Quantum criticality and excitations of a long-range anisotropic XY chain in a transverse field

The critical breakdown of a one-dimensional quantum magnet with long-range interactions is studied by investigating the high-field polarized phase of the anisotropic XY model in a transverse field for the ferromagnetic and antiferromagnetic cases. While for the limiting case of the isotropic long-range XY model we can extract the elementary one-quasiparticle dispersion analytically and calculate two-quasiparticle excitation energies quantitatively in a numerical fashion, for the long-range Ising limit as well as in the intermediate regime we use perturbative continuous unitary transformations on white graphs in combination with classical Monte Carlo simulations for the graph embedding to extract high-order series expansions in the thermodynamic limit. This enables us to determine the quantum-critical breakdown of the high-field polarized phase by analyzing the gap closing including associated critical exponents and multiplicative logarithmic corrections. In addition, for the ferromagnetic isotropic XY model we determined the critical exponents $z$ and $\nu$ analytically by a bosonic quantum-field theory.

Figure 1

Critical exponents as a function of $\alpha$ for the ferromagnetic transverse-field isotropic XY model. We find the system to be in the nearest-neighbor universality class of the isotropic XY model (NN-TFXY) for $\alpha \ge 3$ where the dynamical critical exponent is $z=2$ (blue lines) and the correlation length critical exponent is $\nu =\frac{1}{2}$ (red lines). In a second regime $\alpha <3$ the dynamical (correlation length) exponent varies continuously between $0<z\le 2$ ($\frac{1}{2}\le \nu <\infty$). The exponents $z_{\text{ft}}$ and ${\nu }_{\text{ft}}$ are obtained from the field-theoretical treatment while $z_{\omega }$ and ${\nu }_{\omega }$ are determined by the one-particle dispersion. Both coincide to great accuracy. The second derivative of the critical dynamical exponent $z_{\omega }$ in the inset visualizes that the accuracy of $z_{\omega }$ around $\alpha =3$ strongly depends on the sampling step size $\epsilon$ of the dispersion around the minimum and therefore the floating point precision. Note, the second derivative approaches a Dirac delta function for $\epsilon \to 0$ corresponding to a kink in the exponent.

Figure 2

Characteristic continua and bound states in the 2QP sector. Dark green lines highlight the lower and upper continuum edge determined by Eqs. (24). Upper panel: 2QP continuum for the ferromagnetic transverse-field isotropic XY model with $\lambda =0.4$. Lower panel: Bound state and 2QP continuum of the ferromagnetic isotropic transverse-field XY model in the presence of an additional longitudinal Ising-type coupling with $\kappa =0.4$ and $\lambda =0.4$.

Figure 3

Critical exponents $z\nu$ and critical points ${\lambda }_c$ (inset) as function of $\alpha$ for the anisotropic XY model in a transverse field evaluated for various values of the interpolation parameter $\beta$ (deep red to green color gradient) with ferromagnetic interactions. The data points are averages from highest-order members of families consisting of nondefective DlogPadé extrapolants. When strengthening the long-range coupling the critical point shifts toward zero, i.e., the ordered phase becomes more prominent. For the critical exponent in case of pure Ising-type interactions the system exhibits a nearest-neighbor transverse-field Ising universality (NN-TFIM) for $\alpha \ge 3$ (gray area for $\alpha \ge 2.75$) with $z\nu =1$, a mean-field domain (MF-TFIM) for $\alpha \le \frac{5}{3}$ with $z\nu =\frac{1}{2}$ (light blue areas and gray lines) and an intermediate domain with a continuously varying exponent (cf. Ref. [60] and references therein). The LCE results confirm the presence of these domains for any extrapolation parameter except for $\beta \approx 0$ where the systems jumps to the isotropic behavior as expected (see dark red point at $\alpha =5/3$ for $\beta =0.01$).

Figure 4

Critical exponents $z\nu$ and critical parameters ${\lambda }_c$ (inset) as functions of $\alpha$ for the anisotropic XY model in a transverse field evaluated for various values of the interpolation parameter $\beta$ (red to green color gradient) with antiferromagnetic interactions. The data points are averages from highest-order members of families consisting of nondefective DlogPadé extrapolants. When strengthening the long-range coupling the critical point shifts toward $-\infty$ such that the polarized phase stabilizes. For pure Ising-type interactions Refs. [23, 52] suggest a NN domain for $\alpha \ge \frac{9}{4}$ with $z\nu =1$ (light blue area and gray line) and a second domain for $\alpha <\frac{9}{4}$ with continuously varying exponent. LCE results, that become increasingly challenging to extract for decreasing $\alpha$, do not indicate a second domain and instead suggest the same NN universality class for $\alpha \le \frac{5}{3}$.