Abstract

At quantum critical points (QCPs) quantum fluctuations exist on all length scales, from microscopic to macroscopic, which, remarkably, can be observed at finite temperatures—the regime to which all experiments are necessarily confined. But how high in temperature can the effects of quantum criticality persist? That is, can physical observables be described in terms of universal scaling functions originating from the QCPs? We answer these questions by examining exact solutions of models of systems with strong electronic correlations and find that QCPs can influence physical properties at surprisingly high temperatures. As a powerful illustration of quantum criticality, we predict that the zero-temperature superfluid density, _s(0), and the transition temperature, T_c, of the high-temperature copper oxide superconductors are related by T_cs(0)^y, where the exponent y is different at the two edges of the superconducting dome, signifying the presence of the respective QCPs. This relationship can be tested in high-quality crystals.

Introduction

Do quantum critical points (QCPs)^{1, 2, 3, 4, 5, 6, 7} provide a powerful framework for understanding complex correlated many-body problems? Do they shed light on quantum mechanics of macroscopic systems, providing, for example, a deeper understanding of entanglement? Answers to many such questions require a precise recognition of the experimentally observable regime of quantum criticality. An important class of QCPs are analogues of classical critical points, but in a dimension higher than the actual spatial dimension of the system; that is, a QCP in d spatial dimensions is equivalent to a classical critical point in (d+z) dimensions, where z is the dynamic critical exponent. We shall be primarily concerned with z=1, although an example involving z1 is given below. This extra dimension (imaginary time) is a fundamental consequence of quantum mechanics.

It seems paradoxical that often the region of classical critical fluctuations is small and tends to zero as the temperature, T, tends to zero, whereas the region of quantum critical fluctuations fans out as T increases. The reason is that the quantum critical region is determined by the dominant microscopic energy scale in the hamiltonian, whereas the classical critical region is determined by the ratio of the transition temperature, T_c, to the same scale raised to a positive power (the dimensionality). As this ratio is usually small compared with unity (it is 10^-5 in a conventional superconductor), the classical critical region is small as well. As a classical critical point is driven to zero by tuning a parameter to reach the QCP, T_c itself vanishes as ^-z, provided the spatial correlation length, , at T=0 is large compared with the lattice spacing, further amplifying the contrast between classical and quantum criticalities.

How high in temperature does the effect of a QCP persist? Consider perhaps the simplest possible example of a quantum phase transition described by the hamiltonian of an Ising model in a transverse field⁸ in one dimension,

where the usual Pauli matrices, denoted by , are situated on a one-dimensional lattice labelled by n, of lattice spacing a. Here we have scaled out an overall energy scale denoted by J. When the dimensionless coupling constant ₁1, the ground state is that of a classical one-dimensional Ising model, which is ordered at zero temperature. The transverse field represented by the Pauli matrix ₁ introduces quantum fluctuations and tries to restore the broken Z₂ symmetry. In the extreme limit ₁0, the system is a collection of isolated two-level systems and the ground state of each is the symmetric linear superposition of up and down spin states. It is well known that there is a QCP at the value of ₁=1, belonging to the universality class of the famous two-dimensional classical Ising model solved by Onsager. The scale J enters as soon as we attempt to calculate the free-energy density at finite temperature T, which is (after reducing the units such that the Boltzmann constant k_B=1)

where =1/T, is the dimensionless scaled free energy and is Planck's constant divided by 2. The quantity _k denotes the excitation spectrum of the Jordan–Wigner fermions as a function of wavenumber k (ref. 8),

and e₀ is the ground-state energy density. The quantity is the velocity of the elementary excitations at low energies and is clearly non-universal.

For the validity of a QCP at finite temperatures, the free-energy density should satisfy the constraints of finite-size scaling⁹, namely it should be expressible in terms of a universal scaling function _s. In the neighbourhood of the QCP, this function has as its argument the ratio of two fundamental lengths, and approaches a pure number in the limit T0, when tuned to the exact critical point. The extent to which approximates _s defines the quantum critical scaling regime; _s can be extracted from equation (1) and satisfies

where L=c/T is the thickness in the imaginary time direction and 1/|1-₁| is the correlation length when T=0, that is, when the thickness L=; the exponent =1 for the present model.

If we tune to exact quantum criticality, that is, ₁=1, then it is known that^{10, 11}_s(0)=/12. As this is the best chance for quantum criticality to survive at finite temperatures, we test our free energy at this point. The result is shown in Fig. 1. It is remarkable that scaling holds up to a temperature as great as J/2. A large overall energy scale helps ensure quantum criticality at high temperatures.

Figure 1: The scaled free energy (₁=1, J/T) plotted as a function of 2J/T.

The asymptote as T0 should be the universal number /12, if quantum critical scaling holds at finite temperatures. It clearly does so with sufficient accuracy for temperatures as large as J/2.

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There are always irrelevant operators, however, whose effects must disappear before scaling can be observed. The above hamiltonian was constructed to have no irrelevant operators. We now turn our attention to a modification of the above hamiltonian by adding an irrelevant operator involving a three-spin interaction term. This term is generated in the first step of a real space renormalization group procedure¹²; here we allow its coefficient to be arbitrary. The new hamiltonian is

The only change now is that the transverse field is modified by the nearest-neighbour spins. We have solved this hamiltonian, once again, by Jordan–Wigner transformation. The fermionic spectrum

vanishes at ka=, when ₂=1-₁, and at k=0, when ₂=1+₁, where ₁,₂>0, thus defining two critical lines (see Fig. 2).

Figure 2: The critical lines of the Ising model in a transverse field with three-spin interaction.

There is a region of parameter space near ₁=0 and ₂=1 in which the system will be close to both critical lines, meaning it has low-energy excitations at both k=0 and ka=, with gaps ₀=2J|₁-(₂-1)| and =2J|₁+(₂-1)|, respectively. ₀ collapses along ₂=1+₁ and collapses along ₂=1-₁. In either case, the critical exponent is given by =1. Under the duality transformation described in the Methods section, the multicritical point (₁=0,₂=1) maps onto the isotropic fixed point of the quantum XY model in a transverse field.

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The introduction of the irrelevant operator with the coupling ₂ does not change the universality class because the correlation length exponent is still given by =1 all along the critical lines. It is sufficient to examine the free energy density for ₂=1-₁ and 0₁1:

where . In Fig. 3 we plot the function (₁,₂=1-₁,J/T). The results show that as we move along the critical line starting at (₂=0,₁=1) and ending at (₂=1,₁=0), we can markedly alter the regime of validity of quantum critical scaling at finite temperatures.

Figure 3: The quantity (₁,₂=1-₁, J/T) is plotted as a function of 2J/T for different values of ₁ although maintaining criticality, that is, =0.

From top to bottom, the values of ₁ are 0, 0.02, 0.04, 0.06, 0.08, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0. Five of these curves are shown in colour, with the corresponding excitation spectra displayed in the inset. In every case, the gapless excitations near ka= are responsible for scaling. However, as we approach the multicritical point (₂=1,₁=0), a new set of low-energy modes appears at k=0, with the gap ₀/J=2|₁-(₂-1)|=4₁. does not approach _s=/12 until T₀, which requires arbitrarily low temperatures as ₀0. Precisely at the point (₂=1,₁=0), ₀ collapses and the number of critical modes is doubled, giving rise to a new asymptote _s=/6.

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We analyse another model that is important in the study of antiferromagnetism and high-temperature superconductors, the quantum O(N)-nonlinear -model⁴. Taking d=2, we consider the partition function

where =(0,1,2),n is an N-component unit vector and g₁ is a coupling constant. We obtain an exact expression for the free energy in the limit N, as in ref. 6. The only difference is that we compute the free energy numerically without assuming T/c ( is the ultraviolet cutoff) and follow the same procedure as before: tune to the QCP, g_1c, and look for a breakdown of finite-size scaling. As shown in Fig. 4, scaling is valid up to a temperature T^*c/8. Using spin wave theory and an appropriate lattice regularization⁴, we obtain T^*J/2, where J is the antiferromagnetic exchange constant of the Heisenberg model, which is of order 1,000 K in cuprates.

Figure 4: The scaled free energy density minus the ground-state energy density, (f-e₀)(c) ²/T³, of the quantum O(N)-nonlinear -model in the limit N (dotted line).

In the scaling limit (T0), this quantity approaches the asymptote -0.153. Quantum critical scaling holds up to T^*c/8 with 1% accuracy.

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Does T^* change in the presence of extra irrelevant couplings? We add the simplest irrelevant perturbation that permits an exact solution in the N limit, namely g₂n⁴n, where g₂ is another coupling constant. Using equation (3) given in the Methods section, we have estimated T^* at several critical points (_1c,g_2c) and found that it changes very little as we move along the critical surface. Hence the irrelevant coupling g₂ seems to be truly innocuous, in contrast to the parameter ₂ in our previous model. The difference is not so surprising: in the latter case we added a perturbation that altered the low-energy excitation spectrum—not only by renormalizing the velocity, but, more importantly, by introducing a secondary minimum. The term we added here, however, has no effect whatsoever in the limit frequency 0 and the wavenumber k0, leaving the original low-energy spectrum intact. A different approach to the present model has also suggested T^*J/2 (ref. 13). Our analysis of the leading irrelevant operator and our consideration of the free energy have strengthened this result.

In the cuprate superconductors, quantum criticality leads to a powerful and testable prediction relating the transition temperature, T_c, and the T=0 superfluid density, _s(0). Consider first the case of a pure system with (d+z)-dimensional QCPs at the edges of the superconducting dome at dopings xx_c. In general the universality classes of these transitions are of course different. Scaling^{14, 15} leads to T_c|x-x_c|^z_d+z and _s(0)|x-x_c|^{2_d+z-_d+zd+z} for x close to x_c (with standard definitions of critical exponents). Therefore, T_cs(0)^y, where y is determined by the universality class of the transition in question. If (d+z) is such that hyperscaling holds (below the upper critical dimension d_u), then y=z/(d+z-2), independent of _d+z. Moreover, if d=2, even z cancels out from the formula, resulting in the superuniversal result y=1. Remarkably, this is also the Uemura plot, further elaborated in ref. 16. If (d+z)>d_u, the gaussian fixed point is stable, and y=z_d+z/(2_d+z-_d+zd+z)=z/2 because _d+z=_d+z=1/2, _d+z=0.

As the coupling to nodal fermions is irrelevant¹⁷, the QCP in the underdoped regime belongs to the XY universality class, with z=1 and d_u=4. Asymptotically close to the QCP, the transition must correspond to d=3, but there is a regime farther away in which the layers are effectively decoupled because of weak coupling between them; hence the spatial dimensionality can be set to 2. The details of this crossover depend on various microscopic parameters that are beyond the scope of a scaling theory and must be established empirically. Thus, there is a regime in which Uemura scaling holds. However, the true criticality is given by y=1/2, modulo weak logarithmic corrections. In contrast, the QCP in the overdoped regime reflects a superconductor-to-metal transition, suggesting z>2 and y>1.

Strong disorder and macroscopic inhomogeneity can of course destroy the QCPs and therefore scaling. Of more interest to us is weak disorder that modulates x_c (analogous to modulation of T_c in a classical phase transition), which may not change the scaling arguments. The reason is as follows: a simple extension of the famous Harris criterion to the T=0 quantum problem implies that the exponents of the pure system remain unchanged if _d+z>2/d. If this inequality is violated, there are two possiblities: (i) the system flows to a new fixed point, in which case the scaling arguments remain valid but with new exponents; (ii) the transition is seriously rounded and the QCP itself is destroyed. As in any problem involving disorder, a priori it is difficult to know which of the two possibilities holds. The robustness of the scaling argument is striking, however. When (d+z)<d_u, _d+z cancels out, and even z cancels out in d=2; when (d+z)>d_u, y is simply z/2.

Experiments in high-quality crystals¹⁸ do in fact yield a value of y=0.61, substantially less than unity, in the extreme underdoped regime. Moreover, on the basis of our analysis, a plot of T_c versus _s(0) should rise more steeply in the underdoped regime than in the overdoped regime; several experiments seem to support this assertion at a qualitative level^{19, 20, 21} (an effect known in the literature as the 'boomerang' effect), although more data are necessary, especially because there are other experiments that do not fit into this picture²².

Clearly the role of irrelevant operators is a limiting factor²³ in observing scaling, but, as the term suggests, it is difficult to know the set of all such operators ahead of time. In some cases unexpected subtleties can arise at very low temperatures, as in an experiment involving LiHoF₄ (ref. 24). In this case, the QCP dictated by the electronic spins is forestalled by the hyperfine coupling of electronic and nuclear spins. The true QCP is the one at which both the electronic and nuclear spins are entangled. This entanglement shows up as a transfer of intensity from the magnetic excitation of electronic origin to the soft modes of coupled electronic and nuclear origin in the 10 eV range. The result is an extra quasi-elastic peak at the measured temperatures due to thermal decoherence. In systems such as the cuprates, where the dominant microscopic energy scale (the exchange constant, for example) is very large, such effects are not of much consequence, and the QCP determined by the larger scale provides the correct picture. Another aspect of quantum criticality is the quantum critical fan^{3, 4, 14} that is determined by the crossover temperatures, T_x, generically given by T_x=A|x-x_c|^z_d+z. The amplitude A controls the extent of the fan, but decreasing the product z_d+z clearly narrows it, hence the scaling regime.

The extended transverse-field Ising model seems to be an interesting example, where the QCP of higher symmetry crosses over to a QCP of lower symmetry at low temperatures (see the Methods section). Whether such a mechanism is effective in a real system such as the anomalous normal state of the cuprates is not known²⁵. Can quantum criticality explain linear-in-T resistivity of optimally doped high-T_c superconductors to temperatures as high as 1,000 K? It has been argued that this is improbable²⁶. Nonetheless, we have shown that simple considerations of quantum criticality at the edges of the superconducting dome can lead to very powerful experimental consequences for cuprates that can be precisely tested.

Acknowledgements

We thank E. Fradkin, S. Sachdev, S. L. Sondhi, and A. P. Young for helpful comments on the manuscript and H. M. Rønnow and D. Bonn for the prepublication copies of refs 18,24. This work was supported by the NSF under grant: DMR-0411931.

Competing interests statement:

The authors declare that they have no competing financial interests.

Received 21 March 2005; Accepted 28 July 2005; Published online 29 September 2005.

References

1. Pfeuty, P. & Elliott, R. J. The Ising model with a transverse field. II. Ground state properties. J. Phys. C 4, 2370–2385 (1971). | ISI |
2. Young, A. P. Quantum effects in the renormalization group approach to phase transitions. J. Phys. C 8, L309–L313 (1975). | ISI |
3. Hertz, J. A. Quantum critical phenomena. Phys. Rev. B 14, 1165–1184 (1976). | Article | ISI |
4. Chakravarty, S., Halperin, B. I. & Nelson, D. R. Two-dimensional quantum Heisenberg antiferromagnet at low temperatures. Phys. Rev. B 39, 2344–2371 (1989). | ISI |
5. Millis, A. J. Effect of a nonzero temperature on quantum critical points in itinerant fermion systems. Phys. Rev. B 48, 7183–7196 (1993). | Article | ISI |
6. Chubukov, A. V., Sachdev, S. & Jinwu, Y. Theory of two-dimensional quantum Heisenberg antiferromagnets with a nearly critical ground state. Phys. Rev. B 49, 11919–11961 (1994). | Article |
7. Coleman, P. & Schofield, A. J. Quantum criticality. Nature 433, 226–229 (2005). | ISI |
8. Pfeuty, P. The one-dimensional Ising model with a transverse field. Ann. Phys. (NY) 57, 79–90 (1970). | Article |
9. Privman, V. & Fisher, M. E. Universal critical amplitudes in finite-size scaling. Phys. Rev. B 30, 322–327 (1984). | ISI |
10. Affleck, I. Universal term in the free energy at a critical point and the conformal anomaly. Phys. Rev. Lett. 56, 746–748 (1986). | Article | ISI |
11. Blöte, H. W. J., Cardy, J. L. & Nightingale, M. P. Conformal invariance, the central charge, and universal finite-size amplitudes at criticality. Phys. Rev. Lett. 56, 742–745 (1986). | ISI |
12. Hirsch, J. E. & Mazenko, G. F. Renormalization-group transformation for quantum lattice systems at zero temperature. Phys. Rev. B 19, 2656–2663 (1979). | ISI |
13. Chubukov, A. V., Sachdev, S. & Sokol, A. Universal behavior of the spin-echo decay rate in La₂CuO₄. Phys. Rev. B 49, 9052–9056 (1994). | ISI |
14. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, Cambridge, 1999).
15. Fisher, M. P. A., Weichman, P. B., Grinstein, G. & Fisher, D. S. Boson localization and the superfluid-insulator transition. Phys. Rev. B 40, 546–570 (1989). | Article | ISI |
16. Emery, V. J. & Kivelson, S. A. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434–437 (1995). | ISI |
17. Balents, L., Fisher, M. P. A. & Nayak, C. Nodal liquid theory of the pseudo-gap phase of high-T_c superconductors. Int. J. Mod. Phys. B 12, 1033–1068 (1998). | ISI |
18. Liang, R., Bonn, D. A. & Hardy, W. Lower critical field and superfluid density in highly underdoped YBa₂Cu₃O_6+x single crystals. Phys. Rev. Lett. 94, 117001 (2005). | Article |
19. Niedermayer, C. et al. Muon spin rotation study of the correlation between T_c andn_s/m^* in overdoped Tl₂Ba₂CuO₆₊. Phys. Rev. Lett. 71, 1764–1767 (1993). | Article | ISI |
20. Uemura, Y. J. et al. Magnetic-field penetration depth in Tl₂Ba₂CuO₆₊ in the overdoped regime. Nature 364, 605–607 (1993). | Article | ISI |
21. Bernhard, C. et al. Magnetic penetration depth and condensate density of cuprate high-T_c superconductors determined by muon-spin-rotation experiments. Phys. Rev. B 52, 10488–10498 (1995). | Article | ISI |
22. Panagopoulos, C. et al. Superfluid response in monolayer high-T_c cuprates. Phys. Rev. B 67, 220502 (2003). | Article | ChemPort |
23. Belitz, D., Kirkpatrick, T. R. & Rollbuhler, J. Breakdown of the perturbative renormalization group at certain quantum critical points. Phys. Rev. Lett. 93, 155701 (2004). | Article |
24. Rønnow, H. M. et al. Quantum phase transition of a magnet in a spin bath. Science 308, 389–392 (2005). | Article |
25. Anderson, P. W. In praise of unstable fixed points: the way things actually work. Physica B 318, 28–32 (2002). | ISI |
26. Phillips, P. & Chamon, C. Breakdown of one-paramater scaling in quantum critical scenarios for the high-temperature copper-oxide superconductors. http://arxiv.org/abs/cond-mat/0412179 (2004).
27. Fradkin, E. & Susskind, L. Order and disorder in gauge systems and magnets. Phys. Rev. D 17, 2637–2658 (1978). | ISI |
28. Wei, T.-C., Das, D., Mukhopadyay, S., Vishveshwara, S. & Goldbart, P. M. Global entanglement and quantum criticality in spin chains. Phys, Rev. A 71, 060305 (2005). | Article |
29. Castro Neto, A. H. & Fradkin, E. The thermodynamics of quantum systems and generalizations of Zamolodchikov's c-theorem. Nucl. Phys. B 400, 525–546 (1993). | ISI |

1. Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095-1547, USA

Correspondence to: Sudip Chakravarty¹ e-mail: sudip@physics.ucla.edu

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