Evolution of Quantum Fluctuations Near the Quantum Critical Point of the Transverse Field Ising Chain System CoNb$_2$O$_6$

The transverse field Ising chain (TFIC) model is ideally suited for testing the fundamental ideas of quantum phase transitions, because its well-known $T=0$ ground state can be extrapolated to finite temperatures. Nonetheless, the lack of appropriate model materials hindered the past effort to test the theoretical predictions. Here we map the evolution of quantum fluctuations in the TFIC based on Nuclear Magnetic Resonance (NMR) measurements of CoNb$_2$O$_6$, and demonstrate the finite temperature effects on quantum criticality for the first time. From the temperature dependence of the $^{93}$Nb longitudinal relaxation rate $1/T_1$, we identify the renormalized classical, quantum critical, and quantum disordered scaling regimes in the temperature ($T$) vs. transverse magnetic field ($h_{\perp}$) phase diagram. Precisely at the critical field $h_{\perp}^{c}=5.25 \pm 0.15$ T, we observe a power-law behavior, $1/T_{1} \sim T^{-3/4}$, as predicted by quantum critical scaling. Our parameter-free comparison between the data and theory reveals that quantum fluctuations persist up to as high as $T \sim 0.4 J$, where the intra-chain exchange interaction $J$ is the only energy scale of the problem.

The concept of T = 0 quantum phase transitions has emerged as an overarching theme in strongly correlated electron physics [1][2][3][4][5][6][7][8][9]. The nature of quantum fluctuations near the quantum critical point, however, remains enigmatic [10]. How well does the quantum criticality account for finite temperature properties? How high in temperature does the e↵ect of the quantum critical point persist? [10,11] The dearth of appropriate model materials has not permitted experimentalists to address these fundamental questions even for the transverse field Ising chain (TFIC), a celebrated textbook example of quantum criticality [6]. Here we use NMR (Nuclear Magnetic Resonance) to map the evolution of quantum spin fluctuations in CoNb 2 O 6 [12][13][14], which has recently been proposed as an ideal model material for the TFIC [15,16]. We will demonstrate that the e↵ect of the quantum critical point persists to a remarkably high temperature, T ⇠ 0.4J, where the exchange coupling J is the only energy scale in the problem.
The TFIC Hamiltonian is deceptively simple [6]; where J (> 0 for ferromagnetic Ising chains in CoNb 2 O 6 ) represents the nearest-neighbor spin-spin exchange interaction, is the z(x)-component of the Pauli matrix at the i-th site, and the dimensionless coupling constant g is related to the transverse magnetic field h ? applied along the x-axis as g = h ? /h c ? , where h c ? is the critical field (h c ? = 5.25 ± 0.15 Tesla in CoNb 2 O 6 , as shown below). Since z i and x i do not commute, the classical Ising Hamiltonian for g = 0 becomes the quantum TFIC Hamiltonian for g > 0. The QCP (Quantum Critical Point) is located at g = 1, where the applied field is tuned precisely at h c ? ; a magnetic field greater than h c ? coerces the magnetic moments along its direction and destroys the T = 0 ferromagnetic ground state. See Fig. 1(a) for the generic theoretical phase diagram of the TFIC [6,17]. In spite of its apparent simplicity, the TFIC served as the foundational model for the quantum Monte Carlo simulations [18], and continues to attract attention in quantum ⇤ Electronic address: imai@mcmaster.ca information theory [19].
A major advantage of working with the TFIC as a model system for testing the fundamental ideas of QPT's (Quantum Phase Transitions) is that, in the absence of a transverse magnetic field (g = 0), the thermodynamic properties of the Ising chain can be rigorously solved at arbitrary temperatures [20]. Even in a finite transverse field (g > 0), the TFIC is well understood at T = 0 [18,21], and QC (Quantum Critical) scaling theory extended the T = 0 results to finite temperatures [6,17].
Our primary goals here are to (a) experimentally test the phase diagram of the TFIC in Fig. 1(a) above T = 0, and (b) demonstrate that the e↵ect of the QCP persists to T ⇠ 0.4J, based on NMR measurements of CoNb 2 O 6 . NMR is a powerful low energy probe, and good at probing the physical properties near QCP's [22][23][24][25][26][27][28]. In what follows, we take advantage of the hyperfine interactions between Co electrons and 93 Nb nuclear spins. Based on the measurement of the nuclear spin-lattice relaxation rate 1/T 1 at the non-magnetic 93 Nb sites, we map the evolution of low energy quantum fluctuations of Co spins near the QCP. See Supplementary Information for the details of NMR data acquisition.
We show the crystal structure of CoNb 2 O 6 in Fig. 2 [29]. The Co-O-Co chains propagate along the c-axis, and the easy axis of the Co moments lies within the ac-plane [12,13]. The ferromagnetic super-exchange interaction between the nearest-neighbor Co ions is estimated to be J = 17 ⇠ 23 K, based on ESR [14] and neutron scattering [15] measurements. The inter-chain couplings between adjacent Co chains are antiferromagnetic [12,15], weaker than J by an order of magnitude [14,15], and frustrated [12,16]. This means that the three-dimensional (3D) magnetic long range order induced by inter-chain interactions, which tends to mask the e↵ects of the 1D QCP of the individual Ising chains, is suppressed; the 3D ordering temperature is as low as T 3D c = 2.9 K even in h ? = 0 [12,13]. Combined with the modest J, CoNb 2 O 6 is an ideal model material for the TFIC, but was overlooked for three decades. Very recently, Coldea et al. demonstrated that the application of h ? along the b-axis induces a QPT from the 3D ordered state to a paramagnetic state above the 3D critical field, h c,3D ? = 5.5 Tesla [15,30]. In Fig. 3, we summarize the T and h ? dependences of 1/T 1 . Notice that 1/T 1 varies by more than three orders of magnitude between h ? = 3 and 9 Tesla. Quite generally, 1/T 1 probes the wave vector k-integral within the first Brillouin zone of the dynamical spin structure factor arXiv:1401.6917v1 [cond-mat.str-el] 27 Jan 2014 (a) A generic T h ? phase diagram of the TFIC encompasses three scaling regimes with distinct behaviors of the spin-spin correlation length ⇠: RC (Renormalized Classical, g < 1 hence h ? < h c ? , and ⇠ ⇠ exp(+ /T )), QC (Quantum Critical, ⇠ ⇠ 1/T ), and QD (Quantum Disordered, g > 1 hence h ? > h c ? , and ⇠ ⇠ constant) [6]. The dashed and dotted lines represent the crossover temperature from the QC to RC regime at T ⇠ and from the QC to QD regime at T ⇠ | |, respectively. An isolated 1D Ising chain would exhibit ferromagnetic long range order only at T = 0 below h c ? , but the 3D inter-chain couplings lead to a 3D order at T > 0 up to h c,3D The spin excitation spectrum in the RC regime has two components, the quasi-elastic peak at the origin (represented by a filled dot) and the propagating domain walls (inset). The dispersion of the latter is ✏(k) = J[2 2g cos(k) + O(g 2 )], with an excitation gap = 2J(1 g) [6]. The quasi-elastic peak becomes a Bragg peak when ⇠ diverges toward the 1D ferromagnetic long range order at T = 0. (c) The spin excitation spectrum in the QD regime, ✏(k) = Jg[2 (2/g) cos(k) + O(1/g 2 )] with a gap | | = 2|1 g| [6], arises from the propagation of flipped spins (inset). Unlike the RC regime, there is no quasi-elastic peak.
S(k, ! n ) at the NMR frequency ! n /2⇡ (⇠ 50 MHz): where a hf is the hyperfine coupling between the observed nuclear spin and Pauli matrices. In essence, 1/T 1 measures the strength of Co spin fluctuations at the time scale set by the NMR frequency. Our 1/T 1 data at low In other words, the spectral weight of the Co spin-spin correlation function grows at the quasi-elastic peak located at k = 0 in Fig. 1 Tesla. The Co spin-spin correlation length ⇠ along the chain grows as ⇠ ⇠ exp(+ /T ) in the RC regime [6], where is the gap in the spin excitation spectrum as defined in Fig. 1(b). Accordingly, we expect 1/T 1 ⇠ exp(+ /T ) for T ⌧ . See Supplementary Information for the details of the theory and data analysis of 1/T 1 . In contrast, 1/T 1 observed above h ? ⇠ 5.3 Tesla saturates and begins to decrease with temperature, signaling the opening of a spin excitation gap in the Quantum Disordered (QD) regime in Fig. 1(a). The behavior of 1/T 1 is therefore dominated by the thermal activation of spin excitations across the gap, | |, as defined in Fig. 1(c). Therefore we expect 1/T 1 ⇠ exp( | |/T ) for T ⌧ | |. We have thus identified the 1D QCP of each individual Ising chain as h c ? ⇠ 5.3 Tesla. In Fig. 4(a), we present the exponential fit of 1/T 1 ⇠ exp( /T ) with as a free parameter. The fitting range barely satisfies T < | | near h ? ⇠ 5.3 Tesla, and hence we conducted a scaling analysis of 1/T 1 to improve the accuracy of . We summarize the h ? dependence of in Fig. 4(b). Remarkably, we found that varies linearly   with h ? . This linear behavior is precisely what we expect from the theoretical prediction for the nearest-neighbor quantum Ising chain, = 2J(1 h ? /h c ? ) [6]. From the intercepts of the linear fit with the horizontal and vertical axes, we estimate h c ? = 5.25 ± 0.15 Tesla and J = 17.5 +2. 5 1.5 K, respectively, in excellent agreement with earlier reports [14,15].
We present the colour plot of 1/T 1 in Fig. 3(b). Also shown in Fig. 3(b) is the crossover temperatures, and | |, based on the linear fit in Fig. 4(b). Our colour plot visually captures the crossover from the QC regime to the RC and QD regimes successfully. We are the first to verify the theoretical T h ? phase diagram in Fig. 1(a) for finite temperatures, T > 0, using an actual model material.
Having established the phase diagram of the TFIC in CoNb 2 O 6 , we are ready to test its quantum criticality in the QC regime located between the RC and QD regimes. At the 1D critical field h c ? , we applied QC scaling to eq.(2), and obtained We fitted the 1/T 1 data observed at 5.2 Tesla to eq.(3) with J = 17.5 K, and found |a hf /~| 2 = 7.6 ⇥ 10 14 s 2 . It is important to realize that our experimental data exhibits the expected power-law behavior, 1/T 1 ⇠ T 0.75 , up to ⇠ 7 K, which corresponds to ⇠ 0.4J. Our finding therefore addresses an important and unresolved question that has been facing the strongly correlated electrons community for years: How high in temperature does the e↵ect of the QCP persist? For the TFIC, the quantum fluctuations originating from the zero temperature QCP persist up to as high as T ⇠ 0.4J. Our experimental finding is consistent with the earlier theoretical analysis of the QC scaling function for the TFIC [11].
We mark the upper bound of the QC scaling regime, T ⇠ 0.4J, in Fig. 3(b) with a horizontal arrow. Such a robust quantum criticality observed at finite temperatures above the QCP is in stark contrast with the case of thermally induced classical phase transitions; the critical region of the latter generally narrows as the phase transition temperature approaches zero, and eventually diminishes at T = 0 [10]. Many authors have constructed analogous colour plots for di↵erent parameters (such as electrical resistivity, as an example) for a variety of strongly correlated electron systems, including copper-oxide and iron-pnictide high T c superconductors and heavy Fermion systems [8,9]. The aim of these authors was to build a circumstantial case that quantum fluctuations persist at finite temperatures far above the QCP. Fig. 3b is a rare example for the solvable TFIC Hamiltonian, and the overall similarity to the case of high T c cuprates and other exotic superconductors gives us hope that quantum fluctuations may indeed account for the mechanism of exotic superconductivity.

NMR Measurements of 1/T 1
We grew the CoNb 2 O 6 single crystal from a stoichiometric mixture of cobalt and niobium oxides using a floating zone furnace. We assessed the surface quality and oriented the crystal utilizing Laue x-ray di↵ractometry. Once the material was sectioned into oriented slices along the a, b and c crystallographic directions, these were individually scanned with the Laue di↵ractometer and showed a uniform, single-crystalline structure. A small section of the single crystal was ground into a powder and analyzed using powder x-ray di↵raction which showed only single phase cobalt niobate in the crystal within instrument resolution. The features present in the SQUID magnetometry data matched previously published data on this material. All the pictorial images of the crystal structure in this paper were drawn using VESTA [S1].
For NMR measurements, we cut a piece of single crystal with the approximate dimensions of 4 mm x 2 mm x 5 mm. We glued the crystal to a sturdy sample holder made of machinable aluminum-oxide (MACOR ceramic) with a thickness of ⇠ 3 mm to ensure that the crystal orientation did not change at low temperatures. We found that the strong magnetic torque applied to the crystal by the external magnetic field could easily bend sample holders made of soft materials such as plexiglass or plastic, and introduce noticeable systematic errors below ⇠ 10 K.
We measured the 93 Nb (I = 9/2) nuclear spin-lattice relaxation rate 1/T 1 primarily at the central transition between the I z = + 1 2 and I z = 1 2 energy levels. The central transition is the strongest among all 9 peaks in Fig. 2(d), and hence most advantageous in terms of the signal intensity. We summarize the temperature dependence of the NMR lineshape for the central transition in Fig. S1. We applied an inversion ⇡ pulse prior to the ⇡/2 ⇡ spin echo sequence, and monitored the recovery of the spin echo intensity M (t) as a function of the delay time t. The typical width of the ⇡/2 pulse is ⇠ 1µs. An example of the signal recovery of the central transition observed at 130 K in h ? = 3 Tesla is shown in Fig. S2, in comparison to that observed for a fourth satellite transition between the I z = ± 9 2 and I z = ± 7 2 energy levels on the higher frequency side.
When the relaxation rate exceeds 1/T 1 ⇠ 2 ⇥ 10 3 s 1 , accurate measurements of 1/T 1 using the central transition become increasingly di cult because the recovery curve M (t) is dominated by two extremely fast normal modes, 0.653 e 45t/T1 + 0.215 e 28t/T1 ; the signal intensity, M (t), begins to recover at a time scale comparable to the inversion pulse width. Accordingly, measurements of 1/T 1 using the fourth satellite transition becomes more advantageous in the low temperature, low field regime, because its recovery curve is dominated by slower normal modes, 0.256 e 15t/T1 + 0.279 e 10t/T1 . We present an additional example of the 1/T 1 measurement using the fourth satellite at 2 K and h ? = 5.2 Tesla in Fig. S2.

Theoretical derivations of 1/T 1 in the quantum Ising chain
Here we will summarize the derivations of the theoretical expressions of 1/T 1 in the TFIC. Our notation will be the same as in [S3]. Some results will be specific to the nearest-neighbor Ising model, but most are more generally applicable to the vicinity of the quantum critical point of a generic one-dimensional Ising chain. In general, the NMR relaxation rate is defined by where a hf represents the hyperfine coupling between the nuclear spin and the Pauli matrices , as defined by the hyperfine HamiltonianĤ hf =Î · a hf ·ˆ . We define the correlation function for Pauli matrices, and~= k B = 1 unless noted otherwise.

Renormalized Classical Regime
This region is characterized by an energy gap ⇠ (g c g) and a T = 0 ordered moment N o ⇠ (g c g) 1/8 . The N o represents the ordered moment of an Ising chain at T = 0, and should not be confused with the 3D ordered moment induced by inter-chain couplings. By expressing our results in terms of and N o , they are generally valid beyond the nearest-neighbor model. For the specific case of the nearest-neighbor model, we have = 2J(1 g) and N o = (1 g 2 ) 1/8 . The result for C(x, t) may be found below (4.81) in Ref. [S3], and this leads to Notice that 1/T 1 is expected to diverge exponentially, even though there is an energy gap in the excitation spectrum of the domain-wall quasi-particles. This is because NMR is a low energy probe, and 1/T 1 in the RC regime is dominated by the low frequency spin fluctuations associated with the quasi-elastic mode of the 1D Ising chain induced by ferromagnetic short range order.

Quatum Critical Regime
Here, we have in imaginary time, ⌧ , from (4.106) in Ref. [S3] that where G I (0) = 0.858714569, and the value of Z is a general result upon approaching from the ordered side, valid beyond the nearest-neighbor model. From (4), we have the local susceptibility in imaginary time We evaluate the Fourier transform using (3.12), (3.22), and (3.24) of Ref. [S4], and obtain This gives us

Quatum Disordered Regime
Here we can expect that 1/T 1 diminishes exponentially in the quantum disordered regime due to the excitation gap, | |, and so where now < 0. However there is no explicit computation in the TIFC establishing this, and the pre-factor is unknown. There is no low frequency quasi-elastic mode, unlike the case of the RC regime.

Analysis of the 1/T 1 data
We begin the analysis of 1/T 1 in the QC regime. First, we combine eq. (5) with the results for the nearestneighbor model, = 2J(1 g) and N o = (1 g 2 ) 1/8 , and obtain Z = J 1/4 . We also introduce the hyperfine coupling A hf between the nuclear spin I and electron spin S through the hyperfine HamiltonianĤ hf =Î · A hf ·Ŝ.
That is, a hf = SA hf , and the Co 2+ is in a high-spin state with S = 3/2 due to the e↵ects of the crystal field. Recalling that 1/T 1 measured with an external magnetic field applied along the crystal b-axis probes the fluctuating hyperfine fields along the a-and c-axes, we may rewrite eq.(8) as where we show~and k B explicitly. From the best fit of the 1/T 1 data observed at 5.2 Tesla to eq.(10), we estimated the hyperfine form factor as |a hf /~| 2 2 = 7.6 ⇥ 10 14 (s 2 ), using J = 17.5 K. As shown below, A hf is proportional tō gµ B , whereḡ is the g-factor and µ B represents the Bohr magneton. Therefore |a hf | 2 , and hence 1/T 1 , is proportional to (ḡµ B S) 2 . Physically, this is related to the fact that the fluctuating hyperfine field is generated by Co 2+ magnetic moments.
We can check the consistency of the fit based on the measurements of the NMR frequency shifts, chem is a small temperature independent chemical shift. Thus A where the 93 Nb nuclear gyromagnetic ratio is n /2⇡ = 10.407 MHz/Tesla, and N A is Avogadro's number. In Fig. S3, we plot K (↵) as a function of the molar magnetic susceptibility (↵) measured along the corresponding orientations (see Fig. 2(c)), choosing T as the implicit parameter. From the linear fit, we estimate the slope as dK (↵) d (↵) = 0.386, 0.221, and 0.311 for ↵ = a, b, and c, respectively.
To proceed, we need to relate A (↵) hf /~in eq.(10) with the uniform component A (↵) hf (0)/~in eq.(11). We point out that the Nb-O-Nb chain is inside an isosceles triangle formed by three Co-O-Co chains within the ab-plane, as shown in Fig. S4(a) and (b). For simplicity, we assume that the hyperfine interaction at the 93 Nb site is dominated by the transferred hyperfine fields from two nearby Co-O-Co chains, and their strength, ⇠ A (↵) hf (0)/2~, is comparable. In addition, since we conducted 1/T 1 measurements above 2 K, we assume that the e↵ects of the inter-chain interactions between adjacent Co-O-Co chains are negligibly small. Then the fluctuations of the transferred hyperfine fields from two chains may be considered independent, and hence additive. That is, Combining eqs.(10), (11) and (12), we can estimate the only unknown parameter in our QC scaling analysis, the magnitude of the Co 2+ moment, asḡµ B S ⇠ 3µ B . This is in good agreement with the earlier report, 3.05µ B [S6]. Our estimation ofḡµ B S relies on simplistic assumptions, and hence should be considered provisional. Nonetheless, the overall consistency in our analysis of 1/T 1 and K indicates that the QC scaling theory outlined in the previous section II is valid at a quantitative level.
In the QD regime, the pre-factor of the leading e /T term in eq. (9) is not known at this time, hence we fit the 1/T 1 data to the simple activation form in Fig. 4(a). In the case of the RC regime, we obtained the pre-factor of the leading e +| |/T term as ⇠ 1/T in eq. (3). We found, however, that the observed divergent behavior of 1/T 1 is weaker, perhaps because our experimental range of T and h ? is not deep inside the RC regime, or possibly due to the influence of additional terms in the Hamiltonian neglected in the theoretical calculations. Accordingly, we fit the 1/T 1 data in the RC regime also with the simple exponential form, 1/T 1 / e + /T , as shown in Fig. 4(a).
For the 1/T 1 data measured in h ? ⇠ 5.3 Tesla, our experimental temperature range and the resultant estimated from Fig. 4(a) barely satisfy T < | |. To improve the accuracy of , we constructed the scaling plots of T +0.75 /T 1 as a function of /T in Fig. S5. We first estimated the magnitude of from Fig. 4(a), then made slight adjustments to improve the scaling collapse of the data. The final results of are presented in Fig.  4(b) using N. We note that this procedure changes the value of only by a few K.