Powerful method to evaluate the mass gaps of free-particle quantum critical systems

We present a numerical method for the evaluation of the mass gap and the low-lying energy gaps of a large family of free-fermionic and free-parafermionic quantum chains. The method is suitable for some generalizations of the quantum Ising and XY models with multispin interactions. We illustrate the method by considering the Ising quantum chains with uniform and random coupling constants. The mass gaps of these quantum chains are obtained from the largest root of a characteristic polynomial. We also show that the Laguerre bound, for the largest root of a polynomial, used as an initial guess for the largest root in the method, gives us estimates for the mass gaps sharing the same leading finite-size behavior as the exact results. This opens an interesting possibility of obtaining precise critical properties very efficiently, which we explore by studying the critical point and the paramagnetic Griffiths phase of the quantum Ising chain with random couplings. In this last phase, we obtain the effective dynamical critical exponent as a function of the distance to criticality. Finally, we compare the mass gap estimates derived from the Laguerre bound and the strong-disorder renormalization-group method. Both estimates require comparable computational efforts, with the former having the advantage of being more accurate and also being applicable away from infinite-randomness fixed points. We believe this method is a relevant tool for tackling critical quantum chains with and without quenched disorder.

Figure 1

Schematic recursive procedure Eq. (6) for generating the coefficients of the polynomial for the case $p=1$ and initial conditions $C_{m\le 0}\left(\ell \right)={\delta }_{m,0}{\delta }_{\ell ,0}$. The $\ell \mathrm{th}$ slot in the $M\mathrm{th}$ line represents the coefficient $C_M\left(\ell \right)$ in Eq. (24). Dashed slots represent nonexistent coefficients. The arrows link a parent coefficient to their offsprings in accordance to Eq. (6). (For clarity, only a few arrows are shown.) If only the last $C_{\text{last}}$ coefficients are needed in a given generation, say, $\kappa$, then only $\text{min}\{C_{\text{last}},\overline{M}+1\}$ coefficients are needed in the previous ones (shaded slots).

Figure 2

The relative difference between the numerical $\mathrm{\Delta }$ and analytical ${\mathrm{\Delta }}_{\text{analytic}}$ mass gaps of the quantum critical Ising chain Eq. (3) ($h_i=J_i=1$) with the former being computed from the characteristic polynomial truncated with $C_{\text{last}}$ coefficients Eq. (31). In the top panel, three different chain lengths $L$ and two different numerical precisions are considered for the case without longitudinal surface field $h_s=0$. In the bottom panel, the cases with ($h_s=1$) and without ($h_s=0$) longitudinal fields are shown for different numerical precisions and $L=540$. DP and QP stand for double and quadruple precision in FORTRAN codes, respectively. The solid lines are simple fits of the data.

Figure 3

The finite-size mass gap $\mathrm{\Delta }$ of the quantum critical Ising chain Eq. (3) ($h_i=J_i=1$) with ($h_s=1$) and without ($h_s=0$) a longitudinal surface field for lattice sizes $L=100,200,...,1.2\times {10}^6$. The Laguerre's bound estimate $[{\mathrm{\Delta }}_{\text{LB}}=2/\sqrt{z_{+}}$, see Eq. (25)] is also shown. Linear fits to the entire data are provided in the panel.

Figure 4

The mass gap $\mathrm{\Delta }$ and the associate Laguerre's bound estimate ${\mathrm{\Delta }}_{\text{LB}}$ of the quantum critical chain Eq. (1) with $p=2$ at the isotropic point (${\lambda }_i=1$) and lattice sizes $200<M<1.2\times {10}^7$. The results are divided into three sets of curves according to the value of $Mmod3$. Best linear fits to the entire data are provided in the panel.

Figure 5

The relative difference between the exact mass gap ${\mathrm{\Delta }}_{\text{analytic}}$ and the one $\mathrm{\Delta }$ computed with standard FORTRAN quadruple precision (32 digits) using the truncated polynomial Eq. (31) with $C_{\text{last}}$ terms. The model is the transverse-field Ising chain Eq. (3) with $h_i=1$ and $J_i=J$ and longitudinal surface field $h_s$. Different values of $J$ and $h_s$ are indicated in the panels.

Figure 6

The relative difference between the analytic known gap $\mathrm{\Delta }$ and the upper Laguerre's bound estimate ${\mathrm{\Delta }}_{\text{LB}}$ as a function of the coupling constant $J$ for the transverse-field Ising chain (3) ($h_i=1$, $J_i=J$, and $h_s=0$) and for different system sizes $L$.

Figure 7

The relative difference between the numerical $\mathrm{\Delta }$ and (see text) virtually exact ${\mathrm{\Delta }}_{\text{exact}}$ mass gaps of the quantum critical Ising chain Eq. (3) with quenched disorder ($\delta =0$) when the former is computed from the characteristic polynomial truncated with $C_{\text{last}}$ coefficients Eq. (31). We show the results for chains of sizes $L=20$ (circles), 40 (triangles), and 80 (stars) for five distinct disorder realizations (different colors) and for the cases without ($h_s=0$, top) and with ($h_s=1$, bottom) longitudinal surface fields. The solid lines are simple guides to the eyes.

Figure 8

The CPU time required for computing the mass gap $\mathrm{\Delta }$ as a function of the system size $L$. The chain is critical $\delta =0$ and for $h_s=0$. The data is averaged over ${10}^3$ different disorder realizations. The program was coded in FORTRAN with standard quadruple precision and run on a conventional portable computer. Best power-law fits for the data (solid lines) are provided in the panel.

Figure 9

The top (middle) panel shows the probability distribution $W$ of the relative difference ${\alpha }_{\text{SDRG}}$ (${\alpha }_{\text{LB}}$) between the finite-size gap $\mathrm{\Delta }$ and the corresponding SDRG (Laguerre's bound) estimate ${\mathrm{\Delta }}_{\text{SDRG}}$ (${\mathrm{\Delta }}_{\text{LB}}$) for different system sizes $L$. The chains are critical $(\delta =0)$ and the surface longitudinal field is $h_s=0$ (main panel) or $h_s=1$ (inset). Each curve was constructed using ${10}^6$ distinct disorder realizations. Bottom: The mean values of ${\alpha }_{\text{SDRG}}$ and ${\alpha }_{\text{LB}}$ as a function of $L$. Power-law best fits are provided for ${\overline{\alpha }}_{\text{LB}}$ (solid lines). Solid lines for ${\overline{\alpha }}_{\text{SDRG}}$ are simple guides to the eyes.

Figure 10

The distribution of the rescaled mass gap $\eta$ Eq. (43) of the random transverse-field Ising chain at criticality $\delta =0$ without surface field $h_s=0$ for lattice sizes $L=400$ and 800. Both the exact value of the mass gap $\mathrm{\Delta }$ and the Laguerre bound estimate ${\mathrm{\Delta }}_{\text{LB}}$ are considered. Both distributions are statistically identical within our accuracy. Each curve was obtained from $1.1\times {10}^8$ distinct samples. The dotted line is the analytic strong-disorder renormalization-group estimate Eq. (44) for $h_s=0$ while for $h_s=1$ it is the numerical implementation of the SDRG method for ${10}^6$ chains of size $L=1600$.

Figure 11

Top: The typical value of the mass gap $\overline{\mathrm{\Gamma }}$ Eq. (46) (open circles) and the corresponding Laguerre's bound estimate ${\overline{\mathrm{\Gamma }}}_{\text{LB}}$ ($\times$ symbols) as functions of lattice size $L$ (up to $L=2500$) for the quantum Ising chain Eq. (3) with $h_s=0$ and for various distances from criticality $\delta =1-ln{J}_{\text{max}}$ (different colors), $J_{\text{max}}=2.2,\cdots ,2.7$ from bottom to top. Standard linear fits are restricted to within the dashed box region (solid lines), and the slopes are provided in the figure. Middle: The corresponding finite-size estimator $z(L,L^{\prime })$ Eq. (48) as a function of $L$ for fixed $\mathrm{\Delta }L=25$. As in the top panel, open circles and $\times$ symbols refer to $z$ and $z_{\text{LB}}$, respectively. In both panels, the error bars are smaller than the symbol sizes. Bottom: The estimator $z(L,L^{\prime })$ for system sizes up to $L=6\times {10}^5$, $J_{\text{max}}=2.7$ (corresponding to $\delta =6.7\times {10}^{-3}$), and $h_s=0$ (black circles) and $h_s=1$ (red squares). Solid lines are the fittings to Eqs. (49) and (50) (see text). The inset shows $\overline{\mathrm{\Gamma }}$ (for $L<6\times {10}^5$, circles and squares) from which the main panel was derived, see Eq. (48). In addition, $z_{\text{LB}}(L,L^{\prime })$ (green triangles) is plotted for sizes $6\times {10}^5<L<6\times {10}^6$. Linear fits to the data are provided in the figure. Each data point is obtained by averaging over ${10}^6$ (top and middle) ${10}^4$ (bottom) different samples.

Figure 12

The dynamical exponent $z$ as a function of the distance from criticality $\delta$ in the Griffiths paramagnetic phase $0<\delta <1$ of the random transverse-field Ising chain Eq. (3) for the cases with ($h_s=1$) and without ($h_s=0$) the longitudinal surface field. The dotted line is the analytic prediction Eq. (45).

Figure 13

The mass gap $\mathrm{\Delta }$ and the corresponding Laguerre's bound and SDRG estimates of two randomly chosen configurations of couplings $\left\{J_i\right\}$ of the system Hamiltonian Eq. (3) with $h_s=0$ and $\delta =0$, see Eq. (42). These quantities are computed considering the sequence $\left\{J_i\right\}$ from 1 to $L$; we vary the lattice sizes from $L=50000$ to $300000$.