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arXiv:cond-mat/0502652 · Physical Review B

Deconfinement in the presence of a Fermi surface

Ki‐Seok Kim

Published Dec 7, 2005• 18 citations
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Abstract

U(1) gauge theory of nonrelativistic fermions interacting via compact U(1) gauge fields in the presence of a Fermi surface appears as an effective field theory in low dimensional quantum antiferromagnetism and heavy fermion liquids. We investigate deconfinement of fermions near the Fermi surface in the effective U(1) gauge theory. Our present analysis benchmarks the recent investigation of quantum electrodynamics in two space and one time dimensions $({\mathrm{QED}}_{3})$ by [Hermele et al.Phys. Rev. B 70, 214437 (2004)]. Utilizing a renormalization group analysis, we show that the effective U(1) gauge theory with a Fermi surface has a stable charged fixed point. Remarkably, the renormalization group equation for an internal charge $e$ [the coupling strength between nonrelativistic fermions and U(1) gauge fields] reveals that the conductivity $\ensuremath{\sigma}$ of fermions near the Fermi surface plays the same role as the flavor number $N$ of massless Dirac fermions in ${\mathrm{QED}}_{3}$. This leads us to the conclusion that if the conductivity of fermions is sufficiently large, instanton excitations of U(1) gauge fields can be suppressed owing to critical fluctuations of the nonrelativistic fermions at the charged fixed point. As a result a critical field theory of nonrelativistic fermions interacting via noncompact U(1) gauge fields is obtained at the charged fixed point.

DOI
10.1103/physrevb.72.245106

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U(1) gauge theory of nonrelativistic fermions interacting via compact U(1) gauge fields in the presence of a Fermi surface appears as an effective field theory in low dimensional quantum antiferromagnetism and heavy fermion liquids. We investigate deconfinement of fermions near the Fermi surface in the effective U(1) gauge theory. Our present analysis benchmarks the recent investigation of quantum electrodynamics in two space and one time dimensions (QED3)({\mathrm{QED}}_{3})(QED3​) by [Hermele et al.Phys. Rev. B 70, 214437 (2004)]. Utilizing a renormalization group analysis, we show that the effective U(1) gauge theory with a Fermi surface has a stable charged fixed point. Remarkably, the renormalization group equation for an internal charge eee [the coupling strength between nonrelativistic fermions and U(1) gauge fields] reveals that the conductivity \ensuremathσ\ensuremath{\sigma}\ensuremathσ of fermions near the Fermi surface plays the same role as the flavor number NNN of massless Dirac fermions in QED3{\mathrm{QED}}_{3}QED3​. This leads us to the conclusion that if the conductivity of fermions is sufficiently large, instanton excitations of U(1) gauge fields can be suppressed owing to critical fluctuations of the nonrelativistic fermions at the charged fixed point. As a result a critical field theory of nonrelativistic fermions interacting via noncompact U(1) gauge fields is obtained at the charged fixed point.
Recently, Hermele et al. claimed that the infrared (IR) fixed point of noncompact QED3{\mathrm{QED}}_{3}QED3​ is stable against instanton excitations in the limit of large flavors of massless Dirac fermions [M. Hermele et al., cond-mat/0404751, Phys. Rev. B (to be published October 2004)]. We investigate an effect of nonmagnetic disorder on the deconfined quantum critical phase dubbed U(1)U(1)U(1) spin liquid (U1SL)(U1\mathrm{SL})(U1SL) in the context of quantum antiferromagnet. In the case of weak disorder the IR fixed point remains stable against the presence of both the instanton excitations and nonmagnetic disorder and thus the U1SLU1\mathrm{SL}U1SL is sustained. In the case of strong disorder the IR fixed point becomes unstable against the disorder and the Anderson localization is expected to occur. We argue that in this case deconfinement of spinons does not occur since the Dirac fermion becomes massive owing to the localization.
We study the low-energy effective theory for a non-Fermi-liquid state in 2+12+12+1 dimensions, where a transverse U(1) gauge field is coupled with a patch of Fermi surface with NNN flavors of fermion in the large NNN limit. In the low-energy limit, quantum corrections are classified according to the genus of the two-dimensional surface on which Feynman diagrams can be drawn without a crossing in a double line representation and all planar diagrams are important in the leading order. The emerging theory has the similar structure to the four-dimensional SU(N) gauge theory in the large NNN limit. Because of strong quantum fluctuations caused by the abundant low-energy excitations near the Fermi surface, low-energy fermions remain strongly coupled even in the large NNN limit. As a result, there are infinitely many quantum corrections that contribute to the leading frequency dependence of the Green's function of fermion on the Fermi surface. On the contrary, the boson self-energy is not modified beyond the one-loop level and the theory is stable in the large NNN limit. The nonperturbative nature of the theory also shows up in correlation functions of gauge-invariant operators.
The compact U(1) gauge field occurs in many fractionalized descriptions of low dimensional quantum magnetism and heavy fermion systems. In this respect a fundamental question about the gauge field is whether it is confined or not in the presence of Fermi surface. In this paper we show that the U(1) gauge field with a possible positive anomalous dimension in the presence of Fermi surface has to be in the confined state, although it is strongly coupled to abundant gapless fermionic excitations near the Fermi surface in two spatial dimensions. This means that some quantum spin liquids and/or fractionalized metallic states described in terms of this kind of gauge fields with positive anomalous dimensions are unstable to other possible symmetry-breaking states, as a result, they are not be appropriate candidates for ground states in related systems. The instability obtained shows some novel features of the quantum spin liquids and/or fractionalized metallic states in the presence of Fermi surface which have not been reported in the literature. The result could be useful for further study in the quantum spin liquids and many related slave-particle gauge theories in strongly correlated electron systems.
Compact quantum electrodynamics in 2 + 1 dimensions often arises as an effective theory for a Mott insulator, with the Dirac fermions representing the low-energy spinons. An important and controversial issue in this context is whether a deconfinement transition takes place. We perform a renormalization group analysis to show that deconfinement occurs when N > Nc = 36/pi3 approximately to 1.161, where N is the number of fermion replica. For N < Nc, however, there are two stable fixed points separated by a line containing a unstable nontrivial fixed point: a fixed point corresponding to the scaling limit of the noncompact theory, and another one governing the scaling behavior of the compact theory. The string tension associated with the confining interspinon potential is shown to exhibit a universal jump as N --> Nc-. Our results imply the stability of a spin liquid at the physical value N = 2 for Mott insulators.
We investigate the continuous quantum phase transition from an antiferromagnetic metal to a heavy-fermion liquid based on the Kondo lattice model in two dimensions. We propose that antiferromagnetic spin fluctuations and conduction electrons fractionalize into neutral bosonic spinons and charged spinless fermions at the quantum critical point. This deconfined quantum criticality leads us to establish a critical field theory in terms of the fractionalized fields interacting via emergent U(1) gauge fields. The critical field theory not only predicts non-Fermi liquid physics near the quantum critical point but also recovers Fermi liquid physics away from the quantum critical point.
We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless U(1)U(1)U(1) gauge fluctuations can be stable in two dimensions, at least when the physical SU(2)SU(2)SU(2) spin symmetry is generalized to SU(N)SU(N)SU(N). Equivalently, we show that compact QED3{\mathrm{QED}}_{3}QED3​ has a deconfined phase for a large number of fermion fields, in the sense that monopole fluctuations can be irrelevant at low energies. A precise characterization is provided by an emergent global topological U(1)U(1)U(1) symmetry corresponding to the conservation of gauge flux. Beginning with an SU(N)SU(N)SU(N) generalization of the S=1∕2S=1∕2S=1∕2 square-lattice Heisenberg antiferromagnet, we consider the $\ensuremath{\pi}$-flux spin liquid and, via a systematic analysis of all operators, show that there are no relevant perturbations (in the renormalization-group sense) about the large-NNN spin-liquid fixed point, which is thus a stable phase. We provide a further illustration of this conclusion with an approximate renormalization-group calculation that treats the gapless fermions and the monopoles on an equal footing. This approach directly points out some of the flaws in the erroneous ``screening'' argument for the relevance of monopoles in compact QED3{\mathrm{QED}}_{3}QED3​.