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arXiv:2007.07896 · Physical Review X

Non-Fermi Liquids as Ersatz Fermi Liquids: General Constraints on Compressible Metals

Dominic V. Else, Ryan Thorngren, T. Senthil

Published Apr 6, 2021• 90 citations
arXivHTMLPDF

Abstract

A system with charge conservation and lattice translation symmetry has a well-defined filling ν, which is a real number representing the average charge per unit cell.We show that if ν is fractional (i.e., not an integer), this imposes very strong constraints on the low-energy theory of the system and give a framework to understand such constraints in great generality, vastly generalizing the Luttinger and Lieb-Schultz-Mattis theorems.The most powerful constraint comes about if ν is continuously tunable (i.e., the system is charge compressible), in which case, we show that the low-energy theory must have a very large emergent symmetry group-larger than any compact Lie group.An example is the Fermi surface of a Fermi liquid, where the charge at every point on the Fermi surface is conserved.We expect that in many, if not all, cases, even exotic non-Fermi liquids will have the same emergent symmetry group as a Fermi liquid, even though they could have very different dynamics.We call a system with this property an ersatz Fermi liquid.We show that ersatz Fermi liquids share a number of properties in common with Fermi liquids, including Luttinger's theorem (which is thus extended to a large class of non-Fermi liquids) and periodic "quantum oscillations" in the response to an applied magnetic field.We also establish versions of Luttinger's theorem for the composite Fermi liquid in quantum Hall systems and for spinon Fermi surfaces in Mott insulators.Our work makes a connection between filling constraints and the theory of symmetry-protected topological phases, in particular through the concept of "'t Hooft anomalies."

DOI
10.1103/physrevx.11.021005

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Non-Fermi Liquids as Ersatz Fermi Liquids: General Constraints on Compressible Metals

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A system with charge conservation and lattice translation symmetry has a well-defined filling ν, which is a real number representing the average charge per unit cell.We show that if ν is fractional (i.e., not an integer), this imposes very strong constraints on the low-energy theory of the system and give a framework to understand such constraints in great generality, vastly generalizing the Luttinger and Lieb-Schultz-Mattis theorems.The most powerful constraint comes about if ν is continuously tunable (i.e., the system is charge compressible), in which case, we show that the low-energy theory must have a very large emergent symmetry group-larger than any compact Lie group.An example is the Fermi surface of a Fermi liquid, where the charge at every point on the Fermi surface is conserved.We expect that in many, if not all, cases, even exotic non-Fermi liquids will have the same emergent symmetry group as a Fermi liquid, even though they could have very different dynamics.We call a system with this property an ersatz Fermi liquid.We show that ersatz Fermi liquids share a number of properties in common with Fermi liquids, including Luttinger's theorem (which is thus extended to a large class of non-Fermi liquids) and periodic "quantum oscillations" in the response to an applied magnetic field.We also establish versions of Luttinger's theorem for the composite Fermi liquid in quantum Hall systems and for spinon Fermi surfaces in Mott insulators.Our work makes a connection between filling constraints and the theory of symmetry-protected topological phases, in particular through the concept of "'t Hooft anomalies."
A system with charge conservation and lattice translation symmetry has a well-defined filling ν, which is a real number representing the average charge per unit cell. We show that if ν is fractional (i.e., not an integer), this imposes very strong constraints on the low-energy theory of the system and give a framework to understand such constraints in great generality, vastly generalizing the Luttinger and Lieb-Schultz-Mattis theorems. The most powerful constraint comes about if ν is continuously tunable (i.e., the system is charge compressible), in which case, we show that the low-energy theory must have a very large emergent symmetry group—larger than any compact Lie group. An example is the Fermi surface of a Fermi liquid, where the charge at every point on the Fermi surface is conserved. We expect that in many, if not all, cases, even exotic non-Fermi liquids will have the same emergent symmetry group as a Fermi liquid, even though they could have very different dynamics. We call a system with this property an ersatz Fermi liquid. We show that ersatz Fermi liquids share a number of properties in common with Fermi liquids, including Luttinger's theorem (which is thus extended to a large class of non-Fermi liquids) and periodic "quantum oscillations" in the response to an applied magnetic field. We also establish versions of Luttinger's theorem for the composite Fermi liquid in quantum Hall systems and for spinon Fermi surfaces in Mott insulators. Our work makes a connection between filling constraints and the theory of symmetry-protected topological phases, in particular through the concept of "'t Hooft anomalies."Received 17 August 2020Revised 22 December 2020Accepted 21 January 2021DOI:https://doi.org/10.1103/PhysRevX.11.021005Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAnomaliesFermi surfacePhysical SystemsStrongly correlated systemsTechniquesFermi liquid theoryNon-Fermi-liquid theorySymmetries in condensed matterCondensed Matter & Materials Physics
The discovery of many strongly correlated metallic phases has inspired different routes to generalize or go beyond the celebrated Landau Fermi liquid theory. To this end, from universal consideration of symmetries and anomalies, Else, Thorngren, and Senthil (ETS) have introduced a class of theories called ersatz Fermi liquids which possess a Fermi surface and satisfy a generalized Luttinger's theorem. In this work, we view all such fermion liquids obeying the Luttinger theorem as incompressible quantum Hall liquids in higher-dimensional phase space and use it as the starting point to derive their effective low-energy field theory. The noncommutativity of phase space motivates us to use the Seiberg-Witten map to derive the field theory in an ordinary (commutative) space and naturally leads to terms that correspond to the correct topological Chern-Simons action postulated by ETS in one, two, and three dimensions. Additionally, our approach also reproduces all the nontopological terms that characterize important contributions to the response, including the semiclassical equations of motion. Finally, our derivations of Chern-Simons terms from the Seiberg-Witten map also verify a longstanding conjecture in noncommutative field theory.
A long-standing mystery of fundamental importance in correlated electron physics is to understand strange non-Fermi liquid metals that are seen in diverse quantum materials. A striking experimental feature of these metals is a resistivity that is linear in temperature (T). In this Letter we ask what it takes to obtain such non-Fermi liquid physics down to zero temperature in a translation invariant metal. If in addition the full frequency (ω) dependent conductivity satisfies ω/T scaling, we argue that the T-linear resistivity must come from the intrinsic physics of the low energy fixed point. Combining with earlier arguments that compressible translation invariant metals are "ersatz Fermi liquids" with an infinite number of emergent conserved quantities, we obtain powerful and practical conclusions. We show that there is necessarily a diverging susceptibility for an operator that is odd under inversion and time reversal symmetries, and has zero crystal momentum. We discuss a few other experimental consequences of our arguments, as well as potential loopholes, which necessarily imply other exotic phenomena.
Luttinger's theorem for Fermi liquids equates the volume enclosed by the Fermi surface in momentum space to the electron filling, independent of the strength and nature of interactions. Motivated by recent momentum balance arguments that establish this result in a nonperturbative fashion [M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000)], we present extensions of this momentum balance argument to exotic systems which exhibit quantum number fractionalization focusing on Z2{Z}_{2}Z2​ fractionalized insulators, superfluids and Fermi liquids. These lead to nontrivial relations between the particle filling and some intrinsic property of these quantum phases, and hence may be regarded as natural extensions of Luttinger's theorem. We find that there is an important distinction between fractionalized states arising naturally from half filling versus those arising from integer filling. We also note how these results can be useful for identifying fractionalized states in numerical experiments.
Given the considerable theoretical challenges in understanding strongly coupled metals and non-Fermi liquids, it is valuable to have a framework to understand properties of metals that are universal, in the sense that they must hold in any metal. It has previously been argued that an infinite-dimensional emergent symmetry group is such a property, at least for clean, compressible metals. In this paper, we will show that such an emergent symmetry group has very strong implications for the dynamics of the metal. Specifically, we show that consideration of the hydrodynamics of the associated infinitely many emergent conserved quantities automatically recovers the collisionless Boltzmann equation that governs the dynamics of a Fermi liquid. Therefore the hydrodynamic prediction is that in the low-temperature, collisionless regime where the emergent conservation laws hold, the dynamics and response to external fields of a general spinless metal will be identical to a Fermi liquid. We discuss some potential limitations to this general statement, including the possibility of nonhydrodynamic modes. We also report some interesting differences in the case of spinful metals.
Nonperturbative constraints on many-body physics, such as the famous Lieb-Schultz-Mattis theorem, are valuable tools for studying strongly correlated systems. To this end, we present a number of nonperturbative results that constrain the low-energy physics of systems having conserved dipole moments. We find that for these systems, a unique translationally invariant gapped ground state is only possible if the polarization of the system is integer. Furthermore, if a lattice system also has U(1)\text{U}(1)U(1) subsystem charge conservation symmetry, a unique gapped ground state is only possible if the particle filling along these subsystems is integer. We also apply these methods to spin systems, and determine criteria for the existence of a different type of magnetic response plateau in the presence of a nonuniform magnetic field. Finally, we formulate a version of Luttinger's theorem for one-dimensional systems consisting of dipoles.