Fidelity-mediated analysis of the transverse-field $XY$ chain with the long-range interactions: Anisotropy-driven multi-criticality

As a preliminary survey, by the agency of the fidelity susceptibility (6), we analyze the critical point $H_{c}$ with the fixed interaction parameters to $\sigma=1$ and $\eta=0.5$, for which an elaborated series-expansion result Adelhardt20 is available.

In Fig. 3, we present the fidelity susceptibility $\chi_{F}$ for various values of the transverse field $H$ and the system sizes, ($+$) $N=28$, ($\times$) $30$ and ($*$) $32$, with the fixed interaction parameters, $(\sigma,\eta)=(1,0.5)$. The fidelity susceptibility exhibits a pronounced peak around $H\approx 0.55$, indicating that the transverse-field-driven phase transition separating the order ($H<H_{c}$) and disorder ($H>H_{c}$) phases takes place.

Aiming to estimate the critical point precisely, in Fig. 4, we present the approximate critical point $H_{c}^{*}(N)$ for $1/N^{1/\nu}$ Albuquerque10 with the fixed $(\sigma,\eta)=(1,0.5)$. Here, the approximate critical point $H_{c}^{*}(N)$ denotes the location of the fidelity-susceptibility peak

for each $N$, and as mentioned above, the index $\nu$ describes correlation-length’s singularity, i.e., $\xi\sim|H-H_{c}|^{-\nu}$. The abscissa scale $1/N^{1/\nu}$ Albuquerque10 comes from this formula; actually, the relation $H_{c}^{*}(N)-H_{c}\sim N^{1/\nu}$ follows immediately, because the correlation length has the same scaling dimension as that of the system size Albuquerque10 , $\xi\sim N$. The inverse-correlation-length critical exponent is expressed as

from the scaling formula $\nu=2\nu_{SR}(D)/\sigma$ ($\nu_{SR}(D)$: correlation-length exponent for the short-range $D$-dimensional counterpart) Defenu17 , the $\epsilon$-expansion result $1/\nu_{SR}(D)=2-(4-D)/3$ Amit05 , and Eq. (1). This approximate expression $1/\nu$ (9) is not used in the crossover-scaling analysis in Sec. 2.3, which is the main concern of this paper; note that the multi-critical behavior is not identical to that of the aforementioned transverse-field-driven one.

The least-squares fit to the data in Fig. 4 yields an estimate $H_{c}=0.62665(5)$ in the thermodynamic limit $N\to\infty$. Clearly, the estimate may be affected by the extrapolation errors. In order to appreciate a possible systematic error, we carried out the same extrapolation scheme with an alternative $1/\nu[=\sigma/(2-0.037)/0.63]\approx 0.81$ obtained via Eq. (32) of Ref. Defenu17 and the 3D-Ising result Deng03 . Replacing the abscissa scale with this value $1/\nu=0.81$, we arrived at an alternative one $H_{c}=0.62877$. The deviation from the aforementioned estimate $H_{c}=0.62665$ appears to be $\approx 2\cdot 10^{-3}$, which seems to dominate the aforementioned least-squares-fitting error $5\cdot 10^{-5}$; namely, the deviation between these independent extrapolation schemes indicates an appreciable systematic error. Hence, considering the former $2\cdot 10^{-3}$ as the error margin, we estimate the critical point as

This result appears to agree with the series-expansion result Adelhardt20

for $\sigma=1$ and $\eta=0.5$. This value (11) was read off by the present author from Fig. 3 of Ref Adelhardt20 ; here, the Kac factor ${\cal N}=4\zeta(2)$ (4) has to be multiplied so as to remedy the energy-scale difference.

A few remarks are in order. First, the agreement between the sophisticated-series-expansion estimate $H_{c}\approx 0.627$ [Eq. (11)], and ours $H_{c}=0.6267(20)$ [Eq. (10)] confirms the validity of the fidelity-susceptibility-mediated simulation scheme. Second, as seen from Fig. 3, the back-ground contributions (non-singular part) as to the fidelity-susceptibility peak appear to be rather suppressed. Actually, as argued in the next section, the critical exponent of the fidelity susceptibility is substantially larger than that of the specific heat, and the fidelity susceptibility exhibits a pronounced signature for the criticality. Such a character was reported by the exact-diagonalization analysis for the two-dimensional $XXZ$ magnet with the restricted $N\leq 20$ Yu09 . Last, we explain the reason why only the large system sizes $N\approx 30$ were treated in the extrapolation analysis as in Fig. 4. Because we are considering the crossover critical phenomenon, the series of finite-size data exhibit two types of scaling behaviors, such as the Ising- and $XX$-type singularities, as $N$ increases. In our preliminary survey, the large system sizes $N\approx 30$ were found to capture the desired scaling behavior coherently, at least, for the critical domain undertaken in this simulation study. Because of this reason, the extrapolation scheme in Fig. 4 cannot be straightforwardly replaced with the sophisticated extrapolation sequences.

In this section, we investigate the critical behavior of the fidelity susceptibility, following the analyses in the preceding section. For that purpose, we rely on the scaling theory (7) for the fidelity susceptibility developed in Ref. Albuquerque10 ; afterward, this scaling theory is extended in order to investigate the multi-criticality Riedel69 ; Pfeuty74 around $\eta=0$. According to the scaling argument Albuquerque10 the scaling dimension $\alpha_{F}/\nu$ of the fidelity susceptibility satisfies the relation

with the dynamical critical exponent Defenu17

and the specific-heat critical exponent $\alpha$; namely, the specific heat exhibits the singularity as $C\sim|H-H_{c}|^{-\alpha}$. Notably enough, the scaling dimension $\alpha_{F}/\nu$ of the fidelity susceptibility is larger than that of the specific-heat $\alpha/\nu$, indicating that the former should exhibit a pronounced singularity, as compared to the latter. As a consequence, we arrive at the expression

from the hyper-scaling relation $\alpha=2-(1+z)\nu$ Albuquerque10 , and Eq. (9), (12), and (13). The critical indices associated with the above scaling formula (7) are all fixed, and now, we are able to carry out the scaling analysis of the fidelity susceptibility without any adjustable parameters.

In Fig. 5, we present the scaling plot, $(H-H_{c})N^{1/\nu}$-$N^{-\alpha_{F}/\nu}\chi_{F}$, for various system sizes, ($+$) $N=28$, ($\times$) $30$, and ($*$) $32$, with the fixed $\sigma=1$ and $\eta=0.5$. Here, the scaling parameters, $H_{c}$, $1/\nu$ and $\alpha_{F}/\nu$, are given by Eq. (10), (9), and (14), respectively. We observe that the scaled data collapse into a scaling curve satisfactorily, confirming the validity of the analysis in Sec. 2.1 as well as the scaling argument Defenu17 introduced above.

A number of remarks are in order. First, no adjustable parameter was incorporated in the scaling analysis of Fig. 5. Actually, the scaling parameters, $H_{c}$, $\nu$, and $\alpha_{F}/\nu$, were fixed in prior to undertaking the scaling analysis. Last, the scaling data, Fig. 5, appear to be less influenced by the finite-size artifact, indicating that the simulation data already enter into the scaling regime. It is a benefit of the fidelity susceptibility that it is not influenced by corrections to scaling very severely Yu09 . Encouraged by this finding, we explore the multi-critical behavior via $\chi_{F}$ in the next section.

In this section, we investigate the crossover-scaling (multi-critical) behavior around the $XX$-symmetric point $\eta=0$ via the fidelity susceptibility. Here, we set the decay rate to $\sigma=4/3$, which corresponds to $D=2.5$ according to Eq. (1). As mentioned in Introduction, the $D=2$ Mukherjee11 and $3$ Wald15 ; Henkel84 ; Jalal16 ; Nishiyama19 cases have been studied, and the transient behavior in between remains unclear.

For that purpose, we incorporate a yet another parameter $\eta$ accompanied with the crossover exponent $\phi$. Then, the aforementioned expression (7) is extended to the crossover-scaling formula Riedel69 ; Pfeuty74

with a certain scaling function $g$. Here, the symbol $H_{c}(\eta)$ denotes the critical point for each $\eta$, and the indices $\dot{\nu}$ and $\dot{\alpha}_{F}$ denote the correlation-length and fidelity-susceptibility critical exponents, respectively, right at the multi-critical point $\eta=0$; namely, respective singularities are given by $\xi\sim|H-H_{c}(0)|^{-\dot{\nu}}$ and $\chi_{F}\sim|H-H_{c}(0)|^{-\dot{\alpha}_{F}}$. The former relation together with $N\sim\xi$ immediately yields Riedel69 ; Pfeuty74

because the second argument of the crossover-scaling formula (15), $\eta N^{\phi/\dot{\nu}}$, should be dimensionless (scale-invariant). Hence, the crossover exponent $\phi$ governs the power-law singularity of the phase boundary. As in Eq. (12), these critical indices satisfy the scaling relation Albuquerque10

with the specific-heat and dynamical critical exponents, $\dot{\alpha}$ and $\dot{z}$, respectively, at the multi-critical point.

Before commencing the crossover-scaling analyses of $\chi_{F}$, we fix the set of the multi-critical indices appearing in Eq. (15). At the $XX$-symmetric point, the critical indices for the transverse-field-driven phase transition were determined Zapf14 as $\dot{\alpha}=1/2$, $\dot{z}=2z$ and

This index (18) is taken from Eq. (19) of Ref. Defenu17 , as it means the mean-field value Zapf14 . Hence, from Eq. (13) and (17), we arrive at

The above argument completes the prerequisite for the crossover-scaling analysis. The index $\phi$ has to be determined so as to attain a good data collapse of the crossover-scaling plot, based on the formula (15).

In Fig. 6, we present the crossover-scaling plot, $(H-H_{c}(\eta))N^{1/\dot{\nu}}$-$N^{-\dot{\alpha}_{F}/\dot{\nu}}\chi_{F}$, for various system sizes, ($+$) $N=28$, ($\times$) $30$, and and ($*$) $32$, with $\sigma=4/3$, $1/\dot{\nu}=\sigma$ (18), and $\dot{\alpha}_{F}/\dot{\nu}=3\sigma/2$ (19). Here, the second argument of the crossover-scaling formula (15) is fixed to $\eta N^{\phi/\dot{\nu}}=15.2(\approx 0.15\cdot 32^{\phi/\dot{\nu}})$ under the optimal crossover exponent $\phi=1$, and the critical point $H_{c}$ was determined via the same scheme as that of Sec. 2.1. From Fig. 6, we see that the crossover-scaled data fall into a scaling curve satisfactorily, indicating that the choice $\phi=1$ should be a feasible one.

Setting the crossover exponent to a slightly large value $\phi=1.15$, in Fig. 7, we present the crossover-scaling plot, $(H-H_{c}(\eta))N^{1/\dot{\nu}}$-$N^{-\dot{\alpha}_{F}/\dot{\nu}}\chi_{F}$, for various system sizes $N=28,30,32$; the symbols and the critical indices, $1/\dot{\nu}$ and $\dot{\alpha}_{F}/\dot{\nu}$, are the same as those of Fig. 6. Here, the second argument of the crossover-scaling formula (15) is set to $\eta N^{\phi/\dot{\nu}}=30.5$ with $\phi=1.15$. For such a large vale of $\phi=1.15$, the hilltop data get scattered, as compared to those of Fig. 6. Likewise, in Fig. 8, for a small value of $\phi=0.85$, we present the crossover-scaling plot, $(H-H_{c}(\eta))N^{1/\dot{\nu}}$-$N^{-\dot{\alpha}_{F}/\dot{\nu}}\chi_{F}$, for various system sizes $N=28,30,32$; the symbols and the critical indices, $1/\dot{\nu}$ and $\dot{\alpha}_{F}/\dot{\nu}$, are the same as those of Fig. 6. Here, the second argument of the crossover-scaling formula (15) is set to $\eta N^{\phi/\dot{\nu}}=7.62$ with $\phi=0.85$. For such a small value of $\phi=0.85$, the right-side-slope data seem to split off. Considering that the cases, Fig. 7 and 8, set the upper and lower bounds, respectively, for $\phi$, we conclude that the crossover exponent lies within

This result (20) indicates that the phase boundary for $\sigma=4/3$ ($D=2.5$) rises up linearly, $H_{c}(\eta)\sim|\eta|$, around the multi-critical point $\eta=0$; see Fig. 1. Namely, the multi-criticality is identical to that of $D=3$ Wald15 ; Henkel84 ; Jalal16 ; Nishiyama19 . Hence, it is suggested that the linearity is robust against the dimensionality $D$, and merely, the slope grows up monotonically, as the dimensionality $D$ increases from $D=2$. In other words, no exotic character such as the reentrant behavior predicted by the large-$N$ theory occurs, at least, for the $XY$ magnet.

A number of remarks are in order. First, underlying physics behind the crossover-scaling plot, Fig. 6, differs significantly from that of the fixed-$\eta$ scaling plot, Fig. 5. Actually, the former scaling dimension, $\dot{\alpha}_{F}/\dot{\nu}=3\sigma/2$ (19), is larger than that of the latter, $\alpha_{F}/\nu=\sigma-1/3$ (14). Hence, the data collapse in Fig. 6 is by no means accidental, and accordingly, the crossover exponent $\phi$ has to be adjusted rather carefully. Second, we stress that the critical indices other than $\phi$ were fixed in prior to performing the crossover-scaling analyses. Last, in the $\chi_{F}$-mediated analysis, no presumptions as to the order parameters are made. Because in our study, the crossover between the $XX$- and $XY$-symmetric cases is concerned, it is significant that the quantifier is sensitive to both order parameters in a systematic manner.