A paradigm of spontaneous marginal phase transition at
finite temperature in one-dimensional ladder Ising models

The mathematical signature of phase transitions is the non-analyticity of the system’s thermodynamic free energy $f(T)$ where $T$ denotes temperature. A $k$th-order phase transition means that the $k$th derivative of $f(T)$ starts to be discontinuous at the transition. The exactly solved 1D Ising models on both the single chain Mattis_book_08_SMMS ; Mattis_book_1985 ; Baxter_book_Ising ; 003_Huang_08_book and ordinary $n$-leg ladders with $n=2,3,4$ Mejdani_Ising_ladder offer examples of analytic free energies.

The simplest description of the phase transition phenomenon is the level crossing in the first-order phase transition, such as melting of ice or the boiling of water. As illustrated in Fig. 1a, suppose the free-energy functions of two phases $-g[a(T)]$ and $-g[b(T)]$ cross at the critical temperature $T_{0}$. Then, the free energy of the system taking the lower value of $-g[a(T)]$ and $-g[b(T)]$ is given by

which is nonanalytic at $T_{0}$ with its first derivative with respect to $T$ being discontinuous.

We hypothesize that SMPT can be created by using an analytic function to mimic Eq. (2) and consider here 009_review_Souza_SSC_18_Ising-XYZ-diamond-chain_double-tetrahedral-chain-spin-electron

which satisfies the following two conditions: (i) $a(T)$ and $b(T)$ cross at $T_{0}$, and (ii) $c(T)\ll|a-b|$ except for an ultra-narrow temperature region around $T_{0}$. The width of the crossover $2\delta T$ can be estimated by solving $\frac{|a-b|}{2}=c$ at $T=T_{0}\pm\delta T$. This definition of $\delta T$ is consistent with the crossover width later defined by measuring the order parameter [see Eq. (37)]. Eq. (3) looks the same as Eq. (2) except when zoomed into the ultra-narrow region around $T_{0}$ (see Fig. 1b).

The form of Eq. (3) can result from the 1D Ising model on a single chain. In the thermodynamic limit, the free energy per unit cell $f(T)=-\lim_{N\to\infty}\frac{1}{N}k_{\mathrm{B}}T\ln Z$, where $N$ is the number of unit cells, $Z=\mathrm{Tr}\left(e^{-\beta H}\right)$ is the partition function, and $\beta=1/(k_{\mathrm{B}}T)$ with $k_{\mathrm{B}}$ being the Boltzmann constant. $Z$ of a 1D Ising model can be obtained exactly by using the transfer matrix method Mattis_book_08_SMMS ; Mattis_book_1985 ; Baxter_book_Ising ; 003_Huang_08_book ; Mejdani_Ising_ladder ; 005_Galisova_PRE_15_double-tetrahedral-chain ; 007_Torrico_PRA_16_Ising-XYZ-diamond-chain ; 009_review_Souza_SSC_18_Ising-XYZ-diamond-chain_double-tetrahedral-chain-spin-electron ; 010_Carvalho_JMMM_18_Ising-XYZ-diamond-chain_quantum-entanglement ; 011_Rojas_BJP_20_Ising-Heisenberg-tetrahedral_diamond ; 013_Rojas_PRE_19_previous_4_models ; 014_Rojas_JPC_20_Ising–Heisenberg_spin-1-double-tetrahedral-chain ; 015_Strecka_APPA_20_Ising-diamond-chain ; 015-7_Canova_CzechoslovakJP_04_Ising–Heisenberg_diamond_chain ; 015-8_Canova_JPC_06_Ising–Heisenberg-spin-S-diamond-chain ; 017_Krokhmalskii_PA_21_3-previous-chains_effective_model ; 016_Strecka_book_chapter ; 008_zero-field_Rojas_SSC_16_Ising-Heisenberg_ladder ; 006_zero-field_Strecka_JMMM_16_Ising-Heisenberg-3-leg-tube ; Yin_MPT ; Yin_icecreamcone and is given by

where $\Lambda$ is the transfer matrix, $\lambda_{k}$ the $k$th eigenvalue of $\Lambda$, and $\lambda$ the largest eigenvalue. Thus, $f(T)=-k_{\mathrm{B}}T\ln\lambda$. For the single-chain Ising model, the transfer matrix is of the following form:

with $\lambda=\frac{1}{2}\big{[}a+b+\sqrt{(a-b)^{2}+4c^{2}}\big{]}$. Compared with Eq. (3), $f(T)$ is of the same form when the function $g(x)$ is defined as $g(x)=k_{\mathrm{B}}T\ln(x)$.

However, in the absence of the bias ($h=0$), the Ising model is invariant with respect to flipping all $\sigma_{i}$. By this spin up-down symmetry, $a=b$ and $\lambda=a+c$ in Eq. (7) for single-chain Ising models, including decorated Ising chains. This means the absence of SMPT, which requires the crossing of $a$ and $b$ as $T$ changes.

The nonzero bias $h$ breaks the above invariance and lifts the degeneracy of $a$ and $b$. For the simplest single-chain model, $a=e^{\beta(J+h)}$, $b=e^{\beta(J-h)}$, $c=e^{-\beta J}$. Thus, $a$ and $b$ do not cross at all. How to make $a$ and $b$ cross in decorated single-chain Ising models in the presence of the bias was the subject of the so-called “pseudo transition” 005_Galisova_PRE_15_double-tetrahedral-chain ; 007_Torrico_PRA_16_Ising-XYZ-diamond-chain ; 009_review_Souza_SSC_18_Ising-XYZ-diamond-chain_double-tetrahedral-chain-spin-electron ; 010_Carvalho_JMMM_18_Ising-XYZ-diamond-chain_quantum-entanglement ; 011_Rojas_BJP_20_Ising-Heisenberg-tetrahedral_diamond ; 013_Rojas_PRE_19_previous_4_models ; 014_Rojas_JPC_20_Ising–Heisenberg_spin-1-double-tetrahedral-chain ; 015_Strecka_APPA_20_Ising-diamond-chain ; 015-7_Canova_CzechoslovakJP_04_Ising–Heisenberg_diamond_chain ; 015-8_Canova_JPC_06_Ising–Heisenberg-spin-S-diamond-chain ; 017_Krokhmalskii_PA_21_3-previous-chains_effective_model ; 016_Strecka_book_chapter . It was understood that certain decorations could renormalize the bias to be effectively $T$-dependent and the resulting $h_{\mathrm{eff}}(T)$ changes its sign at the pseudo-critical temperature $T_{0}$ 017_Krokhmalskii_PA_21_3-previous-chains_effective_model . Nevertheless, $h_{\mathrm{eff}}(T)$ vanishes for $h=0$, in agreement with the aforementioned spin up-down symmetry. Therefore, no spontaneous pseudo-transition would occur in those systems.

The extensively studied 1D Ising chain with the first- and second-nearest-neighbor interactions, the $J_{1}$-$J_{2}$ model Fleszar_Baskaran_JPC_85_J1-J2 ; Stephenson_CanJP_70_J1-J2-Ising-chain ; Dobson_JMathP_69_Many‐Neighbored-Ising-Chain ; Dhar_PRE_00_J1-J2-Ising-chain , showcases the emerging of an effective magnetic field $h_{\mathrm{eff}}$ for $h=0$ by the transformation $\sigma_{i}\sigma_{i+1}=\tilde{\sigma}_{i}$. This was inspiring. However, $h_{\mathrm{eff}}=J_{1}$, the nearest-neighbor interaction, is not $T$-dependent, and the transformed model is an ordinary single-chain Ising model in this $T$-independent effective field Fleszar_Baskaran_JPC_85_J1-J2 ; Stephenson_CanJP_70_J1-J2-Ising-chain ; Dobson_JMathP_69_Many‐Neighbored-Ising-Chain . Therefore, the resultant effective $a$ and $b$ do not cross at all.

To find SMPT, we resort to the strategy of dimensionality increase and reduction: First, we increase the transfer matrix from $2\times 2$ to $4\times 4$. Then, we identify the $4\times 4$ transfer matrix that has such high symmetry that it can be block diagonalized and reduced back to a $2\times 2$ matrix Mattis_book_08_SMMS now with the new effective $a$ and $b$ crossing.

We consider the following $4\times 4$ transfer matrix

It can be block diagonalized by the parity-symmetry operations $U$:

and the result is

Thus, the eigenvalues of the $4\times 4$ transfer matrix are $a-u$, $b-v$, and

Then, the task is transformed to how to realize the crossing of $a+u$ and $b+v$.

The $4\times 4$ transfer matrix can be obtained for the 1D Ising model on a 2-leg ladder. The ordinary 2-leg Ising ladder Mejdani_Ising_ladder , the spin-1/2 Ising tetrahedral chain 016_Strecka_book_chapter , and the spin-1/2 Ising-Heisenberg ladder with alternating Ising and Heisenberg inter–leg couplings 008_zero-field_Rojas_SSC_16_Ising-Heisenberg_ladder are known special cases satisfying the symmetry of Eq. (12). Among them, the ordinary 2-leg ladder does not host the SMPT, and the latter two cases with zero-field pseudo-transition have not been analyzed with the above symmetry-based block diagonalization technique, which is vital to systematically looking for more cases of spontaneous ultra-narrow phases crossover. Here, we take the ordinary 2-leg ladder as the parent system and investigate how to properly decorate it with child spins.

Since the model has three spins and $2^{3}=8$ possible states per unit cell, its transfer matrix is $8\times 8$ at a glance. However, the $m=3$ child spins can be exactly summed out as they are coupled only to the $m=1,2$ parent spins on the same rung, yielding the children’s contribution functions

They are translationally invariant, i.e., $\boxed{{}_{\pm}^{\pm}}_{i}=\boxed{{}_{\pm}^{\pm}}$. Then, using the spin up-down symmetry, i.e., ${\boxed{{}_{-}^{-}}}={\boxed{{}_{+}^{+}}}=[2\cosh(2\beta J^{\prime})]^{1/2}$ and ${\boxed{{}_{+}^{-}}}={\boxed{{}_{-}^{+}}}=\sqrt{2}$, we found the $4\times 4$ transfer matrix in the order of $\left({}_{\sigma_{1}}^{\sigma_{2}}\right)=\left({}_{+}^{+}\right),\left({}_{+}^{-}\right),\left({}_{-}^{+}\right),\left({}_{-}^{-}\right)$ to be of the same form as Eq. (12), with $a=e^{\beta(2J+J^{\prime\prime})}\boxed{{}_{+}^{+}}^{2}$, $b=e^{\beta(2J-J^{\prime\prime})}\boxed{{}_{+}^{-}}^{2}$, $u=e^{\beta(-2J+J^{\prime\prime})}\boxed{{}_{+}^{+}}^{2}$, $v=e^{\beta(-2J-J^{\prime\prime})}\boxed{{}_{+}^{-}}^{2}$, and $z=\boxed{{}_{+}^{+}}\boxed{{}_{+}^{-}}$.

The largest eigenvalue of the transfer matrix is

with the household’s frustration functions

which are controlled by the intra-household interactions $J^{\prime}$ and $J^{\prime\prime}$, but independent of the inter-houseld interaction $J$. Here we have used the relationship $\Upsilon_{+}^{2}=\Upsilon_{-}^{2}+4z^{2}$ and $\cosh^{2}(x)=\sinh^{2}(x)+1$. $\lambda$ depends on $J^{\prime}$ and $J^{\prime\prime}$ solely via $\Upsilon_{\pm}$. Note that the equations of (27) and (28) are invariant upon the transformation of $J\to-J$ or $J^{\prime}\to-J^{\prime}$ (i.e., ferromagnetic and antiferromagnetic interactions are interchangeable), but they are not for $J^{\prime\prime}\to-J^{\prime\prime}$. That is, only $J^{\prime\prime}<0$ (antiferromagnetic interaction) introduces frustration.

$\Upsilon_{-}\propto a+u-b-v$ measures the crossing of $a+u$ and $b+v$. It changes sign at $T_{0}$ where $\Upsilon_{-}=0$, i.e.,

This means that $T_{0}$ is determined only by the on-rung interactions $J^{\prime}$, $J^{\prime\prime}$ and independent of the on-leg interaction $J$. For sufficiently large $2|J^{\prime}|/k_{\mathrm{B}}T_{0}$, $2\cosh(2J^{\prime}/k_{\mathrm{B}}T_{0})\simeq e^{2J^{\prime}/k_{\mathrm{B}}T_{0}}\gg 1$,

For example, $k_{\mathrm{B}}T_{0}\approx 0.144$ for $J^{\prime}=3.05$ and $J^{\prime\prime}=-3$ yielding $2\cosh(2J^{\prime}/k_{\mathrm{B}}T_{0})\approx e^{42}\gg 1$. As shown in Fig. 3(a), for $J^{\prime\prime}=0.05-|J^{\prime}|<0$ to keep $k_{\mathrm{B}}T_{0}\simeq 0.144$, $T_{0}$ as a function of $J^{\prime}$ estimated by Eq. (30) (dashed lines) accurately reproduces the exact results of Eq. (29) (solid lines) except for $J^{\prime}<0.15$ where $T_{0}$ increases exponentially; indeed, in the unfrustrated limit of $J^{\prime\prime}=0$, $T_{0}\to\infty$ and no crossover would happen at finite temperature.

It is now clear that SMPT was not found in the ordinary 2-leg ladder without the children. In that case, $J^{\prime}=0$ leads to $\Upsilon_{-}=4\sinh(\beta J^{\prime\prime})$, which does not change sign for given $J^{\prime\prime}$. For $J^{\prime\prime}=0$ as the other limit of no frustration, $\Upsilon_{-}=4\sinh^{2}(\beta J^{\prime})$ does not change sign, either.

Secondly, $(\Upsilon_{-}/\Upsilon_{+})^{2}$ in Eq. (27) has a prefactor of $\sinh^{2}(2\beta J)$, which scales as $2^{1/\alpha}$ in the vicinity of $T_{0}$. So, if Eq. (27) is approximated by neglecting 1 inside $\sqrt{\cdots}$ for the strong frustration of $\alpha\ll 1$,

which becomes non-analytic. The difference between Eq. (27) and Eq. (31) takes place in a region of $(T_{0}-\delta T,T_{0}+\delta T)$, where the crossover width $2\delta T$ can be estimated by $|\Upsilon_{-}/\Upsilon_{+}\sinh(2\beta J)|=1$ at $T_{0}\pm\delta T$, yielding

where Eq. (32) is exact and Eq. (33) is based on Eq. (30). $2\delta T$ has two paths to approach zero: (i) $T_{0}\to 0$, which is not unexpected as zero-temperature phase transition is allowed. (ii) For fixed finite $T_{0}$, which is determined by $J^{\prime}$ and $J^{\prime\prime}$, the width $2\delta T$ approaches zero exponentially as $J$ increases. Using the above example of $k_{\mathrm{B}}T_{0}=0.144$, $2k_{\mathrm{B}}\delta T\simeq 1.6\times 10^{-6}$ and $1.5\times 10^{-12}$ for $J=1$ and $2$, respectively. Note that $\alpha\to 0$ for both cases by Eqs. (25) and (30). Although the proof of nonexistence of any phase transition in 1D Ising models with short-range interactions based on the Perron–Frobenius theorem does not work for infinite-strength interactions Cuesta_1D_PT , similar to zero temperature, it is amazing to realize such an ideal paradigm of SMPT in which $T_{0}$ and $\delta T$ are controlled by different model parameters and for fixed $T_{0}$, $\delta T$ decays exponentially with the single parameter $J$. This issue will be further addressed in Section IV.

The thermodynamical properties can be retrieved from the free energy per trimer $f(T)=-k_{\mathrm{B}}T\ln\lambda$, the entropy $S=-\partial f/\partial T$, and the specific heat $C_{v}=T\partial S/\partial T$.

We compare the free energies per trimer $f(T)$ obtained from using the exact Eq. (27) (Fig. 1b) and the mimicked Eq. (31) (Fig. 1a) for $\alpha=0.05$; they differ within sub-millikelvins for $|J|=300$ K when the slope changes from near 0 to near $-k_{\mathrm{B}}\ln 2$. The SMPT resembles a first-order phase transition with the large latent heat of $k_{\mathrm{B}}T_{0}\ln 2$ per trimer.

Fig. 3b shows a sharp peak in the exact specific heat for $\alpha=0.05$ (blue line), resembling a second-order phase transition. In comparison, the three typical unfrustrated cases, namely (i) $\alpha=3$ or governed by $J^{\prime}$ (orange line), (ii) $J^{\prime}=0$ the ordinary 2-leg ladder governed by $J^{\prime\prime}$ (green line), and (iii) $J^{\prime}=J^{\prime\prime}=0$ the decoupled double chains (red line), do not show such a sharp peak.

Fig. 3c show the temperature dependence of the exact entropy $S=-\partial f/\partial T$ for the same four cases as Fig. 3b. The strongly frustrated case for $\alpha=0.05$ (blue line) shows a waterfall behavior at $T_{0}$, where the entropy falls vertically (within $2\delta T$) from a plateau at $\ln 2$ down to zero. Its low-$T$ zero-entropy behavior is in line with the unfrustrated case for $\alpha=3$ (orange line), while its intermediate-$T$ $\ln 2$-plateau behavior is in line with the ordinary 2-leg ladder case for $J^{\prime}=0$ (green line) where the child spins are completely decoupled from the system and thus yield the specific entropy of $\ln 2$. This offers a hint to the nature of the SMPT: it is an entropy-driven transition, in which the child spins are tightly coupled to the two parent spins on the legs in the low-$T$ region because $|J^{\prime}|>|J^{\prime\prime}|$, but they are decoupled from the outer ones in the intermediate-$T$ region just above $T_{0}$ thanks to the jumping contribution of entropy in terms of $-TS$ to the free energy. Mathematically, in the temperature region near $T_{0}$, one can estimate from Eq. (31) that $\lambda\approx e^{\beta(2J+2J^{\prime}+J^{\prime\prime})}$ below $T_{0}-\delta T$ and $\lambda\approx 2e^{\beta(2J-J^{\prime\prime})}$ above $T_{0}+\delta T$, i.e., the low-$T$ region is controlled by $J^{\prime}$, while the intermediate-$T$ region is controlled by $J^{\prime\prime}<0$ with the decoupled child spin contributing the factor of 2 to $\lambda$ and $\ln 2$ to entropy. Moreover, in the high-$T$ region beyond the $\ln 2$ plateau, the entropy does not continue following the ordinary 2-leg ladder case (green line) but behaves closer to the decoupled double chains for $J^{\prime}=J^{\prime\prime}=0$ (red line). A similar behavior also shows up in the high-$T$ region of the specific heat (Fig. 3a), indicating that frustration effectively decouples the chains. We shall come back to understand this strange behavior soon.

For the revealed three $T$-region behavior of the specific entropy, Eqs. (30) and (32) determine that the SMPT itself is a sole function of $\alpha$. This is demonstrated in Fig. 3d for fixed $\alpha=0.05$ but with different $J^{\prime\prime}$, which directly couples the two legs: The lines overlap in the first two $T$-regions surrounding the waterfall at $T_{0}$. In contrast, in the high-$T$ region, the detail behavior of spin frustration is sensitive to $J^{\prime\prime}$ (and $J^{\prime}$ shown below). It is understandable that as $|J|$ and $|J^{\prime\prime}|$ decreases, the entropy moves up to be closer to the case of the decoupled double chains. What is strange is why the chains with strong inter-chain interactions also appear to be decoupled.

To find an appropriate order parameter that describes the three $T$ regions and the associated transitions, we proceed to calculate the spin-spin correlation functions, which allow us to have detailed site-by-site information:

where $\langle\cdots\rangle_{T}$ denotes the thermodynamical average, $|\mathrm{max}\rangle$ is the eigenvector of the transfer matrix corresponding to the largest eigenvalue $\lambda$ and $|\nu\rangle$ is the eigenvector corresponding to the $\nu$th eigenvalue $\lambda_{\nu}$. So, $C_{mm^{\prime}}(\infty)=\langle\mathrm{max}|\sigma_{0,m}|\mathrm{max}\rangle\langle\mathrm{max}|\sigma_{\infty,m^{\prime}}|\mathrm{max}\rangle$.

As shown in Figs. 4c and 4d, the behavior of intra-chain correlation $C_{11}(L)$ as a function of both $T$ and $L$ (blue lines) is closer to the decoupled double chains of $J^{\prime}=J^{\prime\prime}=0$ (red lines), consistent with the specific heat (Fig. 3b) and the entropy data (Fig. 3c). The data of the unfrustrated yet strongly coupled chains for both $\alpha=3$ (orange lines) and $J^{\prime}=0$ (green lines) cases move to considerably higher temperatures, suggesting that the chain decoupling is related to the energy cost for attempts to break the order.

The conventional choice of the order parameter is the magnetization $\langle\sigma_{i,m}\rangle_{T}=\sqrt{C_{mm}(\infty)}=\langle\mathrm{max}|\sigma_{0,m}|\mathrm{max}\rangle=0$ at finite $T$. This agrees with the theorems about the nonexistence of finite-$T$ phase transition in 1D Ising models with short-range interactions Cuesta_1D_PT . Hence, the magnetization cannot be used as the order parameter for the SMPT. Instead, let us look at the on-rung correlation function $C_{mm^{\prime}}(0)=\langle\mathrm{max}|\sigma_{i,m}\sigma_{i,m^{\prime}}|\mathrm{max}\rangle$:

$C_{12}(0)$, the correlation between two parent spins on the legs, is proportional to the frustration function $\Upsilon_{-}$ and changes sign at $T_{0}$ (exactly zero at $T_{0}$). Because of the exponentially large $\cosh(2\beta J)$ and $\sinh(2\beta|J|)$ near $T_{0}$, $C_{12}(0)\approx\text{sgn}(\Upsilon_{-})\coth(2\beta|J|)\approx+1$ below $T_{0}$ and $-1$ above $T_{0}$, as shown in Fig. 4a (blue line for $\alpha=0.05$). Below $T_{0}$, the blue line overlaps with the unfrustrated $J^{\prime}$-dominated case of $\alpha=3$ (orange line). In the intermediate-$T$ region, it shows a plateau and overlaps with the unfrustrated $J^{\prime\prime}$-dominated ordinary 2-leg ladder case (green line). That is, the two parent spins changes from having like values in the low-$T$ region to having unlike values in the intermediate-$T$ region where the child spins appear to be decoupled from the ladder. So, one rung’s contribution to the free energy for $T<T_{0}$ and $T>T_{0}$ is approximately $-J^{\prime\prime}-2|J^{\prime}|$ and $J^{\prime\prime}-k_{\mathrm{B}}T\ln 2$, respectively. They cross at $k_{\mathrm{B}}T_{0}=2(|J^{\prime}|+J^{\prime\prime})/\ln 2=2\alpha|J|/\ln 2$, in agreement with Eq. (30). What is remarkably new to learn is that for the high-$T$ region, while remaining negative, the blue line of $C_{12}(0)$ does not continue following the green line, which gradually decays to zero as temperature increases. Instead, it decays significantly faster with an inflection point at $T^{*}\approx 0.88$ before gradually decaying to zero. Yet, with $C_{12}(0)\approx-0.82$ at $T^{*}$, it is still far away from the zero red line for the case of the completely decoupled double chains. The two chains seem to be strongly coupled. How can we reconcile this with the effective decoupling seen in the above thermodynamic properties and the intra-chain correlation functions?

The answer is Eq. (36) for $C_{13}(0)$ and $C_{23}(0)$, the correlations between the child spin and the two parent spins on the same rung. They are equal by symmetry. Since $C_{12}(0)$ just told us that spin 1 and spin 2 on the same rung strongly want to have unlike values, it would have been expected in static mean-field theory that $C_{13}(0)$ and $C_{23}(0)$ should have opposite signs; considering that they are required to be equal by symmetry, they should be zero. Indeed, in the intermediate-$T$ region, $C_{13}(0)$ is nearly zero (blue line in Fig. 4b). However, as soon as the temperature increases beyond the intermediate-$T$ region, $C_{13}(0)$ jumps up, meanwhile the magnitude of $C_{12}(0)$ decreases (blue line in Fig. 4a). This means that the child spin starts to re-couple to the parent spins to gain energy at the expense of $C_{12}(0)$, following the astonishing near-linear relationship of Eq. (36). The total energy within one trimer $-J^{\prime\prime}C_{12}(0)-J^{\prime}C_{13}(0)-J^{\prime}C_{23}(0)$ remains nearly unchanged in a wide range of temperatures (dashed line in Fig. 4b), which indicates that the trimers can self-organize to relieve energy cost in their value-changing dynamics. Sizable $C_{13}(0)=C_{23}(0)$ encourages the two parent spins to have like values, which counters negative $C_{12}(0)$. The net effect is decoupling the two legs.

The correlation length of the two-spin quantities $\sigma_{i,m}\sigma_{i,m^{\prime}}$ is infinite at $T_{0}$, since the four-spin correlation function $\lim_{L\to\infty}\langle\sigma_{0,m}\sigma_{0,m^{\prime}}\sigma_{L,m^{\prime\prime}}\sigma_{L,m^{\prime\prime\prime}}\rangle_{T}=C_{mm^{\prime}}(0)C_{m^{\prime}m^{\prime\prime}}(0)$ is not vanishing. Therefore, $C_{mm^{\prime}}(0)$ are bona fide order parameters. To check with the ordinary ladders, Figs. 4a and 4b show that $C_{12}(0)$ and $C_{13}(0)$ can be finite for the other three unfrustrated cases; however, they do not change sign or they are vanishing only at $T=\infty$. This means that using $C_{mm^{\prime}}(0)$ as the order parameters for the ordinary unfrustrated cases still satisfies the theorems that no phase transitions take place at finite temperature in those systems. Therefore, the unconventional order parameters $C_{mm^{\prime}}(0)$ can unify the description of both the ordinary 2-leg ladder system and the unconventional phase transitions in the present frustrated ladder Ising model.

To further verify that $C_{12}(0)$ is a bona fide order parameter, we redefine the crossover width $\delta T$ by measuring the slope of $C_{12}(0)$ at $T_{0}$ as shown in Fig. 6(a):

which differs from the previous definition of $2\delta T$ in Eq. (32) by replacing $\sinh(\frac{2J}{k_{\mathrm{B}}T_{0}})$ with $\cosh(\frac{2J}{k_{\mathrm{B}}T_{0}})$. They are consistent as $\cosh^{2}(\frac{2J}{k_{\mathrm{B}}T_{0}})=\sinh^{2}(\frac{2J}{k_{\mathrm{B}}T_{0}})+1\simeq\sinh^{2}(\frac{2J}{k_{\mathrm{B}}T_{0}})$ for an ultra-narrow phase crossover. Therefore, this nonclassical order parameter, which has a well-defined value space of $[-1,1]$ with the value $0$ meaning $T_{0}$ and its inverse slope at $T_{0}$ meaning $\delta T$, provides an accurate, convenient, and microscopic description of SMPT.

The phase diagram in terms of the unconventional order parameter $C_{12}(0)$ is shown in Fig. 2c. The frustrated 2-leg ladder Ising model can have three phases for $0<\alpha<0.15$: (i) the low-$T$ phase (red zone) is governed by $J^{\prime}$ forcing the two parent spins on the same rung to have like values. (ii) The intermediate-$T$ phase (purple zone) is governed by $J^{\prime\prime}$ forcing the two parent spins on the same rung to have unlike values and the child spin to be decoupled. The abrupt switching between the two phases is an entropy-driven SMPT with large latent heat, resembling the first-order phase transition. (iii) The exotic high-$T$ phase (blue-green zone) is governed by the dynamics of the frustration. The much broader crossover between the intermediate-$T$ phase and the exotic high-$T$ phase reveals itself via the phenomenon of frustration-driven decoupling of the strongly interacted chains. The microscopic mechanism for the decoupling is illustrated in Fig. 2c, where the high energy cost associated with the thermal activated flipping of a parent spin in the strongly interacted double chains can be relieved in the trimer dynamics by flipping the child spin cooperatively. The exact result of sizeable nonzero $C_{13}(0)=C_{23}(0)$ demonstrates that the child spin is not a slave to the mean field generated by the two parent spins, but they have equal rights. The dynamics of the trimers exhibits the art of compromise and finds the way to have created a triple-win workplace in which every neighboring bond gains an optimized share of rewards.