Introduction

The physical properties of a many-body system in the thermodynamic equilibrium are fully determined by the partition function Z(β,λ1,λ2,…,λk), which is a function of the coupling parameters {λk} of the system and the temperature T (or the inverse temperature β ≡ 1/T). The partition function is the summation of the Boltzmann factor over all energy eigen states, i.e., , where the Hamiltonian is characterized by a set of coupling parameters {λk} (e.g., in spin models the magnetic field h = λ1, the nearest neighbor coupling J = λ2 and the next nearest neighbor coupling J′ = λ3, etc.). The normalized Boltzmann factor Z−1exp(−βH) is the probability that the system is in a state with energy H.

At a sufficiently low temperature (β > |λk|−1), the many-body system may have different orders (phases) in different parameter ranges and therefore phase transitions may occur when these parameters are varied (see Fig. 1a). Phase transitions correspond to non-analytic or singularity points of the free energy F (which is related to the partition function by F = −β−1lnZ). For a finite-size system, the partition function is always positive and hence the free energy always analytic for real physical parameters. The partition function can be zero for complex temperature or complex coupling parameters (see Fig. 1b & 1c). Such is the case for Lee-Yang zeros1,2 in the complex plane of magnetic field for a spin lattice and Fisher zeros3 in the complex plane of temperature. At a sufficiently low temperature and in the thermodynamic limit (where the number of particles in the system approaches infinity), the zeros of the partition function can approach to real axes of the coupling parameters, and, hence, phase transitions can occur1,2 (see Fig. 1c). At high temperature, the free energy would be non-analytic only for complex values of temperature and coupling parameters3,4,5 (see Fig. 1b). Therefore, no phase transitions would occur (see Fig. 1a). Physically, this is because, at high temperatures, thermal fluctuations destroy all possible orders and prevents the phase transitions. Phase transitions at high temperatures would be possible if one could access complex parameters, which, however, are generally regarded as unphysical.

Figure 1
figure 1

Phase transitions and zeros of a partition function in the complex plane of a physical parameter.

(a). The schematic phase diagram of a general many-body system. At low temperature, the system has two ordered phases, I and II, separated by a critical parameter λc. At high temperature, the system is in the disordered phase. When we tune the control parameter at low temperatures (as indicated by blue dashed arrow) we cross two phase boundaries (indicated by solid-red lines) and therefore experience two phase transitions. Conversely at high temperature sweeping the parameter in the same range (blue dotted arrow) would not cause a phase transition. (b). The schematic distribution of zeros of the partition function in the complex plane of the parameter for temperature T above the critical point Tc. The two blue points mark the edge singularities. (c). The same as (b) but for the temperature below the critical point (T < Tc). The singularity edges approach to the real axis in the thermodynamic limit.

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Here we present a systematic approach to the complexification of an arbitrary physical parameter, using a quantum evolution that involves a complex phase factor. There are two fundamental laws of physics that involve probabilistic distributions. One is the Boltzmann distribution Z−1exp(−βH), the probability of a system being in a state with energy H at inverse temperature β. The other is the Schrödinger equation that gives exp(−itH), the probability amplitude of a system being in a state with energy H at time t. The unit imaginary number is associated with the time in quantum evolution. Thus a time-dependent measurement of the system

has the form of a partition function with an imaginary coupling parameter it/β. By choosing HI to be the k-th component of , i.e., HI = Hk, the k-th coupling parameter is analytically continued to the complex plane via λk → λk + it/β. One can also realize complexification of the inverse temperature (β → β + it) by choosing HI = H. Such a time-domain measurement, as will be discussed later in this paper, is experimentally realizable, though non-trivial in general. The time-domain measurement in equation (1) is fully determined by the equilibrium-state partition function of the system. The measurement result will be zero when a zero of the partition function in the complex plane is encountered6. Lee-Yang zeoros have been recently observed via such measurement7. Furthermore, critical times may exist corresponding to the singularity points in the complex plane, as indicated by recent studies of central spin decoherence caused by an Ising bath6 and evolution of a quantum spin model8. The measurement in equation (1) provides a systematic approach to accessing different types of zeros in the complex plane for the partition function, such as Lee-Yang zeros1,2 by choosing for a system of spins {Sj}, Fisher zeros3 by choosing HI = H8 and other types of zeros (yet unnamed) by choosing , , etc. More importantly, by complexification of a physical parameter, we have a way to access the singularity points in the complex plane and to reveal phase transitions at high temperature (β−1 ≫ {|λk|}) via time-domain measurement. The high-temperature phase transitions are also inferred from a recent finding that quantum criticality can emerge at high temperatures by long-time quantum evolution9. Note that the phase transitions studied in this paper are not the time-domain or dynamical phase transitions studied in Refs. 6 & 8. Instead, they occur between different parameter regimes, signified by the qualitative changes of time-domain measurement.

High-temperature magnetic phase transitions

Spin systems can have ferromagnetic (FM) or antiferromagnetic (AFM) orders at low temperatures, corresponding to positive or negative coupling (J) between the spins, respectively. Thus an FM-AFM transition would occur if the coupling J varies from positive to negative. At high temperatures, thermal fluctuations destroy the magnetic order and hence no phase transition is expected with changing J. Here we study the Ising spin model to demonstrate the FM-AFM transition in the complex plane of the parameter J. The Hamiltonian for the general Ising model is

where Jij is the coupling between spins σi and σj, h is the magnetic field and the spins σj take values ±1. At low temperatures, when the coupling changes from positive to negative, the Ising model presents a phase transition from the FM order to the AFM order at zero field. Correspondingly, the Lee-Yang zeros in the complex plane of scaled magnetic field z ≡ exp(2βh) exhibit different distributions. Note that the distinct features of Lee-Yang zeros distribution in the complex plane persist even at high temperatures (T ≫ |Jij|).

To be specific, we study the one-dimensional (1D) Ising model with nearest-neighbor coupling J, which can be exactly solved through the transfer matrix method10,11,12 (see Supplementary Information). There is no finite temperature phase transition in the 1D Ising model. The Lee-Yang zeros of the 1D Ising model of N spins have been exactly calculated2. We plot the distribution of Lee-Yang zeros in Fig. 2a. For AFM coupling (J < 0), all the zeros lie on the negative real axis (indicated by the red surface) (Fig. 2a). While for the FM coupling (J > 0), the zeros are distributed on an arc of the unit circle (indicated by the blue surface) (Fig. 2a). At the transition point (J = 0), all the Lee-Yang zeros are degenerate at zn = −1 (indicated by the green solid ball).

Figure 2
figure 2

High-temperature AFM-FM phase transitions revealed by time-domain observation.

(a). Schematic 3D plot of the Lee-Yang zeros in the 1D Ising model. For J > 0, the Lee-Yang zeros are distributed on an arc of the unit circle with two edge singularity points determined by the coupling strength (blue surface). As the coupling strength decreases the arcs shrink and finally the singularity edges merge to a point at z = −1 for J = 0 (indicated by the green solid ball). For J < 0, the Lee-Yang zeros lie on the negative real axis (red surface). (b). The time-domain measurement as a function of time for 1D Ising model with N = 20 spins at β = 1 and J = −1 (AFM). (c). The same as (b) but for J = 1 (FM). (d). Finite size scaling of the time-domain measurement in the 1D Ising model at β = 1 and J = −1 (AFM), the red line is for N = 100 spins, green line for N = 500 and blue line for N = 1000; (e). The same as (d) but for J = 1 (FM).

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To observe the Lee-Yang zeros and the critical behaviors in the complex plane4,5, we study the time-domain measurement in equation (1) by choosing and analytically continuing the magnetic field to the complex plane h → h + it/β. In this case, the time-domain measurement corresponds to decoherence of a quantum probe spin coupled to the Ising model6,7 (see physical realizations). Figs. 2b–e plot the time-domain measurement calculated for different coupling parameters J at fixed inverse temperature β = 1 (which is a high temperature case for the 1D Ising model, since the critical temperature of this model is zero). For AFM coupling J = −1, the time-domain measurement has no zeros (Fig. 2b) and it is a smooth function of time. On the contrary, for the FM coupling J = 1, the time-domain measurement shows a number of zeros (Fig. 2c), which have a one-to-one correspondence to the Lee-Yang zeros6. Approaching the thermodynamic limit, the Yang-Lee edge singularities (the starting and ending Lee-Yang zeros along the arcs)4,5 lead to critical times in the time-domain observation6. To demonstrate this, we perform a finite size scaling analysis on the time-domain measurement and show the scaled results |L(t)|1/N in Figs. 2d & 2e. The profiles of the time-domain measurement in the FM and AFM regions are qualitatively different. For AFM coupling, the scaled measurement is a smooth function of time (Fig. 2d). While for FM coupling, the scaled measurement presents sudden changes at critical times corresponding to the Yang-Lee edge singularities (Fig. 2e). The profiles of the time-domain measurement in the FM and AFM regions cannot be smoothly transformed into each other, which is the signature of the onset of a high-temperature phase transition.

We further study the high-temperature AFM-FM phase transition in the time-domain measurement for a two-dimensional (2D) Ising model. Specifically, we consider a 2D Ising model in a square lattice with nearest neighbor coupling J. This model under zero field is exactly solvable11,12 and has a finite-temperature phase transition at βC ≈ 0.44/|J|. Fig. 3 shows the time-domain measurement in the 2D Ising model calculated for different coupling parameters J at a fixed high temperature, β = 0.3 < βC. For the AFM coupling J = −1, the partition function has no Lee-Yang zeros on the unit circle (|z| ≡ |exp(2βh)| = 1)13 and therefore the time-domain measurement has no zeros (Fig. 3a). While for the FM coupling J = 1, the time-domain measurement presents a number of zeros (equal to the number of spins) (Fig. 3b), corresponding to the Lee-Yang zeros along the unit circle. We do finite-size scaling analysis in Fig. 3c and Fig. 3d for AFM coupling and FM coupling, respectively. It is clear that the time-domain measurement is a smooth function of time for the AFM coupling, while it presents sudden changes at critical times corresponding to the Yang-Lee edge singularities for the FM coupling (Fig. 3d). Thus a phase transition with varying the coupling constant J occurs at a temperature higher than the critical temperature (β < 0.44/|J|).

Figure 3
figure 3

High-temperature AFM-FM phase transitions of the 2D Ising spin model in a square lattice revealed by time-domain observation.

a. The time-domain measurement as a function of time for the 2D Ising model in a N = 4 × 5 square lattice, β = 0.3 and J = −1 (AFM). b. The same as (a) but for J = 1 (FM). c. Finite size scaling of the time-domain measurement with β = 0.3 and J = −1 (AFM), the green line is for N = 4 × 100 spins, the blue line is for N = 6 × 100 and red line for N = 8 × 100. d. The same as (c) but for J = 1 (FM).

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Renormalization group theoretic analysis

The renormalization group (RG) method, a powerful tool for studying conventional phase transitions, can be applied to the phase transitions in the complex plane of physical parameters. Since the phase of any complex number is defined modulo 2π, the RG flows of complex parameters can present novel topological structures.

As an example we first consider the 1D Ising model and define the dimensionless parameters K0 = βJ and h0 = βh. The renormalization of the model can be exactly formulated by blocking two neighboring spins into one (Fig. 4a)14. By continuation of the dimensionless external field to a purely imaginary value h0 = iτ0/2 (the physical external field is zero), the exact RG flow equations become14

where the coupling K1 remains real after renormalization. Since τ is defined modulo 2π, the parameter space can be identified with the surface of an infinitely long cylinder with unit radius. The original system corresponds to the parameter curve K = K0 and −π < τ0 ≤ π and therefore its winding number (W#), defined to be the number of times the parameter curve wraps around the cylinder, is 1 (Fig. 4b).

Figure 4
figure 4

Renormalized parameters of the 1D Ising model with a purely imaginary field.

(a), Real-space renormalization scheme. In each renormalization step the spins on every other site are traced out, which effectively combines pairs of neighboring spins into block spins with renormalized coupling and external field. (b), The un-renormalized parameters. The spin coupling K and evolution time τ (i.e., imaginary part of the external field) form a space identical to the surface of an infinitely long cylinder. When the imaginary field τ is varied at a fixed value of K = K0, the curve winds about the cylinder once and so the winding number W# = 1. (c–h), Under RG flow, the original curve depicted in (a) would transform differently in the different parameter regimes of K0. (c), (e) and (g) in turn show the parameters for the FM case with K0 = 1/8 after one, two and three steps of renormalization. (d), (f) and (h) show the corresponding results for the AFM case with K0 = −1/8. The vertical dashed lines indicate τ = 0 mod(2π), which are identified as the same line when represented on an infinitely long cylinder. The winding numbers can be directly inferred from the number of times the renormalized curve crosses this line.

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The parameter curve is renormalized according to the RG flow equations. Since K1(τ0,K0) = K1(τ0,−K0) in Eq. (3), the distinct behaviors of the FM and AFM Ising chains are encoded entirely in different RG flows of τ. This is illustrated in Figs. 4c–h. Figs. 4c, 4e & 4g show that after successive renormalization, the winding number in the FM case (K0 = 1/8) becomes 2, 4, 8,.... On the contrary, the winding number in the AFM case (K0 = −1/8) is zero after renormalization (Figs. 4d, 4f & 4h). The RG flow equations become trivial at the phase transition point K0 = 0, which corresponds to the infinite temperature limit and the winding number remains unchanged () after the renormalization. In summary, the winding numbers for different couplings after k steps of renormalization are

The different topologies of the RG flows demonstrate unambiguously the high-temperature phase transition with varying the coupling parameter.

We further consider the RG flows of the 2D Ising model in a square lattice. By continuation of the external field to a purely imaginary value of h = iτ/2, the approximate RG flow equations read15 (See Supplementary Information for derivation)

Fig. 5 presents the RG flows of the parameters in the FM and AFM cases. Figs. 5a, c & e present the renormalized parameters under one, two and three applications of the renormalization transformation for the FM case (K0 = 1/8). Figs. 5b, d & f present the results for the AFM case (K0 = −1/8). The winding numbers of the different cases after k steps of renormalization are

where [x] is the integer part of x. The winding number of the parameters under RG reflects the different topologies intrinsic to the RG flow equations in the different parameter regimes.

Figure 5
figure 5

Topological difference between the renormalization group flows of the FM and AFM 2D Ising models.

(a), (c) & (e) present in turn the parameters (spin coupling and imaginary external field) after one, two and three applications of the renormalization transformation in the FM case (K0 = 1/8). (b), (d) & (f) are the same as (a), (c) & (e) in turn but for the AFM case (K0 = −1/8). The renormalized parameters for the FM and AFM cases have different winding numbers (determined by how many periods (2π) of time the parameter curves are spanned over).

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Transverse-field Ising model

The models considered above are all classical models in which different components of the Hamiltonian commute. A natural question arises about whether the high-temperature phase transitions with complex parameters would exist also for quantum models. To address this question, we study the 1D transverse-field Ising model. The model contains N spin-1/2 with nearest neighbor interaction (λ1) along the x-axis and under a transverse field (λ2) along the z-axis, described by the Hamiltonian

where is the Pauli matrix of the j-th spin along the x/y/z-axis. This model is exactly solvable16. It has a quantum phase transition between a magnetic ordered phase for |λ1| > |λ2| and a disordered phase for |λ1| < |λ2| at zero temperature, but has no finite-temperature phase transition for any parameters on the real axis. By defining the dimensionless magnetic field h = λ2/λ1, the Lee-Yang zeros are determined by Re(h)2 + Im(h)2 = 1 + [(n + 1/2)π/β]2, |Re(h)| ≤ 1. Therefore, the zeros are located on circles and have cutoff at singularity edges Re(h) = ±1, Im(h) = ±|(n + 1/2)π/β| (see Fig. 6a). When the temperature approaches zero (|β| → ∞), the Lee-Yang zeros are on the unit circle. When the temperature is high (|β| ≪ 1), the radii of the circles are . Therefore, the zeros are distributed, approximately, along horizontal lines with interval |π/β|. The fact that the Lee-Yang zeros exist only in the parameter range of |h| ≤ 1 indicates that the time-domain measurement of the system would present phase transitions between the two parameter regions, |λ2| > |λ1| and |λ2| < |λ1|. Specifically, the time-dependent measurement can be devised as L(t) = Z−1Tr[exp(−βH−itH2)], which is the partition function with a complex external field (λ2 → λ2 + it/β). The contour plot of the time-domain measurement as a function of external field and time are presented in Fig. 6b. To demonstrate the phase transitions more clearly, we plot in Fig. 6c the time-domain measurement as a function of external field for different times. Since the zeros are bounded in the range |h| ≤ 1 with Yang-Lee edge singularities at Re(h) = ±1, Im(h) = ±|(n + 1/2)π/β|, the time-domain measurement has a sharp change when we tune the parameter from |h| ≤ 1 to |h| > 1 at times Im(h) = ±|(n + 1/2)π/β| (see Fig. 6c).

Figure 6
figure 6

Finite-temperature phase transitions of a transverse field Ising spin chain revealed by time-domain observation.

The one-dimensional model contains 100 spins. The temperature is such that β = 0.5. (a). Lee-Yang zeros in the complex plane of the external field. (b). Contour plot of the time-domain measurement, lg|L|, as a function of the external field Re(h) and the time Im(h). (c). lg|L| plotted as functions of the external field Re(h) for different times Im(h) = 3.10 (red-dashed line), 3.14 (black-solid line), 3.26 (blue-dashed line) and 3.35 (green-dashed line). The phase transitions at Re(h) = ±1 are observed.

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Physical realization

The time-domain measurement in equation (1) resembles the Loschmidt echo17,18, or equivalently, decoherence of a central spin coupled to the system. Thus we may implement the time-domain measurement by coupling the system to a central spin through the probe-system interaction (Sz ≡ |↑〉〈↑| − |↓〉〈↓|) and measuring the central spin coherence. Essentially, the coherence of the probe spin is a complex phase factor associated with a real Boltzmann probability for each state of the system. Therefore the probe spin coherence measurement amounts to continuation of a physical parameter to the complex plane. If we initialize the central spin in a superposition state |↑〉 + |↓〉 and the system in a thermal equilibrium state described by the canonical density matrix, ρ = Z−1Tr[exp(−βH)], the probe spin coherence is

If [HI,H] = 0, the time-domain measurement L(t) = Z−1Tr[exp(−βH)exp(−iHIt)], which is equal to the probe spin coherence with a probe-bath coupling , i.e., H↑ = −H↓ = HI/2. Or if [HI,H] ≠ 0 but [[HI,H],H] = [[HI,H], HI] = 0, the time-domain measurement can also be factored as L(t) = Z−1Tr[exp(−βH)exp(−iHIt)] (See Supplementary Information for details) and can be implemented by the probe spin coherence with a modified probe-bath coupling H↓ = HI/2 − H&H↑ = −HI/2 − H. If [H,HI] = isHI, the time-domain measurement can be written as L(t) = Z−1Tr[exp(−it sin(s)HI/s)exp(−βH)] (See Supplementary Information for details), which can be implemented by probe spin decoherence with a probe-bath coupling H↓ = sin(s)HI/2s − H&H↑ = −sin(s)HI/2s − H. In the general case, the time-domain measurement in equation (1) can be written as , with , and , where are the time-ordering and anti-ordering operators, respectively. If one initializes the bath in a canonical state ρ = exp(−βH′)/Tr[exp(−βH′)], the time-domain measurement can be implemented by probe spin decoherence with probe-bath coupling , up to a normalization factor (see Supplementary Information for details). The physical realization of the modified Hamiltonians H′ and is non-trivial.

Note that the time-domain measurement in equation (1) is similar to the measurement of the characteristic function of the work distribution in a quantum quench19,20,21, which plays a central role in the fluctuation relations in non-equilibrium thermodynamics22,23,24,25,26. The probe spin decoherence realization of the time-domain measurement can also be related to the quench dynamics where the evolution of the system can be controlled under different Hamiltonians for different periods of time27.

Summary

We have shown that the quantum evolution of a system originally in thermodynamic equilibrium is equivalent to the partition function of the system with a complex parameter. By choosing different forms of coupling we have a systematic way to realize the continuation of an arbitrary physical parameter to the complex plane. The time-domain measurement allows us to study a variety of zeros of the partition function. More importantly, we can access the singularity points of thermodynamic functions in the complex plane of physical parameters and therefore observe phase transitions at high temperatures. The physical realization of the time-domain measurement may be nontrivial but in principle it can be implemented by probe spin decoherence or quantum quench experiments. This discovery makes it possible to study thermodynamics in the complex plane of physical parameters.

Methods

The 1D Ising spin model was exactly diagonalized by the transfer matrix method10,11. The evaluation of the partition function after the transformation becomes a trivial problem of diagonalization of a 2 × 2 matrix. The probe spin coherence (which has been formulated in terms of partition functions6) was similarly calculated. Similarly, by applying the transfer matrix method, the 2D Ising model without magnetic field was mapped to a 1D Ising model with a transverse field10,11. For a 2D Ising model in a finite magnetic field, it was mapped by the transfer matrix method to a 1D Ising model with both longitudinal field and transverse field11, which was then numerically diagonalized. Therefore, the partition function and hence the probe spin coherence for the 2D Ising model in finite magnetic field were obtained. We derived the exact RG equations of the 1D Ising model and approximate RG equations for the 2D Ising model in square lattice for real parameters14,15. By analytic continuation we obtained the RG equations in the complex plane of the physical parameters and analyzed the RG flows of the complex parameters. The 1D Ising model with a transverse magnetic field was exactly solved16 and the partition function and time-domain measurement were then calculated.

Full methods are included in the Supplementary Information .