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NQS2017 @ YITP, Kyoto Nov. 22, 2017 Field theory for symmetry protected topological states in quantum antiferromagnets University of Geneva Shintaro Takayoshi ST, P. Pujol, and A. Tanaka, Phys. Rev. B 94 , 235159 (2016). ST, K. Totsuka and A.

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Field theory for symmetry protected topological states in quantum antiferromagnets

University of Geneva Shintaro Takayoshi

ST, P. Pujol, and A. Tanaka, Phys. Rev. B 94, 235159 (2016). ST, K. Totsuka and A. Tanaka, Phys. Rev. B 91, 155136 (2015).

  • Nov. 22, 2017

NQS2017 @ YITP, Kyoto

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Introduction

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Symmetry protected topological (SPT) phase Short range entangled state can be nontrivial by imposing some symmetry Trivial (direct product) state

Local unitary transformation

Long range entangled state Gapped phases Short range entangled state

anyonic exciation, intrinsic topological order (e.g., FQHE) w/o any symmetry

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Introduction

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Pollmann, et al., 2010, 2012

Ground state of S=1 Heisenberg antiferromagnet Haldane phase (Affleck-Kennedy-Lieb-Tasaki state) Haldane phase is protected by A) rotation around spin axes (Dihedral symmetry) B) Time-reversal symmetry C) Parity symmetry (Bond-centered inversion) Discussion by matrix product state (MPS)

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Example: Heisenberg antiferromagnets

S=1 S=2

Tonegawa et al., 2011 Chen et al., 2003

Large-D (trivial) phase ( -basis)

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(1+1)D antiferromagnets and NLSM

Effective field theory: O(3) nonlinear sigma model + theta term

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S: integer S: half-odd integer gapless gapped

Haldane, 1983

What is the field theoretical difference (S=odd/even)?

  • See the ground state wave functional.

Pollmann et al.,2010, 2012

S=odd: SPT S=even: trivial Matrix product state discussion

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SLIDE 6

Planar limit of NLSM

Take the easy-plane config. (planar limit) Theta term vanishes? -> No. Origin: staggered (AF) summation over spin Berry phase.

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Average weight of vortex (coarse grained)

Sachdev, 2002

Planar limit -> space-time vortex contributes

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Ground state wave functional

Strong coupling limit

Path integral formalism Xu-Senthil, 2013

Initial and final imaginary time Winding number of the planar spin config. p.b.c.

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Spin config.

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Edge state

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p.b.c.

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Dual field theory and SPT breaking

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S = even and odd are continuously connected by changing . Staggered field should be prohibited. Phase is locked at

  • dd-S even-S

separated

Dual vortex field theory (and low fugacity expansion)

  • > sine-Gordon model

staggered field -> staggered mag.

  • dd-S even-S
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SLIDE 10

Magnetization plateau

The above discussion is also valid for magnetization plateaus just by replacing .

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ST, K. Totsuka and A. Tanaka, Phys. Rev. B 91, 155136 (2015).

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(2+1)D AKLT states

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monopole number at dual site

Haldane, 1988 Sachdev-Vojta, 2000

at site(x,y) Spatially isotropic VBS state (S=even). Berry phase from monopoles (tunneling of skyrmion).

  • cf. (1+1)D

1 0 1 0

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(2+1)D AKLT states

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Average weight of vortex (coarse grained) Shift

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(2+1)D AKLT states

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Wrapping (skyrmion) # of the spin snapshot configuration Path integral formalism theta term Edge state Ground state wave functional

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1D-2D analogy

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S = 2,6,… : SPT S = 4,8,… : trivial Dual monopole theory: x- and y- dimerization breaks SPT. SO(3) + translational symmetry would protect SPT.

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Strange Correlator

  • Definition
  • Idea

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: Ground state : Trivial (direct product) state Usual two-point correlator No topological effect Strange correlator Effects from the topo. term nonzero or power-law decay: SPT

  • Exp. decay: trivial

Y.-Z. You et al., 2014

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1D strange correlator

Imaginary time correlator of a particle on a ring with flux

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Relabel

Aharonov-Bohm phase Nonzero at : SPT (i) case (S=even) (ii) case (S=even)

  • exp. decay: trivial
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Relabeling of coordinate

2D strange correlator

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Strange correlator

  • > two point correlator in (1+1)d NLSM + theta term

S=2,6,… half-odd integer spin chain (gapless) power-law decay: SPT S=4,8,… integer spin chain (gapped)

  • exp. decay: trivial

Strange correlator correctly detects SPT states.

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Summary

  • We described the SPT properties in AKLT-VBS

states using an effective field theory, especially NLSM + topo. Term.

  • SPT can be distinguished by looking at the

ground state wave functional. In (2+1)d, monopole Berry phase is important.

  • We calculate the strange correlator in one and

two dimensions, and confirm that SPT phase can be detected.

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