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arXiv:1912.09391 · Physical review. B./Physical review. B

Metallic states beyond the Tomonaga-Luttinger liquid in one dimension

Wenjie Ji, Xiao-Gang Wen

Published Nov 4, 2020• 7 citations
arXivHTMLPDF

Abstract

In this paper, we propose strongly correlated gapless states (or critical states) of spin-1/2 electrons in $1+1$ dimensions, such as the doped ferromagnetic and antiferromagnetic spin-1/2 Ising chains. We find that the metallic phases in the doped ferromagnetic and antiferromagnetic Ising chain are different strongly correlated gapless phases, despite the fact that the two phases have the same symmetry. The doped antiferromagnetic Ising chain has a finite energy gap for all charge-1 fermionic excitations even without pairing caused by the attractive interaction, resembling the pseudogap phase of underdoped high ${T}_{c}$ superconductors. Applying a transverse field to the ferromagnetic and antiferromagnetic metallic phase can restore the ${Z}_{2}$ symmetry, which gives rise to two distinct critical points despite the fact that the two transitions have exactly the same symmetry-breaking pattern. We also propose chiral metallic states. Some of these gapless states are strongly correlated in the sense that they do not belong to the usual Tomonaga-Luttinger phase of fermions, i.e., they cannot be smoothly deformed into noninteracting fermion systems with the same symmetry. Our nonperturbative results are obtained by noting that gapless quantum systems have emergent categorical symmetries (i.e., noninvertible gravitational anomalies), which are described by multicomponent partition functions that are modular covariant. This allows us to calculate the scaling dimensions and quantum numbers of all the low-energy operators for those strongly correlated gapless states. This demonstrates an application of emergent categorical symmetries in determining low-energy properties of strongly correlated gapless states, which are hard to obtain otherwise.

DOI
10.1103/physrevb.102.195107

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In this paper, we propose strongly correlated gapless states (or critical states) of spin-1/2 electrons in 1+11+11+1 dimensions, such as the doped ferromagnetic and antiferromagnetic spin-1/2 Ising chains. We find that the metallic phases in the doped ferromagnetic and antiferromagnetic Ising chain are different strongly correlated gapless phases, despite the fact that the two phases have the same symmetry. The doped antiferromagnetic Ising chain has a finite energy gap for all charge-1 fermionic excitations even without pairing caused by the attractive interaction, resembling the pseudogap phase of underdoped high Tc{T}_{c}Tc​ superconductors. Applying a transverse field to the ferromagnetic and antiferromagnetic metallic phase can restore the Z2{Z}_{2}Z2​ symmetry, which gives rise to two distinct critical points despite the fact that the two transitions have exactly the same symmetry-breaking pattern. We also propose chiral metallic states. Some of these gapless states are strongly correlated in the sense that they do not belong to the usual Tomonaga-Luttinger phase of fermions, i.e., they cannot be smoothly deformed into noninteracting fermion systems with the same symmetry. Our nonperturbative results are obtained by noting that gapless quantum systems have emergent categorical symmetries (i.e., noninvertible gravitational anomalies), which are described by multicomponent partition functions that are modular covariant. This allows us to calculate the scaling dimensions and quantum numbers of all the low-energy operators for those strongly correlated gapless states. This demonstrates an application of emergent categorical symmetries in determining low-energy properties of strongly correlated gapless states, which are hard to obtain otherwise.
We study quantum phases and phase transitions in a one-dimensional interacting fermion system with a Lieb-Schultz-Mattis (LSM) type anomaly. Specifically, the inversion symmetry enforces any symmetry-preserving gapped ground state of the system to be a Kitaev chain, following an LSM type theorem that we prove. Alternatively, via the Jordan-Wigner transformation, this system describes a spin system whose gapped ground states must break either the inversion or the Ising symmetry associated with fermion parity. We obtain a phase diagram using analytical methods and variational matrix product state simulations and study the critical behaviors of the quantum phase transitions therein using entanglement entropy, energy variance, and finite-size scaling of order parameters. In particular, we observe a continuous phase transition between two ordered phases that are beyond the Ginzburg-Landau-Wilson paradigm, in analogy to the deconfined quantum critical points in two spatial dimensions. We show this type of a 1D deconfined quantum critical point is described by the Tomonaga-Luttinger liquid theory and extract its Luttinger parameter and critical exponents. We also identify a gapless phase that emerges from a deconfined quantum critical point between two ordered phases, which cannot be described by a U(1) Luttinger liquid.
We show that a doped spin-1/2 ladder with antiferromagnetic intra-chain and ferromagnetic inter-chain coupling is a symmetry protected topologically non-trivial Luttinger liquid. Turning on a large easy-plane spin anisotropy drives the system to a topologically-trivial Luttinger liquid. Both phases have full spin gaps and exhibit power-law superconducting pair correlation. The Cooper pair symmetry is singlet dxyd_{xy}dxy​ in the non-trivial phase and triplet Sz=0S_z=0Sz​=0 in the trivial phase. The topologically non-trivial Luttinger liquid exhibits gapless spin excitations in the presence of a boundary, and it has no non-interacting or mean-field theory analog even when the fluctuating phase in the charge sector is pinned. As a function of the strength of spin anisotropy there is a topological phase transition upon which the spin gap closes. We speculate these Luttinger liquids are relevant to the superconductivity in metalized integer spin ladders or chains.
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We show that a doped spin-1/2 ladder with antiferromagnetic intra-chain and ferromagnetic inter-chain coupling is a symmetry protected topologically non-trivial Luttinger liquid. Turning on a large easy-plane spin anisotropy drives the system to a topologically-trivial Luttinger liquid. Both phases have full spin gaps and exhibit power-law superconducting pair correlation. The Cooper pair symmetry is singlet dxy in the non-trivial phase and triplet Sz=0 in the trivial phase. The topologically non-trivial Luttinger liquid exhibits gapless spin excitations in the presence of a boundary, and it has no non-interacting or mean-field theory analog even when the fluctuating phase in the charge sector is pinned. As a function of the strength of spin anisotropy there is a topological phase transition upon which the spin gap closes. We speculate these Luttinger liquids are relevant to the superconductivity in metalized integer spin ladders or chains.
States with gapless degrees of freedom are typically more complicated and less well- understood than systems possessing a gap. In this thesis, we study strongly-correlated systems described by gapless fermions. In most of the systems we consider, the fermionic excitations are emergent, i.e. they are not adiabatically connected to the electrons which constitute the fundamental building blocks of the system in question.\nWe start by proposing several novel states of matter which we claim are relevant to strongly-correlated systems. The rst state we present is a fractionalized Fermi liquid on the surface of a topological Kondo insulator. A Kondo insulator is a material which becomes insulating at low temperatures as a result of strong electron-spin interactions. When spin- orbit coupling is present, this insulator might be topological and, consequently, host robust gapless surface modes. Given the strong interactions and the decreased dimensionality of the surface, we propose that the spins and electrons there may decouple, resulting in the formation of a fractionalized Fermi liquid.\nWe next argue that quantum electrodynamics in 2+1 dimensions (QED3) with Nf = 4 fermion avours may describe a continuous, decon ned phase transition connecting the 120◦ coplanar Néel phase and the √12 × √12 valence bond solid phase of the triangular lattice antiferromagnet (AF). In addition to being a critical point, QED3 is also believed to describe a critical phase of matter called the Dirac spin liquid. Regardless of whether QED3 is manifest as a phase or a critical point, impurities and imperfections are always present in the real world, and it is therefore important to understand what e ects this may have. We show that when QED3 is perturbed by weak disorder, under certain circumstances, it ows to a new critical point in which both interactions and disorder are present.\nConclusively identifying a system described by QED3 has proven to be a di cult task, not only experimentally, but in numerical studies as well. Since QED3 is a critical theory, its spectrum on a torus is a universal quantity dependent only on low-energy degrees of freedom, such as the torus area. It follows that this data is accessible via exact diagonalization and can serve as an identifying signature of the theory. To this end, we calculate the QED3 torus spectrum using path integral methods.\nWe conclude this thesis with a comprehensive study of gapped Z2 spin liquids of the square lattice antiferrogmagnet, using a critical theory of Dirac fermions coupled to an SU(2) gauge eld as a starting point. There are many di erent Z2 spin liquid groundstates which preserve the same symmetry group. Even if one knows that a system has a Z2 spin liquid groundstate, it is by no means obvious which Z2 spin liquid is being realized. By starting from a gapless theory, the set of possibilities can be reduce considerably. Further, using certain recently proposed dualities, comparisons with theories formulated using bosons can be made.
Though most fermionic Mott insulators order at low temperatures, ordering is ancillary to their insulating behavior. Our emphasis here is on disentangling ordering from the intrinsic strongly correlated physics of a doped half-filled band. To this end, we focus on the two-dimensional Hubbard model. Because the charge gap arises from on-site correlations, we have been refining the nonperturbative approach of Matsumoto and Mancini [Phys. Rev. B 55, 2095 (1997)] which incorporates local physics. Crucial to this method is a self-consistent two-site dynamical cluster expansion which builds in the nearest-neighbor energy scale J. At half-filling, we find that the spectral function possesses a gap of order U and is devoid of any coherent quasiparticle peaks although ordering or charge fractionalization are absent. At low temperatures, local antiferromagnetic correlations emerge. In the doped case, we find that the Fermi surface exceeds the Luttinger volume. The breakdown of Luttinger's theorem in the underdoped regime is traced both to the dynamically generated Mott gap as well to a nonvanishing of the imaginary part of the self-energy at the Fermi level. Spectral weight transfer across the Mott gap also emerges as a ubiquitous feature of a doped Mott insulator and suggests that high- and low-energy scales are inseparable. Additionally in the underdoped regime, we find that a pseudogap exists in the single-particle density of states as well as in the heat capacity. The pseudogap (which is set by the energy scale t2/U){t}^{2}/U)t2/U) is argued to be a ubiquitous feature of a lightly doped Mott state and simply represents the fact that hole transport involves double occupancy. In analogy with the Mott gap and antiferromagnetism, we propose that ordering may also accompany the formation of a pseudogap. We suggest a current pattern within a one-band model that preserves translational but breaks time-reversal symmetry along the canonical x and y axes but not along $x=\ifmmode\pm\else\textpm\fi{}y$ that is consistent with the experimental observations. Finally, we show that the Hall coefficient in a doped Mott insulator must change sign at a doping level x<1/3.x<1/3.x<1/3. The sign change is tied to a termination of strong correlation physics in the doped Mott state.