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.^[1][2]

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,^[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]

Introduction

For an equilibrium system in the canonical ensemble, all statistical information about the system is encoded in the partition function,

${\displaystyle Z=\sum _{i}e^{-\beta E_{i}},}$

where the sum runs over all possible microstates, and ${\displaystyle \beta =1/(k_{B}T)}$ is the inverse temperature, ${\displaystyle k_{B}}$ is the Boltzmann constant and ${\displaystyle E_{i}}$ is the energy of a microstate. The moments ${\displaystyle \langle E^{n}\rangle }$ of the energy statistics are obtained by differentiating the partition function with respect to the inverse temperature multiple times,

${\displaystyle \langle E^{n}\rangle ={\frac {1}{Z}}\partial _{-\beta }^{n}Z={\frac {\sum _{i}E_{i}^{n}e^{-\beta E_{i}}}{\sum _{i}e^{-\beta E_{i}}}}.}$

From the partition function, we may also obtain the free energy

${\displaystyle F=-\beta ^{-1}\log[Z].}$

Analogously to how the partition function generates the moments, the free energy generates the cumulants of the energy statistics

${\displaystyle \langle \!\langle E^{n}\rangle \!\rangle =\partial _{-\beta }^{n}(-\beta F).}$

More generally, if the microstate energies ${\displaystyle E_{i}(q)=E_{i}(0)-q\Phi _{i}}$ depend on a control parameter ${\displaystyle q}$ and a fluctuating conjugate variable ${\displaystyle \Phi }$ (whose value may depend on the microstate), the moments of ${\displaystyle \Phi }$ may be obtained as

${\displaystyle \langle \Phi ^{n}\rangle ={\frac {1}{Z}}\beta ^{-n}\partial _{q}^{n}Z(q)={\frac {1}{Z}}\beta ^{-n}\partial _{q}^{n}\sum _{i}e^{-\beta E_{i}(q)}={\frac {\sum _{i}\Phi _{i}^{n}e^{\beta E_{i}(0)+\beta q\Phi _{i}}}{\sum _{i}e^{\beta E_{i}(0)+\beta q\Phi _{i}}}},}$

and the cumulants as

${\displaystyle \langle \!\langle \Phi ^{n}\rangle \!\rangle =\beta ^{-n}\partial _{q}^{n}[-\beta F(q)].}$

For instance, for a spin system, the control parameter may be an external magnetic field, ${\displaystyle q=h}$, and the conjugate variable may be the total magnetization, ${\displaystyle \Phi =M}$.

Phase transitions and Lee–Yang theory

Illustration of how the zeros closest to the real axis (red circles) in the complex plane of the control parameter ${\displaystyle q}$ may move, with increasing system size ${\displaystyle N}$, towards the (real) critical value ${\displaystyle q^{*}}$ (filled circle) for which a phase transition takes place in the thermodynamic limit.

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, ${\displaystyle Z(q)}$ is a finite sum of exponential functions and is thus always positive for real values of ${\displaystyle q}$. Consequently, ${\displaystyle F(q)}$ is always well-behaved and analytic for finite system sizes. By contrast, in the thermodynamic limit, ${\displaystyle F(q)}$ may exhibit a non-analytic behavior.

Using that ${\displaystyle Z(q)}$ 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 ${\displaystyle q}$. 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 ${\displaystyle q=q^{*}}$.

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).

Examples

Connections to fluctuations

The Lee–Yang zeros may be connected to the cumulants of the conjugate variable ${\displaystyle \Phi }$ of the control variable ${\displaystyle q}$.^[11][12] For brevity, we set ${\displaystyle \beta =1}$ in the following. Using that the partition function is an entire function for a finite-size system, one may expand it in terms of its zeros as

${\displaystyle Z(q)=Z(0)e^{cq}\prod _{k}(1-q/q_{k}),}$

where ${\displaystyle Z(0)}$ and ${\displaystyle c}$ are constants, and ${\displaystyle q_{k}}$ is the ${\displaystyle k}$:th zero in the complex plane of ${\displaystyle q}$. The corresponding free energy then reads

${\displaystyle -F(q)=\log[Z(q)]=\log[Z(0)]+cq+\sum _{k}\log[1-q/q_{k}].}$

Differentiating this expression ${\displaystyle n}$ times with respect to ${\displaystyle q}$, yields the ${\displaystyle n}$:th order cumulant

${\displaystyle \langle \!\langle \Phi ^{n}\rangle \!\rangle =\partial _{q}^{n}[-F(q)]=-\sum _{k}{\frac {(n-1)!}{(q_{k}-q)^{n}}},\quad n>1.}$

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

${\displaystyle \langle \!\langle \Phi ^{n}\rangle \!\rangle =-(n-1)!\sum _{k}{\frac {2\cos(n\arg\{q_{k}-q\})}{|q_{k}-q|^{n}}},\quad n>1,}$

where the sum now runs only over each pair of zeros. This establishes a direct connection between cumulants and Lee–Yang zeros.

Moreover, if ${\displaystyle n}$ is large, the contribution from zeros lying far away from ${\displaystyle q}$ is strongly suppressed, and only the closest pair ${\displaystyle q_{0}}$ of zeros plays an important role. One may then write

${\displaystyle \langle \!\langle \Phi ^{n}\rangle \!\rangle \simeq -(n-1)!{\frac {2\cos(n\arg\{q_{0}-q\})}{|q_{0}-q|^{n}}},\quad n\gg 1.}$

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]

${\displaystyle {\begin{bmatrix}2{\text{Re}}[q-q_{0}]\\|q-q_{0}|\end{bmatrix}}={\begin{bmatrix}1&-{\frac {\kappa _{n}^{(+)}}{n}}\\1&-{\frac {\kappa _{n+1}^{(+)}}{n+1}}\end{bmatrix}}^{-1}{\begin{bmatrix}(n-1)\kappa _{n}^{(-)}\\n\kappa _{n+1}^{(-)}\end{bmatrix}},\qquad \kappa ^{\pm }\equiv {\frac {\langle \!\langle \Phi ^{n\pm 1}\rangle \!\rangle }{\langle \!\langle \Phi ^{n}\rangle \!\rangle }}.}$

Experiments

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. In one experiment in 2015, the Lee–Yang zeros were extracted experimentally by measuring the quantum coherence of a spin coupled to an Ising-type spin bath.^[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.^[14] Furthermore, in 2018, there was an experiment determining the dynamical Fisher zeros of the Loschmidt amplitude, which may be used to identify dynamical phase transitions.^[15]

See also

• Lee–Yang theorem

References