Relationship between phase transitions and topological changes in one-dimensional models

We address the question of the quantitative relationship between thermodynamic phase transitions and topological changes in the potential energy manifold analyzing two classes of one dimensional models, the Burkhardt solid-on-solid model and the Peyrard-Bishop model for DNA thermal denaturation, both in the confining and nonconfining version. These models, apparently, do not fit [M. Kastner, Phys. Rev. Lett. 93, 150601 (2004)] in the general idea that the phase transition is signaled by a topological discontinuity. We show that in both models the phase transition energy $v_c$ is actually noncoincident with, and always higher than, the energy $v_{\theta }$ at which a topological change appears. However, applying a procedure already successfully employed in other cases as the mean field ${\varphi }^4$ model, i.e., introducing a map $\mathcal{M}:v\to v_s$ from levels of the energy hypersurface $V$ to the level of the stationary points “visited” at temperature $T$, we find that $\mathcal{M}\left(v_c\right)=v_{\theta }$. This result enhances the relevance of the underlying stationary points in determining the thermodynamics of a system, and extends the validity of the topological approach to the study of phase transition to the elusive one-dimensional systems considered here.

Figure 1

Sketch of the on-site pinning potential $V_p^{\left(1\right)}\left(q\right)$ for a given choice of the control parameter $L∕R$.

Figure 2

Plots of the on-site pinning potentials $V_p^{\left(2\right)}\left(q\right)$ (full line) and $V_p^{\left(3\right)}\left(q\right)$ (dashed line).

Figure 3

Full line, critical temperature (in reduced units $k_B{T}_c∕U_0$) as a function of the control parameter $\xi$ from Eq. (23) for the $\zeta =0$ case. Dashed line, temperature $T_J$ (in reduced units) at which the “underlying saddle” jumps from minimum to saddle as a function of $\xi$ (see Sec. 5B).

Figure 4

Critical temperature (in reduced units $k_B{T}_c∕U_0$) as a function of the control parameter $\xi$ from Eq. (28) for the indicated $\zeta$ value, $\zeta =0$ (full line), 0.5 (dashed line), 1.5 (dotted line), 3.5 (dashed-dotted line), and 7.5 (dashed-dotted-dotted line).

Figure 5

Critical temperature (in reduced units $k_B{T}_c∕U_0$) as a function of the control parameter $\zeta$ from Eq. (28) for the indicated $\xi$ value, $\xi =2$ (full line), 1 (dashed line), 0.5 (dotted line), 0.25 (dashed-dotted line).

Figure 6

Inverse temperature dependence (in reduced units $\beta U_0$) of the equilibrium potential energy for $\xi =1$ and for the indicated $\zeta$ value, $\zeta =0$ (full line), 0.5 (dashed line), 1.5 (dotted line), 3.5 (dashed-dotted line), and 7.5 (dashed-dotted-dotted line). The thick-dotted line represents the $\beta$ dependence of the potential energy in the high temperature phase.

Figure 7

Temperature dependence (in reduced units $k_{B}T∕U_0$) of the equilibrium potential energy $v$ (full symbols) and energy $v_s$ of underlying saddles (open symbols) for the PB model defined by Eq. (4) with $\xi =0.05$ (upper panel) and $\xi =0.5$ (lower panel). Also indicated in the figure are the values of topological change energy $v_{\theta }$ (horizontal dotted lines), phase transition energy $v_c$ (horizontal dotted-dashed lines) and transition temperature $T_c$ (vertical full line) for $\xi =0.05$ ($v_{\theta }∕U_0=0$, $v_c∕U_0\simeq 0.61$, and $k_B{T}_c∕U_0\simeq 1.22$) and for $\xi =0.5$ ($v_{\theta }∕U_0=0$, $v_c∕U_0\simeq 1.59$, and $k_B{T}_c∕U_0\simeq 3.20$). Dashed lines are the $T$ dependence of the potential energy in the high $T$ phase, $v\left(T\right)=k_{B}T∕2$.

Figure 8

Temperature dependence (in reduced units $k_{B}T∕U_0$) of the equilibrium potential energy $v$ (full symbols) and energy $v_s$ of underlying saddles (open symbols) for the SPB model defined by Eq. (5) with $\xi =0.05$ (upper panel) and $\xi =0.5$ (lower panel). Also indicated in the figure is the value of the topological change energy $v_{\theta }∕U_0=0$ (dotted line) for both cases.

Figure 9

Map $\mathcal{M}:v\to v_s$ defined minimizing the pseudopotential $W={\mid \nabla V\mid }^2$ in the PB and SPB models for $\xi =0.05$ (upper panel) and $\xi =0.5$ (lower panel). Also reported are the corresponding $v_{\theta }$ (dotted lines) and $v_c$ (dotted-dashed lines) for $\xi =0.05(v_{\theta }=0,v_c\simeq 0.61)$ and $\xi =0.5(v_{\theta }=0,v_c\simeq 1.59)$, evidencing the identity $\mathcal{M}\left(v_c\right)=v_{\theta }$.