Computational Approaches to Quantum Many-Body Problems ISSP , U of Tokyo, July 17, 2019 Deconfined Quantum Criticality in the 2D J-Q model Anders W Sandvik Boston University and Institute of Physics, Chinese Academy of Sciences, Beijing Bowen Zhao Boston University) Hui Shao (Beijing Normal University) Wenan Guo (Beijing Normal University)
Main points Large-scale QMC studies of the 2D S=1/2 J-Q model - “designer Hamiltonian” for deconfined quantum criticality VBS AFM 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
2D quantum antiferromagnets Quantum phase transitions out of the Neel state Simplest case: Dimerized Heisenberg models X H = J ij S i · S j strong interactions h ij i weak interactions S = 1 / 2 spins Singlet formation on strong bonds ➙ Néel - disordered transition Ground state (T=0) phases ∆ = spin gap s Transition in 3D O(3) Universality class
More complex non-magnetic states; 1 spin per unit cell � H = J S i · S j g × · · · + � i , j ⇥ • non-trivial non-magnetic ground states are possible, e.g., ➡ resonating valence-bond (RVB) spin liquid ➡ valence-bond solid (VBS) Non-magnetic states often have natural descriptions with valence bonds � √ = ( ↑ i ↓ j − ↓ i ↑ j ) / 2 j i 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)
Field theory description; brief summary Standard low-energy theory of quantum antiferromagnet Z 2 [ c 2 ( ∂ r φ ) 2 + ( ∂ τ φ ) 2 + m 0 φ 2 + u o ( φ 4 )] d d rd τ 1 S = 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 The VBS state corresponds to a certain condensation of topological defects - requires a description beyond 𝝌 4 theory Graph:Senthil et al. Neel vector described by spinors z; φ = z ∗ α σ αβ z β - coupled to U(1) gauge field where hedgehogs correspond to monopoles - VBS on square lattice arises from condensation of quadrupled monopoles Murthy & Sachdev 1991, Read & Sachdev 1991 Nature of the Neel - VBS transition remained unknown…
Intriguing hints from numerics V OLUME 89, N UMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 D ECEMBER 2002 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. Scalapino 2 ˚ 1 QMC study of 2D S=1/2 XY model with plaquette flip (partial ring exchange) X X H � � J B ij � K P ijkl ; VBS pattern for K/J = 10 h ij i h ijkl i j � S y i S y B ij � S � i S � j � S � i S � j � 2 � S x i S x j � ; 0.25 ρ s 2 > <M p P ijkl � S � i S � j S � k S � l � S � i S � j S � k S � 0.20 l ; 2 > ρ s , 50*<M P 0.15 First-order transition would be expected for superfluid (XY magnet) 0.10 to VBS transition 0.05 No discontinuities detected 0.00 6.5 7.0 7.5 8.0 8.5 9.0 K/J Motivated re-examination of the field theory
Deconfined quantum criticality O(3) transition with suppressed topological defects in MC simulations - changes universality class Motrunich and Vishwanath 2004 (+ earlier work in particle physics) Topological defects may be “dangerously irrelevant” at τ the 2D Neel - VBS transition - universality of defect suppressed O(3) - topological defects relevant in VBS state only Z Graph:Senthil et al. Senthil, Vishwanath, Balents, Sachdev, Fisher (2004) + 1 � Z | ( @ µ � iA µ ) z ↵ | 2 + s | z ↵ | 2 + u ( | z ↵ | 2 ) 2 ( ✏ µ ⌫ l @ ⌫ A l ) 2 d 2 r d ⌧ S z = 2 e 2 0 = ⟨ ⃗ S i · ⃗ � S j ⟩ • non-compact (defect-free) CP 1 model • large-N calculations for SU(N) CP N-1 theory Continuous transition found for large N - violation of Landau rule - expected first-order transition between ordered states Is the transition really continuous for N=2 (small N)? - exotic aspects: emergent U(1) symmetry, two divergent lengths,…
VBS states from multi-spin interactions The Heisenberg interaction is equivalent to a singlet projector 4 − ⇤ S i · ⇤ C ij = 1 S j • 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 The “J-Q” model with two projectors is (Sandvik, PRL 2007) � � H = − J C ij − Q C ij C kl � ij ⇥ � ijkl ⇥ • 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 Ground state projection Finite-temperature SSE X f β f α h β | ( � H ) m | α i β n X tr { e − β H } = n ! h α | ( � H ) n | α i αβ n open time boundaries capped by valence bonds (2-spin singlets) Trial state can conserve relevant periodic time boundary conditions ground state quantum numbers - β = aL (a~1, for z=1 criticality) (S=0, k=0,...) - or increase β until T=0 convergence m of order L*N - check for T=0 convergence
Phase transition in the J-Q model VBS AFM Staggered magnetization M = 1 ( − 1) x i + y i ⌅ ⌅ � S i N i Dimer order parameter (D x ,D y ) 1 N 0.8 D x = 1 X ( − 1) x i S i · S i +ˆ x 0.6 N U s i =1 0.4 N D y = 1 X ( − 1) y i S i · S i +ˆ 0.2 y N i =1 0 0 0.02 0.04 0.06 0.08 0.1 Binder cumulants: 1 L = 32 0.8 h M 4 ✓ z i ◆ U s = 5 1 � 1 L = 64 0.6 h M 2 z i 2 L = 128 2 3 U d L = 256 0.4 h D 4 i ✓ ◆ 1 � 1 L = 512 U d = 2 0.2 h D 2 i 2 2 0 0 0.02 0.04 0.08 0.1 0.06 U s → 1, U d → 0 in AFM phase J/Q U s → 0, U d → 1 in VBS phase Behaviors of crossing points → exponents
Anomalous scaling behavior First-order scenario: Prokofe’v, Svistunov, Kuklov, Troyer, Deng,… (2008-2013) Jiang, Nyfeler, Chandrasekharan,Wiese (2008) Anomalous scaling of winding numbers z = 1 , β ⇥ L � � W 2 ⇥ � W 2 x ⇥ + � W 2 y ⇥ + � W 2 τ ⇥ = ρ s ⇥ L − 1 , χ ⇥ L − 1 2 βρ s + 4 N = β χ � ⇤ W 2 ⌅ = constant Linear divergence (first-order)? Multiplicative log correction? (Sandvik, PRL 2010) Jiang et al. (2008)
Drifts in exponents with system size Harada et al (PRB 2013) Shao, Guo, Sandvik (Science 2016) - data collapse with different - Binder ratio for (L,2L) pairs sets of system sizes From Binder ratio 0.07 0.06 2.5 SU(2) 2 g* 2 1/ ν 0.05 2 x 1.5 1.5 1 0.04 1 1.50 2.5 SU(3) 2 1.49 R 1 * 2 1/ ν 2 x 1.5 1.48 1.5 1 1 1.47 2.0 2.5 SU(4) 2 2 1/ ν∗ 1.5 1.5 1/ ν 2 x 1.5 1 1 Square: 1/ ν Honeycomb: 1/ ν 1.0 2 x m 2 x m 0.5 0.5 2 x Ψ 2 x Ψ 0 0.05 0.1 0.15 0 0 1/L 16 32 64 128 256 ν = 0 . 45 ± 0 . 01 L max
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) Wang, Nahum, Metlitski, Xu, Senthil, PRX 2017 Comment: The DQCP phenomenology would still be valid - large length scale; almost deconfined spinons - spin dynamics show good agreement with theory - 4 gapless points, spinon continuum Sandvik, Kotov, Sushkov, PRL 2011 Suwa, Sen, Sandvik, PRB 2016 Ma et al., PRB 2018 II. The transition is truly continuous [Shao, Guo, Sandvik (Science 2016)] - anomalous scaling behaviors reflect two divergent length scales - new physics beyond the original DQCP proposal
Dynamic signatures of deconfined quantum criticality PHYSICAL REVIEW B 98 , 174421 (2018) Editors’ Suggestion 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 Meng 1,7,8 � − Q � � Planar J-Q model: � P ij + � S z i S z H JQ = − J P ij P kl P mn j ⟨ ij ⟩ ⟨ ijklmn ⟩ Spin structure factor S(q, 𝜕 ) VBS AFM Close to critical point: Good agreement with mean-field fermionic parton theory ( 𝜌 -flux) = ↑ ↓ ϵ k = 2 ( sin 2 ( k x ) + sin 2 ( k y ) ) 1 / 2 S i = 1 2 f † i σ f i � i( f † x f i + ( − ) x f † H MF = y f i ) + H.c. i + ˆ i + ˆ i Deconfinment manifest on large length scales close to the phase transition
The VBS order parameter Dimer order parameter D x = 1 X ( � 1) x S x,y · S x +1 ,y , N x,y Dy D y = 1 X ( � 1) y S x,y · S x,y +1 , N x,y Dx Collect histograms P(D x ,D y ) with QMC Two possible types of order patterns distinguished by histograms plaquette columnar = + Finite-size fluctuations - amplitude - angular
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