Quantum criticality at infinite temperature revealed by spin echo

Quantum criticality at infinite temperature revealed by spin echo Shao-Wen Chen*, Zhan-Feng Jiang* & Ren-Bao Liu Department of Physics and Centre for Quantum Coherence, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China * These authors contribute equally. Criticality occurs widely at transitions between qualitatively different states of matters. Quantum criticality due to diverging quantum fluctuations [1], which can be observed by varying system parameters at zero temperature, is particularly important as it indicates the emergence of new orders of quantum matters [-8]. However, at temperature higher than the system's interaction strength thermal fluctuations will dominate over the quantum fluctuations and hence conceal the quantum criticality. Extremely low temperatures are required for quantum criticality to occur in many interesting systems. For example, for nuclear spins in solids [5] and cold atoms in optical lattices [6-8], temperatures of or even Kelvin are required [5, 9]. Here we show that quantum criticality can be observed at high or even infinite temperature by measuring the echo signal of a probe spin coupled to a quantum many-body system, because the spin echo can remove the thermal fluctuation effect [1] and therefore reveal the quantum fluctuation effect. We also establish the correspondence between the probe coherence time and the inverse temperature required for the onset of quantum criticality. In particular, quantum criticality that would occur below or even Kelvin can be detected by spin echo with coherence time longer than milliseconds or seconds, respectively. This discovery provides a new route to a vast land of quantum matters [-8], via trading long coherence time of a quantum probe for extremely low temperature. Criticality is an important phenomenon in physics. It occurs at continuous phase transitions in which a new order of matter, such as ferromagnetism and superconductivity, emerges with lowering the temperature of a macroscopic system. At the phase transition point the thermal fluctuation diverges leading to the critical behaviour. Quantum criticality [1] refers to critical phenomena that accompany quantum phase transitions at zero temperature. At zero temperature, a macroscopic system stays in its ground state. By tuning a parameter in the system Hamiltonian H(), the ground state, may change dramatically at a critical point c. Thus the quantum fluctuation diverges at the quantum phase transition point. Quantum criticality is important as it signatures emergence of new quantum matters and new physics [-8]. Naturally, observation of quantum criticality requires temperature T much lower than the interaction strength in the systems, which is extremely low in many interesting physical systems (such as to Kelvin for nuclear spins in solids [5] and cold atoms in optical lattices [9]). Such limitation excludes many new classes of quantum matters and hence new physics from experimental investigation. The discovery reported here is inspired by the famous Lee-Yang "unit circle" theorem [11] in statistical physics. The theorem states that the thermodynamic functions of certain macroscopic systems have singularities distributed along a circle in the complex plane of inverse temperature ( 1 T ) (see Fig. a for illustration). When the temperature T is lowered from infinity 1

(or is increased from zero) through the singularity circle on the real axis, a phase transition and hence criticality occur. An intriguing question is: Would criticality occur if the inverse temperature is increased from zero through the singularity circle on the imaginary axis? In theoretical physics study, time is sometimes taen as an imaginary inverse temperature and vice versa. These considerations motivate us to conjecture that even if the temperature is infinite (), quantum criticality may be detected if the system response is measured at a time long enough after initiation. The ey is to devise a time-dependent measurement that is sensitive to quantum fluctuation in macroscopic systems. A previous study has proposed Loschmidt echo as a probe of quantum criticality [1], which we briefly explain below. One can couple a system with a Hamiltonian H() H H 1 to a probe spin-1/ with interaction z z g H, where g is a small coupling constant and is the Pauli 1 matrix of the probe spin along the z-axis. If the probe spin is initially prepared in a superposition state as and the system in the ground state,, the coupled probe and system will evolve to, ( t), ( t) with gt ih, ( t) e,. The Loschmidt echo, ( ), ( ) x y L t t t i decays with time t, corresponding to free-induction decay (FID) of the probe spin polarization in the x-y plane, i.e., spin decoherence. The probe spin decoherence is greatly enhanced at the critical point c since the quantum states of the system at different sides of the critical point evolve dramatically differently. Nuclear magnetic resonance experiments that simulate quantum phase transitions at effective zero temperature have shown such enhancement of FID at the state crossover of a three-nucleus system [13]. The effect of dynamical decoupling control over the probe spin has also been studied [14]. A series of theoretical wors have established the relation between quantum phase transitions and the fidelity [15, 16], in this case the overlap between the ground states for two slightly different parameters, g, g. The fidelity theory has also been extended to finite temperature transitions [17]. The Loschmidt echo or FID, however, lie other conventional measurement of quantum criticality, requires that temperature be much lower than the interaction strength of the system. In magnetic resonance spectroscopy, spin echo can be used to eliminate the effect of thermal fluctuation (or inhomogeneous broadening) [1] with the decay of echo signal induced mostly by quantum fluctuation. In spin echo, the probe spin is flipped ( ) at a time, and the spin coherence is measured at t. The thermal fluctuation due to probability distribution of the local field (gh 1 ) imparted by the system to the probe spin will have opposite effects before and after the probe spin flip and therefore cancel out in spin echo. Then the quantum fluctuation taes effect. This motivates us to use spin echo to study quantum criticality at high or even infinite temperature. We first formulate the echo signal of a probe spin-1/ coupled to a macroscopic system (a bath) at finite temperature. The bath at thermal equilibrium with inverse 1 H temperature is described by a density matrix Z e with the normalization factor Z Tr e H. To have conclusive results, we choose an exactly solvable model

(schematic shown in Fig. 1a), namely, the one-dimensional Ising model in a transverse field with Hamiltonian N x x z j j1 j 1 H H H, N (1) j1 j1 where N is the number of spins in the chain of a ring configuration (with N j j) and x / y / z is the Pauli matrix of the jth spin along the x/y/z-axis. The probe-bath interaction j is z z HI g H1 B, where B gh1 is the local field noise on the probe spin which causes decoherence. The probe strength is chosen to scale with the bath size as g ~1 N which is for a large bath so that the bath is only wealy perturbed ihgt ihgt by the probe. The FID L t Tr e e FID and the echo signal Tr ih g ih g ih g ih g LSE e e e e. The spin chain model has no phase transition at finite temperature and has quantum criticality between a ferromagnetic order for and a paramagnetic order for. This model has been used previously to demonstrate the effect of quantum criticality on Loschmidt echo [1]. A previous study on spin echo for this model [18], however, missed the quantum criticality features at high temperature due to inadequate approximation. Figure 1b shows that FID of the probe spin is greatly enhanced at the quantum critical point when the temperature is zero ( ), which is consistent with previous study. The sharp dip at the critical point, however, is blurred with increasing temperature (see Supplementary Figure S1) and disappears at infinite temperature (Fig. 1c). In contrast, the spin echo signal (Fig. 1d) presents enhanced decoherence at the critical point even at infinite temperature ( ). The feature is pronounced when t. The above-mentioned phenomena can be understood from the energy level structure and noise spectrum of the spin chain bath. The fluctuation of the local field B gh has both thermal and quantum components. At finite temperature, the bath 1 has probability P Z 1 exp E n in the energy eigenstate n, of eigen energy n E n. The local field fluctuation has the correlation function C t B t B B t B with O P n, O n, denoting the iht iht expectation value of an operator O and Ot e Oe is determined by the noise spectrum exp n n. The probe spin decoherence S C t i t dt. The thermal n n is solely due to the thermal fluctuation part Sth P n, B n, B distribution in different eigenstates which yield different local field. Such thermal fluctuation is static and presents a sharp zero-frequency pea in the noise spectrum. Generally, the local field operator H 1 does not commute with the bath interaction Hamiltonian H. Thus transitions between different eigenstates by elementary excitations lead to quantum fluctuation. The quantum fluctuation is dynamical and has a S Q E E P n n m n, B m, m, B n,. At zero nm temperature the thermal fluctuation vanishes but the quantum fluctuation still exists due to elementary excitations from the ground state. At finite temperature the thermal and spectrum 3

quantum fluctuations coexist and the quantum fluctuation also includes contributions of transitions from the excited states. At high temperature, the thermal fluctuation is usually much stronger than the quantum fluctuation. As the thermal fluctuation is static its effect on the probe spin decoherence can be removed by spin echo. Then the decoherence is determined by the dynamical quantum fluctuation. In the long time limit, the decoherence would be mostly due to the low-frequency noise caused by low-energy or long-wavelength excitations in the bath, which are particularly important in quantum criticality. The excitation energy as a function of wavevector in our specific spin chain 1 cos. The excitation has a finite energy gap except for model is the critical point 1 (Fig. 1e). The quantum fluctuation spectrum is gapless at the critical point (Fig. 1f) and has a low-frequency cut-off for 1 (Fig. 1g). The quantum fluctuation is suppressed to some extent as temperature increases but never vanishes. Gapless fluctuation emerging at the critical point is responsible for the decoherence enhancement in the long time limit. We further explore the correspondence between the time and inverse temperature required to observe the quantum criticality. Fig. 1d shows that the decoherence enhancement at the critical point is visible only for t (the dimensionless interaction strength in the spin chain bath taen as unity). This is actually similar to conventional detection of quantum criticality where the inverse temperature is required to be. 1 z Fig. b presents the magnetic susceptibility of the spin chain N i i as a function of the inverse temperature and the external field strength (without coupling to the probe spin, i.e., g ). The quantum criticality feature is visible for large (e.g., resembling the echo signal as a function of time and field strength in Fig. 1d. Figure c compares the susceptibility for various inverse temperatures with the echo signal for various times, as functions of the field strength. The sharp features at the critical point are pronounced at similar values of and t. Furthermore, Fig. d demonstrates that the time required for the echo signal to display enhanced decoherence decreases with increasing the inverse temperature. This can be understood from the fact that the quantum fluctuation is stronger at lower temperature as shown in Figs. 1f & 1g. Actually, if the measurement time is long enough, even the FID would display a sudden transition at the critical point, as shown in Fig. e. For large time (t ), a sudden change of decoherence is seen at the critical point. This transition, however, is far beyond any feasible measurement since the thermal fluctuation has already caused complete decoherence and the remaining coherence at such a long time is as little as 1. Thus, the role of spin echo in detecting quantum criticality at high temperature can be viewed as a prolonging of the probe coherence time by the removal of the thermal fluctuation effects. The time-inverse temperature correspondence enables utilization of long coherence time to detect quantum criticality and hence new quantum matters which would occur at extremely low temperature. Physics that would emerge at 1 or 1 Kelvin [5, 9] now is within the reach of a probe with respective coherence time of milliseconds or seconds, even in solid-state systems at room temperature. A broad range of physical systems may be considered for studying quantum criticality at high temperature. Such systems should satisfy the following conditions. First, the system is relatively isolated from the environment within the time of measurement; second, there is a parameter such as an external field that is tuneable 4

across a quantum critical point; third, there exists a quantum probe with coherence time that can be extended (by spin echo, e.g.) to be much longer than the inverse interaction strength in the bath; and fourth, the probe-bath interaction is much weaer than the intra-bath interaction so that the bath dynamics is not strongly perturbed (otherwise the transition would occur in a finite region, as shown in Fig. S4 in Supplementary Information). A few examples of such systems are cold atoms in optical lattices or traps, defect spins in diamond, donor spins in silicon, and nuclear spins in large molecules, where probes with coherence time from milliseconds to ten seconds are available [19-]. Two specific systems are illustrated in Figs. 3a & 3b. The probe is a nitrogen-vacancy (NV) centre spin beneath a surface of high-purity diamond. The spin coherence of a single NV centre can be controlled and measured by optically detected magnetic resonance techniques [3]. In high-purity diamond, the NV centre spin has long coherence time (~ ms) []. The bath spins are electron spins of 16 O atoms encapsulated in 1 C 6 cages on the 16 O-terminated diamond surface [4]. The endohedral C 6 fullerenes can either be self-organized into a two-dimensional lattice (Fig. 3a) or be placed by atomic force microscopy techniques into a periodic ring (Fig. 3b). A wea magnetic field B is applied along the z-axis (normal to surface) to tune the bath across the critical point. The dipolar interaction between the spins depends inverse-cubically on distance and the distance between bath spins is much less than the probe-bath distance, so the wea probe condition is satisfied. The spin lattice in Fig. 3a may have non-trivial quantum orders such as spin liquids [4], and the spin ring in Fig. 3b resembles the Heisenberg-Ising (XXZ) model which has a quantum phase transition between the ferromagnetic and paramagnetic orders [5]. We numerically simulate the spin-ring system (see Method Summary and Supplementary Information). To simplify the simulation, we replace the spin-1 of the O atoms with spin-1/. In calculation for spins on a ring of radius 15 nm as shown in Figs. 3c-3g, we adopted the nearest-neighbor coupling approximation for interaction between bath spins (justification shown in Supplementary Information). As shown in Fig. 3c, the ring of coupled bath spins (with the coupling to the NV centre dropped) has two nearly degenerate ground states for magnetic field below the critical value ( BC. Gauss) [5], corresponding to spontaneous ferromagnetic ordering with the spins aligned along the ring in parallel. For field above the critical value, the ground state degeneracy is lifted and the bath is in a paramagnetic phase. This can be indeed seen from the long-range correlations of the spins (Fig. 3d), which presents a sharp decrease near the critical point at zero temperature. The transition region has finite width as the bath is finite. As temperature increases to above Kelvin, however, the transition is smeared out and eventually disappears at temperature approaching the interaction strength between neighboring bath spins (~.3 Kelvin or 3.1 s ). The FID of the NV spin at zero temperature is enhanced near the critical point (Fig. 3e). The decoherence, however, is far from complete and presents oscillation features, which is a finite-size effect (the oscillation period near the critical point is proportional to the number of spins in the ring). As temperature increases, the thermal fluctuation and hence the decoherence are enhanced, but the transition near the critical point is smeared out. At infinite temperature, the decoherence presents no critical feature except that a revival structure appears in the paramagnetic phase but is absent in 5

the ordered phase (Fig. 3f). Such revival, however, is an effect of finite and uniform probe strength (see Supplementary Information Fig. S for further discussion). While the revivals of FID signals provide an interesting detection of quantum criticality in finite systems at high temperature, the strong thermal fluctuation in general macroscopic systems would, however, conceal the quantum criticality. The spin echo signal (Fig. 3g) shows a sudden change at the critical point even at infinite temperature. Unlie the Ising spin chain model studied in Figs. 1 and, the current model presents a step feature rather than a dip in the spin echo decay. This is because the quantum fluctuation is large in the ordered phase due to the anisotropic dipolar interaction between the bath spins. In realistic experiments disorder may exist. Numerical chec shows that the quantum criticality feature is robust against disorder of % dislocation (Supplementary Information Fig. S6). Novel quantum matters and new physics may emerge in many possible physical systems and interacting models [-8]. While it can be extremely difficult to lower the temperature to the quantum critical regime, now we propose a new route to the wonderland of quantum matters, by pushing along the imaginary axis to the long-time limit in the complex plane of inverse temperature. Spin echo, by largely removing the thermal fluctuation effect, can prolong the coherence time of a quantum probe. It is conceivable that longer coherence time and therefore richer physics can be brought into the reach by applying many-pulse dynamical decoupling control over the probe [14, 6-8]. Therefore it can be envisaged that dynamical decoupling become a useful tool to study many-body correlations in baths, beyond its existing applications in noise spectrum measurement [9] and high-sensitivity metrology [3]. METHODS SUMMARY The Hamiltonian in Eq. (1) for the spin chain model can be exactly diagonalized by the standard Jordan-Wigner transformation and Bogoliubov transformation [1]. The Hamiltonian after the transformation becomes a free-fermion system H b b 1, where () is the energy-wavevector dispersion of the elementary excitations. The decoherence and noise spectra can be therefore calculated. The spins on diamond surface and the NV centre spin have dipolar interactions. In calculation for Figs. 3c-3g, only the dipolar interaction between the nearest neighbours in the bath is considered. Numerical chec confirms that inclusion of the full interaction between all spins does not significantly change the results. The Zeeman z energy of the spins is HZ ebsi. The NV centre spin has a zero-field splitting H z S i. Since the zero-field splitting GHz is much greater than the probe-bath interaction (~ MHz), the flipping between the NV centre spin states and 1 (quantized along the z-axis) is negligible. So the interaction terms containing x y S and S are dropped from the Hamiltonian H dip, which results in a so-called pure dephasing model in which the probe-bath interaction provides a local field fluctuation along the z-axis and causes decay of the centre spin polarization in the x-y plane. The transition 1 of the NV centre spin is used as a pseudo-spin-1/ to probe the quantum criticality in the bath. The energy levels of the bath spin ring are calculated by 6

exact diagonalization of the Hamiltonian. The decoherence and the spin correlation are calculated by exact numerical simulation. References 1. Sachdev, S. Quantum Phase Transitions (Cambridge University Press, New Yor, 1999).. Wen, X. G. Quantum orders and symmetric spin liquids. Phys. Rev. B 65, 165113 (). 3. Levin, M. & Wen, X. G. Photons and electrons as emergent phenomena. Rev. Mod. Phys. 77, 871-879 (5). 4. Balents, L. Spin liquids in frustrated magnets. Nature 464, 199-8 (1). 5. Oja, A. S. & Lounasmaa, O. V. Nuclear magnetic ordering in simple metals at positive and negative nanoelvin temperatures. Rev. Mod. Phys. 69, 1 136 (1997). 6. Lewenstein, M. et al. Ultracold atomic gases in optical lattices: Mimicing condensed matter physics and beyond. Adv. Phys. 56, 43-379 (7). 7. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 8, 885 964 (8). 8. Micheli, A., Brennen, G. K. & Zoller P. A toolbox for lattice-spin models with polar molecules. Nature Physics, 341-347 (6). 9. Ho, T. L. & Zhou, Q. Intrinsic heating and cooling in adiabatic processes for bosons in optical lattices. Phys. Rev. Lett. 99, 144 (7). 1. E. Hahn, Spin echoes. Phys. Rev. 8, 58-594 (195). 11. Lee, T. D. & Yang, C. N. Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model. Phys. Rev. 87, 41 419 (195). 1. Quan, H. T., Song, Z., Liu, X. F., Zanardi, P. & Sun, C. P. Decay of Loschmidt Echo Enhanced by Quantum Criticality. Phys. Rev. Lett. 96, 1464 (6). 13. Zhang, J., Peng, X., Rajendran, N. & Suter, D. Detection of quantum critical points by a probe qubit. Phys. Rev. Lett. 1, 151 (8). 14. Rossini, D. et al. Bang-bang control of a qubit coupled to a quantum critical spin bath. Phys. Rev. A 77, 511 (8). 15. Zanardi, P. & Paunovic, N. Ground state overlap and quantum phase transitions. Phys. Rev. E 74, 3113 (6). 16. You, W.-L., Li, Y. W. & Gu, S.-J., Fidelity, dynamic structure factor, and susceptibility in critical phenomena. Phys. Rev. E 76, 11 (7). 17. Zanardi, P., Quan, H. T., Wang, X. & Sun, C. P., Mixed-state fidelity and quantum criticality at finite temperature. Phys. Rev. A 75, 319 (7). 18. Yi, X. X., Wang, H. & Wang, W. Hahn echo and criticality in spin-chain systems. Eur. Phys. J. D 45, 355-36 (7). 19. Büning, G. K. et al. Extended coherence time on the cloc transition of optically trapped Rubidium. Phys. Rev. Lett. 16, 481 (11).. Balasubramanian, G. et al. Ultralong spin coherence time in isotopically engineered diamond. Nature Materials 8, 383-387 (9). 7

1. Tyryshin, A. M. et al. Electron spin coherence exceeding seconds in high-purity silicon. Nature Materials (11) doi:1.138/nmat318.. Ladd, T. D. e al. Coherence time of decoupled nuclear spins in silicon. Phys. Rev. B 71, 1441 (5). 3. Jelezo, F., Gaebel, T., Popa, I. Gruber, A. & Wrachtrup, J. Observation of coherent oscillations in a single electron spin. Phys. Rev. Lett. 9, 7641 (4). 4. Strobel, P., Riedel, M., Ristein, J. & Ley, L. Surface transfer doping of diamond. Nature 43, 439 (4). 5. Dmitriev, D. V., Krivnov, Y. Ya. & Ovchinniov, A. A. Gap generation in the XXZ model in a transverse magnetic field. Phys. Rev. B 65, 1749 () 6. Du, J. et al. Preserving spin coherence in solids by optimal dynamical decoupling. Nature 461, 165-168 (9). 7. de Lange, G., Wang, Z. H., Ristè, D., Dobrovitsi, V. V. & Hanson, R. Universal dynamical decoupling of a single solid-state spin from a spin bath. Science 33, 6-63 (1). 8. Ryan, C. A., Hodges, J. S. & Cory, D. G. Rubust decoupling techniques to extend quantum coherence in diamond. Phys. Rev. Lett. 15, 4 (1). 9. Bylander, J. et al. Noise spectroscopy through dynamical decoupling with a superconducting flux qubit. Nature Physics 7, 565 57 (11). 3. Zhao, N., Hu, J. L., Ho, S. W., Wan, J. T. K. & Liu, R. B. Atomic-scale magnetometry of distant nuclear spin clusters via nitrogen-vacancy spin in diamond. Nature Nanotechnology 6, 4 (11). Acnowledgements We than Nan Zhao and Sen Yang for discussion on some specific physical systems. This wor was supported by Hong Kong RGC/GRF CUHK48 & CUHK441, Chinese University of Hong Kong Focused Investments Scheme, National Natural Science Foundation of China Project 11851, and Hong Kong RGC/CRF HKU8/CRF/11G. Author Contributions R.B.L. conceived the idea, designed the models, formulated the theories, and wrote the paper. S.W.C. studied the 1D Ising model. Z.F.J. studied the diamond system. All authors analyzed the results and commented on the manuscript. Author Information The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to R.B.L. (rbliu@phy.cuh.edu.h). 8

Figure 1 Quantum criticality in a spin chain detected by decoherence of a probe spin. a, Schematic of a probe spin (at the centre) wealy coupled to a spin chain bath. An external magnetic field is applied in the vertical direction to tune cross the quantum criticality in the bath. b & c, Free-induction decay of the probe spin coherence as a function of time and the external field strength for the bath initially at zero and infinite temperature, respectively. A sharp feature presents at the critical point () at zero temperature but disappears at infinite temperature. d, Spin echo signal of the probe spin as a function of time and the external field strength for the bath initially at infinite temperature. A sharp feature presents at the critical point for time. e, Energy of elementary excitations in the spin chain bath as a function of wavevector for various external field strength. f & g, Noise spectra of quantum fluctuations at various temperatures (indicated by the inverse temperature ) for external field and, respectively. At the critical point (), the noise spectrum is gapless. In the calculation shown in (b-g), the number of spins in the bath is N and the probe-bath coupling gn. 9

Figure Time-inverse temperature correspondence in emergence of quantum criticality. a, Schematic of Lee-Yang circle of singularity points in the complex plane of inverse temperature for a thermodynamic function of a macroscopic system. The imaginary inverse temperature can be interpreted as time. b, Susceptibility of the spin chain bath as a function of the external field and inverse temperature. c, Susceptibility of the bath (upper panel) and probe spin echo signals (lower panel) as functions of the external field for various inverse temperature and echo time t, respectively. Criticality becomes pronounced at large t or. d, Contour plot of spin echo signal at the critical point as a function of the time and inverse temperature. For symmetric plot, calculation is also carried out for negative time and temperature (which can be regarded as for non-equilibrium bath states). e, Free-induction decay of probe spin coherence as a function of the external field for various time t. The coherence at t is amplified by. The model used in the calculation shown in (b-e) is the same as in Fig. 1. 1

Figure 3 Quantum criticality of realistic spin systems at high temperature. a & b, Schematic of the physical system made of a monolayer (a) or ring (b) of endohedral 16 O@ 1 C 6 fullerenes on a diamond surface with an NV centre beneath. A magnetic field B is applied along the z-axis. c, The lowest energy levels of a ring of spin-1/ s as functions of the external magnetic field. The inset zooms in the levels near the critical point (about. Gauss). d, Long range correlation between two bath spins at the two ends of a diameter, S1 n1s N 1 n N 1, with n j denoting the direction tangential to the ring at the position of the jth spin. The correlation is plotted as functions of the external magnetic field for various temperature T,,, Kelvin corresponding to the inverse temperature (written in dimension of time),,, and s in turn. e & f, FID of the NV spin at zero and infinite temperature, respectively. g, Spin echo of the NV spin at infinite temperature. In calculation for (c-g), the ring contains equally separated spins, the ring has radius 15 nm and the NV centre is 5 nm below the centre of the ring. 11

SUPPLEMENTARY INFORMATION for Quantum criticality at infinite temperature revealed by spin echo Shao-Wen Chen, Zhan-Feng Jiang and Ren-Bao Liu I. Full Methods and Supplementary Equations A. Exact solution of the transverse-field Ising model The Hamiltonian of the one-dimensional transverse-field Ising model coupled to a centre spin probe is with N z z H H g, (S1) j1 N x x z j j j 1 j1 j H. (S3) The Hamiltonian is bloc diagonalized and can be written as H H H, by which the spin bath is driven by different Hamiltonians H H ( g) depending on the probe spin state. We prepare the probe spin on a coherent state, and then the initial state of the probe-bath system is J if the bath is initially in a pure state J. At time t, the system evolves to an entangled state coherence of the centre spin is iht iht e J e J. Therefore, the iht iht LFID J e e J thermal equilibrium state, the coherence of the centre spin is. If the bath is initially in a iht iht LFID Tr e e. (S4) The one-dimension transverse-field Ising model can be exactly solved. By applying Jordan-Wigner transformation [1] x i y z a, (S5) z j 1 aa j j, j j i j i j the Hamiltonian is transformed to be a free-fermion one as N H a a a a a a N, (S6) j j j 1 j 1 j j j1 where a j and a j are the annihilation and creation fermion operators at the jth site, respectively. Here, we choose the boundary condition according to the parity N z ( P j ) of the system, namely, c N 1 c for P 1. 1 j1 1

By Fourier transformation exp to a spinless fermion system where c and a c ij N, the spin system is mapped j H cos c c isin c c c c N, (S7) c are creation and annihilation operators of fermions with wavevector. This Hamiltonian can be diagonalized. The vacuum state of the th mode is, by a Bogoliubov transformation. The corresponding transformation between the fermion operators is b u iv c b u iv c, b iv u c b iv u c where u cos, v sin. After the transformation, one gets a diagonalized fermion Hamiltonian with dispersion with tan sin / cos 1, The transformed vacuum state of the th mode is, Ising chain is (S8) H b b (S9) ( ) 1 cos. (S1). The ground state of the, since all quasi-particles have positive energy. The excited states are obtained by applying the creation operators b to the ground state. The transformation between the Foc states corresponding to different sets of fermion operators is,, u iv 1,1 1,1 iv u.,1 1,1 1, 1 1, The coherence of the centre spin in the case of free-induction decay is FID,, > (S11) L ( t) Tr U ( t) U ( t), (S1) with being the th component density matrix of the initial bath state 13

T, 1, 1 1,1 e 1,1, 1e e,1 e,1 e 1, 1, evolution operator U, T, A B, 1,1 B A 1,1 () t,1 1,1 1, 1 1, cos, cos, sin, and B sin, sin, t A t i t and the with. Here, we have added an extra index for and to indicate the evolutions driven by different Hamiltonians H. It is straightforward to extend this result to Hahn echo: L ( t) Tr U t U t U t U t. (S13) SE,,,, > The decoherence presented in Figs. 1 & of the main text is calculated by exact diagonalization of the bath Hamiltonian. B. Noise spectrum To better understand the decoherence in free-induction decay and spin echo, we now exam the noise spectrum of the local field fluctuation. Fluctuations of the effective local field N z B g felt by the probe spin cause decoherence of the probe spin. In j 1 j Gaussian noise approximation, the decoherence is determined by the noise correlation C t t B t B t B t B t as [31] 1 1 1 where the modulation function 1 t t L ( t) exp C t1 t f t1 f t dt 1dt, (S14) ffid 1 f t for t, t and f t for t t, t SE 1 t in the case of free-induction decay and SE 1 The quantum fluctuation part gives the noise spectrum S in the case of spin echo. 1exp 8g sin. (S15) 1 exp Q The thermal fluctuation gives the noise at zero frequency with amplitude S exp (S16) 1 exp 64g cos. th 4 Consequently, one gets the decoherence of the probe spin as 14

t t T d F t L ( t) exp Sth f t1 f t dt 1dt exp S Q( ), (S17) t in which exp F t f t i t dt is the filter function. F t t FID 4 sin for free-induction decay and 4 FSE t 16 sin t 4 for spin echo. In the case of spin echo, the thermal fluctuation effect is removed. C. Magnetic susceptibility The magnetic susceptibility is evaluated by 1 e N N 1 1 i1 1 e z i N cos. (S18) D. Model for the diamond spin system In the diamond spin system, the bath contains N electron spins of 16 O atoms, each encapsulated in a 1 C 6 cage (the endohedral cage denoted as 16 O@ 1 C 6 ). The 16 O@ 1 C 6 cages are placed in a ring on the 16 O terminated surface of high-purity diamond. The choice of the isotopes is such that the interference from the nuclear spin effects is minimized. The probe is an NV centre located below the centre of the 16 O ring. A magnetic field B is applied perpendicular to the surface (along the z direction). The NV centre has a spin-1. The 16 O atom has an electron spin-1 and zero nuclear spin. The 1 C 6 cages have no nuclear spin or electron spin. The coupling to the nuclear spins of 13 C in the diamond can be neglected within the timescale considered in the paper. Or we can also assume that the diamond has been 1 C-purified. To simplify the numerical calculation, we use a free electron spin-1/ to model the 16 O atom spin, which does not change the essential physics. The probe-bath Hamiltonian is given by H BS S R 4 R N N N z z e e i S 3, 3 i S j ij Si RijRij S j (S19) i i j j ij where S is the spin-1 of the NV centre electron at R ( x, y, z), S i is the 11 R x, y, z, 1.76 1 rad s T is the gyromagnetic ratio of ith bath spin at i i i i e the electron spin,.87 GHz is the zero-field splitting of the NV centre spin, and R R - R. In the nearest-neighbor approximation (as in calculation for Fig. 3 of the ij i j main text), the coupling between non-neighboring bath spins ( ij 1) is neglected. Since the zero-field splitting is much greater than the probe-bath interaction (~ MHz), the flipping between the NV centre spin state and 1 is negligible. The model is reduced to a pure dephasing model. When the transition 1 of 15

the NV centre spin is used to probe the quantum criticality of the bath, the effective Hamiltonian is given by H () ( 1) H H H 1 1, () H bath, ( 1) H eb H bath Hint N N N z e e i 3 i j ij i ij ij j i1 i j j1 4 R S S S R R S ij N e z int 3 j j j j j j1 4 R R S j, H bath BS 3 R, H S 3 R z, (S) where z j z z j. The NV centre spin is initially in a coherent state 1 in a thermal equilibrium state described by a density matrix normalization factor of FID at time t is Z e H bath Tr[ ], and the bath is initially 1 Hbath Z e with the. The coherence of the NV centre spin in the case and the spin echo signal is ( ) Tr ih () t ih ( 1) L t, FID t e e (S1) ( ) Tr ( 1) () ( 1) () ih ih ih ih L. SE t e e e e The spin correlation in Fig. 3 of the main text is defined between two bath spins at the endpoints of a diameter of the ring, 1 1 N 1 N 1 (S) C 1, N 1 Tr S n S n, (S3) where n 1 and n N 1 are the unit vectors along the tangential directions of the ring at the positions R 1 and R N 1, respectively. The low-energy states of the spin bath are calculated by the Lanczos algorithm [3]. () ih t The time evolution e ih t or e and the density matrix the Chebyshev polynomial expansion [33-35]. ( 1) bath e H are calculated by E. Exact numerical diagonalization Tens of lowest eigenenergies and corresponding eigenstates of H bath are numerically obtained using the standard Lanczos algorithm [3]. The convergence of the solution has been tested. F. Chebyshev polynomial expansion For a time-independent Hamiltonian, the evolution operator U( t) exp ith is 16

expanded by the Chebyshev polynomials of the operator G H / E, where E is a rescaling factor which maes the absolute values of all the eigenvalues of G less than one [33]. After this rescaling, the evolution operator is expanded by U( t) exp( itg) J( t ) ( i) J( t ) T( G), (S4) where t Et is the dimensionless rescaled time, J () t is the th order Bessel function of first ind, and T ( G ) is the Chebyshev polynomials of the operator G, defined as 1 T1 ( G) G, T 1( G) GT ( G) T 1( G). In practice, a truncation at 1.5t already gives a precision of 7 1 or better [33]. At infinite temperature, the density matrix N I is proportional to a unity matrix, and we can approximate the trace Tr[...] by averaging the expectation values over M samples of states M 1 M m... m, where m is a normalized random m1 bath state. In the basis of the direct-product states N m a a a1 Y a of N bath spins, C Y, where C a are independent, uniformly distributed random complex numbers satisfying the normalization condition N a1 C a 1. For a sufficiently large number of bath spins (N > 1), a single realization of C a (M ) is sufficient to calculate decoherence at infinite temperature precisely [34]. To calculate the properties at finite temperature, an evolution operator along exp H. The imaginary-time axis (inverse-temperature axis) is defined evolution operator is again calculated by the Chebyshev polynomial expansion where E G I I T G 1 (S5) exp 1, (S6) is the dimensionless rescaled inverse temperature, and I th order modified Bessel function of first ind. is the Contrary to the case of infinite temperature, to evaluate the trace Tr[...], we need thousands of random samplings to get the converged results, because the puts most of the weight on the eigenstates imaginary-time evolution operator with lower energies [35]. The lower the temperature is, the more samplings are needed. In the case of extreme low temperature, we switch to a direct calculation of exph expen n n, because only several eigenstates with lowest n energies are relevant, which we have obtained by the Lanczos algorithm. 17

G. Disorder effect In realistic experiments, there are possible imperfections in locations of the C 6 cages and the NV centre. To investigate the disorder effect, we simulate some cases in which the positions of the bath spins and the NV centre have uncertainty to some degree. We assume the position of the NV centre obeys the Gaussian distribution in three dimensions, 3/ f R 3 exp R R, where is the standard deviation, and the positions of the C 6 cages obey the Gaussian distribution in the x-y 1 plane, f R exp i xi xi yi y i. We abandon the configurations in which the distance between any two 16 O atoms is less than 1.1nm, because the van der Waals diameter of a C 6 molecule is 1.1 nm [36]. 18

II. Supplementary Figures and Discussions A. Free-induction decay at finite temperature In the FID of the probe spin coupled to the transverse-field Ising model, the sharp dip at the quantum critical point gradually disappears as temperature increases (Fig. S1). Supplementary Figure S1. Free-induction decay evaluated at different finite temperatures. (a) =5 ; (b) =1 ; (c) =5 ; (d) =. The number of spins in the transverse-field Ising model is N and the probe-bath coupling g. 19

B. Revival of free-induction decay in finite system The revival of FID in Fig. 3f of the main text results from the finite and uniform coupling between the probe and the bath spins. It can be understood by considering the limiting case of infinite magnetic field in which the bath eigenstates have all spins quantized along the z-axis. As the probe-bath coupling g is uniform for all the bath spins, the random local field taes integer multiples of g and the decoherence by the thermal fluctuation presents periodic revivals with period /(4g). With decreasing magnetic field, the bath spin quantization along the field direction is less perfect and the revival becomes less pronounced with the period increasing and eventually disappears at the ordered phase. We further chec this picture using the exact solution of the transverse-field Ising model (Fig. S). For macroscopic systems or for non-uniform probe-bath coupling, the revival would be absent. In the study presented in Figs. 1 & of the main text, we consider time scales shorter than the revival period. Supplementary Figure S. Revivals in free-induction decay of a probe spin coupled to the transverse-field Ising model at infinite temperature. The calculation is performed with the number of spins in the bath N and the probe-bath coupling g being (a) N 5, g.1; (b) N, g.1; (c) N, g.. The FID presents revivals in the paramagnetic field with periods /(4g).

C. Noise spectrum method The noise spectra of the transverse-field Ising model are determined by the quasi-particle excitation spectra. At the critical point (), the noise spectra are gapless (Fig. S3 a & b). The spin decoherence is determined by the noise spectrum and the filter functions (Fig. S3 c & d). The spin echo filters out the zero frequency component of the noise. As time increases, the filter function has greater low-frequency pass (Fig. S3 d). The noise spectrum method provides a quite good approximation of the decoherence (Fig. S3 e & f). Supplementary Figure S3. Noise spectrum approach for probe spin decoherence coupled to the transverse-field Ising model. (a) & (b) show respectively the noise spectra of the quantum fluctuation at zero temperature ( ) and finite temperature ( 1) for different external field. (c) & (d) show respectively the filter functions for free-induction decay and spin echo for different times. (e) & (f) compare the decoherence behavior obtained by the noise spectrum approach and the exact solution for and 1, respectively. 1

D. Finite transition region in finite systems The finite probe-bath coupling leads to a finite transition region near the critical point (Fig. S4). To clearly resolve the critical point, it is desirable to have wea probe-bath coupling. Supplementary Figure S4. Spin echo signals of a probe coupled to the transverse-field Ising model at infinite high temperature as functions of external field. The number of spins in the bath is N The probe-bath coupling is (a) g=.1, (b) g=.5. The larger probe strength leads to a broader transition region.

E. Nearest-neighbor dipolar interaction approximation In the main text, we simulate the decoherence of the NV centre coupled to the bath of spins with the nearest-neighbor approximation. To test the validity of this approximation, we compare calculation with the full-interaction Hamiltonian and that with the nearest-neighbor approximation for a bath of 16 spins. The results in Fig. S5 show that the nearest-neighbor approximation describes the system very well except that the critical external field is slightly shifted from.15 Gauss in (a-c) to.11 Gauss in (d-f). Supplementary Figure S5. Test of the nearest-neighbor dipolar interaction approximation. (a), (b) & (c) are in turn the lowest eigenenergies, spin correlations (at various inverse temperature indicated by values of ), and spin echo signal (at infinite temperature) calculated with the full-interaction Hamiltonian. (d), (e) & (f) are similar to (a), (b) & (c) but calculated with the nearest-neighbor dipolar interaction approximation. The dashed lines denote the critical external fields. The ring of 16 O spins has radius of 15 nm and the NV centre is 5 nm below the centre of the ring. 3

F. Disorder effect Figure S6 shows simulations of three random configurations of 16 bath spins on a diamond surface with an NV centre 5 nm beneath the surface. The full interaction Hamiltonian without the nearest-neighbor approximation is used in the calculation. The disorder effect shifts the critical external field and smears the transition in spin echo signals, but some nontrivial ins are still visible in the spin echo signal diagrams indicating the critical fields. The critical fields in the disordered cases are slightly different from the value (.15 Gauss) for the configuration without disorder. Supplementary Figure S6. Disorder effects on quantum criticality in the diamond spin system. (a), (d) & (g) are three configurations showing the (x,y) coordinates of the 16 16 O spins and the NV centre, (b), (e) & (h) are the corresponding lowest energy levels as functions of the magnetic field, and (c), (f) & (i) are the corresponding spin echo signals at infinite temperature as functions of echo time and the magnetic field. The positions of the NV centre and 16 O atoms obey the Gaussian distribution with standard deviation (a).5 nm, (d) 1nm & (g) 1.5 nm. The NV centre is (a) 4.7 nm, (d) 4.8 nm & (g) 5.6 nm beneath the plane of the ring. 4

Supplementary References 31. Cywińsi, Ł., Lutchyn, R. M., Nave, C. P. & Das Sarma, S. How to enhance dephasing time in superconducting qubits. Phys. Rev. B 77, 17459 (8). 3. Saad, Y., Numerical Methods for Large Eigenvalue Problems, nd Edition, SIAM (11). 33. Dobrovitsi, V. V. & De Raedt H. A., Efficient scheme for numerical simulations of the spin-bath decoherence, Phys. Rev. E 67, 567 (3). 34. Zhang, W., Konstantinidis, N., Al-Hassanieh, K. A. & Dobrovitsi, V. V. Modelling decoherence in quantum spin systems. J. Phys.: Condens. Matter 19, 83 (7). 35. Weiße, A. & Fehse, H., Chebyshev Expansion Techniques, Lecture notes in Physics 39, 545 (8). 36. Katz, E. A., Fullerene Thin Films as Photovoltaic Material, in Sōga, T. Nanostructured materials for solar energy conversion, Elsevier (6). 5