Griffiths singularities in the random quantum Ising antiferromagnet: A tree tensor network renormalization group study

The antiferromagnetic Ising chain in both transverse and longitudinal magnetic fields is one of the paradigmatic models of a quantum phase transition. The antiferromagnetic system exhibits a zero-temperature critical line separating an antiferromagnetic phase and a paramagnetic phase; the critical line connects an integrable quantum critical point at zero longitudinal field and a classical first-order transition point at zero transverse field. Using a strong-disorder renormalization group method formulated as a tree tensor network, we study the zero-temperature phase of the quantum Ising chain with bond randomness. We introduce a new matrix product operator representation of high-order moments, which provides an efficient and accurate tool for determining quantum phase transitions via the Binder cumulant of the order parameter. Our results demonstrate an infinite-randomness quantum critical point in zero longitudinal field accompanied by pronounced quantum Griffiths singularities, arising from rare ordered regions with anomalously slow fluctuations inside the paramagnetic phase. The strong Griffiths effects are signaled by a large dynamical exponent $z>1$, which characterizes a power-law density of low-energy states of the localized rare regions and becomes infinite at the quantum critical point. Upon application of a longitudinal field, the quantum phase transition between the paramagnetic phase and the antiferromagnetic phase is completely destroyed. Furthermore, quantum Griffiths effects are suppressed, showing $z<1$, when the dynamics of the rare regions is hampered by the longitudinal field.

Figure 1

The zero-temperature phase diagram of the clean antiferromagnetic Ising chain with $J=1$ in transverse ($\mathrm{\Gamma }$) and longitudinal ($h$) magnetic fields. The critical line, obtained by DMRG and QMC calculations, separates a quantum antiferromagnetic (AFM) phase in weak fields and a quantum paramagnetic (PM) phase in strong fields. The classical multicritical point (CMP) at $(\mathrm{\Gamma }=0,h=2)$ corresponds to a first-order transition between an AFM phase and a ferromagnetic (FM) phase in the absence of quantum fluctuations. The critical line is well described by a quadratic relation: ${\mathrm{\Gamma }}_c\approx 1-0.19h^2$ in the vicinity of the point $(\mathrm{\Gamma }=1,h=0)$, while it is linear, ${\mathrm{\Gamma }}_c\approx 1.4(2-h)$, near the classical multicritical point.

Figure 2

The Binder cumulant $U$ of the order parameter of the clean chain with $J=1$ in the absence of longitudinal fields (a) and in a finite longitudinal field $h=1$ (b), plotted against the strength of the transverse field. The crossing point of the curves for different chain lengths $L$ indicates the critical value ${\mathrm{\Gamma }}_c$ of the transverse field for each case; we find ${\mathrm{\Gamma }}_c=1$ for $h=0$ and ${\mathrm{\Gamma }}_c=0.8$ for $h=1$. The inset of (b) is a scaling plot of the data for $h=1$, using the critical exponent $\nu =1$ of the 2D Ising universality class.

Figure 3

Schematic of the tensor network SDRG algorithm. (a) The system is partitioned into blocks using MPO formalism; the green boxes represent the $W$ tensors, the vertical lines are physical indices, and the horizontal lines represent the bond indices. At each step of RG, the pair of nearest-neighbor blocks (marked with the pink-shaded square) with largest energy gap ${\mathrm{\Delta }}_{\chi }$ is chosen for renormalization, where ${\mathrm{\Delta }}_{\chi }$, as sketched in (b), is the gap between the highest energy of the $\chi$ lowest energy states that would be kept and the lowest-energy multiplet that would be discarded. The $\chi$ lowest energy eigenvectors of the two-block Hamiltonian are used to create an isometric tensor $V$, represented by a three-leg triangle, which satisfies $V^{†}V=\mathbb{1}$ shown in (d). (c) Illustration of the truncation of the two blocks with the largest ${\mathrm{\Delta }}_{\chi }$ into one block in terms of isometric tensors.

Figure 4

(a) The full SDRG algorithm can be seen as a binary tree tensor network with an inhomogeneous structure. The circles represent the ground-state eigenvector of the final block resulting from the SDRG procedure. (b) An example of calculating the ground-state expectation value of some operator which acts only on two sites, such as the two-point correlation function; only those tensors associated with the two sites that the operator acts on need to be considered.

Figure 5

The disorder-averaged Binder cumulant $\overline{U}$ of the random chain in the absence of longitudinal fields. The curves for different sizes are graphed versus the cutoff ${\mathrm{\Gamma }}_0$ of the random transverse fields. The inset is a plot for extracting the location of the disordered quantum critical point by using the crossings of $\overline{U}$ for system sizes $(L,2L)$; from an extrapolation to $1/L\to 0$, we obtain the critical value ${\mathrm{\Gamma }}_0^c=1$.

Figure 6

Distributions of energy gaps (a) and end-to-end correlation functions (c) at the critical point ($h=0,{\mathrm{\Gamma }}_0=1$) for different sizes. (b) and (d) Scaling plots for the data in (a) and (c), respectively; the solid lines in the scaling plots are the analytical results given in Ref. [31]. (e) The mean correlations; the solid red line has a slope of $-1$, in agreement with the theoretical prediction Eq. (18).

Figure 7

Finite-size scaling of the distributions of the first excitation energy ${\varepsilon }_1$ (a) and the second excitation energy ${\varepsilon }_2$ (b) in the Griffiths phase at $h=0$ and ${\mathrm{\Gamma }}_0=3$. The solid lines are plots of the Fréchet distribution in Eq. (39) with the dynamical exponent $z=1.335$, which is the solution of Eq. (37), and a fitting parameter $\alpha =0.07$. The upper figure shows the unscaled distribution of $ln\left({\varepsilon }_1\right)$.

Figure 8

The disorder-averaged Binder cumulant $\overline{U}$ of the random chain in a longitudinal field of strength $h=0.02$, graphed versus the cutoff ${\mathrm{\Gamma }}_0$ of the random transverse fields. (Inset) The disorder average of the squared staggered magnetization as a function of ${\mathrm{\Gamma }}_0$.

Figure 9

(a) The distribution of the log of the first excitation energy ${\varepsilon }_1$ at ${\mathrm{\Gamma }}_0=1h=0.02$, in the vicinity of infinite-randomness quantum critical point. (b) Scaling plot of the data in (a) using the scaling variable $x=ln\left({\varepsilon }_1{L}^z\right)$. The black dashed line indicates the slope of low-energy tail, which gives $z\simeq 0.8$; the black solid line is a plot of the Fréchet distribution given in Eq. (39).

Figure 10

Sketch of the $h-{\mathrm{\Gamma }}_0$ phase diagram of the random-bond Ising antiferromagnetic chain with $J_i\in (0,1]$. The ground state exhibits a quantum phase transition governed by an infinite-randomness fixed point (IRFP) between an antiferromagnetic (AFM) phase at zero-$h$ and a paramagnetic (PM) phase with pronounced quantum Griffiths effects (the dark gray strip along the ${\mathrm{\Gamma }}_0$ axis), resulting from rare ordered regions with anomalously slow fluctuations. In the finite-$h$ regime, the AFM order is destroyed even in the classical limit ${\mathrm{\Gamma }}_0\to 0$, and also the quantum Griffiths effects are weakened.