Thermodynamic holography

The holographic principle states that the information about a volume of a system is encoded on the boundary surface of the volume. Holography appears in many branches of physics, such as optics, electromagnetism, many-body physics, quantum gravity, and string theory. Here we show that holography is also an underlying principle in thermodynamics, a most important foundation of physics. The thermodynamics of a system is fully determined by its partition function. We prove that the partition function of a finite but arbitrarily large system is an analytic function on the complex plane of physical parameters, and therefore the partition function in a region on the complex plane is uniquely determined by its values along the boundary. The thermodynamic holography has applications in studying thermodynamics of nano-scale systems (such as molecule engines, nano-generators and macromolecules) and provides a new approach to many-body physics.

The most famous example of holography is probably the optical hologram, where the three-dimensional view of an object is recorded in a two-dimensional graph 1 . The holographic principle indeed has profound implications in many branches of physics. In electromagnetism, for instance, the electrostatic potential in a volume is uniquely determined by its values at the surface boundary 2 . Density functional theory 3,4 , which is the foundation of quantum chemistry and first-principle calculations 4 , may be viewed as a holography that maps the full ground-state wave function of a many-electron system (a complex function in an enormously high-dimensional space) to the ground state single-particle density (a real function in three-dimensional space). The holographic principle has also been shown relevant in quantum gravity 5 and string theory 6,7 . Recently, the holographic approach has been employed to tackle strongly-correlated systems in condensed matter physics 8 .
The physical properties of a system at thermodynamic equilibrium are fully determined by the partition function β λ λ λ Ξ( , , , …, ) , where the Hamiltonian λ = ∑ H H j j j is characterized by a set of coupling parameters {λ j } (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 β Ξ (− ) − H exp 1 is the probability of the system in a state with energy H. In the following we consider one of the physical parameters of the system, λ λ λ λ ∈ , , …, { } nitrogen-vacancy centre spin decoherence for just one setting of parameters. The holographic principle is applicable for any finite systems of fermions and spins. Because the system size can be arbitrarily large, the physics in systems that approach the thermodynamic limit can be captured. For bosons, however, the validity of theory is verified only in some special cases.

Results
Holography of partition function. The thermodynamic holography states that the partition function of a finite physical system in an area of the complex plane of a physical parameter is uniquely determined by its values along the boundary. This stems from the Cauchy theorem in complex analysis 9 for analytic functions. The analyticity of partition functions is based on the following Theorem (see Fig. 1a for illustration).
Theorem 1. The partition function λ Ξ( ) of a finite physical system is analytic in the whole complex plane of a physical parameter λ.
The proof is based on the fact that the partition function of a finite system can be expanded into Taylor's series in terms of the Hamiltonian and the Taylor's series, being polynomial functions of the parameter λ, are all analytic and converge uniformly (see details in Methods). Here by "finite physical system" we mean that the system has a finite number of discrete basis states. For a finite (but arbitrarily large) system of spins and fermions on lattices, the partition function is analytic in the whole complex plane of physical parameters. For bosons, the partition function can be non-analytical in certain regions 13,14 (e.g., the partition functions of free bosons have singularities along the imaginary axis of frequencies, and for coupled bosons the system can become unstable when the coupling is too strong). The theorem can be generalized to infinite-dimensional systems that satisfy certain conditions (see Methods). However, infinite systems in general cases still need further study.
Here comes the main result of this paper. According to Cauchy theorem and Theorem 1, the partition function satisfies where λ′ is a complex parameter enclosed by the integration contour C in the complex plane of λ where the partition function is analytic (see Fig. 1a). A convenient choice of the boundary (the integration contour) can be a rectangular path (see Fig. 1b) that consists of two straight lines perpendicular to the real axis, whose real parts λ R are respectively λ 1 and λ 2 , and two segments parallel to the real axis, whose imaginary parts are respectively − ∞ and + ∞.   Theorem 2 can be simplified by introducing a constant M − less than the minimum eigenvalue of ∂ λ H (which is just . That leads to Corollary 1. If the partition function is analytic on the half complex plane λ λ ( ( ) ≥ ) R 1 of a physical parameter, the partition function along this line uniquely determines the partition function on that half complex plane, that is, , we have Corollary 2. If the partition function is analytic on the half complex plane λ λ ( ( ) ≤ ) R 2 of a physical parameter, the partition function along this line uniquely determines the partition function on that half complex plane, that is, for any complex parameter λ′ satisfying λ λ −∞ < ( ′) < R 2 . Corollaries 1 and 2 are particularly usefully for boson systems, where the partition functions may not be analytic on the whole complex plane of a parameter. For example, the partition function of free bosons has singularity points along the imaginary axis.
Equations (2-4) establish the holographic principle for partition functions. In practical application, one can make use of the partition functions of weakly interacting systems, which can be easy to calculate, to derive the partition functions of strongly interacting systems, which are usually difficult to calculate since the perturbation method may fail in absence of small parameters.
Possible experimental realization of thermodynamic holography. Recently our group discovered that the partition function of a system with a complex parameter is equivalent to the quantum coherence of a probe spin coupled to the system 10-12 . One can use a probe spin-1/2 x y which is equivalent to the partition function with a complex parameter, λ + itη/β. Now the evolution time serves as the imaginary part of the physical parameter. Recently, Lee-Yang zeros have been observed via such a measurement 12 . In more general cases ( , , it is possible to engineer the probe-bath interaction so that the probe spin coherence still has the form in equation (5) 11 . In terms of the probe spin coherence, equation (2) can be rewritten as Similarly, equation (3) can be rewritten as Note that the probe spin coherence resembles the form of quantum quenches 15 . Therefore the quantum quench dynamics may also be studied using the thermodynamic holography.
Equations ( Thus we can extract the full thermodynamic properties of the system from the probe spin coherence measurement for just a single value of the physical parameter. Note that previously the free energy difference (Δ F) has been related to the work (Δ W) in a non-equilibrium physical process by the Jarzynski equality 16 . The Jarzynski equality is particularly useful for determining free energy differences for small thermodynamic systems such as quantum engines and biomolecular systems [17][18][19] . Using the thermodynamic holography and the Jarzynski equality, we establish a general relation between the probe spin coherence, the work done on small systems, and the free energy changes. This general relation is indeed the foundation of two recent proposals for experimental measurement of the characteristic function of the work distributions 20,21 , which plays a central role in the fluctuation relations in quantum quenches 15 and more generally in non-equilibrium thermodynamics 22 . The power of the thermodynamic holography is that one can obtain free energy change for any parameters using the probe spin coherence measurement for just one value of the parameter instead of quenching the system to various parameters [17][18][19] .
The thermodynamic holography can also be used to determine the probe spin coherence for an arbitrary parameter by the coherence measurement for just one value of the parameter. Choosing a complex parameter λ′ + it′ η/β, we have the probe spin coherence The holographic method can be further simplified in many cases. Since the partition function, , is the sum of exponential function of the Hamiltonian, the probe spin coherence would be a periodic function of time if the energy level differences of a system are quantized in some unit, such as in the spin Ising models 10 . Then in such cases one does not need to measure the probe spin coherence as a function of time from − ∞ to + ∞. Instead, the probe spin coherence in one period of time, from 0 to 2π/η, is sufficient to produce the full information of the partition function. In such cases, equation (9)

Discussion
Thermodynamic holography of a mechanical resonator coupled to a probe spin. To demonstrate the idea of thermodynamic holography, we study an experimentally realizable system as the model example, namely, a nitrogen-vacancy (NV) centre spin coupled to a nano-mechanical resonator 23,24 (see Fig. 2a). This model may also be realized in a superconducting resonator 25  But the partition function is analytic on the half complex plane of frequency with positive real part (See Methods). Hence the holographic principle applies. We shall demonstrate that the free energy difference can be extracted from the probe spin coherence measurement of the NV center. We assume that the NV center is initialized in the superposition state + 1 0 and the mechanical resonator in the thermal equilibrium. Note that in the current case the probe spin coherence is a periodic function of time since the energy levels of the oscillator are equally spaced. So the probe spin coherence in one period of time, from 0 to 2π/η, is sufficient to yield the full information of the partition function. The measurement time of the NV center spin coherence is ~2π/η ≈ 100 μs, which is within the spin coherence time of an NV centre in an isotopically purified diamond 27 . Figure 2b presents the real (red solid line) and imaginary (blue dashed line) parts of the NV centre spin coherence as functions of time for the resonator at temperature 150 mK (βω = 1). From the spin coherence we can obtain the free energy (relative to the value at βω = 1) for arbitrary βω. Figure 2c shows the exponentiated free energy difference (green squares and blue dots) constructed via equation (11) from the NV center spin coherence shown in Fig. 2b. The green squares are obtained with numerical integration of 63 evenly separated data points of the NV center spin coherence during one period of evolution at βω = 1. The error in the numerical integration is π ∼( ) ″( )/( × ) ≤ . f x 2 1 2 63 001 3 2 , where . The results obtained by the holographic method agree with the direct calculation of the free energy (the solid red line in Fig. 2c) within the numerical errors.
The holographic approach can also be used to determine the probe spin coherence for arbitrary βω. The real and imaginary parts of the constructed spin coherence for βω = 2 as functions of time are presented respectively in Fig. 3a,b. The green squares are obtained by equation (12) with numerical integration of 63 evenly spaced data points of the NV center spin coherence at βω = 1 during one period of time. The error in the numerical integration is π ∼( ) ″( )/( × ) ≤ . f x 2 1 2 63 002 3 2 . The blue dots are obtained by numerical integration of 157 evenly spaced data points with numerical error π ∼( ) ″( )/( × ) ≤ . f x 2 1 2 157 0 002 3 2 . The results agree with the direct solution within numerical errors.

Summary
In this work we have established the concept of thermodynamic holography. We prove that the partition function of a finite physical system in an area of the complex plane of a physical parameter is uniquely determined by its values along the boundary. Since the partition function with a complex parameter is equivalent to the probe spin coherence, one can experimentally implement the thermodynamic holography through probe spin coherence measurement for just one physical parameter. Thermodynamic holography may have applications in studying thermodynamics of nano-scale systems (such as molecule engines 28 , nano-generators 29 and macromolecules 17,30 ) and provide a new approach to many-body physics.

Methods
Proof of Theorem 1. The system has a finite number (K) of basis states. We define a series of func- , which are sums of a finite number of polynomial functions of λ. Obviously, all these functions are analytic in the whole complex plane of λ. Considering the parameter in the range λ ≤ Λ, for an arbitrarily small quantity ε, there exists an integer N ε such that i.e., the function series uniformly converge to the partition function in the parameter range λ ≤ Λ. According to the uniform convergence theorem of analytic functions 9 , the partition function is analytic for λ ≤ Λ. Since Λ can be chosen to be arbitrarily large, Theorem 1 is proved.
A trace inequality. Lemma   Proof. For a system with a finite number of basis states, both A and B are finite dimensional matrices. By Hölder's inequality for finite-dimensional matrices 31  Generalization of Theorem 1 to certain infinite-dimensional systems. For a physical system with an infinite number of basis states (such as bosons), we assume that the basis states are discrete (countable), which is always possible since we can confine the system using a sufficiently large box. We can generalize Theorem 1 to infinite-dimensional Hamiltonians that satisfy the specified condition for