Lee–Yang theory

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.

Originally developed for the Ising model, the theory has been extended and applied to a wide range of models and phenomena, including protein folding,[3] percolation,[4] complex networks,[5] and molecular zippers.

[6] The theory is named after the Nobel laureates Tsung-Dao Lee and Yang Chen-Ning,[7][8] who were awarded the 1957 Nobel Prize in Physics for their unrelated work on parity non-conservation in weak interaction.

[9] For an equilibrium system in the canonical ensemble, all statistical information about the system is encoded in the partition function, where the sum runs over all possible microstates, and

may be obtained as and the cumulants as For instance, for a spin system, the control parameter may be an external magnetic field,

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.

is a finite sum of exponential functions and is thus always positive for real values of

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

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

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).

The molecular zipper is a toy model which may be used to illustrate the Lee–Yang theory.

It has the advantage that all quantities, including the zeros, can be computed analytically.

of different ways that a link can be open, the partition function of a zipper with

links reads This partition function has the complex zeros where we have introduced the critical inverse temperature

, the zeros closest to the real axis approach the critical value

, a phase transition takes place at the finite temperature

To confirm that the system displays a non-analytic behavior in the thermodynamic limit, we consider the free energy

In this case, the first derivative of the free energy is discontinuous, corresponding to a first-order phase transition.

may be applied (here we assume that it is uniform and thus independent of the spin indices).

then reads In this case, the partition function reads The zeros of this partition function cannot be determined analytically, thus requiring numerical approaches.

lie on the unit circle in the complex plane of the parameter

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.

:th order cumulant Furthermore, using that the partition function is a real function, the Lee–Yang zeros have to come in complex conjugate pairs, allowing us to express the cumulants as where the sum now runs only over each pair of zeros.

One may then write This equation may be solved as a linear system of equations, allowing for the Lee–Yang zeros to be determined directly from higher-order cumulants of the conjugate variable:[11][12] Being complex numbers of a physical variable, Lee–Yang zeros have traditionally been seen as a purely theoretical tool to describe phase transitions, with little or none connection to experiments.

However, in a series of experiments in the 2010s, various kinds of Lee–Yang zeros have been determined from real measurements.

[13] In another experiment in 2017, dynamical Lee–Yang zeros were extracted from Andreev tunneling processes between a normal-state island and two superconducting leads.