[Paper Review] An Exactly Solvable Constrained XXZ Chain

F C Alcaraz, R. Z. Bariev|ArXiv.org|Apr 2, 1999

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TL;DR

This paper introduces a new family of exactly solvable quantum spin chains by generalizing the XXZ model with a hard-core constraint that prevents spins up from being within distance $ t $ of each other ($ t = 0,1,2,\dots $). Using the coordinate Bethe ansatz, the authors derive exact Bethe ansatz equations and compute critical exponents via finite-size scaling, revealing that the critical behavior at half-filling $ \rho = 1/(t+2) $ is independent of $ \Delta $, with $ X_p = (t+2)^2/4 $, and that the $ \Delta \to -\infty $ limit maps to a free-fermion-like model with effective $ t' = t+1 $. The model exhibits non-trivial criticality even for $ \Delta < -1 $, extending integrability beyond the standard XXZ chain.

ABSTRACT

A new family of exactly solvable models is introduced. These models are generalizations of the XXZ chain where the distance among spins up ($σ^z$-basis) cannot be smaller or equal to t (t=0,1,2,...). The case t=0 recovers the standard XXZ chain. The coordinate Bethe ansatz is applied and the phase diagram is calculated. Exploring the finite-size consequences of conformal invariance the critical exponents are evaluated exactly at the critical regions of the phase diagram

Motivation & Objective

• To generalize the integrable XXZ spin chain by introducing a hard-core constraint that forbids spins up within distance $ t $, for $ t = 0,1,2,\dots $.
• To preserve integrability under this constraint and extend the applicability of the Bethe ansatz to non-local interactions.
• To compute critical exponents in the low-energy limit using finite-size scaling and conformal field theory.
• To explore the phase diagram and critical behavior, especially at half-filling and in the $ \Delta \to -\infty $ limit.
• To establish a mapping between the constrained model and a free-fermion-like system in the strong repulsion limit.

Proposed method

• Generalize the XXZ Hamiltonian by replacing the local fermion exclusion $ P_0 $ with a non-local projector $ P_t $, which forbids two up spins within distance $ \leq t $.
• Express the Hamiltonian in terms of spinless fermions with a modified hopping and density interaction: $ H_t(\Delta) = -\sum_i P_t (c_i^\dagger c_{i+1} + \text{h.c.} + \Delta n_i n_{i+t+1}) P_t $.
• Apply the coordinate Bethe ansatz to the wave function $ \Psi = \sum_{x_1 < \cdots < x_n} a(x_1,\dots,x_n) |x_1,\dots,x_n\rangle $, with amplitudes $ a = \sum_P A_k e^{i\sum k_j x_j} $.
• Derive the Bethe ansatz equations for $ t > 0 $, which reduce to those of an XXZ chain of size $ L' = L - tn $ with twisted boundary conditions.
• Use finite-size scaling of energy levels to extract critical exponents $ X_p $, relating them to conformal field theory via $ X_p = \frac{1}{4}(t+2)^2(\pi - \gamma) $ at half-filling.
• Analyze the $ \Delta \to -\infty $ limit, showing that the system maps effectively to a model with $ t' = t+1 $ and $ \Delta = 0 $, yielding $ X_p = 1/[4(1 - \rho(t+1))^2] $ for $ \rho < 1/(t+2) $.

Experimental results

Research questions

• RQ1 How does imposing a minimum distance $ t $ between spin-up particles affect the integrability and spectrum of the XXZ chain?
• RQ2 What are the exact critical exponents of the constrained XXZ chain, and how do they depend on $ \Delta $, $ t $, and particle density $ \rho $?
• RQ3 Does the critical behavior at half-filling $ \rho = 1/(t+2) $ remain independent of $ \Delta $, and if so, why?
• RQ4 What is the effective low-energy description of the model in the $ \Delta \to -\infty $ limit?
• RQ5 Can the Bethe ansatz be generalized to non-local constraints such as $ t > 0 $, and how does it relate to twisted boundary conditions?

Key findings

• The Bethe ansatz equations for the constrained chain with $ t > 0 $ are equivalent to those of a standard XXZ chain of size $ L' = L - tn $ with twisted boundary conditions.
• At half-filling $ \rho = 1/(t+2) $, the critical exponent is $ X_p = (t+2)^2/4 $, independent of $ \Delta $, due to gap closing rather than interaction effects.
• In the $ \Delta \to -\infty $ limit and $ \rho < 1/(t+2) $, the model maps effectively to a free-fermion system with $ t' = t+1 $, yielding $ X_p = 1/[4(1 - \rho(t+1))^2] $.
• At low density $ \rho \to 0 $, $ X_p = 1/4 $, and at high density $ \rho \to 1/(t+1) $, $ X_p = (t+1)^2/4 $, matching the $ \Delta = 0 $ case due to negligible interaction.
• For $ \Delta < -1 $, the critical exponents are non-trivial and not previously reported, even for the standard XXZ chain ($ t=0 $).
• The model with $ t = -1 $ is also exactly solvable, corresponding to unbounded site occupation, and its Bethe ansatz equations extend to $ t = -1 $, with a Hamiltonian realizable via infinite-dimensional matrices.