Abstract
At low temperature, a thermodynamic system undergoes a phase transition when a physical parameter passes through a singularity point of the free energy. This corresponds to the formation of a new order. At high temperature, thermal fluctuations destroy the order. Correspondingly, the free energy is a smooth function of the physical parameter and singularities only occur at complex values of the parameter. Since a complex valued parameter is unphysical, no phase transitions are expected when the physical parameter is varied. Here we show that the quantum evolution of a system, initially in thermal equilibrium and driven by a designed interaction, is equivalent to the partition function of a complex parameter. Therefore, we can access the complex singularity points of thermodynamic functions and observe phase transitions even at high temperature. We further show that such phase transitions in the complex plane are related to topological properties of the renormalization group flows of the complex parameters. This result makes it possible to study thermodynamics in the complex plane of physical parameters.
Introduction
The physical properties of a manybody system in the thermodynamic equilibrium are fully determined by the partition function Z(β,λ_{1},λ_{2},…,λ_{k}), which is a function of the coupling parameters {λ_{k}} 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 {λ_{k}} (e.g., in spin models the magnetic field h = λ_{1}, the nearest neighbor coupling J = λ_{2}, and the next nearest neighbor coupling J′ = λ_{3}, etc.). The normalized Boltzmann factor Z^{−1}exp(−βH) is the probability that the system is in a state with energy H.
At a sufficiently low temperature (β > λ_{k}^{−1}), the manybody 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 nonanalytic or singularity points of the free energy F (which is related to the partition function by F = −β^{−1}lnZ). For a finitesize 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 LeeYang zeros^{1,2} in the complex plane of magnetic field for a spin lattice and Fisher zeros^{3} 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 occur^{1,2} (see Fig. 1c). At high temperature, the free energy would be nonanalytic only for complex values of temperature and coupling parameters^{3,4,5} (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.
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^{−1}exp(−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 timedependent measurement of the system has the form of a partition function with an imaginary coupling parameter it/β. By choosing H_{I} to be the kth component of , i.e., H_{I} = H_{k}, the kth coupling parameter is analytically continued to the complex plane via λ_{k} → λ_{k} + it/β. One can also realize complexification of the inverse temperature (β → β + it) by choosing H_{I} = H. Such a timedomain measurement, as will be discussed later in this paper, is experimentally realizable, though nontrivial in general. The timedomain measurement in equation (1) is fully determined by the equilibriumstate partition function of the system. The measurement result will be zero when a zero of the partition function in the complex plane is encountered^{6}. LeeYang zeoros have been recently observed via such measurement^{7}. 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 bath^{6} and evolution of a quantum spin model^{8}. 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 LeeYang zeros^{1,2} by choosing for a system of spins {S_{j}}, Fisher zeros^{3} by choosing H_{I} = H^{8}, 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} ≫ {λ_{k}}) via timedomain measurement. The hightemperature phase transitions are also inferred from a recent finding that quantum criticality can emerge at high temperatures by longtime quantum evolution^{9}. Note that the phase transitions studied in this paper are not the timedomain or dynamical phase transitions studied in Refs. 6 & 8. Instead, they occur between different parameter regimes, signified by the qualitative changes of timedomain measurement.
Hightemperature 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 FMAFM 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 FMAFM transition in the complex plane of the parameter J. The Hamiltonian for the general Ising model is where J_{ij} is the coupling between spins σ_{i} and σ_{j}, h is the magnetic field, and the spins σ_{j} 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 LeeYang zeros in the complex plane of scaled magnetic field z ≡ exp(2βh) exhibit different distributions. Note that the distinct features of LeeYang zeros distribution in the complex plane persist even at high temperatures (T ≫ J_{ij}).
To be specific, we study the onedimensional (1D) Ising model with nearestneighbor coupling J, which can be exactly solved through the transfer matrix method^{10,11,12} (see Supplementary Information). There is no finite temperature phase transition in the 1D Ising model. The LeeYang zeros of the 1D Ising model of N spins have been exactly calculated^{2}. We plot the distribution of LeeYang 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 LeeYang zeros are degenerate at z_{n} = −1 (indicated by the green solid ball).
To observe the LeeYang zeros and the critical behaviors in the complex plane^{4,5}, we study the timedomain measurement in equation (1) by choosing and analytically continuing the magnetic field to the complex plane h → h + it/β. In this case, the timedomain measurement corresponds to decoherence of a quantum probe spin coupled to the Ising model^{6,7} (see physical realizations). Figs. 2b–e plot the timedomain 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 timedomain measurement has no zeros (Fig. 2b) and it is a smooth function of time. On the contrary, for the FM coupling J = 1, the timedomain measurement shows a number of zeros (Fig. 2c), which have a onetoone correspondence to the LeeYang zeros^{6}. Approaching the thermodynamic limit, the YangLee edge singularities (the starting and ending LeeYang zeros along the arcs)^{4,5} lead to critical times in the timedomain observation^{6}. To demonstrate this, we perform a finite size scaling analysis on the timedomain measurement and show the scaled results L(t)^{1/N} in Figs. 2d & 2e. The profiles of the timedomain 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 YangLee edge singularities (Fig. 2e). The profiles of the timedomain measurement in the FM and AFM regions cannot be smoothly transformed into each other, which is the signature of the onset of a hightemperature phase transition.
We further study the hightemperature AFMFM phase transition in the timedomain measurement for a twodimensional (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 solvable^{11,12} and has a finitetemperature phase transition at β_{C} ≈ 0.44/J. Fig. 3 shows the timedomain measurement in the 2D Ising model calculated for different coupling parameters J at a fixed high temperature, β = 0.3 < β_{C}. For the AFM coupling J = −1, the partition function has no LeeYang zeros on the unit circle (z ≡ exp(2βh) = 1)^{13} and therefore the timedomain measurement has no zeros (Fig. 3a). While for the FM coupling J = 1, the timedomain measurement presents a number of zeros (equal to the number of spins) (Fig. 3b), corresponding to the LeeYang zeros along the unit circle. We do finitesize scaling analysis in Fig. 3c and Fig. 3d for AFM coupling and FM coupling, respectively. It is clear that the timedomain measurement is a smooth function of time for the AFM coupling, while it presents sudden changes at critical times corresponding to the YangLee 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).
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 K_{0} = βJ and h_{0} = 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 h_{0} = iτ_{0}/2 (the physical external field is zero), the exact RG flow equations become^{14} where the coupling K_{1} 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 = K_{0} and −π < τ_{0} ≤ 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).
The parameter curve is renormalized according to the RG flow equations. Since K_{1}(τ_{0},K_{0}) = K_{1}(τ_{0},−K_{0}) 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 (K_{0} = 1/8) becomes 2, 4, 8,.... On the contrary, the winding number in the AFM case (K_{0} = −1/8) is zero after renormalization (Figs. 4d, 4f & 4h). The RG flow equations become trivial at the phase transition point K_{0} = 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 hightemperature 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 read^{15} (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 (K_{0} = 1/8). Figs. 5b, d & f present the results for the AFM case (K_{0} = −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.
Transversefield 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 hightemperature phase transitions with complex parameters would exist also for quantum models. To address this question, we study the 1D transversefield Ising model. The model contains N spin1/2 with nearest neighbor interaction (λ_{1}) along the xaxis and under a transverse field (λ_{2}) along the zaxis, described by the Hamiltonian where is the Pauli matrix of the jth spin along the x/y/zaxis. This model is exactly solvable^{16}. It has a quantum phase transition between a magnetic ordered phase for λ_{1} > λ_{2} and a disordered phase for λ_{1} < λ_{2} at zero temperature, but has no finitetemperature phase transition for any parameters on the real axis. By defining the dimensionless magnetic field h = λ_{2}/λ_{1}, the LeeYang 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 LeeYang 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 LeeYang zeros exist only in the parameter range of h ≤ 1 indicates that the timedomain measurement of the system would present phase transitions between the two parameter regions, λ_{2} > λ_{1} and λ_{2} < λ_{1}. Specifically, the timedependent measurement can be devised as L(t) = Z^{−1}Tr[exp(−βH−itH_{2})], which is the partition function with a complex external field (λ_{2} → λ_{2} + it/β). The contour plot of the timedomain 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 timedomain measurement as a function of external field for different times. Since the zeros are bounded in the range h ≤ 1 with YangLee edge singularities at Re(h) = ±1, Im(h) = ±(n + 1/2)π/β, the timedomain 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).
Physical realization
The timedomain measurement in equation (1) resembles the Loschmidt echo^{17,18}, or equivalently, decoherence of a central spin coupled to the system. Thus we may implement the timedomain measurement by coupling the system to a central spin through the probesystem interaction (S_{z} ≡ ↑〉〈↑ − ↓〉〈↓) 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^{−1}Tr[exp(−βH)], the probe spin coherence is If [H_{I}_{,}H] = 0, the timedomain measurement L(t) = Z^{−1}Tr[exp(−βH)exp(−iH_{I}t)], which is equal to the probe spin coherence with a probebath coupling , i.e., H_{↑} = −H_{↓} = H_{I}/2. Or if [H_{I}_{,}H] ≠ 0 but [[H_{I}_{,}H],H] = [[H_{I}_{,}H], H_{I}] = 0, the timedomain measurement can also be factored as L(t) = Z^{−1}Tr[exp(−βH)exp(−iH_{I}t)] (See Supplementary Information for details) and can be implemented by the probe spin coherence with a modified probebath coupling H_{↓} = H_{I}/2 − H & H_{↑} = −H_{I}/2 − H. If [H,H_{I}] = isH_{I}, the timedomain measurement can be written as L(t) = Z^{−1}Tr[exp(−it sin(s)H_{I}/s)exp(−βH)] (See Supplementary Information for details), which can be implemented by probe spin decoherence with a probebath coupling H_{↓} = sin(s)H_{I}/2s − H & H_{↑} = −sin(s)H_{I}/2s − H. In the general case, the timedomain measurement in equation (1) can be written as , with , and , where are the timeordering and antiordering operators, respectively. If one initializes the bath in a canonical state ρ = exp(−βH′)/Tr[exp(−βH′)], the timedomain measurement can be implemented by probe spin decoherence with probebath coupling , up to a normalization factor (see Supplementary Information for details). The physical realization of the modified Hamiltonians H′ and is nontrivial.
Note that the timedomain measurement in equation (1) is similar to the measurement of the characteristic function of the work distribution in a quantum quench^{19,20,21}, which plays a central role in the fluctuation relations in nonequilibrium thermodynamics^{22,23,24,25,26}. The probe spin decoherence realization of the timedomain 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 time^{27}.
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 timedomain 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 timedomain 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 method^{10,11}. 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 functions^{6}) 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 field^{10,11}. 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 field^{11}, 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 parameters^{14,15}. 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 solved^{16} and the partition function and timedomain measurement were then calculated.
Full methods are included in the Supplementary Information.
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Acknowledgements
We thank Dr. J. A. Crosse for improving the English of our paper. This work was supported by Hong Kong Research Grants Council  General Research Fund Project 401413, The Chinese University of Hong Kong Focused Investments Scheme, and Hong Kong Research Grants Council  Collaborative Research Fund Project HKU10/CRF/08.
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Department of Physics, Centre for Quantum Coherence, and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
 BoBo Wei
 , ShaoWen Chen
 , HoiChun Po
 & RenBao Liu
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Contributions
R.B.L. conceived the idea, designed project, formulated the theory, and supervised the project. B.B.W. studied magnetic phase transitions in the Ising models and calculated the RG flows of the 2D Ising model. H.C.P. discovered the topological features of the RG flows of the 1D model. S.W. studied the transversefield Ising model. B.B.W. & R.B.L. wrote the manuscript. All authors discussed the results and the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to RenBao Liu.
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