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arXiv:1606.07813 · Physical review. B./Physical review. B

Spin density wave order, topological order, and Fermi surface reconstruction

Subir Sachdev A, Erez Berg, Shubhayu Chatterjee, Yoni Schattner

Published Sep 21, 2016• 28 citations
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Abstract

In the conventional theory of density wave ordering in metals, the onset of spin density wave (SDW) order coincides with the reconstruction of the Fermi surfaces into small ``pockets.'' We present models which display this transition, while also displaying an alternative route between these phases via an intermediate phase with topological order, no broken symmetry, and pocket Fermi surfaces. The models involve coupling emergent gauge fields to a fractionalized SDW order, but retain the canonical electron operator in the underlying Hamiltonian. We establish an intimate connection between the suppression of certain defects in the SDW order and the presence of Fermi surface sizes distinct from the Luttinger value in Fermi liquids. We discuss the relevance of such models to the physics of the hole-doped cuprates near optimal doping.

DOI
10.1103/physrevb.94.115147

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In the conventional theory of density wave ordering in metals, the onset of spin density wave (SDW) order coincides with the reconstruction of the Fermi surfaces into small ``pockets.'' We present models which display this transition, while also displaying an alternative route between these phases via an intermediate phase with topological order, no broken symmetry, and pocket Fermi surfaces. The models involve coupling emergent gauge fields to a fractionalized SDW order, but retain the canonical electron operator in the underlying Hamiltonian. We establish an intimate connection between the suppression of certain defects in the SDW order and the presence of Fermi surface sizes distinct from the Luttinger value in Fermi liquids. We discuss the relevance of such models to the physics of the hole-doped cuprates near optimal doping.
Recent work has used a U(1) gauge theory to describe the physics of Fermi pockets in the presence of fluctuating spin density wave order. We generalize this theory to an arbitrary band structure and ordering wave vector. The transition to the large Fermi-surface state, without pockets induced by local spin density wave order, is described by embedding the U(1) gauge theory in a SU(2) gauge theory. The phase diagram of the SU(2) gauge theory shows that the onset of spin density wave order in the Fermi liquid occurs either directly, in the framework discussed by Hertz, or via intermediate non-Fermi-liquid phases with Fermi surfaces of fractionalized excitations. We discuss application of our results to the phase diagram of the cuprates.
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.
This review describes how topological order associated with the presence of emergent gauge fields can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to topological order using Wegner's quantum [Formula: see text] gauge theory on the square lattice: the topological state is characterized by the expulsion of defects, carrying [Formula: see text] magnetic flux. The interplay between topological order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension D = 3; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking order, along with the presence of emergent gauge fields. The phase diagrams of the [Formula: see text] gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave order, topological order associated with defect expulsion, deconfined emergent gauge fields, reconstructed Fermi surfaces (with 'chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model and photoemission observations of the electron-doped cuprates. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.
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.
This review describes how topological order can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to topological order using Wegner's quantum Z2\mathbb{Z}_2Z2​ gauge theory on the square lattice: the topological state is characterized by the expulsion of defects, carrying Z2\mathbb{Z}_2Z2​ magnetic flux. The interplay between topological order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension D=3D=3D=3; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking order. The phase diagrams of the Z2\mathbb{Z}_2Z2​ gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave order, topological order associated with defect expulsion, reconstructed Fermi surfaces (with `chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.
Insulating quantum spin liquids can undergo a confinement transition to a valence bond solid via the condensation of topological excitations of the associated gauge theory. We extend the theory of such transitions to fractionalized Fermi liquids (FL*): These are metallic doped spin liquids in which the Fermi surfaces only have gauge neutral quasiparticles. Using insights from a duality transform on a doped quantum dimer model for the U(1)-FL* state, we show that projective symmetry group of the theory of the topological excitations remains unmodified, but the Fermi surfaces can lead to additional frustrating interactions. We propose a theory for the confinement transition of Z2{\mathbb{Z}}_{2}Z2​-FL* states via the condensation of visons. A variety of confining, incommensurate density wave states are possible, including some that are similar to the incommensurate ddd-form factor density wave order observed in several recent experiments on the cuprate superconductors.