Spin Chain Ansatz Wavefunctions

• Spin chain ansatz wavefunctions are parameterized quantum many-body states that encode spatial and spin correlations in 1D systems.
• They employ a factorization into charge and spin parts using variational methods and overcomplete valence bond superpositions to capture unique entanglement properties.
• These constructions yield analytic results for observables such as the momentum distribution function and demonstrate universal behavior in coupled chains and Kondo systems.

Spin chain ansatz wavefunctions are explicit, parameterized forms for the quantum many-body states of spin chains and related 1D correlated systems. These ansatzes encode the spatial and spin correlations of a system, often reflecting underlying integrability, symmetry, or coupling to other degrees of freedom. Recent developments involve variational and algebraic constructions that capture regimes of strong quantum entanglement, the decoupling of spin and charge, and emergent phenomena such as edge localization, spin-incoherent (SILL) phases, and connections to integrable quantum field theory.

1. Factorized Variational Ansatz and Spin-Incoherent Regimes

The foundational framework for constructing spin-incoherent states in 1D correlated systems is the factorization of the wavefunction into charge and spin components. For the $t$-$J$ chain at $J \to 0$, this factorizes as

$|g.s.\rangle = |\phi\rangle \otimes |\chi\rangle,$

where $|\phi\rangle$ is the noninteracting spinless fermion ground state (charge sector) and $|\chi\rangle$ is the "squeezed" spin wavefunction, defined on occupied sites only. When the chain is coupled to a spin bath or another chain (as in antiferromagnetically coupled ladders or Kondo systems), the ansatz generalizes to

$$|g.s.\rangle = |\phi\rangle_1 \otimes |\phi\rangle_2 \otimes |S\rangle,$$

where $|S\rangle$ is an equal-amplitude superposition of all possible valence bond (VB) singlet coverings pairing spins on different chains (sublattices) (Soltanieh-ha et al., 2012).

This VB superposition encodes maximal spin entanglement due to the effective infinite "spin temperature" generated by coupling to a bath or weak external spin exchange. The resulting spin-incoherent regime is characterized by highly entangled yet featureless spin correlations, yielding physical observables and entanglement properties distinct from those of coherent Luttinger liquids.

2. Valence Bond Formalism and Overcomplete Basis

The VB formalism, central to describing the spin sector in these ansatzes, involves representing the spin state $|S\rangle$ as a sum over all pairings between two sublattices:

$$|S\rangle = \sum_{\text{VB coverings}} |\text{VB configuration}\rangle.$$

This basis is overcomplete. Although one might expect $N$ singlets to yield entanglement entropy $N\log2$, calculation shows that the entropy is actually $S = \log(N+1)$, due to this overcompleteness—an interpretation akin to maximal entanglement between two large spins of $N/2$ (Soltanieh-ha et al., 2012). The overcompleteness ensures that all possible spin pairings contribute, consistent with the "incoherent" limit where no unique spin state is selected.

3. Physical Observables from Factorized Ansatz Wavefunctions

The factorization property enables analytical results for certain observables:

• Momentum Distribution Function (MDF):

With the ansatz $|g.s.\rangle = |\phi\rangle \otimes |S\rangle$, the MDF reduces to that of noninteracting spinless fermions,

$$n(k) = \frac{1}{L} \sum_{l,\sigma} e^{i k l} \langle f_1^\dagger f_l \rangle,$$

leading to a sharp discontinuity at $2k_F$ (for Fermi momentum $k_F$), in contrast to the power-law singularity of traditional Luttinger liquids (Soltanieh-ha et al., 2012).

• Entanglement Entropy:

The universal result $S = \log(N+1)$ for a chain of $N$ electrons signifies the unique structure of entanglement in the spin-incoherent regime. All nontrivial contributions are from the spin sector, owing to the product structure (Soltanieh-ha et al., 2012).

These results are robust for both the two-chain $t$-$J$ system and $t$-$J$ Kondo chains, and suggest application to a wider class of models with similar spin-charge separation and weak spin-bath coupling.

4. Universality and Generalizations

The variational ansatz formalism for spin-incoherent Luttinger liquids is argued to be universal for a wide family of models:

• Two coupled $t$-$J$ Chains/Ladders: Factorization and VB description capture the ground state when chains are weakly coupled via antiferromagnetic exchange $J'$, introducing incoherence in the spin sector (Soltanieh-ha et al., 2012).
• $t$-$J$ Kondo Chain: The spin-incoherent picture applies when conduction electrons couple to localized impurity spins, and unpaired spins become fully polarized, while the remainder is described as a maximally entangled VB state.
• Systems with Weak Bath Coupling: Any model with spin-charge separation and a weak coupling of spins to a bath (even at $T=0$) generically displays the same spin-incoherent structure.

This universality extends the spin-incoherent paradigm beyond its original finite-$T$ Luttinger liquid context, providing a mechanism for "spin temperature" at $T=0$ via environmental entanglement.

5. Applications and Physical Consequences

The application of spin chain ansatz wavefunctions in these contexts leads to several notable phenomena:

• Sharp Discontinuity in MDF: The appearance of a "Fermi edge" at $2k_F$ and a finite quasi-particle weight $z=1$ highlights a key difference with ordinary Luttinger liquids, where these features are suppressed.
• Spin "Half Fermi Liquid": In the $t$-$J$ Kondo chain, majority (up) and minority (down) spin electrons display different low-energy physics: majority spins show finite quasi-particle weight; minority spins have zero weight and scale as in a regular Luttinger liquid, reminiscent of—but distinct from—FFLO states (Soltanieh-ha et al., 2012).
• Crossover Behavior: The ansatz captures the transition from spin-coherent to spin-incoherent regimes, previously interpreted only as a finite-$T$ effect, now revealed as ground-state physics in the presence of spin-bath coupling.
• Extension to Multi-Chain and Layered Systems: The approach provides a framework for analyzing more general situations (e.g., ladders or multilayer structures), especially in regimes dominated by quantum fluctuations and spin-charge separation.

6. Methodological Implications and Relation to Other Wavefunction Constructions

The ansatz wavefunctions discussed here share commonalities with other approaches for integrable and non-integrable spin systems, including:

• Matrix Product and Tensor Network States: Factorizations and projection strategies are echoed in recent matrix-product-based methods for representing ground states and excitations.
• Spin Chain Bethe Ansatz: While the variational VB approach is distinct from algebraic Bethe ansatz, it shares the principle of exact solvability in the appropriate sector, and in some cases (e.g., $t$-$J$ ladders), complements or generalizes integrable results to coupled/bath-coupled systems.
• Valence Bond and RVB States: The overcomplete superposition of VB coverings links the ansatz directly to the resonating valence bond philosophy in quantum magnetism, connecting to broader classes of quantum spin liquids.

7. Outlook and Future Directions

The variational ansatz wavefunctions developed for spin-incoherent Luttinger liquids and related models provide a blueprint for analyzing nontrivial entanglement and correlation effects in 1D and quasi-1D systems:

• The explicit, exactly solvable structure for densities and entropy supports comparison with numerical and experimental results in coupled chains and cold atom systems.
• Extension to environments with multiple baths, higher dimensionality, and fluctuating coupling is plausible, leveraging the underlying factorization and VB ideas.
• This line of work suggests a unifying framework for understanding how environmental coupling modifies fundamental properties (such as entanglement entropy and quasi-particle discontinuity) in strongly-correlated quantum systems.

In summary, spin chain ansatz wavefunctions—particularly those based on factorization and the VB formalism—have become essential tools for characterizing, analyzing, and generalizing the ground states and excitations of essentially 1D systems subject to strong spin-charge separation and environmental coupling (Soltanieh-ha et al., 2012). They provide exact results, capture universal features of entanglement and quasi-particle spectra, and lay the groundwork for an extended theory of incoherent quantum phases in low-dimensional correlated matter.