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arXiv:1712.08771 · Physical review. B./Physical review. B

Magnetic field induced quantum criticality and the Luttinger sum rule

Yunori Nishikawa, Oliver J. Curtin, Alex C. Hewson, Daniel J. G. Crow

Published Sep 14, 2018• 14 citations
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Abstract

We show that when there is a sudden transition from a small to a large Fermi surface at a field-induced quantum critical point---similar to what may have been observed in some heavy-fermion compounds---an additional term has to be taken into account in the Luttinger-Friedel sum rule. We calculate this additional term for a local model which has a field-induced quantum critical point (QCP) and show that it changes abruptly at the transition, such that it satisfies a generalized Luttinger-Friedel sum rule on each side of the transition, and characterizes the two Fermi-liquid phases separated by the QCP as a discrete (topological) index.

DOI
10.1103/physrevb.98.104419

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We show that when there is a sudden transition from a small to a large Fermi surface at a field-induced quantum critical point---similar to what may have been observed in some heavy-fermion compounds---an additional term has to be taken into account in the Luttinger-Friedel sum rule. We calculate this additional term for a local model which has a field-induced quantum critical point (QCP) and show that it changes abruptly at the transition, such that it satisfies a generalized Luttinger-Friedel sum rule on each side of the transition, and characterizes the two Fermi-liquid phases separated by the QCP as a discrete (topological) index.
The Luttinger Theorem, which relates the electron density to the volume of the Fermi surface in an itinerant electron system, is taken to be one of the essential features of a Fermi liquid. The microscopic derivation of this result depends on the vanishing of a certain integral, the Luttinger integral I L , which is also the basis of the Friedel sum rule for impurity models, relating the impurity occupation number to the scattering phase shift of the conduction electrons. It is known that non-zero values of I L with I L = ± π /2, occur in impurity models classified as singular Fermi liquids. Here we show the same values, I L = ± π /2, occur in an impurity model in phases with regular low energy Fermi liquid behavior. Consequently the Luttinger integral can be taken to characterize these phases, and the quantum critical points separating them interpreted as topological.
The quantum critical point (QCP) in YbRh2Si2YbRh_{2}Si_{2}YbRh2​Si2​ is an enigma for the itinerant view of QCP. In an alternative view, this QCP is intimately linked to the selective Mott localization of the heavy fff electrons. Following a perusal of this unusual QCP, I study an Extended Periodic Anderson Model (EPAM) within DMFT. A quantum phase transition (FQPT), accompanied by a rapid change in the Fermi volume, is found near the quantum-critical end-point of the selective Mott transition in the fff-electron sector. The theory accounts for a wide range of unusual, singular non-Fermi liquid features exhibited at this QCP in YbRh2Si2YbRh_{2}Si_{2}YbRh2​Si2​ in a natural way.
We study correlated quantum impurity models that undergo a local quantum phase transition (QPT) from a strong coupling, Fermi liquid phase to a non-Fermi liquid phase with a globally doubly degenerate ground state. Our aim is to establish what can be shown exactly about such ``local moment'' (LM) phases, of which the permanent (zero-field) local magnetization is a hallmark, and an order parameter for the QPT. A description of the zero-field LM phase is shown to require two distinct self-energies, which reflect the broken symmetry nature of the phase and together determine the single self-energy of standard field theory. Distinct Friedel sum rules for each phase are obtained, via a Luttinger theorem embodied in the vanishing of appropriate Luttinger integrals. By contrast, the standard Luttinger integral is nonzero in the LM phase but found to have universal magnitude. A range of spin susceptibilites are also considered, including that corresponding to the local order parameter, whose exact form is shown to be RPA-like, and to diverge as the QPT is approached. Particular attention is given to the pseudogap Anderson model, including the basic physical picture of the transition, the low-energy behavior of single-particle dynamics, the quantum critical point itself, and the rather subtle effect of an applied local field. A two-level impurity model that undergoes a QPT (``singlet-triplet'') to an underscreened LM phase is also considered, for which we derive on general grounds some key results for the zero-bias conductance in both phases.
Discontinuous quantum phase transitions besides their general interest are clearly relevant to the study of heavy fermions and magnetic transition metal compounds. Recent results show that in many systems belonging to these classes of materials the magnetic transition changes from second order to first order as they approach the quantum critical point (QCP). We investigate here some mechanisms that may be responsible for this change, specifically the coupling of the order parameter to soft modes and the competition between different types of order near the QCP. For weak first order quantum phase transitions general results are obtained. In particular, we describe the thermodynamic behaviour at this transition when it is approached from finite temperatures. This is the discontinuous equivalent of the non-Fermi-liquid trajectory close to a conventional QCP in a heavy-fermion material.
The quantum phase transition from a spin-Peierls phase with a small Fermi surface to a paramagnetic Luttinger-liquid phase with a large Fermi surface is studied in the framework of a one-dimensional Kondo-Heisenberg model that consists of an electron gas away from half filling, coupled to a spin-1/2 chain by Kondo interactions. The Kondo spins are further coupled to each other with isotropic nearest-neighbor and next-nearest-neighbor antiferromagnetic Heisenberg interactions which are tuned to the Majumdar-Ghosh point. Focusing on three-eighths filling and using the density-matrix renormalization-group (DMRG) method, we show that the zero-temperature transition between the phases with small and large Fermi momenta appears continuous, and involves a new intermediate phase where the Fermi surface is not well defined. The intermediate phase is spin gapped and has Kondo-spin correlations that show incommensurate modulations. Our results appear incompatible with the local picture for the quantum phase transition in heavy fermion compounds, which predicts an abrupt change in the size of the Fermi momentum.
Fermi surface reconstruction in cuprates can lead to an abrupt change in the Fermi momentum kFk_FkF​ between different phases. This phenomenon remains subject of debate and is at the heart of an ongoing discussion about the nature of the metallic state in the pseudogap regime. Here we study a minimal model of a kFk_FkF​ changing crossover in the one-dimensional Fermi-Hubbard model, where a tuning of the onsite interaction leads to a crossover between a spin-1/21/21/2 Luttinger liquid with small Fermi momentum and a spinless chargon liquid with large Fermi momentum. We attribute this to an emergent U(1)U(1)U(1) symmetry in the strongly correlated limit, which can be used to derive a modified Luttinger sum rule recovering the large Fermi momentum. We analyse Friedel oscillations at the edge of a system to directly probe the change of Fermi momentum at zero and non-zero temperature. This paves the way for a direct experimental observation of changes of the Fermi momentum using ultracold fermions in a quantum gas microscope, with possible extensions to higher dimensional systems.