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

The transverse field Ising chain 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 transverse field Ising chain based on nuclear magnetic resonance measurements of ${\mathrm{CoNb}}_2{\mathrm{O}}_6$, and we demonstrate the finite-temperature effects on quantum criticality for the first time. From the temperature dependence of the ${}^{93}\mathrm{Nb}$ 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 $h_{\perp }^c=5.25\pm 0.15\mathrm{T}$, 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 $T\sim 0.4J$, where the intrachain exchange interaction $J$ is the only energy scale of the problem.

Popular Summary

Classical phase transitions occur with the addition of heat. In contrast, quantum phase transitions take place at absolute zero temperature when a different kind of physical parameter, such as a magnetic field, is changed. Decades ago, theoretical physicists noted that the magnetically ordered ground state of Ising chains (analogous to ice) melts to form a new state of matter called quantum paramagnets (analogous to water) when a magnetic field is applied along the transverse direction. The two different states of matter compete with each other, and very strong quantum fluctuations are accordingly predicted at the phase boundary at absolute zero temperature, commonly known as a quantum critical point. We map the strength of quantum fluctuations in an Ising chain material near the quantum critical point and demonstrate that quantum fluctuations survive at surprisingly high temperatures.

Over the last two decades, tremendous effort has been devoted to understanding the quantum fluctuations near quantum critical points because these fluctuations may account for the mechanism of exotic superconductivity. However, controversy remains about the highest temperatures that such quantum fluctuations can survive in if one raises the temperature near the quantum critical point. We use nuclear magnetic resonance measurements at 2–295 K of transverse-field Ising chains in ${\mathrm{CoNb}}_2{\mathrm{O}}_6$ to probe the physical properties near the quantum critical point. We find that the quantum critical behavior persists up to approximately 7 K (or 40% of the nearest-neighbor, spin-spin exchange interaction temperature).

The robust quantum criticality in ${\mathrm{CoNb}}_2{\mathrm{O}}_6$ above the quantum critical point is an example of how quantum systems deviate significantly from classical phase transitions whose critical region diminishes in size as the phase-transition temperature goes to zero. Furthermore, the new findings suggest that the concept of quantum criticality may indeed be applicable to research related to the mechanism of exotic superconductivity.

Figure 1

A generic $T-h_{\perp }$ phase diagram of the TFIC encompasses three scaling regimes with distinct behaviors of the spin-spin correlation length $\xi$: RC [$g<1$, hence $h_{\perp }<h_{\perp }^c$, and $\xi \sim \exp (+\mathrm{\Delta }/T)$], QC ($\xi \sim 1/T$), and QD ($g>1$, hence $h_{\perp }>h_{\perp }^c$, and $\xi \sim \text{constant}$) [6]. The dashed and dotted lines represent the crossover temperature from the QC to the RC regime at $T\sim \mathrm{\Delta }$ and from the QC to the QD regime at $T\sim |\mathrm{\Delta }|$, respectively. An isolated 1D Ising chain would exhibit ferromagnetic long-range order only at $T=0$ below $h_{\perp }^c$, but the 3D interchain couplings lead to a 3D order at $T>0$ up to $h_{\perp }^{c,3D}$ ($>h_{\perp }^c$). The filled circle at $T=0$ and the 1D critical field $h_{\perp }^c$ represents the QCP of the individual Ising chain.

Figure 2

(a) The crystal structure of ${\mathrm{CoNb}}_2{\mathrm{O}}_6$. (b) Both magnetic ${\mathrm{CoO}}_6$ and nonmagnetic ${\mathrm{NbO}}_6$ octahedra form a chain along the c axis, as seen from the c-axis direction. The Nb-O-Nb chain is inside an isosceles triangle formed by three Co-O-Co chains. The transverse field $h_{\perp }$ is applied along the b axis. (c) Each Nb site is bonded with two Co-O-Co chains across O sites. (d) Bulk magnetic susceptibility $\chi$ data measured with SQUID in an external magnetic field of 0.01 T.

Figure 3

The temperature dependence of the ${}^{93}\mathrm{Nb}$ NMR line shape observed for the central transition between the $I_z=+\frac{1}{2}$ and $-\frac{1}{2}$ energy levels in $h_{\perp }=5.3\mathrm{Tesla}$ applied along the b axis. We obtained the line shapes using the FFT of the spin-echo signal above 77 K. For the broader line shapes below 77 K, we measured the integral of the spin echo as a function of the frequency. Inset: The ${}^{93}\mathrm{Nb}$ NMR line shape at 295 K observed at 7.507 T using the FFT of spin-echo signals. The largest peak in the middle is the central transition, and four additional pairs of weaker peaks arise from $I_z=m$ to $m+1$ transitions ($m=-9/2$, $-7/2$, $-5/2$, $-3/2$, $+1/2$, $+3/2$, $+5/2$, and $+7/2$).

Figure 4

Examples of the recovery of the spin-echo intensity, $M(t)$, observed for the central and fourth satellite transitions at 130 K in $h_{\perp }=3\mathrm{Tesla}$. For comparison, we normalized the recovery curves by plotting $1-[M(\infty )-M(t)]/A$ as a function of $t$. The solid lines represent the best fit with $1/T_1=1.99\times 10^3{\mathrm{s}}^{-1}$ for the central transition and $1/T_1=1.96\times 10^3{\mathrm{s}}^{-1}$ for the fourth satellite transition, as described in the text. Also plotted is the recovery curve observed for the fourth satellite peak at 2 K in $h_{\perp }=5.2\mathrm{Tesla}$.

Figure 5

The temperature dependence of $1/T_1$ in $h_{\perp }$ applied along the b axis. All dashed lines interconnecting the data points are guides for the eye. The black solid line through the data points measured at 5.2 Tesla represents a power-law fit, $1/T_1\sim 6.2\times 10^3{T}^{-0.75}{\mathrm{s}}^{-1}$.

Figure 6

(a) The two components of the spin excitation spectrum in the RC regime: the quasielastic peak at the origin (represented by a filled dot) and the propagating domain walls, as schematically shown in the inset. The dispersion of the latter (solid curve) is $\epsilon (k)=J[2-2g\cos (k)+O(g^2)]$, with an excitation gap $\mathrm{\Delta }=2J(1-g)$ [6]. The quasielastic peak becomes a Bragg peak when $\xi$ diverges toward the 1D ferromagnetic long-range order at $T=0$. Since NMR is a low-energy probe, our $1/T_1$ data measured below $h_{\perp }^c$ probe the quasielastic mode. (b) The spin excitation spectrum in the QD regime, $\epsilon (k)=Jg[2-(2/g)\cos (k)+O(1/g^2)]$ with a gap $|\mathrm{\Delta }|=2|1-g|$ [6], which arises from the propagation of flipped spins (inset). Unlike the RC regime, there is no quasielastic peak.

Figure 7

Estimation of the gap $\mathrm{\Delta }$. (a) The exponential fit $1/T_1\sim \exp (\mathrm{\Delta }/T)$ for representative values of $h_{\perp }$. (b) Filled circle represents Δ as determined from (a), while filled triangle is based on the scaling analysis. Also shown is a linear fit, $\mathrm{\Delta }=2J(1-h_{\perp }/h_{\perp }^c)$. From the fit, we estimate $h_{\perp }^c=5.25\pm 0.15\mathrm{Tesla}$ and $J=17.5_{-1.5}^{+2.5}\mathrm{K}$.

Figure 8

The scaling plots of $T^{+0.75}/T_1$ as a function of $\mathrm{\Delta }/T$ in (a) the RC regime and (b) the QD regime. For clarity, we normalized the overall magnitude of $T^{+0.75}/T_1$ as unity for the QC regime. The dashed-dotted line in (a) is a guide for the eyes, while the solid line in (b) represents $1/T_1\propto e^{-|\mathrm{\Delta }|/T}$.

Figure 9

A color plot of $1/T_1$. The dashed (dotted) line represents the expected crossover temperature $\mathrm{\Delta }$ ($|\mathrm{\Delta }|$) from the QC to the RC (QD) regime, based on the linear $h_{\perp }$ dependence of $\mathrm{\Delta }$ estimated in Fig. 7. Also shown (grey x) is the 3D ordering temperature $T_c^{3D}$ [28].

Figure 10

The NMR frequency shift $K^{(\alpha )}$ vs the bulk magnetic susceptibility ${\chi }^{(\alpha )}$, with $T$ as the implicit parameter ($\alpha =a$, b, or c). The straight lines are the best linear fits.