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arXiv:2103.05009 · Physical review. B./Physical review. B

Small to large Fermi surface transition in a single-band model using randomly coupled ancillas

Alexander Nikolaenko, Maria Tikhanovskaya, Subir Sachdev A, Ya-Hui Zhang

Published Jun 16, 2021• 15 citations
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Abstract

We describe a solvable model of a quantum transition in a single band model involving a change in the size of the electron Fermi surface without any symmetry breaking. In a model with electron density $1\ensuremath{-}p$, we find a ``large'' Fermi surface state with the conventional Luttinger volume $1\ensuremath{-}p$ of electrons for $p>{p}_{c}$, and a first-order transition to a ``small'' Fermi surface state with a non-Luttinger volume $p$ of holes for $p<{p}_{c}$. As required by extended Luttinger theorems, the small Fermi surface state also has fractionalized spinon excitations. The model has electrons with strong local interactions in a single band; after a canonical transformation, the interactions are transferred to a coupling to two layers of ancilla qubits, as proposed by Zhang and Sachdev [Phys. Rev. Research 2, 023172 (2020)]. Solvability is achieved by employing random exchange interactions within the ancilla layers, and taking the large $M$ limit with $\mathrm{SU}(M)$ spin symmetry, as in the Sachdev-Ye-Kitaev models. The local electron spectral function of the small Fermi surface phase displays a particle-hole asymmetric pseudogap, and maps onto the spectral function of a lightly doped Kondo insulator of a Kondo-Heisenberg lattice model. We discuss connections to the physics of the hole-doped cuprates: the asymmetric pseudogap observed in STM, and the sudden change from incoherent to coherent antinodal spectra observed recently in photoemission. A holographic analogy to wormhole transitions between multiple black holes is briefly noted.

DOI
10.1103/physrevb.103.235138

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We describe a solvable model of a quantum transition in a single band model involving a change in the size of the electron Fermi surface without any symmetry breaking. In a model with electron density 1\ensuremath−p1\ensuremath{-}p1\ensuremath−p, we find a large'' Fermi surface state with the conventional Luttinger volume $1\ensuremath{-}p$ of electrons for $p>{p}_{c}$, and a first-order transition to a small'' Fermi surface state with a non-Luttinger volume ppp of holes for p<pcp<{p}_{c}p<pc​. As required by extended Luttinger theorems, the small Fermi surface state also has fractionalized spinon excitations. The model has electrons with strong local interactions in a single band; after a canonical transformation, the interactions are transferred to a coupling to two layers of ancilla qubits, as proposed by Zhang and Sachdev [Phys. Rev. Research 2, 023172 (2020)]. Solvability is achieved by employing random exchange interactions within the ancilla layers, and taking the large MMM limit with SU(M)\mathrm{SU}(M)SU(M) spin symmetry, as in the Sachdev-Ye-Kitaev models. The local electron spectral function of the small Fermi surface phase displays a particle-hole asymmetric pseudogap, and maps onto the spectral function of a lightly doped Kondo insulator of a Kondo-Heisenberg lattice model. We discuss connections to the physics of the hole-doped cuprates: the asymmetric pseudogap observed in STM, and the sudden change from incoherent to coherent antinodal spectra observed recently in photoemission. A holographic analogy to wormhole transitions between multiple black holes is briefly noted.
Quantum phase transitions of metals involve changes in the Fermi surface, and can be divided into three categories. The first two categories involve symmetry breaking, and lead to a deformation or reconstruction of the Fermi surface. The third category involves a change in the volume enclosed by the Fermi surface without any symmetry breaking: one phase is a Fermi liquid (FL) with the conventional Luttinger volume, while the other phase is a `fractionalized Fermi liquid' (FL*), which has a non-Luttinger volume Fermi surface accompanied by a spin liquid with fractionalized excitations. It is a relatively simple matter to obtain a FL*-FL transition in Kondo lattice models. However, the FL*-FL transition can also be present in single-band Hubbard-like models: this is efficiently described by the `ancilla' method, which shows that the transition is a `flipped' Kondo lattice transition. This single-band FL*-FL transition is argued to apply to the metallic states of the hole-doped cuprates. In the clean limit, the critical properties of the quantum transitions in the three categories are distinct, but all lead to perfect metal transport in the quantum-critical regime. Impurity-induced `Harris disorder', with spatial fluctuations in the local position of the quantum critical point, is a relevant perturbation to the clean critical points. In the presence of Harris disorder, all three categories exhibit strange metal behavior, which can be described by a universal two-dimensional Yukawa-Sachdev-Ye-Kitaev model.
One of the fundamental questions about high-temperature cuprate superconductors is the size of the Fermi surface underlying the superconducting state. By analyzing the single-particle spectral function for the Fermi-Hubbard model as a function of repulsion UUU and chemical potential $\ensuremath{\mu}$, we find that the Fermi surface in the normal state undergoes a transition from a large Fermi surface matching the Luttinger volume as expected in a Fermi liquid, to a Fermi surface that encloses fewer electrons that we dub the ``Luttinger breaking'' phase, as the Mott insulator is approached. This transition into a non-Fermi-liquid phase that violates the Luttinger count occurs at a critical density in the absence of any other broken symmetry. We obtain the Fermi-surface contour from the spectral weight ${A}_{\mathbf{k}}(\ensuremath{\omega}=0)$ and from an analysis of the singularities of the Green's function ReGk(E=0)\mathrm{Re}\phantom{\rule{0.16em}{0ex}}{G}_{\mathbf{k}}(E=0)ReGk​(E=0), calculated using determinantal quantum Monte Carlo and analytic continuation methods. We discuss our numerical results in connection with experiments on Hall measurements, scanning tunneling spectroscopy, and angle-resolved photoemission spectroscopy.
The pseudogap metal phase of hole-doped cuprates can be described by small Fermi surfaces of electronlike quasiparticles, which enclose a volume violating the Luttinger relation. This violation requires the existence of additional fractionalized excitations which can be viewed as fractionalized remnants of the paramagnon. We fractionalize the paramagnon into the bosonic spinons of the spin liquid described by the CP1{\mathbb{CP}}^{1}CP1 U(1) gauge theory, and we present a gauge theory of the bosonic spinons, a Higgs field, and an ancilla layer of fermions coupled to the original electrons. Along with the small Fermi surface pseudogap metal, this theory displays conventional phases: the large Fermi surface Fermi liquid with a low-energy paramagnon mode, and phases with spin density wave order. We describe the evolution of the electronic photoemission spectrum across these quantum phase transitions. We consider both the two-sublattice N\'eel and incommensurate spin density wave phases, and we find that the latter has spiral spin correlations.
We present results of a systematic quantum Monte Carlo study for the single-band Hubbard model. Thereby we evaluated single-particle spectra (PES and IPES), two-particle spectra (spin and density correlation functions), and the dynamical correlation function of suitably defined diagnostic operators, all as a function of temperature and hole doping. The results allow us to identify different physical regimes. Near half-filling we find an anomalous ``Hubbard-I phase,'' where the band structure is, up to some minor modifications, consistent with the Hubbard-I predictions. At lower temperatures, where the spin response becomes sharp, additional dispersionless ``bands'' emerge due to the dressing of electrons/holes with spin excitations. We present a simple phenomenological fit that reproduces the band structure of the insulator quantitatively. The Fermi surface volume in the low-doping phase, as derived from the single-particle spectral function, is not consistent with the Luttinger theorem, but qualitatively in agreement with the predictions of the Hubbard-I approximation. The anomalous phase extends up to a hole concentration of $\ensuremath{\approx}15%,$ i.e., the underdoped region in the phase diagram of high-Tc{T}_{c}Tc​ superconductors. We also investigate the nature of the magnetic ordering transition in the single-particle spectra. We show that the transition to a spin-density wave-like band structure is not accomplished by the formation of any resolvable ``precursor bands,'' but rather by a (spectroscopically invisible) band of spin-3/2 quasiparticles. We discuss implications for the ``remnant Fermi surface'' in insulating cuprate compounds and the shadow bands in the doped materials.
Tractable model for a fractionalized Fermi liquid (FL∗^*∗) on a square lattice
Motivated by the continued interest in Fermi-surface reconstruction without symmetry breaking, we present an analytically tractable microscopic model of a fractionalized Fermi liquid (FL∗^*∗) on a square lattice and discuss its potential relevance to the cuprates. As in ancilla-qubit constructions, the model is related to Kondo lattice systems, but in this case, the conduction electrons interact with a Z2\mathbb{Z}_2Z2​ spin liquid of the Yao--Lee type, with a Majorana Fermi surface. The associated Z2\mathbb Z_2Z2​ gauge theory is static so that the model can be analytically solved to leading-logarithic accuracy. There are two phases: one in which the fractionalized fermions of the spin liquid hybridize with conduction electrons to form a common Fermi surface violating the naive Luttinger count, and one in which they remain decoupled. We discuss the salient features of the small Fermi-surface phase, including analytically derived momentum dependent coherence factors responsible for the appearance of Fermi arcs à la Yang-Rice-Zhang. We further discuss the impact of quantum and thermal fluctuations, including a strong diamagnetic response and a logarithmically divergent Sommerfeld coefficient at the onset of the pseudogap.
We calculate the Fermi surface of electrons in hole-doped, extended ttt-JJJ models on a square lattice in a regime where no long-range antiferromagnetic order is present, and no symmetries are broken. Using the ``spinon-dopon'' formalism of Ribeiro and Wen, we show that short-range antiferromagnetic correlations lead to a reconstruction of the Fermi surface into hole pockets which are not necessarily centered at the antiferromagnetic Brillouin zone boundary. The Brillouin zone area enclosed by the Fermi surface is proportional to the density of dopants away from half-filling, in contrast to the conventional Luttinger theorem, which counts the total electron density. This state realizes a ``fractionalized Fermi liquid'' (FL*), which has been proposed as a possible ground state of the underdoped cuprates; we note connections to recent experiments. We also discuss the quantum phase transition from the FL* state to the Fermi liquid state with long-range antiferromagnetic order.