In statistical mechanics, Lee–Yang theory, sometimes also known as Yang–Lee theory, is a scientific theory which seeks to describe phase transitions in In statistical mechanics, Lee–Yang theory, sometimes also known as Yang–Lee theory, is a scientific theory which seeks to describe phase transitions in large physical systems in the thermodynamic limit based on the properties of small, finite-size systems. The theory revolves around the complex zeros of partition functions of finite-size systems and how these may reveal the existence of phase tran In statistical mechanics, Lee–Yang theory, sometimes also known as Yang–Lee theory, is a scientific theory which seeks to describe phase transitions in large physical systems in the thermodynamic limit based on the properties of small, finite-size systems. The theory revolves around the complex zeros of partition functions of finite-size systems and how these may reveal the existence of phase transitions in the thermodynamic limit. Lee–Yang theory constitutes an indispensable part of the theories of phase transitions. Originally developed for the Ising model, the theory has been extended and applied to a wide range of models and phenomena, including protein folding, percolation, complex networks, and molecular zippers. The theory is named after the Nobel laureates Tsung-Dao Lee and Yang Chen-Ning, who were awarded the 1957 Nobel Prize in Physics for their unrelated work on parity non-conservation in weak interaction. The partition function and the free energy are intimately linked to phase transitions, for which there is a sudden change in the properties of a physical system. Mathematically, a phase transition occurs when the partition function vanishes and the free energy is singular (non-analytic). For instance, if the first derivative of the free energy with respect to the control parameter is non-continuous, a jump may occur in the average value of the fluctuating conjugate variable, such as the magnetization, corres In statistical mechanics, Lee–Yang theory, sometimes also known as Yang–Lee theory, is a scientific theory which seeks to describe phase transitions in large physical systems in the thermodynamic limit based on the properties of small, finite-size systems. The theory revolves around the complex zeros of partition functions of finite-size systems and how these may reveal the existence of phase transitions in the thermodynamic limit. Lee–Yang theory constitutes an indispensable part of the theories of phase transitions. Originally developed for the Ising model, the theory has been extended and applied to a wide range of models and phenomena, including protein folding, percolation, complex networks, and molecular zippers. The theory is named after the Nobel laureates Tsung-Dao Lee and Yang Chen-Ning, who were awarded the 1957 Nobel Prize in Physics for their unrelated work on parity non-conservation in weak interaction. The partition function and the free energy are intimately linked to phase transitions, for which there is a sudden change in the properties of a physical system. Mathematically, a phase transition occurs when the partition function vanishes and the free energy is singular (non-analytic). For instance, if the first derivative of the free energy with respect to the control parameter is non-continuous, a jump may occur in the average value of the fluctuating conjugate variable, such as the magnetization, corresponding to a first-order phase transition. Importantly, for a finite-size system, Z ( q ) {\displaystyle Z(q)} is a finite sum of exponential functions and is thus always positive for real values of q {\displaystyle q} . Consequently, F ( q ) {\displaystyle F(q)} is always well-behaved and analytic for finite system sizes. By contrast, in the thermodynamic limit, F ( q ) {\displaystyle F(q)} may exhibit a non-analytic behavior. Using that Z ( q ) {\displaystyle Z(q)} is an entire function for finite system sizes, Lee–Yang theory takes advantage of the fact that the partition function can be fully characterized by its zeros in the complex plane of q {\displaystyle q} . These zeros are often known as Lee–Yang zeros or, in the case of inverse temperature as control parameter, Fisher zeros. The main idea of Lee–Yang theory is to mathematically study how the positions and the behavior of the zeros change as the system size grows. If the zeros move onto the real axis of the control parameter in the thermodynamic limit, it signals the presence of a phase transition at the corresponding real value of q = q ∗ {\displaystyle q=q^{*}} . In this way, Lee–Yang theory establishes a connection between the properties (the zeros) of a partition function for a finite size system and phase transitions that may occur in the thermodynamic limit (where the system size goes to infinity). where we have introduced the critical inverse temperature β c − 1 = k B T c {\displaystyle \beta _{c}^{-1}=k_{B}T_{c}} , with T c = ε k B log ⁡ g {\displaystyle T_{c}={\frac {\varepsilon }{k_{B}\log g}}} . We see that in the limit N → ∞ {\displaystyle N\rightarrow \infty } , the zeros closest to the real axis approach the critical value β k = β c {\displaystyle \beta _{k}=\beta _{c}} . For g = 1 {\displaystyle g=1} , the critical temperature is infinite and no phase transition takes place for finite temperature. By contrast, for g > 1 {\displaystyle g>1} , a phase transition takes place at the finite temperature T c A similar approach can be used to study dynamical phase transitions. These transitions are characterized by the Loschmidt amplitude, which plays the analogue role of a partition function.