Scaling and commensurate-incommensurate crossover for the , quantum critical point of itinerant antiferromagnets.
Quantum critical points exist at zero temperature, yet, experimentally their influence seems to extend over a large part of the phase diagram of systems such as heavy-fermion compounds and high-temperature superconductors. Theoretically, however, it is generally not known over what range of parameters the physics is governed by the quantum critical point. We answer this question for the spin-density wave to fermi-liquid quantum critical point in the two-dimensional Hubbard model. This problem is in the universality class. We use the Two-Particle Self-Consistent approach, which is accurate from weak to intermediate coupling, and whose critical behavior is the same as for the self-consistent-renormalized approach of Moriya. Despite the presence of logarithmic corrections, numerical results demonstrate that quantum critical scaling for the static magnetic susceptibility can extend up to very high temperatures but that the commensurate to incommensurate crossover leads to deviations to scaling.
Non-Fermi-liquid ground states, electron phase diagrams and phase transitions in model systems Lattice fermion models (Hubbard model, etc.) Specific approaches applied to studies of phase transitions Quantum phase transitions
There are strong indications that quantum critical points, i.e. critical points at zero temperature, influence the physical properties of materials at surprisingly high temperature. But the precise region of temperature over which this influence is felt is currently not well understood. In solvable models of quantum critical behavior,  power law scaling and universality associated with quantum criticality were found up to temperatures of order where is the exchange constant. That is in sharp contrast with classical critical points where scaling is usually observed only in a very narrow range around the critical point. The importance of quantum critical points  has thus come to the fore in the study of numerous materials, including high-temperature superconductors and heavy-fermion materials where quantum phase transitions and power law scaling are observed. 
One particularly relevant case in this context is that of itinerant electrons undergoing a paramagnetic Fermi liquid to spin-density wave (SDW) transition in two dimensions. The Hubbard model is the simplest microscopic model that contains this physics. There is no analog of the Ginzburg criterion that allows us to determine the parameter range where the influence of the quantum critical point is important. In that regime, temperature acts like a finite-size cutoff for the correlation length . In this paper, we quantify the range of temperature where quantum critical scaling is observable in this model, in other words we find out whether details of the Fermi surface (that lead for example to commensurate-incommensurate (C-I) crossovers), logarithmic corrections, or interaction effects, lead to sizable deviations from quantum critical behavior at finite temperature.
For this problem, the dynamical critical exponent is equal to two and the corresponding universality class at the upper critical dimension is ill understood. [3, 2, 4] In particular, the standard Hertz-Millis action for quantum critical phenomena is invalid. [5, 6] More specifically, when the SDW is commensurate at the antiferromagnetic wave vector, it has been suggested that all the coefficients of the Ginzburg-Landau-Wilson action become singular and that the spin susceptibility scaling becomes with . The generic case where the SDW is not commensurate should not have these singularities.
An alternative approach is the self-consistent renormalized theory of Moriya. This theory includes logarithmic corrections. [7, 3] However, it is not adequate to make quantitative predictions for deviations from quantum critical effects in the Hubbard model since it necessitates phenomenological constants as input. In addition, it does not satisfy the Pauli principle. In a theory that satisfies the Pauli principle, there is an interaction independent sum rule on spin and charge susceptibilities  that should be enforced and, in addition, the local moment, , with the component of the local spin, cannot exceed when the filling satisfies and when . There is nothing that imposes these constraints in the theory of Moriya.
Method and model: We use the non-perturbative Two-Particle Self-Consistent (TPSC) approach . This approach respects the Pauli principle, the Mermin-Wagner theorem and conversation laws. It also contains quantum fluctuations in crossed channels that lead to Kanamori-Brückner screening.  It is valid in the weak to intermediate coupling regime and not too deep in the renormalized classical regime where a pseudogap is observed. Numerical results obtained from TPSC in its domain of validity are extremely close to the numerically exact solution obtained (barring statistical errors) with benchmark Quantum Monte Carlo calculations on the Hubbard model. [8, 9, 10, 11, 12, 13]. The approach gives a satisfactory description of the pseudogap in electron-doped cuprates in a wide doping range. [14, 15] It has been shown to be in the universality class, where is the number of components in the vector model.  Since we are looking for deviations from universality and the theory has been benchmarked in non-universal regimes, we argue that our results are reliable for this question, even though we cannot claim to be completely accurate in the regime. Nevertheless, we will demonstrate that TPSC has the same critical behavior as Moriya theory and hence has the same logarithmic corrections. These logarithms have the same functional form as those of the renormalization group asymptotically close to the quantum critical point, but in TPSC and in Moriya theory the mode-mode coupling term does not flow, hence the corrections may differ in the details from the renormalization group.  Quantum critical behavior of the susceptibility and of the self-energy in the closely related spin-fermion model has been discussed by Abanov et al. 
We study the two-dimensional Hubbard model on the square lattice at weak to intermediate coupling,
where are the hopping integrals, are the site index, is the spin label, and are the particle creation an annihilation operators. Each doubly occupied site costs an energy and . The units are such that , and lattice spacing is unity. All the numerical results are presented in units where The dispersion relation is written as:
We concentrate on the behavior of the spin susceptibility. In TPSC, the retarded spin susceptibility is written as:
with the local-moment sum rule that follows from the fluctuation-dissipation theorem
where is the temperature, and double occupancy. We dropped the site index using translational invariance and we used the Pauli principle to write
All the numerical results below are obtained using the Matsubara frequency version of equations (2) to (6) without any approximation, hence they are valid at arbitrary distance from the quantum critical point. Before proceeding, we show however that the quantum critical behavior of TPSC is the same as that of the self-consistent renormalized theory of Moriya and we discuss conditions for scaling.
Analytical results near the quantum critical point. When the correlation length is large, one can expand the denominator of the TPSC spin susceptibility around the wave vectors where the maxima in occur to obtain
Defining as the value of the interaction at the mean-field SDW transition, the other quantities in the previous expression are
In the expression for the spin susceptibility, the denominators are expanded around each of the four incommensurate wave vectors, or only around the wave vector depending on the situation. We checked explicitly that higher powers of do not improve the description of the C-I crossover and are not relevant.
To determine the quantum critical behavior, one subtracts the self-consistency condition Eq.(6) for a value of temperature and filling close to the quantum critical point from the same equation evaluated at that critical point
In the above expression is evaluated at the quantum critical point where (From now on, a subscript means that the quantity is evaluated at the quantum critical point.) One then performs the integrals over momentum in a circular domain with cutoff and then the frequency integrals. To write the final answer, it is useful to follow Moriya et al.  and to define
and dimensionless measures of and
The definition of Eq.(12) and the fact that and are both of the order of the lattice spacing shows that is a temperature of the order of the Fermi energy. With these definitions and a single maximum in the susceptibility, the self-consistency expression takes the form
where terms of order have been neglected on the left-hand side. We defined
and obtained from the second Binet log gamma formula 
with and Euler’s gamma function. The quantity in Eq.(17) measures the deviation from the quantum critical point. Apart from the logarithm, the self-consistency relation Eq.(16) has the same functional form as Eq.(2.8) in Ref.  Logarithmic corrections for that theory are mentioned without proof in Ref. .
For large local moment, , there is an SDW ordered ground state and The case corresponds to the Fermi liquid ground state and to the quantum critical point. The full filling and temperature dependence of is found from the definitions of and . For example in the hole doped case, defining , we have
Thus, depends on but in the critical regime and so we can neglect terms of order . The quantity can then be written in the form
where is a positive number.
The various limiting solutions for the critical behavior of the dimensionless correlation length can be obtained from the self-consistency condition Eq.(16) as follows.  For one must take the limit first, then and since is exponentially small, can be neglected on the left-hand side leading to This is the renormalized classical regime where the correlation length grows exponentially. At the quantum critical point , the same limit of the functions applies and one must find the solution of which is approximatively , as in the renormalization group.  Finally, in the Fermi liquid regime, , the correlation length (and hence ) is finite so the limit must be taken first and which yields At on the Fermi liquid side, there are logarithmic corrections to the dependence of on asymptotically close to the quantum critical point since whose approximate solution is
In all regimes where in the self-consistency Eq.(16) can be neglected (large ) or replaced by a constant in the temperature range of interest, one can write
where the scaling function is the solution of
with , being the typical value of in the range of temperature under study. We have already discussed limiting cases of above. We demonstrate numerically below that in the range logarithmic corrections are negligible so that scaling holds to an excellent approximation, except at the C-I crossover.
Scaling function. When the explored temperature range is limited on a logarithmic scale, or when is large, logarithmic corrections can be neglected. In addition, in the limit where is much smaller than , the scaling function in Eq.(22) depends only on since we are in the limiting case in the equation that defines Eq.(23). This case occurs when the ground state is paramagnetic, , or above the crossover line to the renormalized classical regime that occurs when In such cases, near anyone of the maxima located at the quantity scales as so the spin susceptibility Eq.(8) as a function of an arbitrary scale factor obeys the scaling relation
where the exponents have values , and . In the above equation, will not be important only if the incommensurate peaks are much narrower in momentum space than the inverse correlation length Let for now and drop the dependence on that variable. Following the above discussions on the behavior of the correlation length, the susceptibility on the right-hand side of the last equation should be, within log corrections, a universal function of its arguments but with the overall scale of each argument and of non-universal. Setting, , and choosing such that we find,
where the scale of the function defined by this equation and an overall prefactor in front of the argument are not universal. is the quantity we will focus on, but we note in passing that the general form Eq.(24) with the given exponents implies scaling for the integrated susceptibility.  Non-universal factors such as and that enter the spin susceptibility can have some temperature and filling dependence in TPSC that can in principle lead to deviations to scaling. In renormalization group language, these dependencies are the irrelevant variables whose importance we are trying to gauge to delimit the scaling regime.
Commensurate-incommensurate crossover. In a strict sense, the value of should be fixed at to check quantum critical scaling. However, itself depends on temperature in general. At high temperature equals , becoming incommensurate at low temperature. The susceptibility there shows four symmetry related peaks for the model we consider.  The value of clearly depends on details of the Fermi surface and is thus non-universal. The above scaling form Eq.(24) nevertheless suggests that scaling in the plane as in Eq.(25) should occur when It is not however possible to define in the C-I crossover regime. In that regime, incommensurate peaks necessarily overlap since the second derivative of vanishes at when the crossover begins, reflecting the fact that there is a broad maximum at that is splitting into four overlapping peaks. in the general scaling function Eq.(25) is not negligible in the C-I crossover region. On general grounds then, we expect deviations to scaling there. One may think that a better strategy to prove scaling is to measure the correlation length as a function of an but cannot be determined in the C-I crossover regime for the same above reasons.
From now on, we thus look for scaling with the susceptibility evaluated at its maximum, This is a well defined quantity experimentally and far from the C-I crossover we will have
Numerical results: Let us first verify the scaling at the quantum critical point . Fig. (1) shows a log-log plot of both the interacting (open circles) and noninteracting (open squares) susceptibilities as a function of temperature for two different sets of parameters.
For temperatures larger than hopping one obtains trivial scaling for both the interacting and non-interacting susceptibilities. While the non-interacting susceptibility flattens at lower temperature, the interacting susceptibility shows quantum critical scaling down to the lowest temperature we could reach, namely We will see that the scaling at that comes from the non-interacting susceptibility does not obey the scaling equation Eq.(25). It is also clear from Fig. (1) that deviations to scaling occur in the C-I crossover regime delimited by the vertical red lines. It is remarkable however that the same straight line fits both the commensurate and the incommensurate regimes. This suggests that non-universal scale factors are very similar on either sides of the commensurate-incommensurate transition. The slight upward curvature at the lowest temperatures is not inconsistent with effects of logarithmic corrections.
To verify the full scaling Eq.(25), we plot as a function of on a log-log plot in Fig. 2. We take values of on the Fermi liquid side of . For a given band structure and interaction, it is only when one has found the correct values of the critical that all the curves for different fillings and temperature collapse on the same curve. We found, when , that and for and respectively and for the electron-doped case with More values can be found in the thesis which is the basis for all the results of the present paper. 
The straight line of slope at large in Figs. 2a,c,d. corresponds to the Fermi liquid regime where both the susceptibility and the correlation length are temperature independent, but diverge as one approaches the quantum critical point. In that regime, scales as when logarithmic corrections are negligible. The scaling of corresponds to plateaus on the left of Figs. 2a,c,d. The deviations from a plateau come from the C-I crossover. To show that the scaling is non-trivial, in Fig. 2b we do not multiply the susceptibility by on the vertical axis. The lined up circles that can be caught by the eye correspond to different temperatures for a given filling the fillings closest to being to the left.
Scale factors depending on band structure and interaction strength should not influence the shape of the scaling function. A simple translation in the plane of the curves for different parameters should allow all of them to collapse. In Fig. 2d, we show scaling functions for various parameters but without translation for non-universal factors. One sees that if there were no deviations to scaling associated with the C-I crossover in the plateau region, simple translation would make all the curves nearly collapse. This also shows that logarithms do not have a large influence on scaling in this temperature range.
In Fig. 3, data analogous to those in Fig. 2 are represented by black open circles and are filtered out near the C-I crossover. The missing data is particularly clear in Fig. 3b where we do not scale the vertical axis. If is the temperature where the crossover occurs for a given doping, the data were filtered in the range () for densities . For larger values of , the data is sufficiently far from the C-I crossover that no filtering is required. The remaining data are those beyond the C-I crossover both above (commensurate) and below (incommensurate) . One sees that a plateau is recovered (black open circles) for all three values of the interaction strength appearing in Figs. 3a,c,d as expected in the quantum critical regime.
We now turn to the high temperature limit of the quantum critical scaling. While the black open circles in Fig. 3 are for those for are represented by red crosses. The deviations to scaling for are obvious. Even though the non-interacting susceptibility scales as for as we saw in Fig.1, it does not pollute the scaling associated purely with the quantum critical point. The latter occurs for with the caveat concerning the C-I crossover. The maximum for scaling, , is an important result that applies in the weak to intermediate coupling regime we have considered here. Clearly quantum critical behavior must disappear at so there should be some dependence to the upper temperature cutoff. At the intermediate coupling values that we considered, the temperature range over which quantum critical scaling is observed should be compared to what would have been naively estimated by substituting and in , obtaining respectively and Basically, the upper limit of is essentially the degeneracy temperature for Fermi-Dirac statistics, which is of the same order as . The irrelevant temperature dependencies of all quantities are thus on this scale.
Conclusion. The quantum critical behavior of TPSC for the universality class is the same as that of the self-consistent renormalized theory of Moriya, hence it includes logarithmic corrections. In TPSC there is no adjustable parameter. By explicit numerical calculations away from the renormalized classical regime of the Hubbard model in the weak to intermediate coupling, we have been able to show that logarithmic corrections are not really apparent in the range of temperature and that the maximum static spin susceptibility in the plane obeys quantum critical scaling. However, near the commensurate-incommensurate crossover, one finds obvious non-universal temperature and filling dependence. Everywhere else, the dependence of the non-universal scale factors is relatively weak. Strong deviations from scaling occur at temperatures of order , the degeneracy temperature, reflecting the fact that the temperature dependence of most irrelevant terms is on the scale of the Fermi energy. That high temperature limit should be contrasted with found in the strong coupling case.  In generic cases the upper limit is well above room temperature. In experiment however, the non-universality due to the C-I crossover may make the identification of quantum critical scaling difficult. And since the dependence of is non universal, one may encounter cases where this is in practice impossible.
Electron-doped high-temperature superconductors appear as an ideal system to check quantum critical scaling since they seem well described by the one-band Hubbard model at weak to intermediate coupling. [14, 15] And experiments [22, 23] strongly suggest the presence of a quantum critical point in these materials. In the case of heavy fermions there are examples of SDW-Fermi liquid quantum critical behavior.  However, these are multiband systems where there are additional energy scales, such as the Kondo coherence scale, so our results would apply only in regimes where an effective one-band Hubbard model applies.
A.-M.S.T. would like to particularly thank A. Chubukov and J. Schmalian for informative and stimulating comments on our work, and the Max-Planck Institute for the Physics of Complex Systems for hospitality. We are also indebted to S. Chakravarty, and S. Sachdev for discussions at the Aspen Center for Physics and most importantly to B. Kyung for sharing with us his experience with TPSC on numerous occasions. Numerical calculations were performed on RQCHP computers and on the Elix cluster. The present work was supported by NSERC (Canada), FQRNT (Québec), CFI (Canada), CIFAR, and the Tier I Canada Research chair Program (A.-M.S.T.).
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