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Hierarchy construction for FQH states via anyon condensation

Carolyn Zhang

Harvard University

Aug 15, 2024

With Ashvin Vishwanath and Xiao-Gang Wen, arXiv:2406.12068

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What is the FQHE?

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What is the FQHE?

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Real materials!

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Real materials!

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FQH states as U(1) enriched SMTC

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FQH states as U(1) enriched SMTC

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FQH states as U(1) enriched SMTC

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FQH states as U(1) enriched SMTCs

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FQH states as U(1) enriched SMTCs

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FQH states as U(1) enriched SMTCs

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FQH states as U(1) enriched SMTCs

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What is realized in experiment?

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Example: 1/3 filling

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Example: 1/3 filling

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Example: 1/3 filling

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Example: 1/3 filling

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1/3 filling in experiment

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1/3 filling in experiment

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Other simple FQH states

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FQH states come in families

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FQH states come in families

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FQH states come in families

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Hierarchy state in experiment

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A new approach

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A new approach

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A new approach

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The math

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The physics

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Warm-up: condensable subgroups

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Example: 1/3 filling

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Example: 1/3 filling

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Example: 1/3 filling

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Example: 1/3 filling

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Example: 1/3 filling

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Example: 1/3 filling

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n=1 hierarchy state

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n=1 hierarchy state

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Nonabelian FQH states

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Three candidates for 1/2 filling

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Three candidates for 1/2 filling

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Pfaffian anyon theory

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Pfaffian anyon theory

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Pfaffian U(1) symmetry fractionalization

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Pfaffian U(1) symmetry fractionalization

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Pfaffian quasihole hierarchy

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Pfaffian: $\mathrm{Ising}\times U(1)_8/\mathbb{Z}_2$

\{1,a^2,\psi,\psi a^2,\sigma a,\sigma a^3\}\times\{1,\psi a^4\}

\frac{1}{16}+\frac{1}{16}-\frac{l^2}{2m}=n\qquad n\in\mathbb{Z}

Fusion rules: $\psi\times\psi=1\qquad(\sigma a)\times(\sigma a)=(1+\psi)a^2\qquad \psi\times (\sigma a)=\sigma a$

Spins: $\theta_{\sigma}=e^{2\pi i/16}\qquad \theta_{\psi}=-1\qquad \theta_{a^n}=e^{2\pi i n^2/16}$

Fractional charge given by braiding with $a^2$: $B(a^n,a^2)=e^{2\pi in/4}\to q_{a^n}=\frac{n}{4}$

$\bigg\}\sigma a \bar{b}^l$ is a boson

\frac{l^2}{2m}=n+\frac{1}{8}=\frac{8n+1}{8}\to l=\pm(8n+1),m=4(8n+1)

\sigma a\bar{b}^l\times\sigma a\bar{b}^l=(1+\psi)a^2\bar{b}^{2l}

Solutions: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-8(16n+1)}/\mathbb{Z}_2$

$n=1$: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-136}/\mathbb{Z}_2$

\mathcal{A}=1+\psi\bar{\psi}+a^2\bar{b}^34+a^2\bar{b}^{34}\psi\bar{\psi}+\cdots+\sigma\bar{\sigma} a\bar{b}^{17}+\cdots

\nu_{\mathrm{tot}}=\frac{1}{2}-\frac{1}{2(16n+1)}=\frac{8n}{16n+1}

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Pfaffian quasihole hierarchy

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Pfaffian: $\mathrm{Ising}\times U(1)_8/\mathbb{Z}_2$

\{1,a^2,\psi,\psi a^2,\sigma a,\sigma a^3\}\times\{1,\psi a^4\}

\frac{1}{16}+\frac{1}{16}-\frac{l^2}{2m}=n\qquad n\in\mathbb{Z}

Fusion rules: $\psi\times\psi=1\qquad(\sigma a)\times(\sigma a)=(1+\psi)a^2\qquad \psi\times (\sigma a)=\sigma a$

Spins: $\theta_{\sigma}=e^{2\pi i/16}\qquad \theta_{\psi}=-1\qquad \theta_{a^n}=e^{2\pi i n^2/16}$

Fractional charge given by braiding with $a^2$: $B(a^n,a^2)=e^{2\pi in/4}\to q_{a^n}=\frac{n}{4}$

$\bigg\}\sigma a \bar{b}^l$ is a boson

\frac{l^2}{2m}=n+\frac{1}{8}=\frac{8n+1}{8}\to l=\pm(8n+1),m=4(8n+1)

\sigma a\bar{b}^l\times\sigma a\bar{b}^l=(1+\psi)a^2\bar{b}^{2l}

Solutions: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-8(16n+1)}/\mathbb{Z}_2$

$n=1$: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-136}/\mathbb{Z}_2$

\mathcal{A}=1+\psi\bar{\psi}+a^2\bar{b}^34+a^2\bar{b}^{34}\psi\bar{\psi}+\cdots+\sigma\bar{\sigma} a\bar{b}^{17}+\cdots

\nu_{\mathrm{tot}}=\frac{1}{2}-\frac{1}{2(16n+1)}=\frac{8n}{16n+1}

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Pfaffian quasihole hierarchy

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Pfaffian: $\mathrm{Ising}\times U(1)_8/\mathbb{Z}_2$

\{1,a^2,\psi,\psi a^2,\sigma a,\sigma a^3\}\times\{1,\psi a^4\}

\frac{1}{16}+\frac{1}{16}-\frac{l^2}{2m}=n\qquad n\in\mathbb{Z}

Fusion rules: $\psi\times\psi=1\qquad(\sigma a)\times(\sigma a)=(1+\psi)a^2\qquad \psi\times (\sigma a)=\sigma a$

Spins: $\theta_{\sigma}=e^{2\pi i/16}\qquad \theta_{\psi}=-1\qquad \theta_{a^n}=e^{2\pi i n^2/16}$

Fractional charge given by braiding with $a^2$: $B(a^n,a^2)=e^{2\pi in/4}\to q_{a^n}=\frac{n}{4}$

$\bigg\}\sigma a \bar{b}^l$ is a boson

\frac{l^2}{2m}=n+\frac{1}{8}=\frac{8n+1}{8}\to l=\pm(8n+1),m=4(8n+1)

\sigma a\bar{b}^l\times\sigma a\bar{b}^l=(1+\psi)a^2\bar{b}^{2l}

Solutions: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-8(16n+1)}/\mathbb{Z}_2$

$n=1$: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-136}/\mathbb{Z}_2$

\mathcal{A}=1+\psi\bar{\psi}+a^2\bar{b}^34+a^2\bar{b}^{34}\psi\bar{\psi}+\cdots+\sigma\bar{\sigma} a\bar{b}^{17}+\cdots

\nu_{\mathrm{tot}}=\frac{1}{2}-\frac{1}{2(16n+1)}=\frac{8n}{16n+1}

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Pfaffian quasihole hierarchy

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Pfaffian: $\mathrm{Ising}\times U(1)_8/\mathbb{Z}_2$

\{1,a^2,\psi,\psi a^2,\sigma a,\sigma a^3\}\times\{1,\psi a^4\}

\frac{1}{16}+\frac{1}{16}-\frac{l^2}{2m}=n\qquad n\in\mathbb{Z}

Fusion rules: $\psi\times\psi=1\qquad(\sigma a)\times(\sigma a)=(1+\psi)a^2\qquad \psi\times (\sigma a)=\sigma a$

Spins: $\theta_{\sigma}=e^{2\pi i/16}\qquad \theta_{\psi}=-1\qquad \theta_{a^n}=e^{2\pi i n^2/16}$

Fractional charge given by braiding with $a^2$: $B(a^n,a^2)=e^{2\pi in/4}\to q_{a^n}=\frac{n}{4}$

$\bigg\}\sigma a \bar{b}^l$ is a boson

\frac{l^2}{2m}=n+\frac{1}{8}=\frac{8n+1}{8}\to l=\pm(8n+1),m=4(8n+1)

\sigma a\bar{b}^l\times\sigma a\bar{b}^l=(1+\psi)a^2\bar{b}^{2l}

Solutions: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-8(16n+1)}/\mathbb{Z}_2$

$n=1$: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-136}/\mathbb{Z}_2$

\mathcal{A}=1+\psi\bar{\psi}+a^2\bar{b}^34+a^2\bar{b}^{34}\psi\bar{\psi}+\cdots+\sigma\bar{\sigma} a\bar{b}^{17}+\cdots

\nu_{\mathrm{tot}}=\frac{1}{2}-\frac{1}{2(16n+1)}=\frac{8n}{16n+1}

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Filling 1/2 hierarchies

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Pfaffian: $\mathrm{Ising}\times U(1)_8/\mathbb{Z}_2$

\{1,a^2,\psi,\psi a^2,\sigma a,\sigma a^3\}\times\{1,\psi a^4\}

\frac{1}{16}+\frac{1}{16}-\frac{l^2}{2m}=n\qquad n\in\mathbb{Z}

Fusion rules: $\psi\times\psi=1\qquad(\sigma a)\times(\sigma a)=(1+\psi)a^2\qquad \psi\times (\sigma a)=\sigma a$

Spins: $\theta_{\sigma}=e^{2\pi i/16}\qquad \theta_{\psi}=-1\qquad \theta_{a^n}=e^{2\pi i n^2/16}$

Fractional charge given by braiding with $a^2$: $B(a^n,a^2)=e^{2\pi in/4}\to q_{a^n}=\frac{n}{4}$

$\bigg\}\sigma a \bar{b}^l$ is a boson

\frac{l^2}{2m}=n+\frac{1}{8}=\frac{8n+1}{8}\to l=\pm(8n+1),m=4(8n+1)

\sigma a\bar{b}^l\times\sigma a\bar{b}^l=(1+\psi)a^2\bar{b}^{2l}

Solutions: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-8(16n+1)}/\mathbb{Z}_2$

$n=1$: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-136}/\mathbb{Z}_2$

\mathcal{A}=1+\psi\bar{\psi}+a^2\bar{b}^34+a^2\bar{b}^{34}\psi\bar{\psi}+\cdots+\sigma\bar{\sigma} a\bar{b}^{17}+\cdots

\nu_{\mathrm{tot}}=\frac{1}{2}-\frac{1}{2(16n+1)}=\frac{8n}{16n+1}

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Pfaffian and anti-pfaffian experiments!

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Hierarchies for other states

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Outlook

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Pfaffian: $\mathrm{Ising}\times U(1)_8/\mathbb{Z}_2$

\{1,a^2,\psi,\psi a^2,\sigma a,\sigma a^3\}\times\{1,\psi a^4\}

\frac{1}{16}+\frac{1}{16}-\frac{l^2}{2m}=n\qquad n\in\mathbb{Z}

Fusion rules: $\psi\times\psi=1\qquad(\sigma a)\times(\sigma a)=(1+\psi)a^2\qquad \psi\times (\sigma a)=\sigma a$

Spins: $\theta_{\sigma}=e^{2\pi i/16}\qquad \theta_{\psi}=-1\qquad \theta_{a^n}=e^{2\pi i n^2/16}$

Fractional charge given by braiding with $a^2$: $B(a^n,a^2)=e^{2\pi in/4}\to q_{a^n}=\frac{n}{4}$

$\bigg\}\sigma a \bar{b}^l$ is a boson

\frac{l^2}{2m}=n+\frac{1}{8}=\frac{8n+1}{8}\to l=\pm(8n+1),m=4(8n+1)

\sigma a\bar{b}^l\times\sigma a\bar{b}^l=(1+\psi)a^2\bar{b}^{2l}

Solutions: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-8(16n+1)}/\mathbb{Z}_2$

$n=1$: $\mathcal{D}=\overline{\mathrm{Ising}}\times U(1)_{-136}/\mathbb{Z}_2$

\mathcal{A}=1+\psi\bar{\psi}+a^2\bar{b}^34+a^2\bar{b}^{34}\psi\bar{\psi}+\cdots+\sigma\bar{\sigma} a\bar{b}^{17}+\cdots

\nu_{\mathrm{tot}}=\frac{1}{2}-\frac{1}{2(16n+1)}=\frac{8n}{16n+1}

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Outlook

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