Lee–Yang theory
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 work on parity nonconservation of weak interaction. Introduction. 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 formula_2 is the inverse temperature, formula_3 is the Boltzmann constant and formula_4 is the energy of a microstate. The moments formula_5 of the  energy statistics are obtained by differentiating the partition function with respect to the inverse temperature multiple times, From the partition function, we may also obtain the free energy Analogously to how the partition function generates the moments, the free energy generates the cumulants of the energy statistics More generally, if the microstate energies formula_9 depend on a "control parameter" formula_10 and a fluctuating conjugate variable formula_11 (whose value may depend on the microstate), the moments of formula_11 may be obtained as and the cumulants as For instance, for a spin system, the control parameter may be an external magnetic field , formula_15, and the conjugate variable may be the total magnetization, formula_16. Phase transitions and Lee–Yang theory. 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, formula_17 is a finite sum of exponential functions and is thus always positive for real values of formula_10. Consequently, formula_19 is always well-behaved and analytic for finite system sizes. By contrast, in the thermodynamic limit, formula_19 may exhibit a non-analytic behavior. Using that formula_17 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 formula_10. 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 formula_23. 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. Molecular zipper. 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. The model is based on a double-stranded macromolecule with formula_24 links that can be either open or closed. For a fully closed zipper, the energy is zero, while for each open link the energy is increased by an amount formula_25. A link can only be open if the preceding one is also open. For a number formula_26  of different ways that a link can be open, the partition function of a zipper with formula_24 links reads This partition function has the complex zeros where we have introduced the critical inverse temperature formula_30, with formula_31. We see that in the limit formula_32, the zeros closest to the real axis approach the critical value formula_33. For formula_34, the critical temperature is infinite and no phase transition takes place for finite temperature. By contrast, for formula_35, a phase transition takes place at the finite temperature formula_36. To confirm that the system displays a non-analytic behavior in the thermodynamic limit , we consider the free energy or, equivalently, the dimensionless free energy per link In the thermodynamic limit, we obtain Indeed, a cusp develops at formula_36 in the thermodynamic limit. In this case, the first derivative of the free energy is discontinuous, corresponding to a first-order phase transition. Ising model. The Ising model is the original model that Lee and Yang studied when they developed their theory on partition function zeros. The Ising model consists of spin lattice with formula_24 spins formula_42, each pointing either up, formula_43, or down, formula_44. Each spin may also interact with its closest spin neighbors  with a strength formula_45. In addition, an external magnetic field formula_46 may be applied (here we assume that it is uniform and thus independent of the spin indices). The Hamiltonian of the system for a certain spin configuration formula_47 then reads In this case, the partition function reads The zeros of this partition function cannot be determined analytically, thus requiring numerical approaches. Lee–Yang theorem. For the ferromagnetic Ising model, for which formula_50 for all formula_51, Lee and Yang showed that all zeros of formula_52 lie on the unit circle in the complex plane of the parameter formula_53. This  statement is known as the "Lee–Yang theorem", and has later been generalized to other models, such as the Heisenberg model. Dynamical phase transitions. 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. Connections to fluctuations. The Lee–Yang zeros may be connected to the cumulants of the conjugate variable formula_11 of the control variable formula_10. For brevity, we set formula_56 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 where formula_58 and formula_59 are constants, and formula_60 is the formula_61:th zero in the complex plane of formula_10. The corresponding free energy then reads Differentiating this expression formula_64 times with respect to formula_10, yields the formula_64: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. This establishes a direct connection between cumulants and Lee–Yang zeros. Moreover, if formula_64  is large, the contribution from zeros lying far away from formula_10 is strongly suppressed, and only the closest pair formula_71 of zeros plays an important role. 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: 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. In another experiment in 2017, dynamical Lee–Yang zeros were extracted from Andreev tunneling processes between a normal-state island and two superconducting leads. 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.