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Symmetry-resolved properties of the trace distance in thermalizing SU(2) systems

Haojie Shen, Jie Chen, Xiaoqun Wang

Abstract

We study diagnostics of thermalization in quantum many-body systems with global SU(2) symmetry, where the standard eigenstate thermalization hypothesis (ETH) is generalized to its non-Abelian form. As an eigenstate-level probe, we introduce a symmetry-resolved trace distance constructed from the block structure of the reduced density matrix. This block structure separates spin-sector probabilities from configurational fluctuations within each sector, naturally leading to a decomposition into a probability trace distance and a configurational trace distance. The microcanonical average of the former is bounded by fluctuations of the corresponding spin-sector probabilities within a microcanonical energy window, whereas the latter captures finer intra-sector fluctuations. In non-Abelian thermalizing systems, these spin-sector-probability fluctuations are constrained by the non-Abelian ETH and therefore become exponentially suppressed with system size. Numerical studies of the one-dimensional \(J_1\)--\(J_2\) Heisenberg chain are consistent with this picture and suggest that, in the thermal regime, the trace distance is asymptotically dominated by the configurational trace distance.

Symmetry-resolved properties of the trace distance in thermalizing SU(2) systems

Abstract

We study diagnostics of thermalization in quantum many-body systems with global SU(2) symmetry, where the standard eigenstate thermalization hypothesis (ETH) is generalized to its non-Abelian form. As an eigenstate-level probe, we introduce a symmetry-resolved trace distance constructed from the block structure of the reduced density matrix. This block structure separates spin-sector probabilities from configurational fluctuations within each sector, naturally leading to a decomposition into a probability trace distance and a configurational trace distance. The microcanonical average of the former is bounded by fluctuations of the corresponding spin-sector probabilities within a microcanonical energy window, whereas the latter captures finer intra-sector fluctuations. In non-Abelian thermalizing systems, these spin-sector-probability fluctuations are constrained by the non-Abelian ETH and therefore become exponentially suppressed with system size. Numerical studies of the one-dimensional -- Heisenberg chain are consistent with this picture and suggest that, in the thermal regime, the trace distance is asymptotically dominated by the configurational trace distance.

Paper Structure

This paper contains 2 theorems, 31 equations, 2 figures.

Key Result

Proposition 1

For two neighboring eigenstates labeled by $\alpha$ and $\alpha+1$, the trace distance $D_\alpha^A$ is bounded from below by the change in the spin-sector probabilities and from above by the sum of two contributions defined below.

Figures (2)

  • Figure 1: (a) Finite-size scaling of the sum of variances of the spin-sector probabilities, $\sum_{S_A}\mathrm{Var}_{\mathcal{W}}\!(P_{S_A}^{(\bm{\alpha})})$, and (b) the corresponding window average $\langle D^{A}_{\alpha,\mathrm{prob}}(x=1/2)\rangle_{\mathcal{W}}$. Different colors correspond to different $J_2$ values.
  • Figure 2: Finite-size scaling of $\langle D^{A}_{\alpha}(x)-D^{A}_{\alpha,\mathrm{conf}}(x)\rangle_{\mathcal{W}}$. Different colors correspond to different $J_2$ values.

Theorems & Definitions (8)

  • Definition 1: Subsystem-spin projector
  • proof : Proof of $SU(2)$ invariance
  • Definition 2: Symmetry-resolved reduced density matrix
  • Definition 3: Symmetry-resolved trace distance
  • Proposition 1: Bounds for the trace distance
  • proof
  • Proposition 2: Microcanonical average of $D^{A}_{\alpha,\mathrm{prob}}$
  • proof