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# Itinerant quantum critical point with fermion pockets and hotspots

Edited by Subir Sachdev, Harvard University, Cambridge, MA, and approved July 1, 2019 (received for review January 31, 2019)

## Significance

The present work summarizes major progress in research on the itinerant quantum critical point (QCP). The authors designed a model and developed quantum Monte Carlo simulation to examine itinerant QCPs generated by antiferromagnetic fluctuations. The model has immediate relevance to a wide range of strongly correlated systems, such as cuprate superconductors. Large system size and low temperature are comfortably accessed and quantum critical scaling relations are revealed with high accuracy. At the QCP, a finite anomalous dimension is observed, and fermions at hotspots evolve into a non-Fermi liquid. These results are being observed in an unbiased manner and they could bridge future developments both in analytical theory and in numerical simulation of itinerant QCPs.

## Abstract

Metallic quantum criticality is among the central themes in the understanding of correlated electronic systems, and converging results between analytical and numerical approaches are still under review. In this work, we develop a state-of-the-art large-scale quantum Monte Carlo simulation technique and systematically investigate the itinerant quantum critical point on a 2D square lattice with antiferromagnetic spin fluctuations at wavevector

In the study of correlated materials, quantum criticality in itinerant electron systems is of great importance and interest (1⇓⇓⇓⇓⇓⇓–8). It plays a vital role in the understanding of anomalous transport, strange metal, and non–Fermi-liquid behaviors (9⇓⇓⇓–13) in heavy-fermion materials (14, 15), Cu- and Fe-based high-temperature superconductors (16⇓–18) as well as the recently discovered pressure-driven quantum critical point (QCP) between magnetic order and superconductivity in transition-metal monopnictides, CrAs (19), MnP (20),

The recent development of sign-problem-free quantum Monte Carlo techniques has paved an additional pathway toward sharpening our understanding about this challenging problem (see a concise commentary in ref. 31 and a review in ref. 32 that summarize the recent progress). Via coupling a Fermi liquid with various bosonic critical fluctuations, a wide variety of itinerant quantum critical systems have been studied, such as Ising-nematic (33, 34), ferromagnetic (13), charge density wave (35), spin density wave (36⇓⇓⇓⇓–41), and interaction-driven topological phase transitions and gauge fields (42⇓⇓⇓⇓–47). With the fast development of quantum Monte Carlo (QMC) techniques, in particular the self-learning Monte Carlo (SLMC) (48⇓⇓⇓⇓⇓–54) and elective momentum ultrasize quantum Monte Carlo (EQMC) (41), it now becomes possible to explore larger system sizes than those handled with conventional determinantal quantum Monte Carlo, consequently allowing us to access the genuine scaling behaviors in the infrared (IR) limit for itinerant quantum criticality.

Although many intriguing results and insights have been obtained, for the search of novel quantum critical points beyond the Hertz–Millis–Moriya theory, a major gap between theory and numerical studies still remains. So far, in QMC simulations, among all recently studied itinerant QCPs, either the Hertz–Millis mean-field scaling behavior is found (33, 40) or unpredicted exponents, deviating from existing theories, are observed (13), while theoretically proposed properties beyond the Hertz–Millis–Moriya scaling behaviors still remain to be numerically observed and verified.

In this paper, we aim at improving the convergence between theoretical and numerical studies by focusing on itinerant QCPs with finite ordering wavevector

On the other hand, more recent theoretical developments point out that this conclusion becomes questionable once higher-order effects are taken into account. In particular, 2 different universality classes need to be distinguished, depending on whether *et al*. (23) pointed out explicitly, in this case the Hertz–Millis mean-field scaling law breaks down and a non-zero anomalous dimension emerges. In addition, the critical fluctuations will also change the fermion dispersion near the hotspots, resulting in a critical-fluctuation–induced Fermi surface nesting: i.e., even if one starts from a Fermi surface without nesting, the renormalization group (RG) flow of the Fermi velocity will deform the Fermi surface at hotspots toward nesting (23). This Fermi surface deformation will further increase the anomalous dimension and make the scaling exponent deviate even farther from Hertz–Millis prediction (23, 24), and even modifies the dynamic critical exponent z, as pointed out explicitly by Metlitski and Sachdev (26) and others (28, 29). For

On the numerical side, a QCP with

In this paper, we perform large-scale quantum Monte Carlo simulations to study the antiferromagnetic metallic quantum critical point (AFM-QCP) with

## Model and Method

### Antiferromagnetic Fermiology.

The square lattice AFM model that we designed is schematically shown in Fig. 1*A* with 2 fermion layers and 1 Ising spin layer in between. Fermions are subject to intralayer nearest-, second-, and third-neighbor hoppings

*C* (light blue line), at *B*.

The fermions in *B*, the original FS and FSs from zone folding (shifted by the ordering wavevector

In the simulation, we set *B*.

### Ising Scaling for the Bare Boson Model.

Before presenting our results about the itinerant AFM-QCP, we first discuss the QCP in the pure boson limit without fermions, which serves as a benchmark for the nontrivial itinerant quantum criticality. It is known that the pure boson QCP (**6** explicitly respects the emergent Lorentz symmetry at the Ising critical point.

Without fermions, we can use the standard path-integral scheme to map the 2D transverse-field Ising model to a *A* such a scaling relation is indeed observed with

A similar scaling relation is also observed in the frequency dependence as shown in Fig. 2*B*, where we plot*A* and *B* confirm the QCP for the pure boson part **1** belongs to the

### DQMC and EQMC.

To solve the problem in Eq. **1** we use 2 complementary fermionic quantum Monte Carlo schemes.

The first one is the standard DQMC (13, 63⇓–65) with the SLMC update scheme (48⇓⇓⇓⇓⇓–54) to speed up the simulation. In SLMC, we first perform the standard DQMC simulation on the model in Eq. **1** and then train an effective boson Hamiltonian that contains long-range 2-body interactions both in spatial and in temporal directions. The effective Hamiltonian serves as the proper low-energy description of the problem at hand with the fermion degree of freedom integrated out. We then use the effective Hamiltonian to guide the Monte Carlo simulations; i.e., we perform many sweeps of the effective bosonic model (as the computational cost of updating the boson model is **1** such that the detailed balance of the global update is satisfied. As shown in our previous works (40, 49, 50, 53, 54), the SLMC can greatly reduce the autocorrelation time in the conventional DQMC simulation and make the larger systems and lower temperature accessible.

The other method is the EQMC (41). EQMC is inspired by the awareness that critical fluctuations mainly couple to fermions near the hotspots. Thus, instead of including all of the fermion degrees of freedom, we ignore fermions far away from the hotspots and focus only on momentum points near the hotspots in the simulation. This approximation will produce different results for nonuniversal quantities compared with the original model, such as

In EQMC, because a local coupling (in real space) becomes nonlocal in the momentum basis, one can no longer use the local update as in standard DQMC, as that would cost

In the square lattice model, as shown in Fig. 1*B*, the AFM wavevectors

DQMC and EQMC are complementary to each other; the former provides unbiased results with relatively small systems and the latter, as an approximation, provides results closer to the QCP with finite-size effects better suppressed. One other benefit of EQMC is that it provides much higher momentum resolution close to the hotspots. Fig. 3 depicts the FS of the model in Eq. **1** obtained from *A* and *B*) and EQMC (Fig. 3 *C* and *D*). Fig. 3 *A* and *C* is for *B* and *D* is for *C* and *D*, and the momentum resolution is dramatically improved. For example, in Fig. 3*C*, inside the AFM metallic phase, the gap at hotspots is clearly visualized. And in Fig. 3*D*, at the AFM-QCP the FS recovers the shape of the noninteracting one, and non–Fermi-liquid behavior emerges at the hotspots as shown in the next section. To capture these important physics, EQMC and its higher-momentum resolution play a vital role.

## Results

### Non-Fermi Liquid.

As we emphasized above, the dramatically improved momentum resolution in EQMC enables us to study the fermionic modes on the FS more precisely. We studied the fermion self-energies in the AFM-metal phase and at the AFM-QCP. The results are shown in Fig. 4.

In the AFM-metal phase, although the bands are folded according to Fig. 1*B*, the system remains a Fermi liquid, with a band gap opening up at hotspots. Such expectations are revealed in Fig. 4 *A* and *C*. The Matsubara-frequency dependence of the *B* and *D*,

It is worthwhile to point out that, at the QCP, for fermions at the hotspot, a finite imaginary part in fermion self-energy is observed, which does not seem to decay to 0 as we reduce the frequency. This behavior (a constant term in the imaginary part of the self-energy) is not yet theoretically understood. However, it is consistent with similar QMC studies, where such a finite or constant term always seems to emerge near itinerant QCPs (13, 40, 41).

### Universality Class and Critical Exponents.

In our previous work on triangle lattice AFM-QCP (40) with **5** and **6**) close to the QCP, revealed with

For the square lattice model in this paper, we expect that the dynamic spin susceptibility has the following asymptotic form in the quantum critical region (

We first look at the q dependence of *A*, the momentum *A*. Using EQMC, with the system size as large as *SI Appendix*, *Comparison of EQMC and DQMC at Quantum Critical Region*.

For the frequency dependence in χ, as shown in Fig. 5*B*, we analyze the **10** and the data points fit very well the expected functional form*A*. Although the data are consistent with dynamical exponent

In the absence of fermions,

This difference can be understood in the following way. Between *A*. For *B* and *C*, and in particular the diagram shown in Fig. 6*C* results in logarithmic corrections and is responsible for the breakdown of the Hertz–Millis scaling. However, for

### Comparison with RG Analysis.

On the theory side, perturbative renormalization group calculation has been performed for Heisenberg AFM-QCPs with SU(2) symmetry (23, 26), while the same study for Ising spins has not yet been carefully analyzed to our best knowledge. However, because some of the key features in the RG analysis are insensitive to the spin symmetry (23, 26), many qualitative results will hold and thus here we compare our numerical results with existing theoretical predictions for Heisenberg AFM-QCPs, but it must also be emphasized that agreement at the quantitative level is not expected here because of this difference in symmetry.

In the perturbative renormalization group calculation (23, 26), the anomalous dimension depends on the angle between the hotspot Fermi velocity and the order wavevector

In addition, the RG analysis also predicts that near the QCP the Fermi surface at hotspots will rotate toward nesting (23). This rotation of the Fermi surface will further increase the anomalous dimension and can even renormalize the value of the dynamic critical exponent z (23, 26). As is shown below, our study indeed observed this Fermi surface rotation near the QCP. However, because this RG flow is very slow, our hotspot Fermi surface rotated only by about 0.5^{○} in our simulation before being stopped by a cutoff. For such a small rotation, the resulting increase of the anomalous dimension and the change in the dynamic critical exponent are too weak to be observed.

We calculate the Fermi velocity

According to the RG analysis (23),

## Discussion

If we compare the AFM-QCPs with

In the study of criticality and anomalous critical scalings, the comparison between theory and numerical results plays a vital role. For QCPs in itinerant fermion systems, although non–mean-field scaling beyond the Hertz–Millis theory has been predicted in theory and observed in QMC simulations, it has been a long-standing challenge to reconcile numerical and theoretical results. Our study offers a solid example where an agreement between theory and numerical simulations starts to emerge, which is 1 first step toward a full understanding about itinerant QCPs (32). In particular, to pinpoint the exact value of the frequency exponent and to probe the predicted anomalous dynamical critical exponents (26, 28, 29), lower temperature and frequency range need to be explored. As pointed out in ref. 31, future works along this line are highly desirable and are actively being pursued by us (32).

At the technical level, a combination of DQMC and EQMC methodologies in this work shows a very promising direction in the numerical investigations of itinerant QCPs. Besides the consistency check in ref. 41 for triangular lattice AFM-QCP, the square lattice AFM-QCP investigated here provides the second example of the consistency in DQMC and EQMC in terms of revealing critical properties. Such consistency suggests another pathway for future studies about quantum criticality in fermionic systems, in that one can use DQMC on small systems to provide benchmark results and use EQMC to reveal IR physics at the thermodynamic limit.

## Acknowledgments

We acknowledge valuable discussions with Avraham Klein, Sung-Sik Lee, Yoni Schattner, Andrey Chubukov, Subir Sachdev, and Steven Kivelson on various subjects of itinerant quantum criticality. Z.H.L., G.P., and Z.Y.M. acknowledge funding from the Ministry of Science and Technology of China through the National Key Research and Development Program (2016YFA0300502); the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB28000000); and the National Science Foundation of China under Grants 11421092, 11574359, and 11674370. X.Y.X. is thankful for the support of Hong Kong Research Grants Council through Grant C6026-16W. K.S. acknowledges support from the National Science Foundation under Grant EFRI-1741618 and the Alfred P. Sloan Foundation. We thank the Center for Quantum Simulation Sciences in the Institute of Physics, Chinese Academy of Sciences and the Tianhe-1A and Tianhe-II platforms at the National Supercomputer Centers in Tianjin and Guangzhou for their technical support and generous allocation of CPU time.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: zymeng{at}hku.hk.

Author contributions: X.Y.X., K.S., and Z.Y.M. designed research; Z.H.L., G.P., X.Y.X., K.S., and Z.Y.M. performed research; X.Y.X., K.S., and Z.Y.M. analyzed data; and K.S. and Z.Y.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1901751116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

## References

- ↵
- J. A. Hertz

- ↵
- A. J. Millis

- ↵
- T. Moriya

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- T. Senthil

- ↵
- T. Holder,
- W. Metzner

- ↵
- M. A. Metlitski,
- D. F. Mross,
- S. Sachdev,
- T. Senthil

- ↵
- X. Y. Xu,
- K. Sun,
- Y. Schattner,
- E. Berg,
- Z. Y. Meng

- ↵
- ↵
- A. Steppke et al.

_{1-x}As_{x})2. Science 339, 933–936 (2013). - ↵
- W. Zhang et al.

- ↵
- Z. Liu et al.

- ↵
- Y. Gu et al.

- ↵
- ↵
- J. G. Cheng et al.

- ↵
- M. Matsuda et al.

- ↵
- J. Cheng,
- J. Luo

- ↵
- ↵
- ↵
- M. A. Metlitski,
- S. Sachdev

- ↵
- M. A. Metlitski,
- S. Sachdev

- ↵
- S. Sur,
- S. S. Lee

- ↵
- A. Schlief,
- P. Lunts,
- S. S. Lee

- ↵
- S. S. Lee

- ↵
- A. Schlief,
- P. Lunts,
- S. S. Lee

- ↵
- A. Chubukov

*J. Club Condens. Matter Phys*., 201802 (2018). - ↵
- X. Y. Xu et al.

- ↵
- Y. Schattner,
- S. Lederer,
- S. A. Kivelson,
- E. Berg

- ↵
- S. Lederer,
- Y. Schattner,
- E. Berg,
- S. A. Kivelson

- ↵
- Z. X. Li,
- F. Wang,
- H. Yao,
- D. H. Lee

- ↵
- E. Berg,
- M. A. Metlitski,
- S. Sachdev

- ↵
- Z. X. Li,
- F. Wang,
- H. Yao,
- D. H. Lee

- ↵
- Y. Schattner,
- M. H. Gerlach,
- S. Trebst,
- E. Berg

- ↵
- M. H. Gerlach,
- Y. Schattner,
- E. Berg,
- S. Trebst

- ↵
- Z. H. Liu,
- X. Y. Xu,
- Y. Qi,
- K. Sun,
- Z. Y. Meng

- ↵
- Z. H. Liu,
- X. Y. Xu,
- Y. Qi,
- K. Sun,
- Z. Y. Meng

- ↵
- X. Y. Xu,
- K. S. D. Beach,
- K. Sun,
- F. F. Assaad,
- Z. Y. Meng

- ↵
- F. F. Assaad,
- T. Grover

- ↵
- S. Gazit,
- M. Randeria,
- A. Vishwanath

- ↵
- Y. Y. He et al.

- ↵
- X. Y. Xu et al.

- ↵
- C. Chen,
- X. Y. Xu,
- Y. Qi,
- Z. Y. Meng

- ↵
- J. Liu,
- Y. Qi,
- Z. Y. Meng,
- L. Fu

- ↵
- J. Liu,
- H. Shen,
- Y. Qi,
- Z. Y. Meng,
- L. Fu

- ↵
- X. Y. Xu,
- Y. Qi,
- J. Liu,
- L. Fu,
- Z. Y. Meng

- ↵
- Y. Nagai,
- H. Shen,
- Y. Qi,
- J. Liu,
- L. Fu

- ↵
- H. Shen,
- J. Liu,
- L. Fu

- ↵
- C. Chen et al.

- ↵
- C. Chen,
- X. Y. Xu,
- Z. Y. Meng,
- M. Hohenadler

- ↵
- K. Sun,
- B. M. Fregoso,
- M. J. Lawler,
- E. Fradkin

- ↵
- E. Berg,
- S. Lederer,
- Y. Schattner,
- S. Trebst

- ↵
- H. W. J. Blöte,
- Y. Deng

- ↵
- S. Hesselmann,
- S. Wessel

- ↵
- D. Chowdhury,
- S. Sachdev

- ↵
- Y. C. Wang,
- Y. Qi,
- S. Chen,
- Z. Y. Meng

- ↵
- ↵
- ↵
- R. Blankenbecler,
- D. J. Scalapino,
- R. L. Sugar

- ↵
- J. E. Hirsch

- ↵
- H. Fehske,
- R. Schneider,
- A. Weisse

- F. Assaad,
- H. Evertz

- ↵
- M. A. Metlitski,
- S. Sachdev

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