[Paper Review] Vanishing spin stiffness in the spin-1/2 Heisenberg chain for any nonzero temperature

J. M. P. Carmelo, Tomaž Prosen+1 more|arXiv (Cornell University)|Jul 2, 2014

Quantum many-body systems3 citations

TL;DR

This paper establishes that the spin stiffness of the spin-1/2 Heisenberg XXX chain vanishes exactly at zero spin density (m=0) for any nonzero temperature T>0, using an exact upper bound proportional to m²L in the thermodynamic limit. The result rules out ballistic spin transport in the canonical ensemble at finite temperature, resolving a long-standing controversy by showing stiffness vanishes even though Mazur's inequality suggests otherwise for m≠0.

ABSTRACT

Whether at zero spin density $m=0$ and finite temperatures $T>0$ the spin stiffness of the spin-$1/2$ $XXX$ chain is finite or vanishes remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an exact upper bound on the stiffness within a canonical ensemble at any fixed value of spin density $m$ and show that it is proportional to $m^2 L$ in the thermodynamic limit of chain length $L o\infty$, for any finite, nonzero temperature. Moreover, we explicitly compute the stiffness at $m=0$ and confirm that it vanishes. This allows us to exactly exclude the possibility of ballistic transport within the canonical ensemble for $T>0$.

Motivation & Objective

• To resolve the long-standing controversy over whether spin stiffness vanishes at zero spin density (m=0) in the spin-1/2 Heisenberg XXX chain at finite temperature T>0.
• To provide an exact upper bound on spin stiffness in the canonical ensemble for any fixed m and T>0, showing it scales as m²L in the thermodynamic limit.
• To confirm that spin stiffness vanishes at m=0, excluding ballistic transport in the canonical ensemble at T>0.
• To reconcile conflicting results from different theoretical approaches, including Mazur's inequality and thermodynamic Bethe ansatz (TBA) methods.

Proposed method

• Derives an exact upper bound on spin stiffness using thermal averages and current-current correlation functions in the canonical ensemble.
• Applies the canonical ensemble framework with fixed total spin projection S, ensuring m = m_S = S/L is well-defined.
• Uses the number of pairs sum rule ∑ₙ nMₙ = (L−2S)/2 to relate the distribution of n-pair configurations to effective current carriers.
• Analyzes the spin current in reduced subspaces with different occupancies of n-band particles (n≥1), particularly comparing subspaces with Mₙ=0 for n>1 and those with finite ∑ₙ≥₂ Mₙ.
• Evaluates the current gap Δ_J via solutions of Eqs. (46)–(47) and current expressions (40)–(44), showing Δ_J increases with m_S.
• Demonstrates that the maximum absolute current in the subspace with no n>1 band particles (i.e., Mₙ=0 for n>1) exceeds that in subspaces with finite Mₙ for n>1, due to larger effective current carrier count and larger elementary current per carrier.

Experimental results

Research questions

• RQ1 Does the spin stiffness of the spin-1/2 Heisenberg XXX chain vanish at zero spin density (m=0) for any nonzero temperature T>0?
• RQ2 Can an exact upper bound on spin stiffness be derived in the canonical ensemble that confirms the vanishing of stiffness at m=0 and T>0?
• RQ3 Why do conflicting results arise from Mazur's inequality and other approaches, and how can they be reconciled?
• RQ4 What role do n-pair configurations and effective current carriers play in determining the spin current and stiffness?
• RQ5 How does the current gap Δ_J depend on the spin quantum number S and the distribution of Mₙ across n-bands?

Key findings

• The spin stiffness vanishes exactly at m=0 for any T>0, ruling out ballistic transport in the canonical ensemble.
• An exact upper bound on stiffness is derived as proportional to m²L in the thermodynamic limit L→∞, confirming stiffness vanishes as m→0.
• The maximum absolute spin current is achieved in the reduced subspace with no n>1 band particles (Mₙ=0 for n>1), due to a larger number of effective current carriers and larger elementary current per carrier.
• The current gap Δ_J reaches its minimum value of 2J when M₁ = (L−2S)/2 − 2 and M₂ = 1, corresponding to the largest current-carrying capacity in the subspace with finite n>1 pair occupancy.
• In the (1−m_S)≪1 limit, the current gap Δ_J = 2J∑ₙ(n−1)Mₙ = 2J×δN_carriers, where δN_carriers is the difference in effective current carrier count between subspaces.
• The result confirms that stiffness vanishes at m=0, resolving contradictions between TBA, phenomenological, and nonequilibrium approaches.