Berry phase in the composite Fermi-liquid

We derive the definition of the Berry phase for adiabatic transport of a composite Fermion (CF) in a half-filled composite Fermi-liquid (CFL). It is found to be different from that adopted in previous investigations by Geraedts et al. With the definition, the numerical evaluation of the Berry phase becomes robust and free of extraneous phase factors. We show that the two forms of microscopic wave-functions of the CFL, i.e., the Jain-Kamilla type wave function and the standard CF wave function, yield different distributions of the Berry curvature in the momentum space. For the former, the Berry curvature has a continuous distribution inside the Fermi sea and vanishes outside, whereas for the latter, the Berry curvature is uniform in the whole momentum space. To facilitate an analytic derivation for the latter, we reveal a simple structure of standard CF wave functions by establishing their connections to the Segal-Bargmann transform. We conclude that the CF with respect to both the microscopic wave-functions is not a massless Dirac particle.


Introduction
The ubiquitous presence of the Berry phase is notable in recent theoretical investigations of condensed matter physics. For non-interacting systems, it becomes a unifying concept for characterizing the orbital effects of the spin or other internal degrees of freedom [1], and plays central roles in systems such as topological insulators [2], Dirac/Weyl semimetals [3] and valleytronic materials [4]. Recently, it becomes clear that the Berry phase also plays a role in the theory of composite Fermions (CFs) [5]. Conventionally, the CF is regarded as an ordinary Newtonian particle which interacts weakly and resides in a hidden Hilbert space [6]. A wave function of non-interacting CFs in the hidden Hilbert space can be mapped into a wave function appropriate for describing the physical state of a strongly correlated, fractionally filled Landau level. While the theory of CFs achieves tremendous successes in understanding the fractional quantum Hall effect and related phenomena, the true nature of the CF is still open to debate. The conventional interpretation, as explicated in Halperin-Lee-Read theory of the composite Fermi-liquid (CFL) [7][8][9], suffers from two apparent difficulties: it cannot correctly predict the CF Hall conductance of a half-filled Landau level [10], and an electron filling fraction and its hole conjugate are mapped into different numbers of CF Λ-levels [5]. The difficulties motivate Son to propose that the CF should be a massless Dirac particle [11]. An alternative interpretation, i.e., the CF is neither a Newtonian particle nor a Dirac particle, but a particle subject to a uniformly distributed Berry curvature in the momentum space and the Sundaram-Niu dynamics [12], is also put forward [13][14][15]. The three pictures imply three different distributions of the Berry curvature, i.e., zero, singularly distributed and uniformly distributed, respectively. The clarification of the issue then hinges on the determination of the Berry curvature for CFs.
A "first principles" approach for determining the Berry curvature of CFs should be based on microscopic CF wave-functions. To this end, several attempts have been made. In Ref. [15], the dynamics of the CF Wigner crystal is derived. It shows that the CF is subject to a uniformly distributed Berry curvature in the momentum space. For the half-filled CFL phase, the Berry curvature distribution is also found to be uniform by determining the dynamics of a test (distinguishable) CF added to the CF Fermi sea [14]. A heuristic argument based on the dipole picture of CFs [16] also suggests the same [13,14]. These works may draw criticism for neglecting the particle exchange symmetry in their treatments. It is in this context that the recent works by Geraedts et al. stand out [17,18]. Their calculations are based on a microscopic CFL wave function in its full antisymmetric form. However, a close scrutiny to the works reveals a number of issues. Firstly, the definition of the Berry phase is a prescribed one and is not fully justified. Secondly, the evaluation of the Berry phase based on the definition seems to be not numerically robust, sensitive to the choices of paths and prone to statistical errors. Moreover, there exist extraneous ±π/2 phases preventing direct interpretations of numerical results. Finally, the microscopic CFL wave function adopted for the calculation is of the Jain-Kamilla type [19], which is numerically efficient in implementing the projection to the lowest Landau level (LLL). However, it is unclear whether or not it yields the same result as that from the standard wave function prescribed by the theory of CFs [5].
In this Letter, we solve these issues and determine the distribution of the Berry curvature for CFs. We derive the definition of the Berry phase directly from the time-derivative term in the Schrödinger Lagrangian. It is found to be different from the prescribed one adopted by Geraedts et al. [17,18]. With the definition, the numerical evaluation of the Berry phase becomes robust and free of the extraneous phases. It enables us to numerically determine the distribution of the Berry curvature in the whole momentum space. We show that the two forms of microscopic wave-functions of the CFL, i.e., the Jain-Kamilla wave function and the standard CF wave func-tion, yield different distributions of the Berry curvature in the momentum space. For the former, the Berry curvature has a continuous distribution inside the Fermi sea and vanishes outside, whereas for the latter, the Berry curvature is uniform. To facilitate an analytic derivation for the latter result, we reveal a simple structure of standard CF wave functions by establishing their connections to the Segal-Bargmann transform. Based on these results, we conclude that the CF with respect to both the microscopic wave-functions is not a massless Dirac particle.
Wave functions First, we show the explicit forms of different microscopic CFL wave functions. To unify notations, we use the symbols a i ≡ a ix + ia iy , a * i ≡ a ix − ia iy and a i ≡ (a ix , a iy ) to denote an electron-related variable in its complex form, complex conjugate and vector form, respectively, with the subscript i indexing electrons. Symbols without a subscript (e.g. a ≡ {a i }) denote a list of the variables for all electrons, and symbols in the upper case (e.g. A ≡ i a i ) denote sums of the variables over all electrons. The standard CF wave function of the CFL with a filling fraction ν = 1/m on a torus can be written as (without the Gaussian factor which is a holomorphic function of complex electron coordinates z. The wave function is parameterized in a set of wave vectors k which are quantized as usual on the torus [21]. J(z) is the Bijl-Jastrow factor [6] expressed in terms of the modified sigma function which has the quasi-periodicity [21] σ where L is a period of the torus, and ξ(L) = 1 if L/2 is also a period and −1 otherwise, and A = 2πN φ is the total area of the torus.P LLL denotes the projection to the LLL, which is effectively to replace z * i with an operator 2∂ zi acting on all z i 's [5]. Here, we assume that qB is along the normal direction of the torus, where B is the magnetic field and q is the unit charge of carriers. The total number of magnetic fluxes passing through the torus is N φ = mN e , where N e is the total number of electrons. The unit of length is set to be the magnetic length l B ≡ /eB with B ≡ |B|.
The Jain-Kamilla wave function, which is adopted in Ref. [17] for evaluating the Berry phase, has the form [20] The quantization of k is the same as that in Eq. (4). Definition of the Berry phase Next, we derive the definition of the Berry phase for the many-body CFL system. Both the wave functions are parametrized in k which is interpreted as the list of CF wave vectors. Wave functions with different configurations of k span a Hilbert space. An implicit assumption is that kconfigurations close to the ground state configuration [17] can be corresponded to the low-lying excited states of the CFL [22]. As a result, the semi-classical time-evolution of the k-configuration contains information of the excited states [15]. With the understanding in mind, we can derive the definition of the Berry phase for adiabatic transport of a CF in the k-space from the time-derivative term of the Schrödinger Lagrangian [12,23] by assuming k being time-dependent variables. An alternative and fully quantum approach is to construct a path-integral for CFs by using Ψ k as a basis set. The Berry phase can be obtained from the phase of the overlap matrix Ψ k |Ψ k with k and k being the configurations of two adjacent time slices [24]. The two approaches are equivalent in the adiabatic limit. A complexity is in the necessity of excluding the trivial propagating phase factor (i.e., e ik·z ) from the definition of the Berry phase for a spatially-invariant system. Actually, when defining the Berry phase/connection in the momentum space, we implicitly assume that the position of a particle is fixed when it is transported in the k-space. For a many-body system, unfortunately, Ψ k |z i |Ψ k with an antisymmetric wave function Ψ k does not lead to a useful definition for the position of an individual electron. Nevertheless, we can define a position for the electron associated with a given wave vector k i . This is because a CFL wave function can be related to their unsymmetrized form ϕ k by Ψ k = P (−1) PP ϕ k , whereP denotes a permutation of electron indexes, and P is the parity of the permutation. The explicit form of ϕ k can be obtained from Eq. (1) or (4) by replacing the determinant with a product of its diagonal elements. With ϕ k , there is no ambiguity in pairing an electron coordinate with a wave vector. We can then define the position of an individual electron as [25], and obtain where we drop a non-consequential total-time derivative term and assume that only k 1 is time-dependent. The Berry connection is determined by: where At this point, an ambiguity in the definition of the position of the CF arises [15]. Previous investigations suggest that the position of a CF should be defined as the position of its constituent quantum vortex bundle [13][14][15], which is displaced from its constituent electron by with z e i ≡ z i and n denoting the normal direction of the torus, according to the dipole picture of CFs [16]. As a result, for CFs, the position z i in Eq. (7) and (8) should be interpreted as z v i . We will call the two forms using z e and z v as the electron representation and the CF representation, respectively. It is easy to show that A v k1 = A e k1 − k 1 × n with the superscripts indicating respective representations. We interpret A v k1 as the Berry connection for adiabatic transport of a CF in the momentum space.
The Berry phase for a discrete change of the wavevectors k → k can then be defined as (9) is our definition of the Berry phase for the CFL. It is a definition directly derived from the Schrödinger Lagrangian, therefore a proper definition when the physical consequences of the Berry phase, such as the semi-classical dynamics, the path-integral formalism and wave equation of CFs, are concerned. Compared with the definition adopted in Ref. [17], the difference is in that the "momentum boost operator" (i.e., the factor e −iq·z ) is applied to the unsymmetrized wave function instead of the antisymmetrized one. It is actually more appropriate to call Geraedts et al.'s phase as the scattering phase, i.e., the phase that an electron acquires when scattered by the q-component of a single-body scalar potential. Even for non-interacting systems, the scattering phase is not a reliable predictor for the Berry phase, as evident in the case that a scatter has spin-orbit coupling different from its host. We note that while Geraedts et al.'s phase is not a Berry phase, it does have its own merit (see below).
Berry phase for the JK wave function We apply our definition to the numerical evaluation of the Berry phase for the Jain-Kamilla wave function Eq. (4). We implement a Metropolis Monte-Carlo algorithm similar to that . |D|min is the minimum overlap among steps along a path. For the paths inside the Fermi sea, a hole is transported, and resulting Berry phases are shown with inverted signs. The values marked with * have been scaled by a factor of Ne. U.D. indicates an undeterminable result due to a vanishing overlap. detailed in Ref. [26]. Phases with respect to both our definition and Geraedts et al.'s definition are evaluated for a few representative paths, as shown in Table. I. An immediate observation is that the calculation with our definition is much more robust numerically, as evident from the magnitudes of the overlap. With our definition, the overlap is always close to one and improves when N e is scaled up. For Geraedts et al.'s definition, the overlap is nowhere close to one and further deteriorates for larger N e , and even nearly vanishes for steps along directions perpendicular to the Fermi circle, resulting in poor statistics and undeterminable results. Moreover, our definition yields directly interpretable results, i.e., no subtraction of the extraneous ±π/2 phases noted in Ref. [17] is needed.
It is interesting to observe that the two different definitions actually lead to similar qualitative conclusions. With our definition, the Berry phase of adiabatic transport of a CF around the Fermi circle is converged to π (path a, N e = 110), whereas with Geraedts et al.'s definition, it involves guesswork to reach the same conclusion. We also find that the Berry phase for transport around a unit plaquette outside the Fermi sea (path b2) nearly vanishes. This is consistent with Geraedts et al.'s observation that the phase is independent of the area of the trajectory enclosing the Fermi sea. The consistencies may not be a coincidence. When Geraedts et al.'s phase is properly interpreted as the scattering phase, it does have a physical consequence, i.e., the direction and magnitude of the side jump of a scattered particle [27]. We take the consistencies as an evidence supporting our interpretation, i.e., a CF undergoing physical processes (e.g., the scattering) behaves like an entity with z v 1 as its position. Had we used z e 1 instead, the Berry phase would predict a side jump along the opposite direction.
The distribution of the Berry curvature, both inside and outside the Fermi sea, can now be determined because of the improved numerical robustness. To determine the Berry curvature, we transport a CF or a hole along the edges of a unit plaquette (see path b in Table. I), and the Berry curvature for the plaquette is determined by Ω v = φ v B /S 0 , where S 0 = 2π/N φ is the area of the unit plaquette. The result is shown in Fig. 1. We see that the Berry curvature has a continuous distribution inside the Fermi sea and vanishes outside. The distribution is obviously not the singular one implied by the Dirac interpretation [28].
Uniform background The uniform background of the Berry curvature can be determined analytically by inspecting the quasiperiodicity of the wave function in the k-space [29]. By using Eq. (3) and assuming a fixedk, it is easy to verify that Ψ JK k has an approximated quasiperiodicity in the limit of N e → ∞: Ψ JK k k1+L×n ∝ exp(imL * k 1 /2)Ψ JK k . As a result, we can define a super-Brillouin zone (SBZ) spanned by K a(b) = L a(b) × n with L a and L b being the two unit vectors of the torus. From Eq. (8), the Berry connection has the quasiperiodicity A e k1+K a(b) = A e k1 + m(K a(b) × n)/2. The total Chern number of the SBZ can be determined by C tot = (2π) −1¸Ae k1 · dk 1 with the integral along the boundary of the SBZ, and is equal to −mN φ . The uniform background of the Berry curvature is the average of the Berry curvature in the SBZ, i.e., −m in the electron representation, and 2 − m in the CF representation. The latter confirms Wang's numerical observation [18].
The same consideration immediately leads to the conclusion that the two wave functions Ψ JK k and Ψ CF k must yield different distributions of the Berry curvature. This is because the standard CF wave function Eq. (1) has the different (exact) quasiperiodicity Ψ CF k k1+L×n ∝ exp(iL * k 1 /2)Ψ CF k . The absence of m in the exponential factor is notable. As a result, the total Chern number of the SBZ for Ψ CF k is −N φ . It corresponds to a background Berry curvature equal to −1 and +1 for the electron and the CF representations, respectively. We summarize the results for the background Berry curvature as follows: Segal-Bargmann transform It turns out that the Berry phase with respect to the standard CF wave function Eq. (1) can be obtained analytically. This is because standard CF wave functions have a simple structure which is evident when expressed as a form explicating their connections to the Segal-Bargmann transform [30] is the measure of the Segal-Bargmann space [30,31], and a · b ≡ i a i b i . The Segal-Bargmann transform shown in the square bracket is a unitary transformation that maps a (wave) function ψ (η) in the hidden Hilbert space into a holomorphic function of z and z * , and the CF wave function is obtained from it by a projection z * = 0. All wave functions prescribed by the CF theory can be expressed as such. To obtain a valid CF wave function satisfying the quasiperiodic boundary condition Ψ(z)| zi→zi+L = ξ(L) N φ e L * 2 (z+ 1 2 L) Ψ(z) on a torus [21], the hidden-space wave function ψ(η) should satisfy the quasiperiodic boundary condition ψ(η)| ηi→ηi+L = ξ N φ −mNe (L)e 1−mν 2 L * (ηi+ 1 2 L) ψ(η), exactly the one for a screened effective magnetic field B eff = (1−mν)B as dictated by the theory of CFs [32]. We note that Eq. (11) also leads to an orthogonality condition different from the usual one for the hidden Hilbert space, and will have an effect on its construction.
Berry phase for the CF wave function With the general relation Eq. (11), we can derive an analytic expression for the Berry phase for the standard CF wave function Ψ CF k . According to Eq. (9), we need to determine the matrix element Ψ CF k |e −iq·ẑ |ϕ CF k+q = P (−1) P ϕ CF P k |e −iq·ẑ |ϕ CF k+q , where P k denotes a permutation of the initial configuration k. For the CFL, ϕ CF k can be obtained by using Eq. (11) with an unsymmetrized wave function in the hidden space ψ k (η) = exp(ik · η) = exp[i(k · η * + k * · η)/2]. By applying Eq. (12), we obtain k1 ) · n = ∓1 (in the unit of 1/qB). The effect of particle exchanges is exactly cancelled. The result supports the picture of a uniformly distributed Berry curvature [13][14][15].
Summary In summary, we have determined the Berry phase for the CFL. For both the wave functions, a CF adiabatically transported around the Fermi circle acquires a Berry phase π (−π) in the CF (electron) representation. Since the Berry phase can be interpreted as the anomalous Hall conductance [33] (in the unit of −e 2 /2πh for σ xy [34]), both the wave functions can correctly predict the Hall conductance of CFs for a particle-hole symmetric half-filled Landau level [10], in both its magnitude and sign. At the same time, it is obvious that the CF with respect to both the wave functions is not a massless Dirac particle. The uniform-Berry-curvature picture is actually the correct CF interpretation for the standard CF wave function.
where we have assumed that only k 1 is changed to k 1 + q 1 . We note that Geraedts et al.'s original definition uses an operatorρ q = i e −iq·zi . It is easy to see that the two forms are equivalent except that the matrix element for the latter will be scaled by a factor of N e . For our definition, Ψ k is obtained by modifying the determinant in Eq. (4) to i.e., the column with respect to the changed wave-vector (k 1 ) is modified by the "momentum boost operator." We implement a Metropolis Monte-Carlo algorithm similar to that detailed in Ref. [26]. Specifically, the overlap |D| and phase φ of the matrix elements are evaluated by where N denotes the normalization factor of |Ψ k | 2 , and stands for lattice summation for all z i 's [13,26]. The Markov chains of our Monte-Carlo simulation are sampled by assuming a probability density ∝ |Ψ k | 2 . By using a single Markov chain, we can determine the phases and overlaps with respect to both the definitions, since the two definitions are only different byΨ k .
The numerical results for N e = 13 are presented in Table. S1. For Geraedts et al.'s definition, we compare the results presented in Ref. [26] with our own calculation results. They coincide well within numerical uncertainties. The results with respect to our definition are also shown.  To construct the Markov chains, we try two strategies, i.e., the local and global approaches, for updating configurations. In the local update approach, we only change the position of one electron in each update. The average hopping distance of the electron is adjusted to yield an acceptance rate ∼ 25%. In the global update approach, the positions  of all electrons are updated simultaneously within their respective nearest and next nearest neighbors of the lattice. The two approaches yield results very close numerically, as evident in Fig. S1. Table S2 shows parameters of our simulations, including the lengths of the Markov chains and acceptance rates. Figure S2 shows the typical behavior of convergence. One can see that the result starts to converge after 10 3 N e steps of local updates.