Abstract
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.
Highlights
Phase transitions are characterized by the abrupt change of a many-body system from one state of matter to another as an external control parameter is varied [1,2,3]
We present a direct pathway from the detection of partition function zeros by measuring or simulating fluctuating observables in systems of finite size to the determination of critical points and universal critical exponents of continuous phase transitions [1,2,3]
Unlike most conventional methods, based for instance on Binder cumulants [30,31,32], which require the control parameter to be tuned across the phase transition, we can determine the critical exponents even if the system is away from the phase transition, for example at a fixed high temperature. (In the Appendices, we discuss the statistical aspects of our method, and we compare it with the use of Binder cumulants.) 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 [42,43]
Summary
Phase transitions are characterized by the abrupt change of a many-body system from one state of matter to another as an external control parameter is varied [1,2,3]. The crucial insight of Lee and Yang was that the partition function zeros with increasing system size will approach the real value of the control parameter for which a phase transition occurs These ideas are considered a theoretical cornerstone of statistical physics, and they have found applications across a wide range of topics, including protein folding [8,9], percolation [10,11,12,13], and Bose-Einstein condensation [14,15]. Our method opens an avenue for bottom-up experiments on phase transitions, in which nanoscale structures are carefully assembled, for example by adding single spins to an atomic chain on a surface [44] or by loading individual atoms into an optical lattice one at a time [45], to increase the system size in a controllable manner
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