marshall signs, are difficult to obtain without explicit symmetry breaking in $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$56. In two-dimensional Heisenberg models, energetics are not competitive with state-of-the-art QMC or DMRG, even when the order parameter is qualitatively correct. The quantum-to-classical mapping used to interpret density Jastrow effects is exact only for diagonal observables; it does not control spin off-diagonal or strongly entangled quantities in projected RVB states. Finite-size and boundary-condition effects can also be severe, for example in dimerized states with nearly degenerate momentum sectors (Kaneko et al., 2015).

A broader structural limitation comes from parent-Hamiltonian analysis of pure Jastrow states in the continuum. For a ground state

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$57

the exact parent Hamiltonian in arbitrary spatial dimension generally contains both a two-body potential

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$58

and a three-body term

$$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$59

with $$|\Phi_0\rangle = \exp\!\Big(\sum_{i,j} f_{i,j}\, c^\dagger_{i\uparrow} c^\dagger_{j\downarrow}\Big)\,|0\rangle,$$60. This suggests that apparently simple Jastrow sectors can encode effective many-body interactions that are not transparent at the wavefunction level, and that exact parentage may require more than pairwise couplings except at special integrable points (Beau et al., 2021).

Taken together, these results define a precise scope for Jastrow-Gutzwiller methods. They are highly effective for constructing symmetric variational states that can nevertheless acquire charge, orbital, magnetic, or dimer order after optimization; they interpolate between free-fermion, Haldane-Shastry, RVB, and Mott-correlated regimes; and they admit explicit circuit decompositions in on-site form. At the same time, phases controlled by dynamical gauge fluctuations, nontrivial long-wavelength anisotropies, or more elaborate parent-Hamiltonian structures may lie beyond the reach of naive or purely local Jastrow-Gutzwiller constructions (Tay et al., 2010).