Advances in Diagrammatic Monte Carlo Methods for Quantum Field Theory Calculations in Nuclear, Particle, and Condensed Matter Physics, Trento, October 5-9, 2015

Exploring Quantum Many-Body Physics with Spin Models Anders W Sandvik, Boston University

Simons Foundation: Advancing Research in Basic Science and Mathematics

Advancing Research in Basic Science and Mathematics SIMONS SOCIETY OF FELLOWS

MATHEMATICS & PHYSICAL SCIENCES

Tuesday, October 6, 15

LIFE SCIENCES

AUTIS

1

Outline and main points Quantum spin models: At the forefront of quantum many-body physics throughout the ages! 1D Heisenberg chain + frustration → dimerization (VBS) 2D Heisenberg + coupled-dimer models Served as test beds and motivations for efficient QMC methods + interesting physics (Relatively) new and (very) exciting QMC opportunities

Deconfined quantum criticality

2D continuous Néel - VBS transition QMC-accessible with J-Q spin models

Tuesday, October 6, 15

2

S=1/2 Heisenberg chain Early example of quantum-many body research • influenced development of methods and understanding of N X collective quantum states H=J

i=1

~i · S ~i+1 S

- Bethe’s Ansatz for exact solution (1931); E0 (Hulthen 1938)

- Later work gave dispersion E0(k) (des Closeaux, Person 1962) - Excitations are deconfined spinons (Faddeev, Takhtajan 1981) - Thermodynamics from small-N numerics (Bonner, Fisher 1964) - Correlations from bosonization; Luttinger Liquid (Luther, Peschel 1975) 1/2 ln (r/r0 ) r ⇧ ⇧ C(r) = ⌅Si · Si+r ⇧ ⇤ ( 1) r - Dynamics from numerical BA (Caux et al.,...), QMC, DMRG,....

- Still open issues, active area of research (field theory, numerics,...) - Several experimental realizations

Add other interactions → quantum phase transitions Tuesday, October 6, 15

3

H = J1

N X

N X Frustrated Heisenberg chain ~i · S ~i+1 + J2 ~i · S ~i+2 S S

N N X X i=1~ ~ i=1 ~ ~i+2 H = J1 Si · Si+1 + J2 Si · S i=1

i=1

g = J2 /J1

VBS order the degenerate ground state orderfor for g>0.2411... g>0.2411... doubly • VBS - gap to spin excitations; decaying spin correlations ground state is doublyexponentially degenerate r r/ ⌅ ⌅ C(r) = ⌅ S · S ⇧ ⇤ ( 1) e - gap to spini excitations; exponentially decaying spin correlations i+r -5.5

gcross

- singlet-product state is exact for g=1/2 (Majumdar-Gosh 1969) S=0, k=0 S=0, k=π S=1, k=π

- singlet-product state is exact for g=1/2 (Majumdar-Gosh point) 0.243

-6.0

r/

E

r ⌅ ⌅ C(r) = ⌅Si · Si+r ⇧ ⇤ ( 1) e -6.5

-7.0 0.0

0.242

N = 16 0.2

g

0.4

(b)

(a) 0.6

0.241 0

0.002

0.004

1/N

0.006

Excitations change - level crossing of first and second excited state at gc

2

(Nomura, Okamoto 1992,... Eggert 1996,... Sandvik 2010) - 1/N size shift predicted by conformal field theory (Affleck, Eggert,...)

FIGURE 38. (a) The three lowest energies for a 16-site chain as a function of the coupling ratio g. The crossing point gcross of the k = π singlet and triplet levels can be used as a size-dependent definition for the 2 critical (dimerization) coupling gc . The vertical line indicates the exactly solvable Majumdar-Ghosh point, at which the singlet states are degenerate. (b) The crossing point versus 1/N 2 . The curve is a polynomial fit in 1/N (without a ∝ 1/N term, as the leading correction is ∝ 1/N 2 ).

2D systems; larger lattices needed → QMC in the case of the triplet excitation of the Heisenberg chain the gap closes as 1/N, and the Tuesday, 6, 15 lowestOctober singlet also approaches the ground state as 1/N (as shown in Fig. 33). This being

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QMC algorithms for quantum spins From operators to numbers

P H Ac W c Tr{Ae } c P hAi = ! Tr{e H } Wc

β

τ

- Trotter slicing, discrete imaginary time; Δτ 0 world line methods (Suzuki 1971,...) FIGURE 54. A 1D world line configuration based on the checkerboard decomposition w approximation. Kinetic jumps of the bosons are - Taylor expansion; stochastic Trotter series expansion (SSE) p i (or = 1flips2 of 3a pair 4 of 5 ↑6and7 ↓ spins) 8 s(p across the shaded squares (plaquettes). A time slice of width Δτ consists of two consecutiv The six isolated plaquettes shown to 11 the right correspond to the non-zero matrix el4 (Handscomb 1961,... Sandvik, quettes. Kurkijärvi 1991,...) in the case of a spin model with Heisenberg interactions (for world lines and empty sites 10 0 to ↑ and ↓ spins, respectively) are given by Eq. (244). - Continuous time (take Δτ→0 limit before programming) 9 9 13 8 (Beard, Wiese, 1996, Prokof’ev et al. 1996,...) (kinetic jumps) are allowed only on the7 shaded plaquettes in Fig. 54. More6 - From local updates to loops, worms, directed loops.... 0 “loop” and “directed loop” updates, in6which large segments of several wor 0 be moved simultaneously, are used in 5modern algorithms [31, 191, 33] (wh (Evertz et al. 1993, Beard, Wiese, 1996, 4 discuss in detail below in the context of4 the stochastic series expansion meth Prokof’ev et al. 1996, Sandvik, Syljuåsen 2002) 13 3 Application to the Heisenberg model. It is useful to consider a particu0 2 - Use of valence-bond basis inof the path weights in the Suzuki-Trotter 1 approach. Let us compute the plaq9 elements for the antiferromagnetic Heisenberg interaction; Hi,i+1 = Si · Si+1 ground-state projection 14 0 the boson occupation numbers in (243) are replaced by spin states ↑ and (Liang 1989, Sandvik 2005,...) consider the world lines formingFIGURE between61.the A↑ linked-vertex spins (and note we cou SSEthat configuratio

world lines for the ↓ spins in pictures such asalong Fig.with 54; the they occupy all sites “orientations”, corresponding opera by ↑ world lines and cross those linesand at each diagonal segment). The↔calc flipped, operators are changed, diagonal of 4 ofweanjust 5 the change operator visited twice). vertex involves straight-forward algebra and list results forEvery the six al acted upon by any operator (here the one at i = 1) zero) matrix elements;

Approximation-free ground states for 10 ~ 10 spins Tuesday, October 6, 15

−Δτ Hi j

−Δτ Hi j

−Δτ /4

5

Ground-state projector QMC with valence bonds Liang PRB 1989; Sandvik PRL 2005, Sandvik, Evertz PRB 2010

Poject valence bonds with Hm or exp(-βH)

A

Expectation values (correlation functions) computed using transition graphs

|

|Vl

|Vr

|

Vl |Vr ⇥

Put the spins back in a way compatible with the valence bonds (singlets) and sample in a combined space of spins and bonds Loop updates similar to those in finite-T methods (world-line and stochastic series expansion methods) • “measure” using valence bonds (as before)

Tuesday, October 6, 15

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2D Heisenberg model Sublattice magnetization N X 1 m ~s= N i=1

~ i Si ,

H = J i

= ( 1)

xi +yi

Long-range order: > 0 for N→∞ Quantum Monte Carlo - finite-size calculations - no approximations - extrapolation to infinite size

i,j⇥

Si · Sj

L⨉L lattices up to 256⨉256, T=0

Reger & Young (world-line) 1988

ms = 0.30(2) 60 % of classical value AWS & HG Evertz (valence bonds) 2010

ms = 0.30743(1) Quantum phase transitions out of the Néel state Tuesday, October 6, 15

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Dimerized S=1/2 Heisenberg models • every spin belongs to a dimer (strongly-coupled pair) • many possibilities, e.g., bilayer, dimerized single layer strong interactions weak interactions

Singlet formation on strong bonds ➙ Néel - disordered transition Ground state (T=0) phases = spin gap s

3D classical Heisenberg (O3) universality class confirmed for bilayer, columnar dimers, plaquette pattern,.... Tuesday, October 6, 15

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Are there exceptions to the 3D O(3) Universality? mer (strongly-coupled pair) P H Y Wenzel, S I C ABogacz, L RJanke E V IPRL EW L E T T E R S 008) 2008 P H Y S I C A L R E V I E W L E T T E RS layer, dimerized single layer

EW LETTERS

week ending 19 SEPTEMBER 2008 2.30

0.52 0.48

0.36

2.52 2.50 2.53 2.512.54 α

α

ntum spin (S ¼ 1=2) degrees of spin (Sneighbor ¼ 1=2) degrees of − 2.6 eThe withquantum different nearest are different nearest neighbor ck).lattice (b) −Similar the plaquette 2.8withfor β/ν = 0.545(4) mation. n and thick). (b) Similar for the plaquette − 3.0

2.51

0.44 0.40

2.30

bilayer transition bonds − 1.8➙ Néel - disordered ladder (b) − 2.0 (b) 2.56 pl s jjp − 2.2 2.54 ualization of = thespin J-J 0 model on β/ν 0 = 0.515(4) gap (a) Visualization of the J-J model on − 2.4 2.50

ln( m sz )

0.56

Staggered2.50 dimers (a) 2.45 strong interactions Deviations from 2.45 the expected 2.40 2.40 weakwere interactions exponents observed 2.35 2.35 Q2

J

(a)

ξy /L

J

2.50

ρs ξy Q1 Q2

0.32 2.522.50

α

2.56 2.54

2.52 2.50 2.48

α

J

Q2

J

week 19 SEPTE

2.51 2.53

2.52 2.54 α

ρs ξy Q1 Q2

2.52 2.50

(a) (c) umer formation. 2.48 − 3.2 2.46 0.02 0.04 0.06 0.08 0.10 0.12 2.0 2.5 3.0 3.5 4.0 4.5 (c) 1/L L) were performed to furtherln( op2.46 0.02 0.04 0 data analysis. − 3.5 ighting were performed to further opFIG. 2 strong (color online). (a) The Binder parameter Q2 (b) New universality class or scaling corrections? quantum phase transition, we − 3.0 correlation length !y =L for various lattice sizes fro ics and data analysis. FIG. 2 (color online). (a) The B observables starting from the Tuesday, October 6, 15

9

8

)

g s -

paramagnetic ground state for J % ' J , without symmetry ! breaking of any kind and dominated by intracell singlets # u0 $ 2 %2 1" 2 ! % (Fig. 1). In contrast, for J ≈ J , a semiclassical N´eel state with d 2 2 2 ϕα S = d rdτ c (∇ϕα ) + (∂τ ϕα ) + m0 ϕα + broken SU(2) symmetry is realized. The critical properties of 2 24 % where t FRITZ, DORETTO, WESSEL, WENZEL, BURDIN, AND VOJTA the resulting QPT as function of J /J are the subject of this (1) PHYSICAL REVIEW B 83, 174416 (2011) paper. all mo

Two classes of dimerized models

magnet A. Bond-operator representation in standardcriticality notation. in Here, ϕα (! x ,τ )antiferromagnets is a three-component Cubic interactions and quantum dimerized differen The QPT field between a non-symmetry-breaking paramagnetic An efficient microscopic description of the vector excitations order-parameter describing magnetic fluctuations 0 to the of coupled-dimer models is provided by the bond-operator and a collinear antiferromagnetic phase in an insulating magnet 1 2 3,4 5 6 7 !andwith h. L. Fritz, R. L. Doretto, S. Wenzel, Burdin,Q, M. Vojta nearS. Wessel, the ordering wave S.vector α = x,y,z. For tofield lattic n 1 with SU(2) symmetry is typically described by a quantum Institut f¨ur Theoretische Physik, Universit¨at the zu K¨oaction ln, Z¨ulpicher Straße 77, D-50937 K¨oln, Germany (appropriate simplicity, has been written for real ϕ α ground 4 it 2 (a) theory ofPaulista, the φ 01140-070 type, Instituto de(b)F´ısica Te´orica, for Universidade Estadual S˜ a o Paulo, SP, Brazil ! and isotropictheory Standard time-reversal invariantlow-energy Q) real space; the g exampl 3 Institut f¨ u r Theoretische Physik, Universit¨ a t Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany y, generalization to !other cases is straightforward. The critical J as%2well 4 J " $ # u 1 Institut f¨ u r Theoretische Festk¨ o rperphysik, RWTH Aachen, Otto-Blumenthal-Strasse 26, D-52056 Aachen, Germany 0 h d 2 ! 2 2 2 2 J´ J´ behavior of model (1) is known to be of standard (d + 1)c ϕ S = d + rdτ ( ∇ϕ ) + (∂ ϕ ) + m ϕ 5 α τ α 0 Fig. 1. ´ α α Institute of Theoretical Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland o 2 24 6 O(3)Universit´ universality. Quit Condensed Matter Theory Group, dimensional LOMA, UMR 5798, e de Bordeaux I, F-33405 Talence, France (1) 14–18 7 wecubic show that the spatially anisotropic cubic term Institut f¨ur Theoretische Physik,Below Technische Universit¨ atterm Dresden, D-01062 Dresden, Germany parama also appears (Received 26 January 2011; revised manuscript received 22 March 2011; published 6 May 2011) breakin in standard! notation. Here, ϕα (! x ,τ ) is a three-component (Fig. 1) (c) (d) d In certain Mott-insulating dimerized antiferromagnets, triplet excitations of the paramagnetic phase display I S = iγ d rdτ ϕ ! · (∂ ϕ ! × ∂ ϕ), ! (2) vector order-parameter field magnetic fluctuations 3 0 x describing τ n, broken both three-particle and four-particle interactions. When such a magnet undergoes a quantum !phase transition near the ordering wave vector Q, with α = x,y,z. For J into a magnetically ordered state, the three-particle interaction becomes part of the critical theory provided that the resu c simplicity, theaisspace action has been written forinreal ϕlowα (appropriate -a unless particular reflection the J´lattice ordering wave vector is zero. One microscopic example the staggered-dimer antiferromagnet on J´ with x being particular direction, appears the paper. r ! for time-reversal invariant Q) and isotropic real space; the J the square lattice, for which deviations from O(3) have reported in numerical studies. Using energy fielduniversality theory for 2Dbeen coupled-dimer magnets belonging symmetry of dimers is present r, generalization to other is straightforward. both symmetry arguments and microscopic we show that a nontrivial cubic term arises inbears the The critical B, to classcalculations, B. This cubic interaction ofcases critical fluctuations relevant order-parameter quantum field theory, and we assess itsof consequences a combination analytical behavior model (1)using is known to and beofof standard (d + 1)e some superficial similarity with Berry-phase windingFIG. 1. (Color online) Two-dimensional coupled-dimer magnets ul The cubic and numerical finite-temperature quantum Monte Carlo data for the staggered-dimer term methods. leadsWe toalso a present small correction exponent ω dimensional O(3) universality. An considered in this paper. In all panels, thick (thin) bonds refer to number terms, to be discussed below, however, its prefactor γ 0 c 14–18 antiferromagnet which(open) complement recently spins published results. The canthat be consistently interpreted in termscubic of Belowwith wedata show the spatially anisotropic termof coup circles represent Heisenberg couplings J % (J ), solid slow convergence of leading exponents size is not quantized and the field ϕ is not restricted to unit length. c critical exponents identical to thatconfigurations of the standard dimer. In addition, the singlet in O(3) universality class, but with anomalously large corrections S!i1 (S!i2 ) of each d % ! thatinthe detailed analysis suggests term S3 in J areOur shown. the paramagnetic groundWe states realized forcubic J ' interaction to scaling. argue that the of critical triplons, although irrelevant two cubic spatial dimensions, rectangular lattices (Ld x=to≠2scaling L et al. 2013) h -(a)using y) helps (Jiang 2012, Todo d Staggered-dimer, (b) columnar-dimer, (c) Herringbone-dimer, and space dimensions is irrelevant in the RG sense, albeit S = iγ d rdτ ϕ ! · (∂ ϕ ! × ∂τ ϕ), ! is responsible for the leading corrections due to its small scaling dimension. 3 0 x (a) (2) s (d) bilayer Heisenberg model on the square lattice. The QPTs to with a small scaling dimension. It constitutes the leading % m, /J ) = 2.5196(2) the antiferromagnetic phases are located at (a) (J c DOI: 10.1103/PhysRevB.83.174416 PACS number(s): 75.30.Kz, 75.50.Ee, 75.10.Jm, 05.70.Jk k irrelevant operator at the critical fixed point. Consequently, ) = 1.9096(2) (Refs. 9, 8, and 24), (Refs. 9, 8, 12, and 13), (b) (J % /J c Critical properties are now well understood with x being a particular space direction, appears in the low% % C (Ref. 6). (c) (J /J )c = 2.4980(3) (Ref. 10), (d) (J /J )c = 2.5220(1) the asymptotic critical behavior is of O(3) type, but with belonging magnets s From the analysis in this paper, we conclude that the QPT of models energy field theory for 2D coupled-dimer 11–13 I. INTRODUCTION focusing exclusively on10 of the staggered-dimer model, anomalous corrections to scaling. We show that this scenario Tuesday, October 6, 15 s (a) and (c) belong to class B, while that of (b) and (d) belong to class A. Overview of results

More complex non-magnetic states; systems with 1 spin per unit cell

H = J i,j⇥

Si · Sj + g ⇥ · · ·

• 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

⌅ ⇤i ⇥j )/ 2

The basis including bonds of all lengths is overcomplete in the singlet sector

How can we achieve such ground states? Tuesday, October 6, 15

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Frustrated 2D Heisenberg model

H= i,j⇥

= J1

⌅i · S ⌅j Jij S

= J2

g = J2 /J1

• Ground states for small and large g are well understood ‣ Standard Néel order up to g≈0.45; collinear magnetic order for g>0.6

0

g < 0.45

0.45

g < 0.6

g > 0.6

• A non-magnetic state exists between the magnetic phases ‣ Most likely a VBS (what kind? Columnar or plaquette?) ‣ Some calculations suggest spin liquid 2D frustrated models are challenging: QMC sign problems Tuesday, October 6, 15

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VBS states from multi-spin interactions Sandvik, PRL 2007

The Heisenberg interaction is equivalent to a singlet-projector

Cij =

⇤i · S ⇤j S

1 4

• 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

H=

J

Cij ij⇥

Q

Cij Ckl 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 Testing the deconfined quantum-criticality scenario of Senthil et al. Tuesday, October 6, 15

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carrying S = 1 to the gauge invarianc A µ is unrelated electromagnetic butmeans is an internal order parameter.field, This that the two order-pa N´ebetween elquantum state,ground expressing the spin-wave flu kfield state of H has B T /⇢ s .VBS 0 also obs states and “deconfined” criticality the physical that conveniently describes the couplings the spin quite distinct fields will have long-ranged “statistical” interactio and A is a matter of choice, and the ab µ is, it is useful to have an alternative description pattern of spin polarizatio Read, Sachdev (1989),....,Senthil, Vishwanath, Balents, Sachdev, Fisher (2004) excitations of the As we have noted above, in the transformation connected to antiferromagnet. each other. Consequently will no local theor field 8 there can serve usbe equally well. The dis tates above the N´eel For thein terms equation N´ eel state, expressing theordered spin-wavestate. fluctuations of z ↵ (1) by approaches appears whenfields we move outn sand of more exH = J S · S + g ⇥ · · · includes only the two order-parameter (but i j A µ isdescription a matter of choice, the above theory for the vector rnative is, in and a sense, a purely quantum critical points into other pha i,j⇥ nfield principle: in can serve equally distinction between that the two fields). It The is difficulties force the necessit se: it8 does notusalter anywell. of thethese low-energy h S = N⌧) in some of these phases,where the emergent A= j i⌘(r, 1 co Neel-VBS transition inwe 2Dmove approaches appears when out of theory the N´eel state across ons resonate alternate description which is an conveniently provided and yields an identical low-temperature !i · S !role optional, but essentialimportant characterization = ! S j" • generically continuous quantum critical points into other phases (as we will see later): esulting statein terms the phase. where As we did for S , we can write spinon degrees of freedom. 8 hen expressed of k T /⇢ . The key step r is the position of theory of the a B s i • violating the “Landau rule” in some of these phases, the emergent A µ gauge field is no longer forzz ↵ and A µ by the constraints ofFig. symm ractional spin equation (6) can ctor field 8 in terms of an S = 1 / 2 complex The spinon fields defined in Eq. (1.5) have ordering pattern in 1b " demanding 1st-order transition optional, but an essential characterization of thewhich ‘quantum order’ of now yields or w ↵diVerence . We will ethe↵ Description ="# by most important “gauge” redundancy. Specifically the local phase ron phase. As wewith did spinor for S8 ,field we can write the quantum field Ztheory Section IID. N 2 2 1 olution of the for z(2-component and A by the constraints of symmetry and gauge invariance, two orthogonal vectors complex vector) ↵ µ Sz = i$$r,d%%r d⌧ |(@ µ iA µ )z ↵ | + ⇤ which now gauge redundancy: z of ground states e 8 =yields z↵ ↵ z (3)z →manifold r primary exB. COUPLED-DIMER AN  Z an eVective action for N 1 1,2 nd a valence 2 2 2 2 2 2 This spin model ) a .gaug +and2 (✏ leaves the Néel vector invariant hence S = d r d ⌧ |(@ iA )z | + s|z | + u(|z | ) µ⌫l @ ⌫ Ais l z µ µ ↵ ↵ ↵ Minimiz 2ions Paulibetween matrices. Note that this mapping from the hamiltonian. 2e0 antiferromagnet 2 2 % is the imaginary time coordina of freedom. Here fi = N ordered state has N . WeSec. can VIII). make a space-time-dependent change 2 1 dashed lines to t ee 1 For brevity, we have now used a ‘relativis 1 • non-compact CP action A is a U(1) symmetric gauge 2 fieldare coupled to a U$1% gauge field a &$r the spinons in the hamiltonian, but the field ✓(x, ⌧) (✏ @ A ) . (5) + square-lattice and scaled away the spin-wave velocity vm; µ⌫ ⌫ l l 2 wn -to be obproposed as2ecritical theory Neel and VBS states 0 will & ' , . . . tocan represent t use separating the Greek indices the ,analogue of theundersto spinor be describes VBS stateused when additional terms are added i✓ VOL 4central MARCH 2008 www.nature.com/ nature physics For -brevity, we have now a ‘relativistically’ invariant notation, exchange interacw thesis—subs , %). Our introduce another spinor z ↵ ! e z ↵ space-time indices x , y(4) N-1 theory • SU(N) generalization: large-N calculations for CP ce bond solid and scaled away the spin-wave velocityof v ; the values of the J /g .the A numb by a variety arguments to couplings follow—isofthat criti aks spin rotanged. All physical properties therefore for must the Néel-VBS transition is just theNsim 1+i thetheory transition continuous for N=2 (small N)? MARCH 2008 www.nature.com/naturephysics nature physics VOL 4 Is attice transla2 nder equation (4), and so the quantum rd% Lz, and tinuum action Sz = &dfield Tuesday, October 6, 15

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Critical behavior of the J-QPHYSICAL modelREVIEW B 85, 134407 (2012)

NITE-SIZE SCALING AND BOUNDARY EFFECTS . . .

Staggered magnetization Si

i

N 1 X Si+ˆx Dx = ( 1)xi Si ·  N i=1 0.01

1 Dy = N



J/Q=0.03,C d(rmax)

yi

J/Q=0.10,C d(rmax)RAPID

( 1) Si · SJ/Q=0.03, i+ˆ y

-2

J/Q2 = 0  J/Q2 = 0.03 J/Q2 = 0.0447 J/Q2 = 0.1

COMMUNICATIONS 10

-3

J/Q=0.10, PHYSICAL REVIEW B 80, 180414%R& %2009&

i=1

2

0.00 x,Dy): Emergent U(1) symmetry P(D 0 0.1 0.05 0.15

1/L



10

 

FIG. 12. (Color online) The staggered√part, Eq. (19), of the longtance correlation function (at rmax = 2L) and the total dimer er parameter for the J -Q2 model at J /Q2 = 0.03 and 0.10 on iodic L × L lattices.

10

-1





2

*

2

Cd (rmax),

Dimer order parameter (Dx,Dy)

N X

RAPID COMMUNICATIONS

PHYSICAL REVIEW B 80, 180414%R& %2009&



{ Ó n {

( 1)

-1

2

SOLID…

10

xi +yi ⌅



1 ⌅ M= N 0.02

10

-2

ailable, it would not be possible to unambiguously confirm -3 10 presence of long-range VBS order, even though the order 10 100 ameter here is still above 10% of the maximum value. L Note that correlation function decays

the long-distance

controlled Change in symmetry byDimer length-scale ξ’>ξ P%D , D & for FIG. 4. %Color online& order distribution FIG. 13. (Color online) Size xdependence of the VBS (top) and y ponentially as a function of 1/L in a non-VBS state, i.e., 2 L = 32 systems. N´ eel order parameters the %a& J -Q2qmodel at four different Exponents? changed over time (ν(bottom) has decreased) The leftorder panels are for the J-Q uch faster than the 1/L behavior ofHave the total squared 3 modelof at = 0.0447 should be very close to the coupling ratios. The /Qpapers 2SU%3& ameter. It is therefore also much easier the Sandvik, Melko, Kaul, Damle, Alet, (many 2007-2015) = 0.635 and %b& qto=confirm 0.85, and the Kawashima,.... right panels arepoint for Jthe J-Q 2 quantum-critical value according to the scaling analysis of the spin sence of long-range- order by studying the long-distance First-order scenarion Prokofe’v, Svistunov, Kuklov, Troyer,... (2008-2013) model at %c& q = 0.45 and %d& q = stiffness 0.65. carried out in Ref. 41. The straight lines fitted through the relations. Tuesday, October 6, 15

15

Exponent ν: crossing-point analysis H. Shao, W. Guo, A. W. Sandvik (unpublished)

(1/⌫+!)

0.4080 !

1.50

1.48 1.47

0.4075

1 1 ⇤ ⇤ = ln[s(g , rL)/s(g1.6 , L)] = ln(r) + aL ⇤ ⌫ ⌫ s(g, L) = dR1 (g, 1.4 L)/dg 1/ν’*

0.04

1.49

= R1c + aL

- SSE calculations at 1/T=L 1.0 - Small leading correction; ω ≈ 0.5 0 0.02 0.04 - ν ≈ 0.45 1/L

0.05

R 1*

R1⇤ (L)

Λ*/L

g (L) = gc + aL

0.4085

0.06

! 2.0

1/ν∗

⇤

From Binder ratio

0.07

g*

g*

Binder ratio of the spin order parameter From 2-spinon distance 0.055 hm2sz i R1 = h|msz |i2 0.050 Dimensionless quantity: 0.045 - look at crossing of R1(g,L), R1(g,rL), g=J/Q, g*(L), analyze size 0.4090 dependence (using r=2)

1.5

1.2

1.0 0.06

0

0.05

0.1 1/L

0.15

Figure 2: Crossing-point of (L,(ν=1/3) 2L) pairstransition for the size of the spinon bound st No sign of analysis first-order the Binder ratio (right). The monotonic quantities are fitted with simple power law Tuesday, October 6, 15

16

vanishes when N ! 1 (at least for observables probing distances r * N). to Thespinons QMC loop in updates [20]simulations automatiAccess QMC cally exclude frustrated negative-sign configurations, andvalence-bond this should, thus,basis be the for mostS>0 rapidstates way to approach Extended N¼ 1.2011, Configurations for SJSTAT ¼ 1=2 and S ¼ 1 states are Tang, Sandvik PRL Banerjee, Damle 2010 illustrated in Figs. 2(b) and 2(c). We note that the valencez Consider S =S bond basis with two unpaired spins was used in a pioneer- for eveningN variational spins: N/2-S bonds, 2S unpaired “up” spins study on spinon deconfinement in a VBS - for odd:state (N-2S)/2 2S model upnpaired of a 1Dbonds, frustrated [1]. spins - transition graph has 2S open strings

S=0 S = 1/2

lations of S = 1 del in the re shown Overlaps and matrix elements involve loops and strings FIG. 2 (color online). Illustration of the basis for states with nge model very simple generalizations case (a) S ¼ 0 (even N), (b) Sof ¼ the 1=2 S=0 (odd N), and (c) S ¼ 1 (even m / -e$r=# - loops have 2 states, strings have 1 state show the N). The bonds and unpaired spins of the bra and ket states are shown below and above the line of sites, respectively.

Use to study spinon bound states and unbinding

157201-2

Tuesday, October 6, 15

17

J-Q model deep in VBS phase

transition graphs evolving in imaginary time Tuesday, October 6, 15

18

J-Q model at the critical point

transition graphs evolving in imaginary time Tuesday, October 6, 15

19

Note: The fundamental second length is be the thickness of VBS domain wall - spinon deconfinement follows from it

g*

0

0

0

0.4090

1

0.4085

1

R 1*

0.045

1

0.4080

1

0.4075 1.6 1.4

1/ν∗

Transition is associated with spinon deconfinement

0

0.050

1/ν’*

roduce the energy density eW (L) of states in the winding crossing y onlyΛ/L sampling thosepoints states inconverge the sector better W . Figure 2 bility than P (W ) in forother the system staying in the original winding cases (monotonic) nd W = (2, 0) as a function of m/L2 , respectively, for a analysis shows ν’ = 0.58(2) >ν nel). Slope The corresponding energy density eW (L) converges L), if W "= 0 (upper panel).

Λ*/L

Crossing-point analysis of Λ/L

g*

olumnar VBS state is again projected out after some d by the convergence of the energy densityν’e(L) to the Exponent (confinement lengt) , the probability P (W ) decreases (to W. 0 Guo, for LA.→W.∞). H. Shao, Sandvik (unpublished) the projection time m/N needs to grow as well in order n theDefine thermodynamic we expect that state) the system Λ (size limit, of spinon bound From 2-spinon distance inding number W , and then it is also plausible that the 0.055 as root-mean-square string distance value eW > e0 corresponding to the lowest excited state

1.2 1.0 0

0.02

0.04

0.06

1/L

Figure 3. Snapshots of %Bα (r)! for a Figure periodic2: Crossing-point analysis of (L, 2L) pairs for How two divergent affect other observables? thewhich Binder ratio (right). The monotonic quantities ar systemdo withthe winding number W = (1, 0), lengths in a October 2π domain wall (four separate π/2 domain walls) while two subleading corrections were included in20th 6, 15 ndTuesday,

Quantum criticality with two lengths H. Shao, W. Guo, A. W. Sandvik (unpublished)

Two divergent lengths tuned by one parameter: ⇠ /

⌫

⇠ / Finite-size scaling of some quantity A. Thermodynamic limit: A / ,

⌫0

0



Conventional scenario: /⌫

1/⌫

1/⌫ 0

1/⌫

1/⌫ 0

A( , L) = L f( L , L ) 1/⌫ 1/⌫ 0 1/⌫  , L )!( L ) When L→∞: f ( L

Alternative scenario:

/⌫ 0

A( , L) = L f( L , L ) 1/⌫ 1/⌫ 0 1/⌫ 0  , L )!( L ) When L→∞: f ( L

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

⇢s / L

(z+d 2)

or

⇢s / L

(z+d 2)⌫/⌫ 0

The first scenario has so far been assumed - unexplained drifts in Lρs in J-Q and other models (z=1, d=2) (proposals: first-order transition, strong scaling corrections,...)

Can alternative scaling form resolve the enigma? Tuesday, October 6, 15

21

Thus, we have extracted a rather precise estimate of the exponent ratio ν/ν # ≈ 0.80 ± 0.01, where the error bar is one standard deviation of the slope of the fitted line in Fig. 8 (and we estimate that the error due to very small deviations of q =

Evidence for unconventional scaling in J-Q model VBS domain-wall energy/length 0

1

✏ / (⇠⇠ )

q=1.0 (VBS), β=L/8 q=1.0 (VBS), β=3L/16 q=0.6 (critical), β=L/8 q=0.6 (critical), β=L/4 q=0.6 (critical), β=3L/8

1

✏

Size dependence L-1.80 - Incompatible with L-(1+ν’/ν) - Compatible with L-(1+ν/ν’), ν/v’=0.80

κ

0.1

Other quantities are consistent with the predicted unconventional scaling

0.01

0.01

0.2

0.55

0

a0 L +a1L +a2L 0.2

a0L +a1L

-3.27

0.150

0.2

0

0.2

0

a0L +a1L +a2L

0

b0L +b1L

Lχ*

Lρs*

0.50

0.45

0.145

0.40

1/L -1.05 +a3L FIG. 8. (Color online) VB domain-wall energy per unit length as a function of the inverse system size graphed on a log-log scale. In the case of the strongly ordered VB solid (q = 1), the energy computed with two different inverse temperatures β(L) converges to the same nonzero value as L → ∞, while at the critical point (q = 0.6 ≈ qc ) convergence of the energy for all L with increasing β is demonstrated (the same also holds true at q = 1 if still larger β is used). The converged energy decays as a power-law form ∼L−b . The fitted line shown here has slope b = 1.80 ± 0.01.

-0.58

(a)

(b) 0.140

0.35 0

20

40

L

60

80

0

20

40

0.1

60

Behavior that might be interpreted as flow to first-order transition is due to unconventional scaling!

09442

80

L

3: Size-normalized spin stiffness (a) and long-wavelength susceptibility (b) at their reFinite-temperature similarly affected! e crossing points g ⇤ for system sizes (L, 2L),scaling with fits to is forms with the asymptotic 0 orTuesday, controlled by the exponent 1 ⌫/⌫ ⇡ 0.20 and corrections as indicated in the panels. October 6, 15

22

Transition point from level spectroscopy A. Sen, H. Suwa, A. W. Sandvik (unpublished)

- lowest excitation of the Neel state is a triplet - lowest excitation of a VBS is a singlet (4 quasi-degenerate states) - transition point is associated with level crossing of excited states 0.06 0.045

0.05

0.04

0.04

qc(L)

q = J/(Q+J) → (Q/J)c = 0.0448(1)

0.035

0.03

0

0.02

0.001

0.002

qc( ) = 0.04290(10)

0.01

Almost no corrections to 1/L2 scaling • same form as in 1D critical - VBS transition

0 0

0.002

0.004

0.006

0.008

0.01

1/L2

Scaling form not presently understood (task for field theorists!) Tuesday, October 6, 15

23

Conclusions Quantum spin models are still offering interesting testing grounds for quantum many-body physics, including new phenomena, surprises

Tuesday, October 6, 15

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