Quantum criticality in metals

Quantum criticality in metals has been an ongoing topic of interest in condensed matter physics during the last three decades. While mostly studied in the context of heavy-fermion materials [1-3], the existence of a quantum critical point (QCP) has also been conjectured in cuprate superconductors [4], backed by several experimental hints [5-7]. Despite major experimental and theoretical efforts, the nature of QCPs in metals is still poorly understood and controversial.
In a vast number of experimental studies carried out on a variety of different systems, several features common to these QCPs have been sorted out. One of the most prominent features is a reconstruction of the Fermi surface (FS) when crossing the QCP at T=0, leading to a violation of Luttinger's theorem [8,9] on one side of the QCP. This FS reconstruction leads to a sharp crossover in the Hall coefficient at low temperatures, which has been observed both in heavy-fermion systems [10] and cuprate superconductors [6]. Further, a non-Fermi-liquid (NFL) "strange metal" phase is observed when varying temperature in the vicinity of the QCP. It features a linear in temperature resistivity [7,11], which is to date unexplained.
Driven by these experimental findings, various new approaches to tackle strongly correlated matter have been developed. One of the more successful ones is dynamical mean-field theory (DMFT) [12], where properties of a quantum lattice problem are calculated via an effective quantum impurity problem. In this approach, non-local correlation effects are neglected. However, there are growing indications that these are crucial for driving these systems towards a QCP. To incorporate short ranged non-local correlations, quantum cluster methods [13] like the Cellular DMFT (CDMFT) and the Dynamical Cluster Approximation (DCA) have been developed.
DMFT methods crucially depend on a reliable method to solve quantum impurity problems, which is usually done numerically. The most popular methods are Quantum Monte Carlo (QMC) and Exact Diagonalization (ED) methods, but these methods have some limitations. QMC has problems reaching arbitrary low temperatures and requires analytical continuation of Matsubara spectral data; this is a highly non-trivial task, at low temperatures. ED on the other hand has problems resolving low energy scales and generally comes with inferior data quality compared to QMC. These issues are particularly limiting for studying quantum critical behavior, where access low temperatures and the ability to resolve low energy scales are crucially important.
Both requirements are met by the Numerical Renormalization Group (NRG) [14,15]. This method produces spectral data directly on the real-frequency axis, thus eliminating the need for analytic continuation. The NRG code developed in our group has already been applied in several DMFT studies over the last years with great success. Recent work on quantum criticality in heavy-fermion systems using CDMFT [16] showed that our NRG code can provide insights far beyond the capability of QMC [17] or ED [18,19].
A Master's project on quantum criticality in the Hubbard model in our group would directly build on our previous experience with cluster DMFT plus NRG. The student will use existing code to set up a cluster DMFT plus NRG script to study the Hubbard model. For this purpose, either the DCA with an appropriate patching scheme [20] or the CDMFT [21] methods could be used. The main goal of the project will be to clarify the existence of a QCP for this model and to elucidate its physics.

Mentors: vondelft@lmu.de,  Andreas.Gleis@physik.uni-muenchen.de

[1] H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle
Fermi-liquid instabilities at magnetic quantum phase transitions
Rev. Mod. Phys. 79, 1015 (2007).
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.79.1015

[2] G. R. Stewart
Non-Fermi-liquid behavior in d- and f-electron metals
Rev. Mod. Phys. 73, 797 (2001).
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.73.797

[3] P. Coleman, C. Pépin, Q. Si, and R. Ramazashvili
How do Fermi liquids get heavy and die?
J. Phys. Condens. Matter 13, R723 (2001).
https://iopscience.iop.org/article/10.1088/0953-8984/13/35/202

[4] S. Sachdev
Quantum criticality and the phase diagram of the cuprates
Physica (Amsterdam) 470C, S4 (2010).
https://www.sciencedirect.com/science/article/abs/pii/S0921453409007205

[5] B. Michon, C. Girod, S. Badoux, J. Kačmarčík, Q. Ma, M. Dragomir, H. A. Dabkowska, B. D. Gaulin, J.-S. Zhou, S. Pyon, T. Takayama, H. Takagi, S. Verret, N. Doiron-Leyraud, C. Marcenat, L. Taillefer and T. Klein
Thermodynamic signatures of quantum criticality in cuprate superconductors
Nature 567, 218–222 (2019).
https://www.nature.com/articles/s41586-019-0932-x

[6] S. Badoux, W. Tabis, F. Laliberté, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Béard, D. A. Bonn, W. N. Hardy, R. Liang, N. Doiron-Leyraud, L. Taillefer, and C. Proust
Change of carrier density at the pseudogap critical point of a cuprate superconductor
Nature 531, 210–214 (2016).
https://www.nature.com/articles/nature16983

[7] A. Legros, S. Benhabib, W. Tabis, F. Laliberté, M. Dion, M. Lizaire, B. Vignolle, D. Vignolles, H. Raffy, Z. Z. Li, P. Auban-Senzier, N. Doiron-Leyraud, P. Fournier, D. Colson, L. Taillefer, C. Proust
Universal T-linear resistivity and Planckian dissipation in overdoped cuprates
Nature Physics 15, 142–147 (2019).
https://www.nature.com/articles/s41567-018-0334-2

[8] J. M. Luttinger
Fermi Surface and Some Simple Equilibrium Properties of a System of Interacting Fermions
Phys. Rev. 119, 1153 (1960).
https://journals.aps.org/pr/abstract/10.1103/PhysRev.119.1153

[9] M. Oshikawa
Topological approach to Luttinger's theorem and Fermi surface of a Kondo lattice
Phys. Rev. Lett. 84, 3370–3373 (2000).
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.84.3370

[10] S. Paschen, T. Lühmann, S. Wirth, P. Gegenwart, O. Trovarelli, C. Geibel, F. Steglich, P. Coleman and Q. Si
Hall-effect evolution across a heavy-fermion quantum critical point
Nature 432, 881–885 (2004).
https://www.nature.com/articles/nature03129

[11] L. Prochaska, X. Li, D. C. MacFarland, A. M. Andrews, M. Bonta, E. F. Bianco, S. Yazdi, W. Schrenk, H. Detz, A. Limbeck, Q. Si, E. Ringe, G. Strasser, J. Kono, S. Paschen
Singular charge fluctuations at a magnetic quantum critical point
Science 367, 285 (2020).
https://www.nature.com/articles/nature03129

[12] A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg
Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions
Rev. Mod. Phys. 68, 13 (1996).
http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.68.13

[13] T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler
Quantum cluster theories
Rev. Mod. Phys. 77, 1027 (2005).
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.77.1027

[14] R. Bulla, T. A. Costi, and T. Pruschke
Numerical renormalization group method for quantum impurity systems
Rev. Mod. Phys. 80, 395 (2008).
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.80.395

[15] A. Weichselbaum
Tensor networks and the numerical renormalization group
Phys. Rev. B 86, 245124 (2012).
http://link.aps.org/doi/10.1103/PhysRevB.86.245124

[16] A. Gleis
Cluster Dynamical Mean-Field + Numerical Renormalization Group Approach to Strongly Correlated Systems
Master's thesis (2019).
https://www.theorie.physik.uni-muenchen.de/lsvondelft/publications/pdf/190602_gleis.pdf

[17] D. Tanaskovic, K. Haule, G. Kotliar, and V. Dobrosavljevic
Phase diagram, energy scales, and nonlocal correlations in the Anderson lattice model
Phys. Rev. B 84, 115105 (2011).
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.84.115105

[18] L. De Leo, M. Civelli, and G. Kotliar
T=0 Heavy-Fermion Quantum Critical Point as an Orbital-Selective Mott Transition
Phys. Rev. Lett. 101, 256404 (2008).
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.256404

[19] L. De Leo, M. Civelli, and G. Kotliar
Cellular dynamical mean-field theory of the periodic Anderson model
Phys. Rev. B 77, 075107 (2008).
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.77.075107

[20] E. Gull, M. Ferrero, O. Parcollet, A. Georges, and A. J. Millis
Momentum-space anisotropy and pseudogaps: A comparative cluster dynamical mean-field analysis of the doping-driven metal-insulator transition in the two-dimensional Hubbard model
Phys. Rev. B 82, 155101 (2010).
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.82.155101

[21] K. Haule and G. Kotliar
Strongly correlated superconductivity: A plaquette dynamical mean-field theory study
Phys. Rev. B 76, 104509 (2007).
https://link.aps.org/doi/10.1103/PhysRevB.76.104509