Determination of universal critical exponents using Lee-Yang theory

Lee-Yang zeros are points in the complex plane of an external control parameter at which the partition function vanishes for a many-body system of finite size. In the thermodynamic limit, the Lee-Yang zeros approach the critical value on the real axis, where a phase transition occurs. Partition function zeros have for many years been considered a purely theoretical concept; however, the situation is changing now as Lee-Yang zeros have been determined in several recent experiments. Motivated by these developments, we here devise a direct pathway from measurements of partition function zeros to the determination of critical points and universal critical exponents of continuous phase transitions. To illustrate the feasibility of our approach, we extract the critical exponents of the Ising model in two and three dimensions from the fluctuations of the total energy and the magnetization in lattices of finite size. Importantly, the critical exponents can be determined even if the system is away from the phase transition. Moreover, in contrast to standard methods based on Binder cumulants, it is not necessary to drive the system across the phase transition. As such, our method provides an intriguing perspective for investigations of phase transitions that may be hard to reach experimentally, for instance at very low temperatures or at very high pressures.

Figure 1

Ising lattice and Fisher zeros. (a) The Ising model, here in $d=3$ dimensions with linear size $L=20$ and $N=L^d=8000$ lattice sites. The color of each site denotes the orientation of its spin, blue (up) or red (down). (b) From the energy fluctuations, we find the leading partition function zeros in the complex plane of the inverse temperature using Eq. (8). The inverse temperature is $\beta J=0.23$, where $J$ is the coupling between neighboring spins. No magnetic field is applied. The Fisher zeros approach the critical inverse temperature ${\beta }_c$ with increasing system size, $L=4,5,...,10$. Importantly, from the scaling of the Fisher zeros, we can determine the critical exponents as shown in Fig. 2.

Figure 2

Fisher zeros and critical exponents. (a) The leading Fisher zeros (blue circles) for the Ising model with $d=2$ are extracted from the energy cumulants of order $n=6,7,8,9$. With increasing system size, the Fisher zeros approach the critical inverse temperature ${\beta }_{c}J\simeq 0.4404$ (red circle), which is close to the exact result ${\beta }_{2\mathrm{D}}J=ln(1+\sqrt{2})/2\simeq 0.4407$. The simulations were carried out at a temperature above the phase transition, $\beta J=0.35$ (black circle). For the Ising model in Fig. 1 with $d=3$, we find ${\beta }_{c}J=0.22169$, which is close to the best numerical estimate of ${\beta }_{3\mathrm{D}}J\simeq 0.22165$. The critical inverse temperatures are determined in panel (c). (b) The extracted critical exponents $\nu$ from the finite-size scaling of the imaginary parts are close to the known values for the Ising model, ${\nu }_{2\mathrm{D}}=1$ (exact) and ${\nu }_{3\mathrm{D}}\simeq 0.630$ (numerics) [49]. (c) In the thermodynamic limit, the imaginary part of the zeros vanishes, and the real parts approach the critical values indicated with red circles in panels (a) and (c).

Figure 3

Lee-Yang zeros and critical exponents. (a) The leading Lee-Yang zeros (blue circles) for the Ising model with $d=2$ are extracted from the magnetization cumulants of order $n=6,7,8,9$. Above the critical temperature, $\beta =0.8{\beta }_c$, the Lee-Yang zeros remain complex in the thermodynamic limit (pair of red circles). For the sake of clarity, these results have been shifted horizontally away from the line $\mathrm{Re}[h/J]=0$. At the critical inverse temperature, $\beta ={\beta }_c$, the Lee-Yang zeros approach the critical field $h_c=0$ (red circle) with increasing system size. We note that the perpendicular approach to the real-axis shows that the system exhibits a first-order phase transition as a function of the magnetic field [6, 7, 53, 54]. (b) Finite-size scaling of the imaginary parts of the Lee-Yang zeros and extraction of the ratio of critical exponents $\mathcal{B}/\nu$ for $d=2,3$. (c) Determination of the convergence points of the Lee-Yang zeros (red circle) for $d=2,3$. For $d=3$, the real part also vanishes (not shown).

Figure 4

Monte Carlo simulations and error estimates. (a) The mean location of the leading partition function zeros obtained from $m=10$ Monte Carlo simulations with ${10}^5$ measurements per site are shown for increasing system sizes, $L=4,5,...,11$. The standard errors are denoted by red ellipses. The partition function zeros (blue circles) are extracted from the energy fluctuations of order $n=5,6,7,8$ at the inverse temperature $\beta =0.8{\beta }_c$ which is indicated by a black circle. (b) The imaginary parts of the partition function zeros are depicted in a log-log plot as a function of lattice size $L$ together with the errors bars that increase with the system size. The slope of the plot delivers the critical exponent $\nu \approx 1.004$. (c) Determination of the convergence points of the partition function zeros (red circle). The real part of the zeros moves toward the critical temperature ${\beta }_{c}J\approx 0.4419$, which is close to the exact value for the Ising model ${\beta }_{c}J\approx 0.4407$, whereas the imaginary parts vanish in the thermodynamic limit, signaling a sharp phase transition.

Figure 5

Binder cumulants. (a) The Binder parameter as a function of the inverse temperature for different system sizes. The Binder parameter is evaluated at $\beta =(0.7,0.8,0.85,0.9,0.95,0.97,1.03,1.05,1.1,1.15,1.2,1.3)\times {\beta }_c$. The intersection point (red circle) that is extrapolated to the thermodynamic limit yields the critical temperature ${\beta }_{c}J\approx 0.4434$. The estimation of the critical temperature highly depends on the proximity of the measurements to the critical point, and collecting more statistics in the vicinity of ${\beta }_c$ improves the accuracy. The gray frame corresponds to the data that we utilize to determine the critical exponents and critical temperature with our method. (b) Collapse of the Binder cumulant for different system sizes. The estimate of the critical point from (a) is used to tune the critical exponent $\nu \approx 1.07$ so that all data collapse onto a single curve.