Entanglement Spectrum, Critical Exponents, and Order Parameters in Quantum Spin Chains

We investigate the entanglement spectrum near criticality in finite quantum spin chains. Using finite size scaling we show that when approaching a quantum phase transition, the Schmidt gap, i.e., the difference between the two largest eigenvalues of the reduced density matrix ${\lambda }_1$, ${\lambda }_2$, signals the critical point and scales with universal critical exponents related to the relevant operators of the corresponding perturbed conformal field theory describing the critical point. Such scaling behavior allows us to identify explicitly the Schmidt gap as a local order parameter.

Figure 1

Transverse Ising model. Top-Left: Schmidt gap at the critical point ($J=B_z$) as a function of the chain size. The numerical data (symbols) are compared to a fitting from Eq. (2) (dashed) and to a simple power law (dotted). Top-Right: Entanglement spectrum: the first 8 eigenvalues ${\lambda }_i$ are shown as a function of $J/B_z$ for $L=12000$. Bottom-Left: Schmidt gap as a function of $J/B_z$ for different lengths. Bottom-Right: FSS analysis of $\Delta \lambda$ for $L=768$, 1536, 3072, 6000, 12 000.

Figure 2

Ising model with longitudinal perturbation, DMRG calculations. Left: The first 6 Schmidt eigenvalues as a function of $B_x/B_z$ in logarithmic scale for $L=384$. The critical point corresponds to $B_x\to 0$. Right: FSS analysis of $\Delta \lambda$ for $L=48$, 96, 192, 384.

Figure 3

Spin-1 model. Left: The entanglement spectrum for $\theta =0$ and as a function of $D$ nearby the Néel-Haldane quantum phase transition (for clarity only the first 6 eigenvalues are depicted). Dotted, dashed and continuous lines correspond to spin chains of length $L=192$, 384, and 768, respectively. The location of the critical point obtained after FSS analysis of the Schmidt gap is $D=-0.315$. Right: FSS analysis of the Schmidt gap together with a schematic plot of the critical points marking the transition between Néel (N), Haldane(H), and Large-D (L) phases for $\theta =0$ as a function of D.