Fractionalized Metals and Superconductors in Three Dimensions
Abstract
We study threedimensional metals with nontrivial correlation functions and fractionalized excitations. We formulate for such states a gauge theory, which also naturally describes the fractional quantization of chiral anomaly. We also study fractional superconductors in this description. This formulation leads to the “threedimensional chiral Luttinger liquids” and fractionalized Weyl semimetals, which can arise in both fermion and boson models. We also propose experiments to detect these fractionalized phases.
pacs:
73.43.f, 71.10.Hf, 74.20.Mn, 71.45.d, 71.70.dIntroduction.
Fractionalization of quantum numbers Jackiw and Rebbi (1976); Su et al. (1979, 1980); Heeger et al. (1988); Fradkin and Hirsch (1983); Laughlin (1999); Kivelson (2001); Fradkin (2013); Luther and Emery (1974); Steinberg et al. (2008); Le Hur et al. (2008); Qi et al. (2008a); Ran et al. (2008); Qi and Zhang (2008); Lee et al. (2007); Chamon et al. (2008); Senthil et al. (2004); Seradjeh et al. (2009); Bramwell and Gingras (2001); Castelnovo et al. (2011) such as electrical charge is an intriguing phenomenon in condensed matter physics. Fractionalization in weakly interacting systems is usually associated with topological solitons, a simplest example being the SuSchriefferHeeger solitonSu et al. (1979, 1980); Heeger et al. (1988); Fradkin and Hirsch (1983) in polyacetylene. Fractionalized excitations can also arise in strongly correlated systems, in which strong interaction among electrons plays an essential role in fractionalization. In fact, one of the best known examples of charge fractionalization is the elementary excitations (quasiparticles or quasiholes) carrying charge in the th Laughlin stateLaughlin (1983); Tsui et al. (1982); v. Klitzing et al. (1980), which describes a fractional quantum Hall state, one of the prototypes of strongly correlated systems. The onedimensional (1D) chiral Luttinger liquidsWen (1990a, b, 1992); Chang et al. (1996) (CLLs) at the edge of the Laughlin states are fractionalized metals, for which the electron Green’s function follows a nontrivial powerlawWen (1990a, b, 1992); Chang et al. (1996)
(1) 
where is a velocity parameter. This correlation is in sharp contrast with expectation of Landau’s Fermi liquid theory.
Charge fractionalization is a much more subtle problem for strongly correlated systems in spatial dimensions higher than one (), due to absence of powerful analytical tools such as bosonization. Describing fractionalized metals in analogous to the 1D chiral Luttinger liquids is one of the motivations of the present work. A deeper motivation, as we now explain, is to describe a class of quantum phenomena we dubbed “fractional anomalies”. This was the initial motivation of this work.
In quantum mechanics, an “anomaly” refers to the failure of a symmetry of the classical action to be a symmetry of the full quantum theory. The earliest example is the chiral anomalyPeskin and Schroeder (1995); Adler (1969); Bell and Jackiw (1969), which implies chiral current nonconservation in the presence of nontrivial gauge field backgrounds. Remarkably, chiral anomaly also has significant implications in condensed matter systemsQi et al. (2008b); Volovik (2003); Wang et al. (2011); Ryu et al. (2012); Wang et al. (2014); Wang and Wen (2013); Son and Spivak (2012), where it is deeply related to topological states such as topological insulators and topological superconductorsQi and Zhang (2010); Hasan and Kane (2010); Qi and Zhang (2011); C. L. Kane and E. J. Mele (2005a); B.A. Bernevig and S.C. Zhang (2006); C. L. Kane and E. J. Mele (2005b); B. A. Bernevig et al. (2006); König et al. (2007); Fu and Kane (2006); Moore and Balents (2007); Qi et al. (2008b); Fu et al. (2007); Roy (2009); Wang and Zhang (2012); Schnyder et al. (2008); Kitaev (2009); Wang et al. (2010a); Chen et al. (2012); Wang et al. (2010b); Vishwanath and Senthil (2013); Xu and Senthil (2013). As a simplest example, the onedimensional edge of 2D integer quantum Hall edge states has the chiral anomaly for electrical current^{1}^{1}1We have the factor here because we are concerned with electrical current. For the current associated with particle number, the factor is instead of .
(2) 
the superscript and subscript ‘’ appears because low energy fermions have definite chirality, which is taken to be righthanded, i.e., they all move towards the right direction. Here the integer is the number of these edge modes. According to the bulkedge correspondence, is also equal to the Chern number of the twodimensional bulk of quantum Hall insulators. In this example the origin of chiral anomaly is transparent: The transverse current (Hall current) in the bulk adds or removes charges at the edge, which is regarded as charge noncnservation by an edge observer.
On one hand, there have been various arguments pointing to the quantization (nonrenormalization) of chiral anomaly. On the ohter hand, the existence of fractional quantum Hall effects has deepened our understanding of chiral anomaly. For the Laughlin state with filling factor , the chiral edge state has the fractional chiral anomalyWen (2004) 1990a, b, 1992); Chang et al. (1996). , which is consistent with the chiral Luttinger liquid theoryWen (
In 3D, the integer chiral anomaly reads and , or more compactlyPeskin and Schroeder (1995),
(3) 
where and is the number of modes of righthanded and lefthanded chiral fermions, respectively, and the chiral current . Note that in Eq.(3) we have the factor because refers to electrical current (for the current associated with particle number this factor should be ). The chiral anomaly have interesting physical implications for Weyl semimetals (and Weyl superconductors) Wan et al. (2011); Volovik (2003); Burkov and Balents (2011); Zyuzin et al. (2012); WitczakKrempa and Kim (2012); Hosur et al. (2012); Aji (2011); Liu et al. (2012); Xu et al. (2011); Wang et al. (2012); Halász and Balents (2012); Kolomeisky and Straley (2012); Jiang (2012); Delplace et al. (2012); Meng and Balents (2012); Garate and Glazman (2012); Grushin (2012); Son and Spivak (2012); Wang and Zhang (2013); Gorbar et al. (2013); Liu et al. (2014); Li and Wu (2012); Xu et al. (2013) (“Weyl fermions” is another name of chiral fermions).
By analogy with 1D fractional chiral anomaly, the “fractional chiral anomaly” for righthanded chiral fermions in 3D reads
(4) 
being a noninteger rational number. We shall not repeat the same equation for lefthand chiral fermions. However, such a fractional anomaly can never be achieved in any noninteracting picture or Landau’s Fermi liquid theory. In fact, free fermions always have integer anomaly, and modest interaction cannot renormalize this integer quantization. Invalidating the Landau’s Fermiliquid theory, certain strong interactions may allow fractional quantization, however, a simple formulation of how this actually happens is lacking. As we shall show, the fractional chiral anomaly is naturally realized in the fractionalized metals studied in this paper.
With these motivations, we studied fractionalized (semi)metals in 3D with nontrivial exponents in correlation functions. The elementary excitations carry a fraction of electron charge. In our formulation an important role is played by the Higgs mechanism, which gives mass to gauge bosons, such that their effects are suppressed at low energy. The results of this paper is applicable to 3D “chiral Luttinger liquids” at the surface of 4D quantum Hall states, and to fractionalized Weyl semimetals, which in principle can be realized in experiments.
gauge theory formulation of threedimensional fractionalized metals.
The Hamiltonian of electron in a periodic lattice system can be generally written as
(5) 
where the subscript refer to all microscopic degrees of freedom including site, spin, and orbital, is the particle number operator, and and are temporal and spatial components of a classical electromagnetic potential added for later convenience.
To investigate possible fractional phases, the trickWen (1991a); Maciejko et al. (2010); Swingle et al. (2011) we will use is to write the electron operator in terms of parton operators
(6) 
where the parton operators () are fermionic. The electron operator is invariant under the local gauge transformation , where . We will see shortly that this implies that each parton is coupled to an emergent gauge fieldYang and Mills (1954); Wilson (1974) with lattice gauge potential to be defined below. We also note that this parton approach can work if the original fermions are replaced by bosons , the only modification being that becomes even integer.
Within the mean field approximation, the quadratic Hamiltonian for the partons reads
(7)  
where
and similarly for . In Eq.(7) we have added the Lagrange multiplier () to ensure the constraints for the physical Hilbert space, being the generators of . Writing the meanfield Hamiltonian in Eq.(7) more compactly, we have
(8) 
where . It is readily understood that s are dynamical variables with their effective Lagrangian , whose explicit form does not concern us here. We can write the matrix as
(9) 
in which is a Hermitian matrices describing the ‘phase’ fluctuations of around the mean field value , while the amplitude fluctuation of is ignored. Similarly, we can split the Lagrange multiplier as , and being the vacuum expectation value and the fluctuation of , respectively. We can see that can be regarded as the temporal component of an emergent gauge field, whose spatial components are .
Now let us discuss the dynamics of the gauge potentials in the long wavelength limit, which are referred to as “” (), or more compactly as “” when there is no confusion. The most important problem now is whether the gauge bosons are massless or massive, namely, whether the effective action contains mass terms for . In fact, the gauge bosons can become massive by the Higgs mechanism, the vacuum expectation playing the role of “Higgs fields”. To be precise, the gauge bosons will generally be massive if there exists certain loop given by for which , where the flux , being the path ordering. According to this general criterion, there are several scenariosWen (2004) associated with different types of :

“Generic flux”. For any group generator , there exists certain loop for which the commutator . All gauge bosons are massive in this case.

Trivial flux. We have for an arbitrary loop , in other words, for all and all . All gauge bosons are massless in this case.

Coplanar flux. For some of of the group generators, say , the relation ( for any ) is satisfied; while for other group generators, say , there exists at least one choice of such that . The gauge bosons associated with the first class of group generators are massless, while those associated with the second class are massive.
Similar scenarios emerge in the gaugefield formulation of spin liquidWen (2004, 1991b); Lee et al. (2006). Without going into technical details, here we simply provide an intuitive argument for these scenarios (not a proof). In the continuum limit, the effective Lagrangian generally contain a term , where denotes the background field determined by . When this term is nonvanishing, it can be regarded as a mass term for , therefore, the Higgs mechanism of nonAbelian gauge field does not require matter fields (partons in our context); gauge fields themselves can trigger the Higgs mechanism because they carry gauge chargeWen (2004) (Unlike the Abelian gauge field theory, in which the gauge boson is charge neutral).
Here let us focus on the first scenario, namely that all gauge bosons acquire an mass from the Higgs mechanism. An explicit ansatz realizing this scenario will be given shortly. In this case the gauge bosons can only mediate shortrange interactions, thus they do not cause infrared divergences.
If the original electron system has bands, namely, there are microscopic electronic states (referred to as ) in each unit cell, then the meanfield Hamiltonian Eq.(8) for partons has bands.
Suppose that the parton meanfield Hamiltonian has valleys near (), where the band structure is that of righthanded chiral fermions (Weyl fermions). Within the valley around , the two low energy bands are described approximately by the Hamiltonian , where is the abbreviation for , is the projection operator to the two low energy bands, and
(10) 
where ( or ) play the role of Fermi velocities, and the Pauli matrices () act on the two low energy bands. In the following we will take all and () without affecting main physical conclusions. At , the Fermi surface shrinks to a point. By dimensional analysis, various short range interactions such as (indices omitted) are irrelevant in the renormalization group sense. The electromagnetic interaction is marginally irrelevant^{2}^{2}2It can generate logarithmical corrections to the powerlaws of fermion Green’s functions, which we shall ignore at this stage.. Therefore, at low energy or long distance the Green’s function of partons is just that of the free fermions. The elementary excitations are partons (and holes of partons) carrying charge , which is a direct manifestation of fractionalization.
In the momentum space, the parton Green’s function is denoted as , where and . In the low energy limit it is given by , with being the contribution of the th valley. In our simplest ansatz it reads . Fouriertransformed to the real space, the parton Green’s function is
(11) 
where . The electron Green’s function is
(12) 
where the determinant is calculated regarding the color index ( ) as the row (column) index of matrix , the indices being fixed. Because in the long time limit, it readily follows that in this limit
(13) 
Similarly, we have in the long distance limit. The powerlaw behavior of Green’s function, with the exponent quantized as an integernot , is reminiscent of the chiral Luttinger liquids in 1D. As a comparison, the 1D chiral Luttinger liquid at the edge of th Laughlin state hasWen (2004) .
Since each parton carries electrical charge , the coefficient of chiral anomaly in Eq.(4) is
(14) 
which can be readily obtained from the triangular diagramPeskin and Schroeder (1995), with parton propagators extracted from Eq.(10). Due to the nonrenormalization property of chiral anomaly, modest modifications of Hamiltonian cannot change this quantized .
Sofar we have formulated a selfconsistent theory of fractionalized metals in 3D, for which the Green’s function follows a nontrivial powerlaw, and the chiral anomaly is fractionally quantized. Now we would like to discuss the physical applications of this formulation. The most direct application can be found at the surface of 4D fractional quantum Hall effectsZhang and Hu (2001); Bernevig et al. (2002); Li et al. (2013); Wang and Zhang (2014). From our formulation it is clear that fractionalized metal (“3D chiral Luttinger liquid”) is a possible and consistent scenario in certain regime. Our formulation can also be applied to Weyl semimetals with equal number of modes of righthanded and lefthanded chiral fermions. For Weyl semimetals the nontrivial exponents in Green’s function can be measured in tunneling experiments, which provide information of the electron density of states.
Now let us study two simple examples. We consider singleband lattice boson models, and write the (hardcore) boson operator as . The meanfield ansatz reads , being the shorthand notation for . The first example we consider is , which gives nonzero masses to all three gauge bosons because none of () commutes with for a generic . According to Eq.(14), we have fractional anomaly
(15) 
Such a fractional anomaly can be realized at the boundary of a 4D (boson) quantum Hall insulators. The parton propagator for this model is exactly given by Eq.(11). The boson Green’s function can be obtained from Eq.(12), with the simplification in this special case that the indices are absent. Explicitly, we have
(16) 
which implies in the limit.
The second example of parton meanfield Hamiltonian is
One dimensional fractionalized chiral modes propagating along dislocations.
One way to detect the charge fractionalization in the fractionalized Weyl semimetals is to induce an energy gap and then create dislocations.
For simplicity, let us suppose that the mean field Hamiltonian Eq.(8) has one valley (righthanded fermion) around , where the low energy Hamiltonian is , and another valley (lefthanded fermion) at momentum , described by . Now let us add an external magnetic field . In the presence of this magnetic field, the energy spectra are with , where . For the mode, the sign before is determined by the chirality ( for righthanded, for lefthanded). Therefore, the effective Hamiltonian for the zeroth Landau level can be written compactly as , where the Pauli matrix refers to the two chiralities. Note that is independent on . Due to perfect nesting of Fermi surface, an infinitesimal interaction can dynamically generate a massYakovenko (1993); Yoshioka and Fukuyama (1981); Wang and Zhang (2013); Yang et al. (2011) , where , , and is a slowly varying variable. The (charge or spin) density vary asWang and Zhang (2013) , in which determines the locations of peaks and troughs of the density wave.
Suppose that there is a line dislocation , such that for a loop around we have , thus the peaks and troughs shift by one wavelength by making a circle around [see Fig.1]. In Ref.Wang and Zhang (2013), it was shown for a similar problem (integer case) that there is a chiral mode along , which is analogous to the edge state of integer quantum Hall effect with Hall conductance . By similar calculation, it can be shown for the present problem that there is chiral modes along with “Hall conductance” . In this sense the fractionalized chiral mode along is analogous to the 1D chiral Luttinger liquid. Measuring this fractional “Hall conductance” will be one way to confirm the fractionalized metals in 3D.
Fractional superconductors in 3D.
In the previous sections we have focused on the case with parton chemical potential . When , an infinitesimal attractive interaction can induce Cooper instability, and the ground state is a superconductor. Let us consider a Weyl valley around . For a shortrange interaction , the superconducting gap is
(17) 
where and refer to the two low energy degrees of freedom, is the density of states of partons at the Fermi level, and is a numerical coefficient of order of unity. More complicated scenarios such as color superconductorsRajagopal and Wilczek (2000); Anglani et al. (2014) can also arise in our description, which will be left for future studies. Here we would like to focus on modelindependent physical consequences.
One of the physical predictions is about the Josephson effectJosephson (1962); Anderson and Rowell (1963). In fact, the parton Cooper pairs carry charge , therefore, we expect fractional Josephson effects. In the presence of a voltage between two fractional superconductors connected by a weak link, alternating tunneling current with frequency
(18) 
can be observed (In this formula we have restored the Planck constant , which has been set to unity in our previous presentation). We would like to mention that the fractional Josephson effect has also been studied in 2D systemsCheng (2012); Vaezi (2013); Lindner et al. (2012); Clarke et al. (2013) in different approaches.
Conclusions.
In the present paper we have studied fractionalized (semi)metals in 3D through an gauge theory. We have found for them powerlaw Green’s functions with quantized exponents. This formulation will be useful to 3D chiral Luttinger liquids and fractionalized Weyl semimetals. This gaugetheoretical formulation resembles the standard model of particle physics, in which the chiral fermions are coupled to and gauge fields with , the being suppressed at low energy due to the Higgs mechanism. In the field of condensed matter, we believe that fractional anomalies will provide much information about fractional topological states of quantum matter. In addition to the significance of their own right, the fractionalized (semi)metals can also be regarded as the “mother states” of gapped fractional topological states. In future it will be fruitful to establish explicit manybody Hamiltonians for the scenarios proposed in the present work, and to explore the implications of fractional anomalies in depth.
The author would like to thank XiaoGang Wen, Yi Li, Congjun Wu, and especially ShouCheng Zhang for insightful discussions. The author is supported by NSFC under Grant No. 11304175 and Tsinghua University Initiative Scientific Research Program ( No. 20121087986).
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