Characterizing entanglement at finite temperature: how does a "classical" paramagnet become a quantum spin liquid?

Abstract

Quantum spin liquids (QSL) are phases of matter which are distinguished not by the symmetries they break, but rather by the patterns of entanglement within them. Although these entanglement properties have been widely discussed for ground states, the way in which QSL form at finite temperature remains an open question. Here we introduce a method of characterizing both the depth and spatial structure of entanglement, and use this to explore how patterns of entanglement form as temperature is reduced in two widely studied models of QSL, the Kitaev honeycomb model, and the spin-1/2 Heisenberg antiferromagnet on a Kagome lattice. These results enable us to evaluate both the temperature at which spins within the high-temperature paramagnet first become entangled, and the temperature at which the system first develops the structured, multipartite entanglement characteristic of its QSL ground state.

Figures (4)

• Figure 1: Evolution of a quantum spin liquid (QSL) out of a high temperature paramagnet. (a) Two--step scenario: on cooling, quantum effects enter at a temperature $T_{\sf sep}$, comparable with the interactions enforcing local spin correlations. Structured entanglement, characteristic of the QSL ground state, forms at a lower temperature, $T_{\sf GME}$. This entanglement is multipartite in nature, and confined to cycles of spins. (b) For the Kagome antiferromagnet (KAF), the shortest relevant cycle is the 6--link "bow--tie", highlighted in green. (c) For the Kitaev honeycomb model (KHM), the shortest relevant cycle is a 6--link hexagonal ring.
• Figure 2: Evolution of entanglement as a function of temperature in the spin-1/2 Heisenberg antiferromagnet on a Kagome lattice (KAF). (a) Results for the depth of entanglement ${\mathcal{S}}_k [\rho_n(T)]$ [Eq. (\ref{['eq:depth.of.entanglement']})] present in the 6--link "bow--tie" cycle illustrated in Fig. \ref{['fig:kagome']}. Entanglement on this cycle is first detected at a temperature temperature $T_{\bowtie}^{\sf sep} = 1.26\ J$. Multipartite entanglement of the type needed to support a QSL is first deteched at a temperature $T^{\bowtie}_{\sf GME} = 0.57\ J$. (b) Corresponding results for genuine tripartite entanglement ${\mathcal{S}}_{\sf GTE} [\rho_n(T)]$, calculated through Eq. (\ref{['eq:S.restricted.partitions']}). An example of the type tripartition considered is shown in the legend. (c) Heat capacity $c(T)$, showing how changes in entanglement correlate with changes in thermodynamic properties. Results are taken from thermal pure quantum state (TPQ) calculations for a cluster of 24 spins, as described in the end matter.
• Figure 3: Evolution of entanglement as a function of temperature in the Kitaev model on a honeycomb lattice (KHM). (a) Results for the depth of entanglement, ${\mathcal{S}}_k [\rho_n(T)]$ [Eq. (\ref{['eq:depth.of.entanglement']})] present in the six--link hexagonal cycle illustrated in Fig. \ref{['fig:honeycomb']}. Entanglement on this cycle is first detected at a temperature $T_{\hexagon}^{\sf sep} = 0.18\ J$. Genuine multipartite entanglement (GME) is detected at a lower temperature $T_{\hexagon}^{\sf GME} = 0.018\ J$. (b) Corresponding results for the entanglement ${\mathcal{S}}_\lambda [ \rho_{n}(T)]$, associated with different geometrical partitions $\mathcal{P}_\lambda = \mathscr{P}^{(m)}_{\hexagon}$ [Eq. (\ref{['eq:S.restricted.partitions']}), End Matter]. Tripartite entanglement is detected via $\mathscr{P}^{(3)}_{\hexagon}$ at a temperature $T_{\hexagon}^{\sf GTE} =0.096 \ J$. (c) Expectation value of plaquette operator $\langle \mathcal{W}_p \rangle$ [Eq. (\ref{['eq:W_p']})] associated with $\mathbb{Z}_2$ gauge flux. (d) Heat capacity $c(T)$ [Eq. \ref{['eq:C.of.T']}], showing two--peak structure characteristic of the Kitaev spin liquid. Results are taken from thermal pure quantum state (TPQ) calculations for a cluster of 24 spins, as described in End Matter.
• Figure 4: Clusters of the used in calculations for the Kagome lattice antiferromagnet (AF) and the Kitaev model on a honeycomb lattice. (a) 24--site cluster on Kagome lattice, showing location of 6-bond cycle used to calculate results for entanglement given in the main text. (b) 24--site cluster with full symmetry of honeycomb lattice, showing location of 6--bond cycle considered in the main text. In both cases, periodic boundary conditions are imposed, shown here through the dashed boundary of the cluster.