Calculation of multi-fractal dimensions in spin chains

Abstract

It was demonstrated in Atas & Bogomolny (2012 Phys. Rev. E 86, 021104) that the ground-state wave functions for a large variety of one-dimensional spin- models are multi-fractals in the natural spin-z basis. We present here the details of analytical derivations and numerical confirmations of this statement.

Keywords: ground-state wave function; multi-fractality; spin chains.

Figure 1.

Tree representation of the binomial cascade. The sum of the elements over a line is equal to one.

Figure 2.

(a) Binomial measure for N=12 steps of the binomial cascade for and . (b) Fractal dimensions for the same measure. Horizontal dashed lines indicate the limiting values (2.10). (c) Singularity spectrum for the same measure.

Figure 3.

Key idea of Jordan–Wigner transformation: an up spin |↑〉 (resp. down spin |↓〉) is equivalent to the absence n_f=0 (resp. the presence n_f=1) of a fermion. (Online version in colour.)

Figure 4.

(a) Coefficients of the GS wave function for the QIM with λ=1 and N=11 versus the binary code (2.5). (b) The same but for the XY model with λ=0.4, γ=1.4 and N=12.

Figure 5.

(a) Plot of the logarithm of χ² (4.15) as a function of q for fit (4.14) for the Ising model in critical field λ=1 and non-critical field λ=1.6. Inset: as a function of for λ=0.4,1,1.6 (top to bottom, respectively). (b) Numerical comparison of the fractal dimensions for the critical QIM obtained by the Lanczos technique with N=3 to 18 (red lines) and exact diagonalization with N=3 to 11 (blue squares). (Online version in colour.)

Figure 6.

Contour of integration in the calculation of the GS for the XY model. Black filled circles indicate poles on the unit circle. Red open circles are square root singularities of function f(z) in the case λ²+γ²<1. (Online version in colour.)

Figure 7.

(a) Fractal dimensions of the GS for the XY model in the factorizing field λ=0.8 and γ=0.6 obtained by the Lanczos method for N=5 to 18 are indicated by black circles. The red solid line corresponds to exact equation (5.23) with θ=π/6. Inset: relative errors of moments of GS wave function (5.27) for these parameters: black squares (full lines) correspond to q=2, red circles (dashed lines) to q=2.5 and blue rhombuses (dotted lines) to q=3.5. (b) Plot of the logarithm of χ² (4.15) as a function of q for the XY model in factorizing field λ=0.8 and γ=0.6, and generic field λ=0.4 and γ=1.4 for the fit (4.14). Inset: as a function of for these values of parameters. (Online version in colour.)

Figure 8.

(a) Wave function of the XXZ model at Δ=−1/2 and N=13. (b) Wave function for the XYZ model in zero field with Δ=−0.5, γ=0.6 and N=12.

Figure 9.

Plot of the logarithm of χ² (4.15) as a function of q for the XXZ model in the combinatorial point Δ=−1/2 for three different fits in (6.7). Red line (upper curve) corresponds to the fit f₁. Black line (middle curve) is obtained by the fit f₂. Blue line (lower curve) is for the fit f₃. Inset: as a function of where with odd N=3 to 19. (Online version in colour.)