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Interplay of order and disorder in two-dimensional critical systems with mixed boundary conditions

E. Eisenriegler

TL;DR

The paper investigates how local ordering at one point affects local disorder at another in the two-dimensional critical Ising model with mixed boundary conditions, formalized via the universal cumulant $\langle \sigma({\bf r}_1) \varepsilon({\bf r}_2)\rangle^{(\rm cum)}$. By leveraging the operator-product expansion, boundary-operator expansion, and explicit exact results across geometries (upper half-planes, equilateral triangle, and square), it reveals universal short-distance behavior and how zero lines (where $\langle \sigma\rangle=0$) induce discontinuities or cusps in the cumulant. The study demonstrates the impact of boundaries and corners on the interdependence of order and disorder, showing sign changes and magnitude scaling that depend only on local order parameters and derivatives, and it provides exact mappings between geometries to extend results. The findings highlight the utility of OPE/BOE/COE techniques in predicting boundary-critical phenomena and offer precise benchmarks for simulations in constrained two-dimensional critical systems.

Abstract

In spin systems such as the Ising model, the local order and disorder can be characterized by the order-parameter and energy density profiles $\langle σ({\bf r}_1) \rangle$ and $\langle ε({\bf r}_2) \rangle$, respectively. Does increasing the order at ${\bf r}_1$ always decrease the disorder at ${\bf r}_2$? Does increasing the disorder at ${\bf r}_2$ always decrease the order at ${\bf r}_1$? The answer to these questions is contained in the cumulant response function $\langleσ({\bf r}_1) \, ε({\bf r}_2) \rangle^{(\rm cum)}$. This correlation function vanishes in the unbounded bulk but not in systems with fixed-spin boundary conditions. Using the universal operator-product expansion of $σ({\bf r}_1) \, ε({\bf r}_2)$ and exact results for the Ising model, we analyze $\langleσ({\bf r}_1) \, ε({\bf r}_2) \rangle^{(\rm cum)}$ in two-dimensional critical systems defined on the $x-y$ plane with mixed $+$ and $-$ boundary conditions. Particularly interesting behavior is found when either of the operators $σ$ or $ε$ is located on a ``zero line" in the $x-y$ plane, along which $\langleσ({\bf r})\rangle$ vanishes. Results for half-plane, triangular, and rectangular geometries are presented.

Interplay of order and disorder in two-dimensional critical systems with mixed boundary conditions

TL;DR

The paper investigates how local ordering at one point affects local disorder at another in the two-dimensional critical Ising model with mixed boundary conditions, formalized via the universal cumulant . By leveraging the operator-product expansion, boundary-operator expansion, and explicit exact results across geometries (upper half-planes, equilateral triangle, and square), it reveals universal short-distance behavior and how zero lines (where ) induce discontinuities or cusps in the cumulant. The study demonstrates the impact of boundaries and corners on the interdependence of order and disorder, showing sign changes and magnitude scaling that depend only on local order parameters and derivatives, and it provides exact mappings between geometries to extend results. The findings highlight the utility of OPE/BOE/COE techniques in predicting boundary-critical phenomena and offer precise benchmarks for simulations in constrained two-dimensional critical systems.

Abstract

In spin systems such as the Ising model, the local order and disorder can be characterized by the order-parameter and energy density profiles and , respectively. Does increasing the order at always decrease the disorder at ? Does increasing the disorder at always decrease the order at ? The answer to these questions is contained in the cumulant response function . This correlation function vanishes in the unbounded bulk but not in systems with fixed-spin boundary conditions. Using the universal operator-product expansion of and exact results for the Ising model, we analyze in two-dimensional critical systems defined on the plane with mixed and boundary conditions. Particularly interesting behavior is found when either of the operators or is located on a ``zero line" in the plane, along which vanishes. Results for half-plane, triangular, and rectangular geometries are presented.

Paper Structure

This paper contains 23 sections, 137 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. Upper half plane with a uniform boundary condition +
  3. Upper half plane with a $+-$ boundary condition
  4. Boundary condition $-+-$
  5. $h_1 =i$
  6. $h_2 =i$:
  7. Behavior of $\langle \sigma \, \epsilon \rangle_{-+-}^{\rm cum}$ for arbitrary points on the symmetry axis
  8. Equilateral triangle with $-+-$ boundary condition
  9. ${\cal W} \times {\cal W}$ square with vertical boundaries - and horizontal boundaries $+$
  10. SUMMARY AND CONCLUDING REMARKS
  11. OPERATOR EXPANSION OF THE PRODUCT $\sigma \times \epsilon$ UP TO FOURTH ORDER
  12. Derivation by comparing with the four-point function in the bulk
  13. Checking against the three-point function
  14. Applying the OPE in the upper half plane with uniform boundary condition +
  15. Applying the OPE in the upper half plane with mixed boundary condition $+-$
  16. ...and 8 more sections

Figures (2)

  • Figure 1: Dimensionless response function $R_{+-} \equiv R_{+-}(X_1,X_2)$ along a line parallel to the $x$ axis in the upper half $x,y$ plane with $+-$ boundary condition as given in Eq. (\ref{['se+-cumy0']}). The red, orange, and grey curves in panel (a) show the $X_2$ dependence for $X_1$ fixed at $0, \, -0.2, \, -1.4$ while in panel (b) they show the $X_1$ dependence for $X_2$ fixed at $0, \, -0.25, \, -1$. For fixing $X_1$ and $X_2$ at 0 the upward and downward jumps, $4 \, {\rm sign} X_2$ and $-3 \, {\rm sign X_1}$ appearing in panels (a) and (b), respectively, are consistent with Eqs. (\ref{['x1zero']}) and (\ref{['x2zero']}) since $|\partial_x \, \langle \sigma (x, y_0) \rangle_{+-}|_{x=0} = 2^{1/8} / y_0^{(1/8)+1}$, see Eq. (\ref{['sig+-']}). On decreasing the fixed positions of $X_1$ and $X_2$ in panels (a) and (b), respectively, the corresponding $X_2$ and $X_1$ dependencies tend towards the $|X_2 -X_1|$ dependence for a uniform + boundary given in Eq. (\ref{['+epssigprime']}). In particular, the "disordering enhances order" regions adressed in paragraph (iii) of the Introduction that appear in (b) as $R_{+-}>0$ for $X_1 < 0$, decrease and vanish as the fixed locations $X_2$ of $\epsilon$ decrease from 0 via -0.25 to -1, so that $\langle \sigma \epsilon \rangle^{(\rm cum)}$ becomes negative for all $X_1 < 0$. The complexity of this process is displayed in more detail in panel (c) which shows the $X_1$-dependences for $X_2$ fixed at $0, \, -0.25, \, -0.36, \, -0.39$, and $-0.42$ in red, orange, green, blue, and purple: All curves with $X_2$ fixed must approach zero as $X_1 \to - \infty$. While for $X_2 =0$ (red curve) the approach is from above, see the remark below (\ref{['half+-onepoint']}), for all $X_2 < 0$ the approach is from below.
  • Figure 2: The dimensionless response function $R_{\rm tri}$ on the vertical midline of the triangle defined in Eq. (\ref{['tildeR']}) and evaluated by means of (\ref{['sigepstrafoprime']}). The red, green, and blue curves in panel (a) show the dependencies of $R_{\rm tri}$ on $Y_2$ for $Y_1$ fixed at $0.65$, at $Y_0 = 0.743$, and at $0.85$. The response $R_{\rm tri}$ of the disorder to the up ordering at $Y_1 = Y_0$ (green curve) is negative and positive at $Y_2 < Y_0$ and $Y_2 > Y_0$ since there the original order is in the up and down direction, respectively. For $Y_2$ close to $Y_0$ the $Y_2$-dependence reflects the asymptotic behavior given in Eq. (\ref{['shorttri1at0']}) with an upward discontinuity where $R_{\rm tri}$ jumps from $-(2 S_0)^{9/8} \equiv -(3^{3/4} 2^{2/3})^{9/8} = - 4.249$ to $(2 S_0)^{9/8}$. Unlike panel (a) in FIG 1 the $Y_2$-dependence is not antisymmetric about the discontinuity, and near the base line $Y=0$ and the corner $Y=Y_{\rm C} \equiv 3.196$ of the triangle it attains the behavior determined by Eqs. (\ref{['y2to0']}) and (\ref{['y2toyC']}). This decreases linearly from zero and approaches zero with the fifth power in the distance from the corner, respectively. For $Y_1 \not= Y_0$ (red and blue curves in panel (a)) the short distance singularity is $\propto |Y_2 - Y_1|^{-1} \,{\rm sign} (Y_1 -Y_0)$ and the prefactors of the near baseline and corner behaviors depend on $Y_1$ according to Eqs. (\ref{['y2to0']}) and (\ref{['y2toyC']}). Panel (b) shows the $Y_1$-dependence of the up order induced by the disorder imposed at $Y_2$ for $Y_2$ fixed at 0.65 (red), at $Y_0$ (green), and at 0.85 (blue). For $Y_2 = Y_0$ there is a downward discontinuity in the $Y_1$-dependence and the ratio of the discontinuities in panels (a) and (b) has the universal value of -4/3. Both for $Y_2 =Y_0$ and $Y_2 \not= Y_0$ the power law behaviors of $Y_1$ near the base line and the corner have exponents $2-(1/8)$ and $6-(1/8)$, respectively, with $Y_2$-dependent amplitudes according to Eqs. (\ref{['y1to0']}) and (\ref{['y1toyC']}). Note the $Y_1 < Y_0$ regions with ${\rm R}_{\rm tri} >0$ and the $Y_1 > Y_0$ regions with ${\rm R}_{\rm tri} < 0$ where the magnitude of the order is enhanced by the disordering.