Deconfined Quantum Criticality in the 2D J-Q model Anders W Sandvik - - PowerPoint PPT Presentation

Anders W Sandvik Boston University and Institute of Physics, Chinese Academy of Sciences, Beijing

Deconfined Quantum Criticality in the 2D J-Q model

Computational Approaches to Quantum Many-Body Problems ISSP , U of Tokyo, July 17, 2019

Bowen Zhao Boston University) Hui Shao (Beijing Normal University) Wenan Guo (Beijing Normal University)

Main points

AFM VBS

Large-scale QMC studies of the 2D S=1/2 J-Q model

• “designer Hamiltonian” for deconfined quantum criticality

Emergent U(1) symmetry of the near-critical VBS

• associated with a length scale 𝜊’, diverging faster than 𝜊

Anomalous finite-size scaling in J-Q model

• proposal: fundamental role of the second length scale
• phenomenological scaling function can explain all results

Transition in 3D O(3) Universality class Singlet formation on strong bonds ➙ Néel - disordered transition Ground state (T=0) phases

∆ = spin gap

s

Quantum phase transitions out of the Neel state

weak interactions strong interactions

H = X

hiji

JijSi · Sj S = 1/2 spins

Simplest case: Dimerized Heisenberg models

2D quantum antiferromagnets

• non-trivial non-magnetic ground states are possible, e.g.,

➡ resonating valence-bond (RVB) spin liquid ➡ valence-bond solid (VBS)

H = J

• i,j⇥

Si · Sj + g × · · · More complex non-magnetic states; 1 spin per unit cell

Non-magnetic states often have natural descriptions with valence bonds

= (↑i↓j − ↓i↑j)/ √ 2

i j

VBS breaks lattice symmetries spontaneously

• degenerate ground state
• different from dimerized H

(unique ground state)

• The Néel - VBS transition may be “beyond LGW”
• Deconfined quantum-critical point (DQCP)

Standard low-energy theory of quantum antiferromagnet S = Z ddrdτ 1

2[c2(∂rφ)2 + (∂τφ)2 + m0φ2 + uo(φ4)]

Can describe Neel to featureless paramagnetic transition

• VBS pattern or topological order cannot be captured by 𝝌

Topological defects (hedgehogs) in field configurations:

• suppressed in the Neel state
• proliferate in the quantum paramagnet

τ

Graph:Senthil et al.

The VBS state corresponds to a certain condensation of topological defects

• requires a description beyond 𝝌4 theory

Murthy & Sachdev 1991, Read & Sachdev 1991

Neel vector described by spinors z;

• coupled to U(1) gauge field where hedgehogs correspond to monopoles
• VBS on square lattice arises from condensation of quadrupled monopoles

φ = z∗

ασαβzβ

Nature of the Neel - VBS transition remained unknown…

Field theory description; brief summary

Striped Phase in a Quantum XY Model with Ring Exchange

A.W. Sandvik,1,2 S. Daul,3,* R. R. P . Singh,4 and D. J. Scalapino2

1

˚

VOLUME 89, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 2002

Intriguing hints from numerics

Bij S

i S j S i S j 2Sx i Sx j Sy i Sy j;

Pijkl S

i S j S k S l S i S j S k S l ;

H J X

hiji

Bij K X

hijkli

Pijkl; QMC study of 2D S=1/2 XY model with plaquette flip (partial ring exchange) First-order transition would be expected for superfluid (XY magnet) to VBS transition

6.5 7.0 7.5 8.0 8.5 9.0 K/J 0.00 0.05 0.10 0.15 0.20 0.25 ρs, 50*<MP

2>

ρs <Mp

2>

No discontinuities detected

Motivated re-examination of the field theory

VBS pattern for K/J = 10

Deconfined quantum criticality

τ

Graph:Senthil et al.

Motrunich and Vishwanath 2004 (+ earlier work in particle physics)

O(3) transition with suppressed topological defects in MC simulations

• changes universality class

Senthil, Vishwanath, Balents, Sachdev, Fisher (2004)

Topological defects may be “dangerously irrelevant” at the 2D Neel - VBS transition

• universality of defect suppressed O(3)
• topological defects relevant in VBS state only

Sz =

Z d2rd⌧  |(@µ iAµ)z↵|2 +s|z↵|2 +u(|z↵|2)2

• Z

 + 1 2e2 (✏µ⌫l@⌫Al)2

• Is the transition really continuous for N=2 (small N)?
• exotic aspects: emergent U(1) symmetry, two divergent lengths,…
• non-compact (defect-free) CP1 model
• large-N calculations for SU(N) CPN-1 theory

Continuous transition found for large N

• violation of Landau rule
• expected first-order transition between ordered states

= ⟨⃗ Si · ⃗ Sj⟩

The Heisenberg interaction is equivalent to a singlet projector

Cij = 1

4 − ⇤

Si · ⇤ Sj

VBS states from multi-spin interactions

• we can construct models with products of singlet projectors
• no frustration in the conventional sense (QMC can be used)
• correlated singlet projection reduces the antiferromagnetic order

+ all translations and rotations

(Sandvik, PRL 2007)

The “J-Q” model with two projectors is

H = −J

• ij⇥

Cij − Q

• ijkl⇥

CijCkl

• Has Néel-VBS transition, appears to be continuous
• Not a realistic microscopic model for materials
• “Designer Hamiltonian” for VBS physics and Néel-VBS transition

Use to test the deconfined quantum-criticality scenario

T>0 and T=0 QMC simulations

tr{e−βH} = X

n

βn n! hα|(H)n|αi

periodic time boundary conditions Finite-temperature SSE

• β = aL (a~1, for z=1 criticality)
• or increase β until T=0 convergence
• pen time boundaries capped by

valence bonds (2-spin singlets) Ground state projection Trial state can conserve relevant ground state quantum numbers (S=0, k=0,...)

X

αβ

fβfαhβ|(H)m|αi

m of order L*N

• check for T=0 convergence

0.02 0.04 0.06 0.08 0.1

J/Q

0.2 0.4 0.6 0.8 1

Ud

L = 32 L = 64 L = 128 L = 256 L = 512 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1

Us

Dimer order parameter (Dx,Dy)

Dx = 1 N

N

X

i=1

(−1)xiSi · Si+ˆ

x

Dy = 1 N

N

X

i=1

(−1)yiSi · Si+ˆ

y

⌅ M = 1 N

• i

(−1)xi+yi ⌅ Si

Staggered magnetization

Phase transition in the J-Q model

AFM VBS

Ud = 2 ✓ 1 1 2 hD4i hD2i2 ◆ Us = 5 2 ✓ 1 1 3 hM 4

z i

hM 2

z i2

◆

Binder cumulants: Us → 1, Ud → 0 in AFM phase Us → 0, Ud → 1 in VBS phase Behaviors of crossing points → exponents

First-order scenario: Prokofe’v, Svistunov, Kuklov, Troyer, Deng,… (2008-2013)

Jiang, Nyfeler, Chandrasekharan,Wiese (2008)

Jiang et al. (2008)

Linear divergence (first-order)?

Anomalous scaling of winding numbers W 2⇥ = W 2

x⇥ + W 2 y ⇥ + W 2 τ ⇥

= 2βρs + 4N β χ

z = 1, β ⇥ L ρs ⇥ L−1, χ ⇥ L−1 ⇤W 2⌅ = constant

Multiplicative log correction?

(Sandvik, PRL 2010)

Anomalous scaling behavior

1 1.5 2 2.5 1 1.5 2 2x 1/ν SU(2) 1 1.5 2 2.5 1 1.5 2 2x 1/ν SU(3) 0.5 1 1.5 2 2.5 16 32 64 128 256 0.5 1 1.5 2 2x 1/ν Lmax SU(4)

Square: 1/ν 2xm 2xΨ Honeycomb: 1/ν 2xm 2xΨ

Harada et al (PRB 2013)

• data collapse with different

sets of system sizes Drifts in exponents with system size

0.04 0.05 0.06 0.07

g*

1.47 1.48 1.49 1.50

R1*

0.05 0.1 0.15

1/L

1.0 1.5 2.0

1/ν∗

From Binder ratio

Shao, Guo, Sandvik (Science 2016)

• Binder ratio for (L,2L) pairs

ν = 0.45 ± 0.01

Competing scenarios for the phase transition

• I. The transition is weakly first-order
• anomalous scaling behaviors reflect “runaway RG flow”
• a critical point may exists outside accessible space

(in a fractal dimensionality or in the complex plane) Comment: The DQCP phenomenology would still be valid

• large length scale; almost deconfined spinons
• spin dynamics show good agreement with theory
• II. The transition is truly continuous
• anomalous scaling behaviors reflect two divergent length scales
• new physics beyond the original DQCP proposal

[Shao, Guo, Sandvik (Science 2016)]

Wang, Nahum, Metlitski, Xu, Senthil, PRX 2017 Sandvik, Kotov, Sushkov, PRL 2011 Suwa, Sen, Sandvik, PRB 2016 Ma et al., PRB 2018

• 4 gapless points, spinon continuum

Dynamical signature of fractionalization at a deconfined quantum critical point

Nvsen Ma,1 Guang-Yu Sun,1,2 Yi-Zhuang You,3,4 Cenke Xu,5 Ashvin Vishwanath,3 Anders W. Sandvik,1,6 and Zi Yang Meng1,7,8

PHYSICAL REVIEW B 98, 174421 (2018)

Editors’ Suggestion

HJQ = −J

• ⟨ij⟩

Pij + Sz

i Sz j

− Q

• ⟨ijklmn⟩

PijPklPmn

Planar J-Q model:

Spin structure factor S(q,𝜕)

Deconfinment manifest on large length scales close to the phase transition

Dynamic signatures of deconfined quantum criticality

AFM

VBS

Close to critical point: Good agreement with mean-field fermionic parton theory (𝜌-flux) =

↑ ↓

Si = 1 2f †

i σfi

HMF =

• i

i(f †

i+ˆ xfi + (−)xf † i+ˆ yfi) + H.c.

ϵk = 2(sin2(kx) + sin2(ky))1/2

The VBS order parameter

Dx = 1 N X

x,y

(1)xSx,y · Sx+1,y, Dy = 1 N X

x,y

(1)ySx,y · Sx,y+1,

Dimer order parameter

Dx Dy

Collect histograms P(Dx,Dy) with QMC

= +

Two possible types of order patterns distinguished by histograms

columnar plaquette

Finite-size fluctuations

• amplitude
• angular

Emergent U(1) symmetry of columnar VBS order

0.2 0.4 0.6 0.8 1

φ/2π

0.154 0.156 0.158 0.160 0.162 0.164 0.166

P(φ)

L=128 L=64 L=32

L = 12 L = 24 max

Strong columnar VBS when J/Q3=0

L = 64 L = 128 max

J-Q2 model with J/Q2=0

• weak columnar VBS
• very large angular fluctuations

Lou, Sandvik, Kawashima, PRB (2009), Sandvik, PRB (2012)

i j

Jx

i j

Jy

i j k l m n

Qx

i j k l m n

Qy

J-Q3 model Jx=Jy, Qx=Qy Realize stronger VBS order with J-Q3 model

Dangerously irrelevant perturbations

Cross-over from XY ordering to Zq ordering at length scale 𝝄’ DQC scenario has two divergent length scales on VBS side

• correlation length and emergent U(1) length
• due to dangerously-irrelevant perturbation which causes VBS
• known in many classical systems (e.g., 3D clock models)

ξ ∝ (g − gc)−ν ξ0 ∝ (g − gc)ν0

H = −J X

hiji

cos(Θi − Θj) − h X

i

cos qΘi q = 6

Jose, Kadanoff, Kirkpatrick, Nelson, PRB 1977

h is dangerously irrelevant

• does not change critical point
• changes ordered state

Fixpoints: P = paramagnet X = 3D XY critical point Y = XY symmetry breaking Q = Zq symmetry breaking

Okubo et al, PRB 2015

RG flows can be observed in MC simulations

MC simulations of classical 3D clock model

q = 6

H = −J X

hiji

cos(Θi − Θj)

Restricted to q clock angles

mx = 1 N

N

X

i=1

cos(Θi)

my = 1 N

N

X

i=1

sin(Θi)

Standard order parameter (mx,my) Probability distribution P(mx,my) shows cross-over from U(1) to Zq for T<Tc

Lou, Balents, Sandvik, PRL 2007

Can be quantified with “angular order parameter”: 𝝌q > 0 only if q-fold anisotropy Finite-size scaling of 𝝌q can be used to extract length scale 𝜊’ > 𝜊

φq = Z 2π dθ cos(qθ)P(θ)

→ global angle θ

Relevant field accessed through the Binder cumulant: Um = 2 hm4i hm2i2 Angular order parameter 𝝌q reflects the dangerously irrelevant field Entire RG flow can be explained by phenomenological scaling function with two relevant arguments:

φq = LyqΦq(δL1/ν, δL1/ν0

q)

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 4
• 2

2 4 6 8 10

• 10

3D, q=6 (a) L=32 L=48 L=64 L=96 L=128

Okubo et al. (PRB 2015) The exponent 𝜉’ can be directly extracted from 𝜒q when it is large

L = 2, 3, . . .

MC RG flows in the plane (Um,𝜒q)

• Shao, Guo, Sandvik, arXiv:1904.13640

0.02 0.04 0.06 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 G XY NG Z6 U φ6 T = 1.0 T = Tc T = 3.0

DQCP: In the field theory the VBS corresponds to condensation of topological defects (quadrupoled monopoles on square lattice)

Dx

Dx Dy Dy

Analogy with 3D clock models: The topological defects should be dangerously irrelevant

r

AF

U(1) SL VBS DQCP

Graph from Matthew Fisher

Fugacity of topological defects 𝞵4 MC RG flows for J-Q3 model

• Dx, Dy → angle 𝜾

work in progress….

• preliminary results

L = 4, 6, . . .

• 0.02

0.02 0.04 0.06 0.08 0.1 0.12 0.14

• 1
• 0.5

0.5 1 D4 UD-Um J/Q=0.0667 J/Q=0.2667 J/Q=0.3667 J/Q=0.4667 J/Q=0.5667 J/Q=0.6067 (J/Q)c=0.6667 J/Q=0.7667 J/Q=0.8667 J/Q=1.8667 J/Q=2.8667

J = Q3 model

Two divergent lengths tuned by one parameter: Finite-size scaling of some quantity A. Thermodynamic limit: A ∝ δκ

A(δ, L) = L−κ/νf(δL1/ν, δL1/ν0)

Conventional scenario

f(δL1/ν, δL1/ν0) → (δL1/ν)κ

When L→∞: Alternative scenario

A(δ, L) = L−κ/ν0f(δL1/ν, δL1/ν0) f(δL1/ν, δL1/ν0) → (δL1/ν0)κ

When L→∞: The first scenario has so far been implicitly assumed

• can the drifts be explained using 𝜉 ≈ 0.45, 𝜉’ ≈ 0.98, 𝜉/𝜉’ ≈ 0.46 ?

Two-length scaling hypothesis [Shao, Guo, Sandvik (Science 2016)]

Example: Spin stiffness: κ=ν(z+d-2). At criticality:

ρs ∝ L−(z+d−2) ρs ∝ L−(z+d−2)ν/ν0

• r

At the critical point: Equivalent view:

A = ξ−κ/ν

Replace 𝜊 by L: A = L−κ/ν

A = L−κ/ν0

• r, replace 𝜊’ by L, 𝜊 by L𝜉/𝜉’:

ξ ∼ δν, ξ0 ∼ δν0

Some new preliminary results supporting a continuous phase transition removed. Will be published soon.

Main points

AFM VBS

Large-scale QMC studies of the 2D S=1/2 J-Q model

• “designer Hamiltonian” for deconfined quantum criticality

Emergent U(1) symmetry of the near-critical VBS

• associated with a length scale 𝜊’, diverging faster than 𝜊

Anomalous finite-size scaling in J-Q model

• proposal: fundamental role of the second length scale
• phenomenological scaling function can explain all results

The second length scale must exist also in the AFM state!

• Candidate: the (𝜌,0) triplet mode may scale as 𝜺-𝜉’

Three related recent papers….

H = J

• i,j⇥

Si · Sj

2 3 4 5 ω/J 50 100 150 200 S(q,ω) Experiment SAC(L=48) 2 3 4 5 ω/J

Excitation anomaly in the S=1/2 Heisenberg model

• quantum fluctuations reduce, but do not destroy order
• low-energy excitations are spin waves (magnons)

15 Energy (meV) 10 5

x

3 2 1 ( /2, /2) (0, 0) (0, 0)

a

/J ω ( , 0) π ( , 0) π ( , 0) π ( , ) π π ( , ) π π ( /2, /2) π π ( / 2 , / 2 ) π π ( /2, /2) π π

High-energy (~J) excitations are non-trivial: signs of spinon deconfinement

PHYSICAL REVIEW X 7, 041072 (2017) Nearly Deconfined Spinon Excitations in the Square-Lattice Spin-1=2 Heisenberg Antiferromagnet

Hui Shao,1,2,* Yan Qi Qin,3,4 Sylvain Capponi,6,2 Stefano Chesi,1 Zi Yang Meng,3,5,† and Anders W. Sandvik2,1,‡

1

ARTICLES

PUBLISHEDONLINE:DECEMBER2014|DOI:10.1038/NPHYS3172

• B. Dalla Piazza1*, M. Mourigal1,2,3*, N. B. Christensen4,5, G. J. Nilsen1,6, P. Tregenna-Piggott5,
• T. G. Perring7, M. Enderle2, D. F. McMorrow8, D. A. Ivanov9,10 and H. M. Rønnow1,11
• Fractional excitations in the square-lattice

quantum antiferromagnet

• quantum antiferromagnet
• n CuðDCOOÞ2 · 4D2O,

interpreted as deconfined spinons,

Reduced (𝜌,0) excitation energy + large continuum; precursor to DQCP

• J-Q model demonstrates mechanism of deconfinement

• Appears that there is an O(4) point (the transition point)
• No sign of conventional AFM, PSS coexistence
• Exact emergent symmetry or only up to some length scale?

J Q (b)

AFM-PSS (plaquette-singlet solid)

• Q terms only on every second plaquettes

H = J X

hiji

Pij Q X

ijkl2⇤0

(PijPkl + PikPjl),

Pij = 1

4 − Si · Sj

QMC simulations show first-order transition with emergent O(4) symmetry

• PSS [Ising] and AFM [O(3)] order parameters combine to 4D vector

Distribution P(|mz|,|mp|)

• mz = 1 AFM component
• mp = PSS order param.

Checker-board J-Q (CBJQ) model

J Q (b)

• B. Zhao, P. Weinberg, A. W. Sandvik

Nature Physics 2019

Disordered J-Q model: ‘random-singlet’ state

Lu Liu, Hui Shao, Yu-Cheng Lin, Wenan Guo, AWS, PRX 8, 041040 (2018)

Random J-Q model Quantum phase transition between antiferromagnet and ‘disordered critical spin liquid’ (1/r2 spin correlations) Quantum-critical T>0 scaling Role of domain walls in mediating spinon-spinon interactions Possible realization in random J1-J2 material Sr2CuTe1-xWxO6