Investigating anisotropic quantum Hall states with bi-metric geometry

TL;DR

This paper develops a bi-metric effective theory for anisotropic fractional quantum Hall states, linking geometric responses to anisotropy and verifying predictions through numerical simulations, advancing understanding of nematic order in quantum Hall systems.

Contribution

It introduces a novel bi-metric formalism for anisotropic FQH states and establishes a relationship between shift, Hall viscosity, and a new anisospin coupling.

Findings

Derived a relationship between shift, Hall viscosity, and anisospin.

Numerically verified Hall viscosity predictions using DMRG.

Clarified the role of Berry phase coefficient in nematic order.

Abstract

We construct a low energy effective theory of anisotropic fractional quantum Hall (FQH) states. We develop a formalism similar to that used in the bi-metric approach to massive gravity, and apply it to describe abelian anisotropic FQH states in the presence of external electromagnetic and geometric backgrounds. We derive a relationship between the shift, the Hall viscosity, and a new quantized coupling to anisotropy, which we term "anisospin". We verify this relationship by numerically computing the Hall viscosity for a variety of anisotropic quantum Hall states using the density matrix renormalization group (DMRG). Finally, we apply these techniques to the problem of nematic order and clarify certain disagreements that exist in the literature about the meaning of the coefficient of the Berry phase term in the nematic effective action.

Tables

Table 1. Table 1: This table, combined with Eqs.(26)-(27) in the main text, shows how we can determine the values of s N subscript 𝑠 𝑁 s_{N} and ς 𝜍 \varsigma used in Fig. 1

$\nu$	$N$	$S_N$	$s_N$	$\varsigma$
$1$	$1$	$1$	$\frac{1}{2}$	$0$
$2$	$2$	$3$	$\frac{3}{2}$	$0$
$\frac{1}{3}$	$1$	$3$	$\frac{1}{2}$	$1$
$\frac{2}{5}$	$1$	$4$	$\frac{1}{2}$	$\frac{3}{2}$

Equations

$$H=\frac{1}{2m}{V}^{AB}{\delta }_A^i{\delta }_B^j\sum _q{\pi }_i^q{\pi }_j^q,$$

$${\delta }_A^i\to E_A^i,{\delta }_B^j\to E_B^j$$

$$J^i{}_j=-\frac{1}{2}\sum _q\left(\left\{x_q^i,{\pi }_j^q\right\}-B{ϵ}_{jk}{x}_q^i{x}_q^k\right),$$

$$T^i{}_j=-i\left[H,J^i{}_j\right].$$

$$(\eta {)}^i{}_j{}^k{}_{\ell }=\underset{\omega \to 0}{\lim }\frac{1}{V{\omega }^{+}}\left(⟨\left[T^i{}_j,J^k{}_{\ell }\right]⟩+⟨⟨T^i{}_j,T^k{}_{\ell }⟩⟩\right),$$

$$⟨⟨A,B⟩⟩={\int }_0^{\infty }dt{e}^{i{\omega }^{+}t}⟨\left[A(t),B(0)\right]⟩,$$

$${\tilde{\pi }}_{\alpha }^q={\Lambda }^A{}_{\alpha }{\delta }_A^i{\pi }_i^q,{\tilde{x}}_q^{\alpha }={\lambda }^{\alpha }{}_A{\delta }_i^A{x}_q^i.$$

$$H=\frac{1}{2m}\sum _{q\alpha }{\tilde{\pi }}_{\alpha }^q{\tilde{\pi }}_{\alpha }^q,$$

$${\pi }_i^q={\lambda }^{\alpha }{}_i{\tilde{\pi }}_{\alpha }^q,x_q^i={\Lambda }^i{}_{\alpha }{\tilde{x}}^{\alpha }$$

$$J^i{}_j={\Lambda }^i{}_k{\lambda }^{\ell }{}_j{\tilde{J}}^k{}_{\ell },$$

$$T^i{}_j$$

$$={\Lambda }^i{}_k{\lambda }^{\ell }{}_j\left(-i\left[H,{\tilde{J}}^k{}_{\ell }\right]\right),$$

$$={\Lambda }^i{}_k{\lambda }^{\ell }{}_j{\tilde{T}}^k{}_{\ell },$$

$${\tilde{T}}^i{}_j={\delta }^{ik}\frac{1}{2m}\sum _q\left\{{\tilde{\pi }}_k^q,{\tilde{\pi }}_j^q\right\}$$

$$(\eta {)}^i{}_j{}^k{}_{\ell }$$

$$=-\frac{1}{4}\mathcal{S}\rho \left({\delta }^i{}_{\ell }{\Lambda }^k{}_{\gamma }{ϵ}^{\gamma }{}_{\beta }{\lambda }^{\beta }{}_j-{\delta }_j{}^k{\Lambda }^i{}_{\alpha }{ϵ}^{\alpha }{}_{\zeta }{\lambda }^{\zeta }{}_{\ell }\right)$$

$${\Lambda }^k{}_{\gamma }{\Lambda }^{\ell }{}_{\beta }{ϵ}^{\gamma \beta }=\det (\Lambda ){ϵ}^{k\ell }={ϵ}^{k\ell }$$

$${\Lambda }^k{}_{\gamma }{ϵ}^{\gamma }{}_{\beta }={ϵ}^{km}{\lambda }^{\gamma }{}_m{\delta }_{\gamma \beta }$$

$$(\eta {)}^i{}_j{}^k{}_{\ell }=-\frac{1}{4}\mathcal{S}\rho \left({\delta }^i{}_{\ell }{ϵ}^{km}{v}_{jm}-{\delta }_j{}^k{ϵ}^{im}{v}_{\ell m}\right),$$

$${\eta }_{AB}^H$$

$$=-\frac{1}{8}\mathcal{S}\rho \left({ϵ}^{ji}{ϵ}_{iA}{ϵ}_{kB}{ϵ}^{km}{v}_{jm}-{ϵ}^{k\ell }{ϵ}_{iA}{ϵ}_{kB}{ϵ}^{im}{v}_{\ell m}\right)$$

$$=\rho \frac{\mathcal{S}}{4}{v}_{AB},$$

$$\eta (\omega {)}^i{}_j{}^k{}_{\ell }={\eta }_0(\omega {)}^{\alpha }{}_{\beta }{}^{\gamma }{}_{\zeta }{\Lambda }^i{}_{\alpha }{\lambda }^{\beta }{}_j{\Lambda }^k{}_{\gamma }{\lambda }^{\zeta }{}_{\ell },$$

$${\eta }_0(\omega {)}^i{}_j{}^k{}_{\ell }$$

$$S=\int dt{d}^{2}x\left[i{\psi }^{†}{D}_0\psi +\frac{1}{2m}{\hat{V}}^{AB}{D}_A{\psi }^{†}{D}_B\psi \right].$$

$$\frac{1}{2m}{V}^{AB}{D}_A{\psi }^{†}{D}_B\psi ⟶\frac{1}{2m}{\hat{G}}^{ij}{D}_i{\psi }^{†}{D}_j\psi .$$

$$S=\int dt{d}^{2}x\sqrt{\hat{g}}\left[i{\psi }^{†}{D}_0\Psi +\frac{1}{2m}{\hat{G}}^{ij}{D}_i{\psi }^{†}{D}_j\psi \right].$$

$$W=\frac{N}{4\pi }\int AdA+\frac{N^2}{4\pi }\int Ad\hat{\omega }+\dots ,$$

$$D_{\mu }\to {\partial }_{\mu }+i{A}_{\mu }+i\alpha C_{\mu }.$$

$$S=S_0+S_{\mathrm{c}}+S_{\mathrm{q}},$$

$$S_0$$

$$S_{\mathrm{c}}$$

$$S_{\mathrm{q}}$$

$$Q(\mathbf{x})={\psi }^{†}\left(\begin{array}{cc}{D}_x^2-D_y^2& D_x{D}_y+D_y{D}_x\\ D_x{D}_y+D_y{D}_x& -D_x^2+D_y^2\end{array}\right)\psi ,$$

Full text

Supplementary Material for Investigating anisotropic quantum Hall states with bi-metric geometry

Andrey Gromov

gromovand@uchicago.edu

Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637

  

Scott D. Geraedts

Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544

  

Barry Bradlyn

bbradlyn@princeton.edu

Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544

I Anisotropy in a microscopic model

We consider a microscopic model of non-interacting electrons in magnetic field with an anisotropic band mass tensor $(m_{*}^{-1})^{AB}=\frac{1}{2m}V^{AB}$. We will examine the stress response in the IQH regime in two ways. First, we will show directly from the microscopic Hamiltonian that the Hall viscosity is given by Eq. (23) of the main text. Second, we will derive by direct integration the generating functional of the model, and show that it agrees with Eq. (17) of the main text.

I.1 Viscosity from the Kubo Formula

The Hamiltonian for noninteracting electrons in a magnetic field with this anisotropic band mass tensor is given in flat space by

[TABLE]

where $q$ is a particle index, and $\pi^{q}_{i}=p_{i}^{q}-A_{i}(\mathbf{x}^{i})$ is the kinetic momentum of particle $q$. We will follow the formalism of Ref. Bradlyn et al. (2012) to compute the viscosity tensor from the flat space Hamiltonian. We assume that the coupling to the background metric is minimal, in the sense that the geometry only enters through the replacement

[TABLE]

in Eq. (1). For the sake of simplicity, we will consider only non-compressive metric perturbations. We first identify the strain generators

[TABLE]

which generate area-preserving deformations of the system. The continuity equation for momentum density implies that the (traceless part of) the integrated stress tensor is given by

[TABLE]

Note that this relationship is precisely a consequence of our choice of geometric coupling. It would be modified if, for instance, we inserted into the Hamiltonian a direct coupling to the tensor $C_{\mu}$ from the main text.

Using this Ward identity, we can write the non-compressive contributions to the viscosity tensor (compressive contributions will vanish at zero frequency) as

[TABLE]

where we have defined the retarded correlation function

[TABLE]

and all expectation values are taken with respect to the flat-space Hamiltonian Eq. (1). While we can proceed directly to evaluate this expression, it is easier to first perform a canonical transformation. Recalling that $V^{AB}=\Lambda^{A}{}_{\alpha}\Lambda^{B}{}_{\alpha}$, and $\lambda^{\alpha}{}_{A}\Lambda^{A}{}_{\beta}=\delta^{\alpha}_{\beta}$, we define

[TABLE]

To lighten the notational load, we will exploit the fact that in flat space, index types ($i,A,\alpha$) can be treated equivalently; we will thus from here on out suppress Kronecker $\delta$ symbols which serve only to change index type, with the understanding that the first, upper index of $\Lambda$ is always of $SO(2)$ type ($A$), and the first, upper index of $\lambda$ is always of $\widehat{SO}(2)$ ($\alpha$) type. The constraint $\det(V)=1$ ensures that this transformation is canonical. Under this transformation, the Hamiltonian can be written

[TABLE]

which we recognize as equivalent to the isotropic Landau Hamiltonian. Furthermore, making use of the inverse transformation

[TABLE]

we see that the strain generators can be expressed as

[TABLE]

where $\tilde{J}$ is given by Eq. (3) with $\pi,x$ replaced by $\tilde{\pi},\tilde{x}$. After employing the Ward identity of Eq. (4), we see that the stress tensor is given in terms of the “tilde”-basis as

[TABLE]

where we recognize

[TABLE]

as the operator expression for the isotropic stress tensor, in terms of $\tilde{\pi}$. With this observation, we have mapped the problem of evaluating Eq. (5) back to the problem of evaluating the viscosity for an isotropic system. Using the known result for the viscosity of an integer quantum Hall system, we find immediately that

[TABLE]

where $\mathcal{S}=\nu$ is the shift for $\nu$ filled Landau levels. Noting that

[TABLE]

implies

[TABLE]

we can simplify Eq. (14) to find

[TABLE]

where $v_{\ell m}=\lambda^{\alpha}{}_{\ell}\lambda^{\alpha}{}_{m}$.

Applying formula Eq. (22) from the main text for the contracted Hall tensor, we thus find

[TABLE]

and so we deduce that the isospin $\varsigma=\frac{\mathcal{S}}{2}$ for noninteracting electrons. Because of our choice of coupling to the metric, we also find $\xi=0$.

Finally, note that although we considered only non-compressive perturbations for clarity, The canonical transformation Eq. (7) implies that the full, frequency dependent viscosity tensor $\eta(\omega)^{i}{}_{j}{}^{k}{}_{\ell}$ for the anisotropic electron system (including the compressive contributions) is given by

[TABLE]

where, using Ref. Bradlyn et al., 2012, we have

[TABLE]

for the frequency-dependent viscosity of the isotropic system.

I.2 Generating Functional

In this Section we will directly integrate out the electronic degrees of freedom in the model of non-interacting electrons with anisotropic band mass, in a magnetic field filling $N$ Landau levels. In flat space the model is described by the action

[TABLE]

In curved space one has to replace the covariant derivatives via $D_{A}=E_{A}^{i}D_{i}$ (see, for example, Bradlyn and Read (2015); Gromov and Abanov (2015)), where $E_{A}^{i}$ is the inverse spatial vielbein. Then

[TABLE]

The full action in curved space is given by

[TABLE]

We have used $\det V=1$ to replace $\sqrt{g}$ by $\sqrt{\hat{g}}$ and $D_{\mu}=\partial_{\mu}+iA_{\mu}$.

To integrate out $\psi$ we note that the coupling of $\psi$ to $\hat{g}$ is identical to the coupling of an isotropic system to the ambient metric. Thus, we can use the results of Ref. Abanov and Gromov (2014) for the effective action. We find

[TABLE]

which implies $\varsigma=N$ and every Landau level contributes $\varsigma_{N}=N-\frac{1}{2}$.

Coupling to the tensor $C_{\mu}$ can be included via the modified covariant derivative.

[TABLE]

Since this coupling is the same order in derivatives as the other coupling, we cannot discard it in principle. This coupling leads to the appearence of $\xi=\frac{N}{2}\alpha$ in Eq. (17) of the main text.

When inter-election interactions $S_{\rm int}$ are introduced, the couplings to $g$ and $\hat{g}$ are no longer identical. This leads to the splitting of the shift into $s$ and $\varsigma$ discussed in the main text.

II Additional Numerical Data

In this section we present some additional details about the data in Fig. 1 of the main text. The data in that figure is chosen so that the Hall viscosity has aconstant piece which is equal to $2s_{N}$, and a piece linear in $\alpha$ with slope $2\varsigma$. There is also an additional term which goes as $\alpha^{2}-1$ with coefficient $2\xi$. All this can be seen from Eq.(23) of the main text. In principle, $\xi$ could also depend on $\alpha$, here we are assuming that its dependence on $\alpha$ is constant in the range of $\alpha$ studied. We can use Eq.(26)-(27) to predict the values of $\varsigma$ and $s_{N}$, how we do this is summarized in Table 1. Note that these values are the ones used when the anisotropy is in the interaction, when the anisotropy is in the kinetic part of the Hamiltonian $s_{N}$ and $\varsigma$ are interchanged, as described in the main text.

In Fig. 1 we find deviations from the straight lines determined by $s_{N}$ and $\varsigma$, which we attribute to non-zero $\xi$. We might ask if these deviations are, in fact, not from a non-zero $\xi$, but instead arise from the finite circumference of the cylinder. (As explained in the main text, by going to bond dimensions up to $\chi=8000$ we can reduce finite-bond-dimension errors to $\approx 10^{-3}$, much smaller than the deviations from a straight line seen in the figure). To obtain Fig. 1 for each value of $\alpha$ we fit $\eta_{H}$ vs $L^{2}$ to a straight line, using the system sizes $L=10-20$. After doing this for each value of $\alpha$ we obtain the data in Fig. 1 which we fit to obtain $\xi$. In Fig. 1 we investigate the finite-size dependence of our $\xi$ values at $\nu=1/3$ by performing the same procedure, but instead of using data at all the sizes we obtain we only use three data points at $L-0.5,L,L+0.5$, for a range of $L$. We see that the data does oscillate as a function of $L$. However, at the largest $L$ we can access it seems that the oscillations on the value of $\xi$ we obtain are smaller than its absolute value, leading us to believe that the non-zero value of $\xi$ we obtain is not a finite-size effect.

III Anisotropy from Nematic Order

As we explained in the main text, the tensor $V^{AB}$ can be generated dynamically. In this Section we will examine a particular way of generating this tensor from the quenched configuration of a nematic order parameter in the isotropic phase. In particular, we will explain how our formalism arises in the context of Ref. Maciejko et al. (2013) and Ref. You et al. (2014), and elucidate an implicit assumption that led to an apparent disagreement.

In what follows we will utilize the computation of Ref. You et al. (2014). Our starting point is a system of interacting, non-relativistic electrons with the interaction given by a repulsive Coulomb potential and an attractive quadrupolar interaction:

[TABLE]

where

[TABLE]

where $\rho(\mathbf{x})=\psi^{\dagger}(\mathbf{x})\psi(\mathbf{x})$ is the electron density and $Q(\mathbf{x})$ is a quadrupolar operator

[TABLE]

and $D_{i}=\partial_{i}+iA_{i}$.

Our goal is to derive the effective action coming from (26) and from it compute the Hall viscosity. To do so, we minimally couple (26) to the spatial metric $g_{ij}$ by replacing $\delta^{AB}\rightarrow\delta^{AB}e_{A}^{i}e_{B}^{j}$, introducing a factor of $\sqrt{g}$ in the spatial integration measure, and replacing $|\mathbf{x}-\mathbf{x^{\prime}}|$ with $d(\mathbf{x},\mathbf{x^{\prime}})$ - the geodesic distance between $\mathbf{x}$ and $\mathbf{x^{\prime}}$ evaluated with respect to the metric $g_{ij}$.

Next, we perform the standard flux attachment Lopez and Fradkin (1991); Read (1989); Cho et al. (2014) by introducing a statistical gauge field $a_{i}$. We also introduce a Hubbard-Stratonovich field $\mathbf{M}=(M_{1},M_{2})$ to decouple the quadrupolar interaction $S_{\rm q}$. Then, after doing a mean-field approximation for the attached flux, we find the following effective action:

[TABLE]

where we have introduced the composite fermion field $\Psi$, and suppressed external background fields as arguments.

The first term $S_{\rm n}[\mathbf{M}]$ describes the gapped dynamics of the nematic order parameter. We will not need an exact form of this term, it is sufficient to know that it is gapped. The scale at which these massive degrees of freedom become important is the gap of the zero momentum GMP mode, which is of the order of the Coulomb gap. The latter is assumed to be infinite in our low energy theory. Thus we will drop this term for the remainder of the Letter.

The second term $\tilde{S}_{0}[\Psi,\mathbf{M},a]$ describes the dynamics of free composite fermions which we will choose to fill the lowest Landau level

[TABLE]

where $\hat{G}^{ij}$ is given by Eq. (10) of the main text with $V^{AB}$ given in terms of $\mathbf{M}$ as follows

[TABLE]

and $\tilde{D}_{i}$ is the covariant derivative that describes the interaction of the composite fermion spin with geometry and anisotropy, which we now discuss. Indeed, Eq. (32) requires extra explanation and is, in part, the origin of the disagreement between Ref. Maciejko et al. (2013) and Ref. You et al. (2014). The introduction of a new covariant derivative $\tilde{D}_{i}$ is necessary to ensure that flux attachment is an exact transformation even in curved space Lee and Wen (1994); Cho et al. (2014). However, there exists more than one choice of $\tilde{D}_{i}$ that will accomplish this task. In fact, the most general $\tilde{D}_{i}$ would distribute the coupling between geometry and anisotropy according to $\tilde{D}_{i}=D_{i}+ia_{i}+ip_{1}\omega_{i}+ip_{2}\hat{\omega}_{i}$, with $p_{1}+p_{2}=p$ and $2p_{1},2p_{2}\in\mathbb{Z}$. The values of $p_{1}$ and $p_{2}$ must be in general determined from a microscopic analysis, which is beyond the scope of the present paper. Thus we will keep the coupling general and use the general $\tilde{D}_{i}$ in the remainder of this Section.

Integrating out the composite fermions and the statistical gauge field $a$ we find

[TABLE]

we have suppressed purely gravitational terms. The extra $\frac{1}{2}\hat{\omega}$ comes from integrating out the composite fermions, which unambiguously couple to $\hat{g}$.

To analyze this result we first turn off all external fields, leaving

[TABLE]

where $\varsigma=\frac{2p_{2}+1}{2}$. This term has been obtained in Ref. Maciejko et al. (2013) with anisospin $\varsigma=\bar{s}=\frac{2p+1}{2}$ and in Ref. You et al. (2014) with $\varsigma=\frac{1}{2}$. These are two opposite cases when $p_{2}$ is set to be either $p$ or [math] correspondingly. We see that both scenarios are plausible depending on the value of $p_{2}$, i.e. depending on how exactly the spin of a composite fermion couples to anisotropy. We are not aware of a physical criterion that would select a particular value of $p_{2}$ in the situation when the nematic order parameter is pinned to a fixed configuration. Most likely, this ambiguity points to the fact that the coefficient of the Berry phase term is not universal.

Turning the metric back on we find that Hall viscosity tensor is

[TABLE]

where $\lambda={\mbox{\rm}det}(\lambda^{\alpha}_{A})$ which reduces to $\eta^{H}_{AB}=\frac{2p+1}{4}\bar{\rho}\delta_{AB}$ in the isotropic limit, as it should be. Finally, we note that $\xi=0$ in the flux attachment computation. It is possible to accommodate a non-zero $\xi$ by coupling the original fermions (27) to $C$, however such coupling appears to be unnatural.