Abstract
Universality classes of nonunitary critical theories in twodimensions are characterized by the central charge. However, experimental determination of the central charge of a nonunitary critical theory has not been done before because of the intrinsic difficulty that complex parameters usually occur in nonunitary theory, which is not physical. Here we propose to extract the effective central charge of the nonunitary critical point of a twodimensional lattice model from the quantum coherence measurement of a probe spin which is coupled to the lattice model. A recent discovery shows that quantum coherence of a probe spin which is coupled to a bath is proportional to the partition function of the bath with a complex parameter. Thus the effective central charge of a nonunitary conformal field theory may be extracted from quantum coherence measurement of a probe spin which is coupled to a bath. We have applied the method to the YangLee edge singularity of the twodimensional Ising model and extracted the effective central charge of the YangLee edge singularity with good precision and tested other predictions of nonunitary conformal field theory. This work paves the way for the first experimental observation of the effective central charge of nonunitary conformal field theory.
Similar content being viewed by others
Introduction
Scale invariance is one of the most intriguing features of statistical mechanics^{1,2}. Local scale invariance (conformal invariance) at a critical point has been demonstrated to be remarkably powerful in two dimensions (2D)^{3,4}. Universality classes of critical phenomena in twodimensions are characterized by a single dimensionless number c, termed the central charge of the Virasoro algebra^{5}. It was demonstrated that^{6} unitarity constrains the values of c < 1 to be quantized, c = 1 − 6/[m(m + 1)] with m = 3, 4, 5, … . For such theories, the conformal dimensions of the primary fields are given by the Kac formula^{7}.
Nonunitary conformal field theory is often characterized by a negative effective central charge^{8,9,10}. However, how to experimentally extract the effective central charge of the nonunitary critical point has not been known before because of the intrinsic difficulty that complex parameter occurs in nonunitary theory, which is not physical. It was found that the free energy at a conformal invariant critical point is linearly related to the effective central charge of the corresponding conformal field theory^{8,11,12}. Thus experimental measurement of the free energy with a complex parameter is essential for extracting the effective central charge of a nonunitary conformal field theory. A recent discovery shows that the partition function with a complex parameter is related to the central spin decoherence^{13,14}. This implies that the effective central charge of a nonunitary theory may be extracted by measuring the quantum coherence of a probe spin coupled to the 2D critical theory.
In this article, we propose to extract the effective central charge of a nonunitary critical point from the quantum coherence measurement of a probe spin which is coupled to the critical model. We show that the effective central charge of the nonunitary conformal field theory is related to the quantum coherence of the probe spin. We have applied the method to the YangLee edge singularity of the twodimensional Ising model, and showed that the central charge of the YangLee edge singularity could be extracted with good precision from central spin coherence measurement. Furthermore, we show that other predictions of nonunitary conformal field theory could be tested by measuring the central spin coherence.
Results
Fundamentals of Conformal Minimal Models
The simplest conformal field theory are those having finite number of independent fields, so called minimal conformal field theory^{4}. The central charges of the conformal minimal models satisfy^{3,6}
where p and p′ are coprime integers. For p − p′ > 1, there always exist two integers r_{0}, s_{0} satisfying that r_{0}p − s_{0}p′ = 1, and the corresponding theory has a negative conformal dimension^{6},
In such case, the associated conformal minimal model is nonunitary^{8,9,10}. We shall briefly review some important predictions of the nonunitary conformal field theory.
Free Energy at Nonunitary Conformal Field Theory in TwoDimensions
Let us consider a parallelogram with vertices at (0, 2π, 2πτ, 2π(1 + τ)). The complex number τ with Im τ > 0 is the modular parameter of the parallelogram. A torus with periodic boundary condition can be constructed from a finite cylinder of length 2π Im τ by joining the ends and performing a twist around the axis by 2π Re τ. Thus we have the partition function on the torus^{2,4}
Here q = e^{2πiτ}, \(\bar{q}={e}^{2\pi i/\tau }\) and \(({L}_{0},{\bar{L}}_{0})\) are the generators of Virasoro algebras. Modular invariance gives the behaviour of the partition function as q → 0^{11,12}
Here c_{eff} = c for the unitary case and \({c}_{{\rm{eff}}}=c24{h}_{{r}_{0}{s}_{0}}=1\frac{6}{pp^{\prime} }\) for nonunitary theory.
Thus one has the free energy at critical point in an L × L′ rectangular lattice with periodic boundary condition^{8},
Here x ≡ L′/L. For systems with c_{eff} > 0, if the area of a two dimensional lattice S = L′ × L is fixed and the shape varies, F is in Equation (5) is maximal for a square lattice L′ = L(x = 1). Thus there is a thermodynamic driving force for elongation of the domain and this may be understood due to an attraction of the walls of the rectangle with force per unit length inversely proportional to the square of their separation. From this perspective, the tendency to elongation should be a general geometric effect. From Equation (5), the free energy per particle is,
Here the first term depends on the area and is nonuniversal. However the second term depends on the effective central charge c_{eff}, which is universal for conformal field theory. Equation (6) provides a method to the determination of central charge c_{eff} using finitesize scaling methods of the free energy at conformal invariant critical point. However, nonunitary conformal field theory is usually related to a complex parameter. This implies that we have to measure the free energy with a complex parameter in order to test the predictions of nonunitary conformal field theory. Recently, the author and his collaborators found that the central spin coherence and the partition function with a complex parameter are deeply related^{13,14}. For completeness, we shall review briefly the method in the next section.
Partition Function with a Complex Parameter and Central Spin Decoherence
To extract the central charge of nonunitary conformal invariant critical point, we should design a method for measuring the free energy with a complex parameter. Previous investigations show that the central spin coherence and partition function are deeply related^{13,14,15,16,17,18}, which we briefly explain below. Let us consider a general manybody system with Hamiltonian,
where H_{0} and H_{1} are two competing Hamiltonians and λ is a control parameter of the system. We introduce a probe spin1/2 (all termed central spin1/2) coupled to the manybody system (bath), with probe bath interaction H_{ I } = −ηS_{ z } ⊗ H_{1} and η being a coupling constant between the probe spin and bath and \({S}_{z}=\uparrow \rangle \langle \uparrow \downarrow \rangle \langle \downarrow \) being the Pauli matrix of the probe spin along z direction. If we initialize the probe spin in a superposition state as \(\uparrow \rangle +\downarrow \rangle \) and the bath at thermal equilibrium state with inverse temperature β = 1/T(k_{ B } = 1) described by ρ_{0} = e ^{−βH(λ)}/Z(β, λ) with Z(β, λ) = Tr[e ^{−βH(λ)}] being the partition function of the bath. Then the probe spin and bath evolve in time together under Hamiltonian H + H_{ I }. At time t, the quantum coherence of the probe spin, defined by 〈S_{+} (t)〉 ≡ 〈S_{ x }(t)〉 + i〈S_{ y }(t)〉, has an intriguing form as^{13,14},
The denominator in the above equation is nonzero for real temperature β and real control parameter λ. The numerator resembles the form of a partition function but with a complex control parameter, λ + iηt/β. The evolution time t plays the role of the imaginary part of the control parameter. Equation (8) establishes the relation between partition function with a complex parameter and the central spin coherence, which leads to the first experimental observation of LeeYang zeros^{16,17}.
In experiment, the central spin coherence can be measured from the following steps:

(1)
Initialize the system in the equilibrium state described by the Gibbs density matrix ρ_{0} = e^{−βH(λ)}/Z(β, λ) and the central spin in the \(\downarrow \rangle \) state;

(2)
Apply a π/2 pulse along the y direction to the central spin, which then transforms the central spin in a coherent superposition state as \((\uparrow \rangle +\downarrow \rangle )/\sqrt{2}\);

(3)
The central spin and the system evolve together in time for a time interval t and the evolution is governed by the total Hamiltonian H(λ) + H_{ I };

(4)
Apply a π/2 pulse to the central spin along the y direction;

(5)
Measure the average value of the central spin along z and y directions respectively, namely, 〈S_{ z }(t)〉 and 〈S_{ y }(t)〉, which are the real and imaginary part of the quantum coherence of the central spin 〈S_{+} (t)〉.
Combining Equation (6) and Equation (8), the free energy at a nonunitary critical point can be written as
Here λ + iηt_{ c }/β is the critical point of the nonunitary conformal field theory in twodimensions. In Equation (9), \(\mathop{\mathrm{lim}}\limits_{L^{\prime} \to \infty }\frac{\mathrm{ln}({Z}_{LL^{\prime} }(\beta ,\lambda ))}{LL^{\prime} }\) is free energy per particle away from critical point and thus can be considered as a constant. Therefore we have
Equation (10) is the central result of this work. It implies that the central spin coherence at the critical point of a nonunitary conformal field theory is related to the effective central charge of the nonunitary conformal field theory. Therefore, one may be able to experimentally extract the effective central charge of a nonunitary conformal field theory by measuring the quantum coherence of a probe spin which is coupled to a critical 2D lattice.
The effective central charge of a nonunitary conformal field theory can be extracted experimentally from the following steps. Firstly, one measures the quantum coherence of a probe spin coupled to a 2D lattice with different sizes as a function of time 〈S_{+} (t)〉. Secondly, doing finitesize scaling analysis^{18}, one can get the critical point of the conformal field theory, that is λ + iηt_{ c }/β. This can be achieved by finite size scaling analysis^{18}. Third, getting quantum coherence at the critical point 〈S_{+} (t_{ c })〉 for different lattice sizes and fitting these data points with a linear function of 1/L^{2}, the coefficient of 1/L^{2} tells us the effective central charge of the nonunitary conformal field theory according to Equation (10). Because central spin coherence is directly experimentally measurable^{19,20,21,22,23}, we can extract the effective central charge of nonunitary critical point in twodimension from central spin coherence measurement according to above procedures. To demonstrate the feasibility of the proposal, we shall study the YangLee edge singularity^{24,25} in twodimensional Ising model, which is a typical example of nonunitary conformal invariant critical point as pointed out by Cardy^{26}.
Besides, numerical methods such as the thermodynamic Bethe ansatz approach^{27,28,29} has been used successfully to compute the free energy of both unitary and nonunitary critical points. Recently, the effective central charge also plays a role is in the study of the entanglement entropy of nonunitary conformal field theories^{30,31,32}. This provides a new physical quantity that can be evaluated numerically and from which the effective central charge may be extracted.
Discussion
YangLee Edge Singularity in Twodimensional Ising Model
To illustrate the above idea, we study the YangLee edge singularity in twodimensional (2D) Ising model. The Hamiltonian of the square lattice Ising model with N columns and M rows is
Here σ_{i,j} = ±1 is the spin in the site (i, j) and J is the ferromagnetic coupling constant between nearestneighbor spins in the lattice and h is the magnetic field experienced by all the spins and periodic boundary conditions are applied. We set J = 1 and take it as the basic unit of the energy scale. The 2D Ising model at zero magnetic field h = 0 has been exactly solved by Onsager in 1944^{33} and there is a finite temperature phase transitions at β_{ c } = In(1 + 5)/2^{33,34}. For 2D Ising model under a finite magnetic field, there is no exact solution available but one can map the problem into 1D quantum Ising model with both longitudinal and transverse field by transfer matrix method^{35,36}.
At any temperature above the critical temperature, β < β_{ c }, the 2D Ising model is critical for a purely imaginary magnetic field h_{ c } = ±ih_{ c }(β). This critical point with complex magnetic field is termed the YangLee edge singularity^{24,25}. As pointed out by Cardy^{26}, the YangLee edge singularity at 2D Ising model is a nonunitary conformal invariant critical point and corresponds to the minimal conformal field theory with p = 5, p′ = 2, c = −22/5. There are only two primary fields, the identity I with h_{11} = 0 and the scalar field with h_{12} = −1/5. The only modular invariant partition function build out of these two fields is^{8,9}
As q → 0, we have,
Here c_{eff} = c − 24h_{12} = 1 − 6/pp′ = 2/5 for the YangLee edge singularity in 2D lattices.
Since YangLee edge singularity occurs only for purely imaginary magnetic field for ferromagnetic Ising model, we have from Equation (10)
Here 〈S_{+} (t_{ c })〉 = Z_{ MN }(β, ih_{ c }(β)/Z_{ MN }(β, 0) is the central spin decoherence at YangLee edge singularity point and t_{ c } = βh_{ c }(β)/η.
First, we study the YangLee edge singularity of the 2D rectangular lattice Ising model with M × N spins at inverse temperature β = 0.2. The YangLee edge critical point for β = 0.2 located at t_{ c } = 0.257, which is obtained from the finite size scaling method at YangLee edge singularity^{18}. We then show the central spin coherence at the YangLee edge singularity in 2D Ising model at β = 0.2 as a function of N for different lattice sizes when M = 80 fixed in Fig. 1(a). From Equation(14), we know that for a rectangular lattice with M × N spins, if M × N, the logarithm of the central spin coherence per spin at the YangLee edge singularity is a linear function of 1/N^{2} with the slop being πc_{eff}/6. We then make a linear fit of the logarithm of the central spin coherence per bath spin at the YangLee edge singularity as a function of 1/N^{2} in Fig. 1(b) and found that the slope is 0.210 ± 0.001. From Equation(14), we know that the effective central charge c_{eff} ≈ 0.40 ± 0.01. Thus the estimated effective central charge agrees with the exact solution c_{eff} = 2/5 perfectly.
Since YangLee edge singularity occurs for any temperature above the critical temperature of the bulk system β < β_{ c }. To test the claim, we study the YangLee edge singularity at a different temperature β = 0.3. From the finite size scaling analysis, we know that the YangLee edge critical point for β = 0.3 located at t_{ c } = 0.085^{18}. Figure 2 (a) presents the central spin coherence at the YangLee edge singularity in 2D Ising model at β = 0.3 as a function of N for different lattice sizes when M = 80 fixed. We then make a linear fit of the logarithm of the central spin coherence per spin at the YangLee edge singularity as a function of 1/N^{2} in Fig. 2 (b) and found that the slope is 0.205 ± 0.001. From Equation (14), we know that the effective central charge c_{eff} ≈ 0.39 ± 0.01. Thus the estimated effective central charge agrees with the exact solution c_{eff} = 2/5.
We further test the effect of elongation of the 2D lattices in nonunitary conformal field theory corresponding to the twodimensional critical point. Because of the intimate link between the central spin coherence and the free energy, it is sufficient to test the central spin coherence due to the elongation effect. Figure 3 shows the logarithm of the central spin coherence per bath spin at the YangLee edge singularity of 2D Ising model at β = 0.2 for 2D rectangular lattices with lattice sizes being 2 × 50, 4 × 25, 5 × 20, 10 × 10 (with same area M × N = 100) as a function of the logarithm of the ratio of two edges of the rectangular lattice. First, one can see that the logarithm of the central spin coherence per bath spin is invariant under the modular transformation, interchanging of M and N, i.e. x ↔ 1/x. Second, the logarithm of the central spin coherence per bath spin of a 2D domain with a fixed area is maximum for square x = 1 and decreases as x becomes larger or smaller. This means there is a thermodynamic driving force for elongation of the a rectangular domain at the nonunitary conformal field theory.
Summary
In summary, we show that the effective central charge of the nonunitary conformal invariant critical point of a twodimensional lattice model can be extracted from the quantum coherence measurement of a probe spin which is coupled to the twodimensional lattice model. Furthermore, we show that the other predictions by the nonunitary conformal field theory can also be tested from measuring the quantum coherence of the probe spin. Thus measuring the quantum coherence of a single probe spin provides a practical approach to studying the nonunitary conformal invariant twodimensional interacting manybody systems and pave the way for the first experimentally verification of the predictions of nonunitary conformal field theory.
Methods
The 2D Ising model without magnetic field can be mapped to a 1D Ising model with a transverse field by the transfer matrix method^{35}. For a 2D Ising model in finite magnetic field, it was mapped by transfer matrix method to a 1D Ising model with both longitudinal field and transverse field^{36}, which can be numerically diagonalized. Therefore the partition function and hence the probe spin coherence for the 2D Ising model in finite magnetic field can be obtained.
References
Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena (Clarendon Press, Oxford, 1971).
Cardy, J. L. Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, 1996).
Belavin, A. A., Polyakov, A. M. & Zamolodchikov, A. B. Infinite conformal symmetry in twodimensional quantum field theory. Nucl. Phys. B 241, 333 (1984).
Henkel, M. Conformal Invariance and Critical Phenomena (SpringerVerlag Berlin Heidelberg,1999).
Virasoro, M. A. Subsidiary Conditions and Ghosts in Dual Resonance Models. Phys. Rev. D 1, 2933 (1970).
Friedan, D., Qiu, Z. & Shenker, S. Conformal Invariance, Unitarity, and Critical Exponents in Two Dimensions. Phys. Rev. Lett. 52, 1575 (1984).
Kac, V. G. Group Theoretical Methods In Physics, edited by Beiglbock and Bohm, Lecture Notes in Physics Vol. 94 (SpringerVerlag, New York, 1979).
Itzykson, C., Saleur, H. & Zuber, J. B. Conformal invariance of nonunitary 2dmodels. Europhys. Lett. 2, 91 (1986).
Itzykson, C. & Zuber, J. B. Twodimensional conformal invariant theories in a torus. Nucl. Phys. B 275, 580 (1986).
Wydroa, T. & McCabeb, J. F. Tests of conformal field theory at YangLee edge singularity. AIP Conference Proceedings. 1198, 216 (2009).
Blote, H. W. J., Cardy, J. L. & Nightingale, M. P. Conformal Invariance, the Central Charge, and Universal FiniteSize Amplitudes at Criticality. Phys. Rev. Lett. 56, 742 (1985).
Affleck, I. Universal Term in the Free Energy at a Critical Point and the Conformal Anomaly. Phys. Rev. Lett. 56, 746 (1985).
Wei, B. B. & Liu, R. B. LeeYang Zeros and Critical Times in Decoherence of a Probe Spin Coupled to a Bath. Phys. Rev. Lett. 109, 185701 (2012).
Wei, B. B., Chen, S. W., Po, H. C. & Liu, R. B. Phase transitions in the complex plane of physical parameters. Sci. Rep. 4, 5202 (2014).
Wei, B. B., Jiang, Z. F. & Liu, R. B. Thermodynamic holography. Sci. Rep. 5, 15077 (2015).
Peng, X. H. et al. Experimental Observation of LeeYang Zeros. Phys. Rev. Lett. 114, 010601 (2015).
Ananikian, N. & Kenna, R. Imaginary magnetic fields in the real world. Physics. 8, 2 (2015).
Wei, B. B. Probing YangLee edge singularity by central spin decoherence. New J. Phys. 19, 083009 (2017).
Childress, L. et al. Coherent dynamics of coupled electron and nuclear spin qubits in diamond. Science 314, 281 (2006).
Hanson, R. et al. Coherent dynamics of a single spin interacting with an adjustable spin bath. Science 320, 352 (2008).
Bluhm, H. et al. Dephasing time of GaAs electronspin qubits coupled to a nuclear spin bath exceeding 200μs. Nature Phys. 7, 109 (2010).
Li, Y. et al. Intrinsic spin fluctuations reveal the dynamical response function of holes coupled to nuclear spin baths in (In,Ga)As quantum dots. Phys. Rev. Lett. 108, 186603 (2012).
Zhao, N. et al. Sensing single remote nuclear spins. Nature Nanotech. 7, 657 (2012).
Kortman, P. J. & Griffiths, R. B. Density of Zeros on the LeeYang Circle for Two Ising Ferromagnets. Phys. Rev. Lett. 27, 1439 (1971).
Fisher, M. E. YangLee Edge Singularity and φ3 theory. Phys. Rev. Lett. 40, 1610 (1978).
Cardy, J. L. Conformal invariance and YangLee edge singularity. Phys. Rev. Lett. 54, 1354 (1985).
Zamolodchikov, A. B. Thermodynamic Bethe ansatz in relativistic models: Scaling 3state Potts and LeeYang models. Nucl. Phys. B 342, 695 (1990).
Klassen, T. K. & Melzer, E. Purely elastic scattering theories and their ultraviolet limit. Nucl. Phys. B 338, 485 (1990).
Klassen, T. K. & Melzer, E. The thermodynamics of purely elastic scattering theories and conformal perturbation theory. Nucl. Phys. B 350, 635 (1991).
Bianchini, D. et al. Entanglement entropy of nonunitary conformal field theory. J. Phys. A: Math. Theor. 48, 04FT01 (2015).
Bianchini, D. & Ravanni, F. Entanglement entropy from corner transfer matrix in Forrester Baxter nonunitary RSOS models. J. Phys. A: Math. Theor. 49, 154005 (2016).
Couvreur, R., Jacobsen, J. L. & Saleur, H. Entanglement in nonunitary quantum critical spin chains. Phys. Rev. Lett. 119, 040601 (2017).
Onsager, L. Crystal statistics. I. A Twodimensional model with an orderdisorder transition. Phys. Rev. 65, 117 (1944).
Kramers, H. A. & Wannier, G. H. Statistics of the Twodimensional Ferromagnet. Part. I. Phys. Rev. 60, 252 (1941).
McCoy, B & Wu, T. T. The twodimensional Ising model (Harvard University Press, Cambridge,1973).
Schultz, T. D., Mattis, D. C. & Lieb, E. H. Twodimensional Ising model as a soluble problem of many fermions. Rev. Mod. Phys. 36, 856 (1964).
Acknowledgements
B.B.W. was supported by National Natural Science Foundation of China (Grants No. 11604220) and the Startup Funds of Shenzhen University (Grants No. 2016018).
Author information
Authors and Affiliations
Contributions
B.B.W. conceived the idea, performed the research and wrote the manuscript.
Corresponding author
Ethics declarations
Competing Interests
The author declares no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wei, BB. Probing Conformal Invariant of Nonunitary TwoDimensional Systems by Central Spin Decoherence. Sci Rep 8, 3080 (2018). https://doi.org/10.1038/s41598018213607
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598018213607
This article is cited by

Fluctuation relations for heat exchange in the generalized Gibbs ensemble
Frontiers of Physics (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.