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Stable Higgs mode in anisotropic quantum

Ying Su,1 A. Masaki-Kato,2, 3 Wei Zhu,4 Jian-Xin Zhu,1 Yoshitomo Kamiya,5, ∗ and Shi-Zeng Lin1, † 1Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Computational Condensed Physics Laboratory, RIKEN Cluster for Pioneering Research (CPR), Wako, Saitama 351-0198, Japan 3Computational Materials Science Research Team, RIKEN Center for Computational Science (R-CCS), Kobe, Hyogo 650-0047, Japan 4Westlake Institution of Advanced Study, Westlake University, Hangzhou 300024, China 5School of Physics and Astronomy, Shanghai Jiao Tong University 800 Dongchuan Road, Minhang District, Shanghai 200240, China (Dated: July 7, 2020) Low- excitations associated with the amplitude fluctuation of an order parameter in condensed mat- ter systems can mimic the Higgs , an elementary in the , and are dubbed as Higgs modes. Identifying the condensed-matter Higgs mode is challenging because it is known in many cases to decay rapidly into other low-energy bosonic modes, which renders the Higgs mode invisible. Therefore, it is desirable to find a way to stabilize the Higgs mode, which can offer an insight into the stabilization mechanism of the Higgs mode in . In quantum magnets, magnetic order caused by spontaneous symme- try breaking supports transverse (magnons) and longitudinal (Higgs modes) fluctuations. When a continuous symmetry is broken, the Goldstone magnon mode generally has a lower excitation energy than the Higgs mode, causing a rapid decay of the latter. In this work, we show that a stable Higgs mode exists in anisotropic quantum magnets near the between the dimerized and magnetically ordered phases. We find that an easy axis anisotropy increases the magnon gap such that the magnon mode is above the Higgs mode near the quantum critical point, and the decay of the Higgs mode into the magnon mode is forbidden kinematically. Our results suggest that the anisotropic quantum magnets provide ideal platforms to explore the Higgs physics in condensed matter systems.

I. INTRODUCTION quantum transition point in 2D, whereas the peak re- mains sharp in 3D. This is consistent with the intuition that The in is modeled by a damping of the Higgs mode is stronger in lower dimensions as gauged bosonic condensate, which is responsible for gener- a result of the quantum fluctuations of the Goldstone modes. ating mass for other elementary . Higgs-like excita- One may then argue that it is necessary to consider three di- tions also emerge in condensed matter systems as a conse- mensional systems in order to have a stable Higgs mode. [15] quence of spontaneous symmetry breaking. [1] Examples in- clude charge density [2], superconductors [3–6], quan- In magnetic materials, continuous symmetry can be lifted tum magnets [7,8] and cold condensates in a optical by anisotropy. For instance, the rotation symmetry can × lattice [9–11]. In these systems, the Higgs mode is the collec- be reduced to U(1) Z2 symmetry by either an easy plane tive amplitude fluctuation of the complex order parameter or anisotropy in XY-like systems or an easy axis anisotropy in vector fields, and is usually gapped. Ising-like systems. The rotation symmetry can also be lifted When a continuous symmetry is broken, there exist gap- by an external magnetic field. The reduced symmetry there- less Goldstone modes in addition to the massive Higgs mode. fore can stabilize the Higgs mode in quantum magnets, as will Quantum fluctuations therefore may induce a decay of the be discussed below. The magnons carry spin quantum number ± Higgs mode into the low-lying Goldstone modes, which S m = 1 while the Higgs mode carries spin quantum number causes damping of the Higgs mode. The question is whether S h = 0. A Higgs mode with energy Eh can decay into a pair ± the Higgs mode remains stable. Podolsky et al.[12] addressed of magnons (k1, k2) with S m = 1 constrained by the energy this question by using a field theoretical approach and found and momentum conservation laws, also known as the kine- that the imaginary part of the longitudinal susceptibility as- matic condition Em(k1) + Em(k2) = Eh(k1 + k2). Some or all sociated with the Higgs mode diverges at low frequency ω of magnon branches are gapped in magnets with reduced sym- as 1/ω for two dimensional (2D) systems and log(1/|ω|) for metry, which in turn mitigates the decay of the Higgs mode by three dimensional (3D) systems, which can obscure the spec- reducing the phase space satisfying the kinematic condition in arXiv:2007.02498v1 [cond-mat.str-el] 6 Jul 2020 tral peak of the Higgs mode. This motivates the authors in a part of, or even entire, region of the Brillouin zone. Espe- Refs. [13, 14] to propose a scalar susceptibility, where a well cially, when a quantum undergoes a continuous quan- defined spectral peak corresponding to the Higgs mode ap- tum into the quantum paramagnetic state by pears despite of the strong damping. The scalar susceptibility tuning an external parameter, such as pressure, the magnitude is argued to be identified in the . The of the is suppressed continuously down to spectral peak of the scalar susceptibility is broadened near the zero at the quantum critical point (QCP). The gap of the Higgs mode becomes small near the transition point, and therefore the decay into magnon modes is suppressed when the Higgs mode has lower energy than the magnon modes. Recently, a ∗ [email protected] stable Higgs mode with long lifetime was detected by inelastic † [email protected] scattering measurement in the two dimensional quan- 2 tum magnet C9H18N2CuBr4 with an easy axis anisotropy near a quantum critical point. [16] The authors constructed an ef- fective spin Hamiltonian for C9H18N2CuBr4, and derived the dispersion relation of the magnon and Higgs modes using the XY mean-field bond operator approach. The decay of the Higgs mode in C9H18N2CuBr4 was also investigated recently using quantum Monte Carlo simulations [17]. To investigate the role of spin anisotropy on the stability of /J xy xy the Higgs mode, in this work, we study the Higgs mode in J an anisotropic bilayer quantum antiferromagnetic Heisenberg model for spin S = 1/2 with an easy axis anisotropy by em- ploying the bond operator method, field theoretical approach and quantum Monte Carlo simulation. By combining these Dimer methods, we can show clearly how the spin anisotropy sup- Ising presses the damping of the Higgs mode. The bilayer Heisen- berg model is relevant for several quantum magnets including BaCuSi2O6 [18–20] and Sr3Ir2O7 [21, 22]. We note that the J /J collective excitations including the Higgs mode in an isotropic z model have been considered in Ref. [23]. Upon increasing the interlayer antiferromagnetic interaction, the system under- FIG. 1. Schematic phase diagram of the anisotropic quantum magnet goes a quantum phase transition from the Neel´ order to the model that includes three different phases: interlayer dimer order, nonmagnetic dimerized phase by forming interlayer spin sin- Ising and XY AFM order. The spin texture of the dimer, Ising and glet. Upon reaching the critical point from the magnetically XY AFM are sketched. ordered state, the magnitude of the moment vanishes and the Higgs mode becomes gapless. However, because of the easy axis anisotropy, the magnon modes remain gapped. There ex- anisotropic antiferromagnetic interaction with an Ising-like ists a region where the dispersion of the magnon modes are exchange anisotropy Jz ≥ Jxy described by the first two terms above the Higgs mode, which prevents the decay of Higgs and an antiferromagnetic (AFM) inter-layer coupling in the mode into the magnon modes and therefore the Higgs mode is last term. long lived. We note in passing that the decay of Higgs mode is In this model, three limits can be identified: (i) When already forbidden when the magnon gap is larger than half of J  Jz and Jxy, the AFM interlayer coupling stabilizes sin- the Higgs gap. By tuning the magnon gap, the present model glets between aligned spins in different layers. These singlets allows to investigate the lifetime of the Higgs mode as a func- condense and stabilize a singlet dimer phase. (ii) For Jz  J tion of the magnon gap. and Jxy in the Ising limit, each layer orders antiferromagnet- In the remainder of the paper, we will employ the mean ically with spins aligned along the z direction and staggered field bond operator approach to construct the phase diagram between layers that forms the Neer´ order. (iii) In the XY limit and derive the Higgs and magnon dispersion relations. Then Jxy  J and Jz, where the spins order antiferromagnetically we will use the field theoretical approach to study the decay in the xy plane. The phase diagram of the model and the cor- of Higgs mode into magnon modes. In the region where such responding spin structures in the three limits are sketched in a decay is prohibited because of the the kinematic condition, Fig.1. Here we focus on the phase boundary between the the Higgs mode becomes the lowest lying mode with a very dimer and AFM phases for Jz ≥ Jxy. By gradually reducing sharp spectral peak. Finally, we will present the results of J/Jz, there is a phase transition from the dimer phase to the our quantum Monte Carlo simulation to study the Higgs and AFM phase. To describe this phase transition, we start with magnon modes near the quantum critical point. The paper is the dimer phase in which spin singlets are stabilized along then concluded with a summary. the vertical bonds between two layers, as shown in Fig.1. The bond operator representation is introduced to describe the dimerized spins by one singlet operator si and three triplet op- II. BOND OPERATOR APPROACH erators ti,α with α = x, y, z as

We consider the anisotropic bilayer quantum antiferromag- † 1 s |0i = √ (|↑↓i − |↓↑i) , (2) netic Heisenberg (or XXZ) model defined on a square lattice. i 2 The model Hamiltonian is † 1 X X X t |0i = − √ (|↑↑i − |↓↓i) , (3) x x y y z z i,x 2 H = Jxy [S l,iS l, j + S l,iS l, j] + Jz [S l,iS l, j] + J S1,i · S2,i. h i h i i l, i j l, i j i t† |0i = √ (|↑↑i + |↓↓i) , (4) (1) i,y 2 1 where S is the quantum spin 1/2 operator and l = 1, 2 † l,i ti,z |0i = √ (|↑↓i + |↓↑i) . (5) is the layer index. Here we assume a nearest neighbor 2 3

We choose si and ti,α (here i labels the interlayer dimers) to be 1.4 bosonic operators satisfying the commutation relation (a) 1.2 h i h i h i s , s† = δ , t , t† = δ δ , s , t† = 0. (6) magnon i j i j i,α j,β i j αβ i j,α 1.0 J

For each vertical bond, it can be either in the singlet or one )/ k † P † ( 0.8 of the triplet states, and we have si si + α=x,y,z ti,αti,α = 1 for all i’s. The two spins S1, j and S2, j at the two ends of the j-th 0.6 vertical bond can be expressed in term of the bond operators 0.4 1   Higgs S α = s†t + t† s − i t† t , (7) 1, j 2 j j,α j,α j αβγ j,β j,γ 0.2 1   0 1 2 3 4 5 6 S α = −s†t − t† s − i t† t , (8) 2, j 2 j j,α j,α j αβγ j,β j,γ kx=ky=k (b) where αβγ is the Levi-Civita tensor, and the summation over repeated indices is assumed. 1.5 The Hamiltonian H can be re-expressed in term of these bond operators as J magnon ))/

, 1.0 Jxy Jz J

H = (Hx + Hy) + Hz + HJ, (9) =( k

2 2 4 (

0.5 Higgs X  † †   † †  Hα = si ti,α + ti,α si s j t j,α + t j,α s j hi ji (10) 0.0 X † † 0 0 0 − αβγti,βti,γαβ γ t j,β0 t j,γ , 0.25 0.30 0.35 0.40 0.45 0.50 hi ji Jz/J

  X X  † †  FIG. 2. (a) Dispersion of the Higgs and magnon mode for J = 0.1J HJ = −3s si + t ti,α . (11) xy  i i,α  and J = 0.255J, and (b) the corresponding gap at the wavevector G i α z 0 for Jz/Jxy = 3. In the dimerized phase, the si boson condenses. We can † replace the operator si and si by a real numbers ¯. Further- more, we replace the local constraint on each vertical bond The phase transition point can also be determined by con- by a global one P s† s + P t† t = N with N being the sidering the magnetically ordered phase. In this phase, both i i i i,α i,α i,α d d s t 1 P † i and z,i condense. Because of the intralayer AFM number of dimers, from which we obtains ¯'1 − i t ti,α 2Nd ,α i,α interaction, the ordering wave vector for the tz,i boson is under the Holstein-Primakoff expansion [24]. Here the collec- G0 = (π, π). The wave function can be approxi- tive excitation in the dimerized phase is the triplet excitation, Q mated by |φAFMi = i |φii with which can be obtained by expanding H to the quadratic order in ti,α. The dispersion of the triplet excitation is given by 1  † †  |φii = √ si + λ exp (iG0 · ri) ti,z |0i , (14) q 1 + λ2 2 ξα,k = 2Jxy JAk + J , (12) where λ is a variational parameter to be determined later. [25] † We can introduce a new basiss ˜i |0i = |φii. Then the ground for α = x, y and † state corresponds to the condensation ofs ˜i boson. The other q ˜† † 2 three operators in this new basis are given by ti,x/y = ti,x/y and ξz,k = 2Jz JAk + J , (13)

† 1  † †  where A = cos k + cos k . The higher order terms that are t˜ = √ −λ exp (−iG0 · ri) s + t . (15) k x y i,z 2 i i,z responsible for the damping of the Higgs mode are not con- 1 + λ sidered here, but will be included effectively in the field theo- Here λ can be determined by minimizing the Hamiltonian. retical treatment below. The spin anisotropy splits the other- Again we approximate the local constraint fors ˜i and t˜i,α by wise triply degenerate triplet excitations into two modes with the global one as in the case of dimerized phase. H can be one having double degeneracy. The gap of tz triplet excita- expanded to the second order in t˜i,α, H = H0 + H1 + H2, with tion first vanishes at (Jz/J)c = 1/4 and G0 = (π, π) indicating 2 2 a phase transition into the antiferromagnetic magnetically or- −4Ndλ Jz Nd λ − 3 H0 = + J, (16) dered phase with spins pointing in the ± z direction. (1 + λ2)2 4 1 + λ2 4

" 2 # J 4Jz(1 − λ ) X  †  The gap of the Higgs mode vanishes at the transition point. As H = − λ exp (iG · r ) t˜ + t˜ . 1 1 + λ2 (1 + λ2)2 0 i i,z i,z shown in Fig.2, for a strong anisotropy γ = J /J , the Higgs i z xy (17) mode can lie below the magnon continuum. In this case, the decay of the Higgs mode into magnon continuum is expected The ground state condition requires that the terms linear in t˜ p i,α to be suppressed and therefore the Higgs mode is stabilized. vanishes, which yields λ = (4Jz − J)/(4Jz + J). This λ also The magnon and Higgs modes cease to exist when the system minimizes H0 simultaneously. is tuned to the dimerized phase. The phase diagram can be obtained from the staggered The gaps of the Higgs and magnon modes can be estimated magnetization in the Ising limit, Jz/Jxy → ∞. The magnon carries quantum 1 X spin number S m = ±1, and it corresponds to a single spin flip. M (G ) = hφ | (S α − S α ) exp(iG · r ) |φ i . α 0 N AFM 1,i 2,i 0 i AFM The energy cost is Em = 2Jz + J/2. The Higgs mode has d i quantum spin number S h = 0, and therefore it corresponds (18) to flip a pair of antiferromagnetically aligned spins between q different layers. Its energy cost is E = 4J . This simple 2 − 2 h z Here Mx = My = 0 and Mz(G0) = 16Jz J /4Jz vanishes estimate of the magnon and Higgs gaps agrees well with the continuously at Jz/J = 1/4 upon decreasing Jz, therefore results in Fig.2 (b) when the system is in the well developed the quantum phase transition is of second order. This mean- magnetically ordered phase (large Jz/J region). field critical point is independent of Jxy as long as Jxy ≤ Jz. We proceed to calculate the dynamic spin structure factor The phase transition point is consistent with the previous es- that can be accessed experimentally, timate based on the triplet excitation gap. This consistency X α α is achieved by the approximation scheme used here. First we χα(ω, q) = hS l (−ω, −q)S l (ω, q)iQ, (22) replace the local constraint by the global one, known as the l=1,2 Holstein-Primakoff approximation (HPA) [24]. Within HPA, where h... iQ denotes quantum average. Knowing the disper- hsii = hs˜ii = 1, therefore the HPA neglects the suppression of the amplitude of the singlet condensate due to the triplet sion for the magnon and Higgs modes, χα can be obtained quantum fluctuations. Secondly, we have introduced a ro- straightforwardly ! tated basis to describe the magnetically ordered phase. [25] At 1 1 − Jz/J = 1/4, the ground state described by Eq. (14) is the same χα = Ck + + , (23) ω + i0 + ωM ω + i0 − ωM as the dimerized phase. Therefore, the rotated basis connects !" !# continuously to the un-rotated one upon varying Jz. Alterna- 1 J J Ck = 4 + 2J + 8Jz + Ak Jxy − 4 , (24) tively, one can introduce a chemical potential µ to impose the 64ωM Jz Jz local constraint by adding a term −µ(s† s + P t† t − 1) i i α=x,y,z i,α i,α for α = x, y and 0+ represents a positive infinitesimal number. to the Hamiltonian [26]. In the magnetically ordered phase, For χz, we have one can assume the condensation of si and tα bosons without ! introducing the rotated basis. This approximation, however, J2 1 1 χ = − . (25) does not yield the same transition point by treating the dimer- z 8J ω ω + i0+ + ω ω + i0+ − ω ized and magnetic phase separately. z H H H The second order contribution H2 is The magnon (Higgs) excitation appears in the transverse (lon- J2 X     gitudinal) susceptibility. No damping has been taken into ac- ˜ ˜† ˜ ˜† H2 = ti,z + ti,z t j,z + t j,z count here so the spectral density is a delta function. 32Jz hi ji The Higgs peak in χα can be smeared out severely in the ! J X 1 J     presence of decay, especially in low dimensional systems. To xy ˜ ˜† ˜ ˜† + + η ti,α + ηti,α t j,α + ηt j,α detect the Higgs mode, singlet bond susceptibility was intro- 2 2 8Jz hi ji;α=x,y;η=± duced and was shown to exhibit a sharp Higgs peak despite     J X † J X † of the strong damping [23]. The singlet bond susceptibility is + 2J − t˜ t˜ + 2J + t˜ t˜ . (19) z 2 i,z i,z z 2 i,α i,α analogous to the scalar susceptibility introduced in Ref. 12. i i;α=x,y,z The singlet bond susceptibility is defined as The magnon dispersion associated with the operator t˜† in i,x/y χ ω, q hB −ω, −q B ω, q i , the AFM phase can be obtained by the Bogoliubov transfor- B( ) = ( ) ( ) Q (26) mation and is with B = S · S . It can be calculated s i 1,i 2,i !2 J A JJ 2 k xy 2 (J + 2J ) ωM = + 2Jz + − (JxyAk) . (20) χ = z δ(q)δ(ω) + (16J2 − J2)χ (ω, q − G ). (27) 2 4Jz B 2 z z 0 64Jz The Higgs mode corresponds to the excitation of t˜ boson and i,z The first term accounts for the static dimer correlation at Q = its dispersion is given by 0. There is a G momentum shift between χ and χ because s 0 B z 2 ! of the Neel´ order. Approaching the quantum critical point, the J Ak q ωH = 4Jz 4Jz + . (21) 2 2 8Jz spectral density of χB vanishes as 16Jz − J . 5

(0) (0) be expanded as χσσ (q) = χ̟̟ (q) = Π̟(q) =

S = S0 + SA + SC, (30) Z   1 D  2  2 2 2 2 A 2 ... S0 = d x ∂µσ + ∂µπ + m r σ + π , (31) Σσ(q) = = + + 0 2g Λ 2     2  m2 Z r σ3 + σπ2 σ2 + π2  S 0 D   ... A = d x  √ +  , (32) χσσ (q) = + + + 2g Λ  N 4N    m2 r2 − 1 Z √ 0 D  2 2 SC = d x 2 Nrσ + σ + π , (33) FIG. 3. Feynman diagrams describing the self energy Σσ(q) under the 4g Λ random phase approximation of the polarization bubble Ππ(q) and the full susceptibility of Higgs mode χσσ(q). The and dashed where S0 is the free field action with anisotropy, SA collects lines represent the bare susceptibility of Higgs and magnon modes, the anharmonic contributions, and SC is the counterterm. The respectively. bare susceptibility of Higgs and magnon modes from S0 are g g χ(0) (q) = , χ(0)(q) = , (34) III. FIELD THEORETICAL APPROACH σσ 2 2 2 ππ 2 q + r m0 q + A/2

In the mean-field bond operator approach, we have shown where rm0 is the renormalized mass of the Higgs mode and that the magnon modes can be gapped due to the magnetic q denotes a D-dimensional momentum. Under the analyt- ical continuation q2 → q2 − (ω + i0+)2, the zeroth order anisotropy and the magnon energy can be even larger than that q of the long-wavelength Higgs mode. The question is how the (0) 2 2 2 Higgs and magnon dispersions are ωH (q) = q + m r and magnon gap affects the lifetime of the Higgs mode. Here we 0 (0) p 2 proceed to calculate the lifetime of the Higgs mode by con- ωM (q) = q + A/2, respectively. The dispersions can be sidering the decay of the Higgs mode into magnon modes. A viewed as the expansion of Eq. (20) and (21) around G0 up to a renormalization factor. Then we can make the correspon- more convenient method is a field theoretical approach based p on an effective action. We generalize the calculations in Ref. dence that A ' 2J/Jxy − 8 and rm0 ' J/Jz − 4 around the 12 by including the spin anisotropy. We consider an action of mean-field QCP (Jz/J)c = 1/4 . The Higgs gap, rm0, vanishes the relativistic O(N) field theory with anisotropy, which de- at the QCP at g = gc, consistent with the previous bond opera- scribes various condensed matter systems. For example, the tor approach. Because the magnon gap remains nonzero with (0) √ case with N = 3 describes the long wavelength fluctuation in ωM (0) = A/2, the Higgs mode is below the magnon mode the anisotropic Heisenberg model. The Euclidean time action in the long wavelength√ limit near the QCP considered here. of the model reads as To ensure Φg = (r N, 0) is a stable ground state, the ex- h i Z  2 N  pectation of σ must vanish, σ = 0. This means that the sum 1  m  2 A X  S = dD x (∂ Φ)2 + 0 |Φ|2 − N + Φ2 , of all one-particle irreducible (1PI) diagrams with one σ ex- 2g  α 4N 2 i  Λ i=2 ternal leg mush vanish. In the large N  1 limit, the cancella- (28) tion of the two√ leading-order√ 1PI diagrams originated from the 2 p term rσπ / N in SA and 2 Nrσ in Sc yields r = 1 − g/gc where Φ is a N-component vector field, which can be with parametrized by Φ = (Φ1, π) with π being the (N − 1)- " #−1 component vector. D = d + 1 is the space-time dimension. Z dDk 1 g A > 0 is the hard axis anisotropy, which ensures the saddle c = D 2 √ Λ (2π) k + A/2 point solution hΦ i = N and hπi = 0. We do not write the 1  4π 2  √ √ , D = 3 anisotropy in the easy axis anisotropy form −AΦ /2 because  2 (35) 1  Λ +A/2− A/2 the saddle point solution depends on A in this case. Here m0 is  =  the bare mass and Λ is the ultraviolet cutoff wavevector, both  8π2  √ √ , D = 4  Λ+ Λ2+A/2 of which depend on the microscopic details of the systems. g  2 A √  Λ Λ +A/2− 2 ln is a parameter which controls the strength of quantum fluctu- A/2 ations. There exists a quantum phase transition at g = g and c where√ the integral is due to the π loop contribution in the term the system orders when g < g . Because of the anisotropy 2 c rσπ / N (see AppendixA). In the limit Λ  A, gc = 4π/Λ A > 0, the phase transition exists at d≥1. 2 2 for D = 3 and gc = 8π /Λ for D = 4. Here gc depends only The fluctuations of the field in the ordered phase can be on the ultraviolet cutoff but not the easy axis anisotropy. parametrized as The full Higgs mode susceptibility is given by the Dyson √ equation Φ = (r N + σ, π), (29) g where r is responsible for the suppression of the order param- χ (q) = , (36) σσ 2 2 2 − eter due to the quantum fluctuation. The action Eq. (28) can q + m0r gΣσ(q) 6

(a) (b) (c)

(d) (e) (f)

FIG. 4. (a) and (b) Real and imaginary parts of the self energy of Higgs mode at q = 0 and for D = 2 + 1, N = 100, Λ = 3m0, and g = 0.9gc. Here we consider three different anisotropies with A = 0, 0.02, and 0.3 [see Eq. (28)]. (c) The corresponding spectral function at q = 0. (d)-(f) The intensity plot of the spectral functions of the Higgs mode for A = 0, 0.02, and 0.3, respectively. Here the red and white dash lines are the bare dispersion of Higgs and magnon modes. The decay of the Higgs mode to the magnon modes smears the spectral peak of the Higgs mode in (d) and (e). Because the magnon mode is above the Higgs mode in (f), the spectral peak of the Higgs mode is sharp.

√ 2 where Σσ(q) is the self-energy that collects all the 1PI dia- larization bubble from the rσπ / N term in SA under the grams. In the one-loop order, we consider the dominant po- second order perturbation expansion

  q  4 2 4 2  √1 −1 2A m r Z D m r  cot 2 , D = 3 0 d k 1 0  4π q2 q Ππ(q) = =  q q  (37) D 2   2    2  2A+q2 q2 2 Λ (2π) k + A/2 (k + q) + A/2 2  1 2Λ − −1  16π2 1 + log A 2 q2 tanh 2A+q2 , D = 4

that describes the decay of one Higgs mode into two magnon which is a Lorentzian function. For q = 0, the spectral peak q modes, as shown in Fig.3. The loop integral can be evaluated is centered at ω = r2m2 − gRe[Σ (ω )] and its width is by using the Feynman parameterization (see AppendixB). c 0 σ c The decay of one Higgs mode into other Higgs modes is neg- Γσ = 2gIm[Σσ(ωc)]. When twice of the magnon gap is above the Higgs mode for A > 2r2m2, the decay of the Higgs mode ligible in the large N limit and has no contribution in the low 0 frequency region [12]. The other one-loop tadpole diagrams into the magnon mode is absent. Since the Higgs mode be- from σ4 and 2σ2π2 in S are cancelled out by (r2 −1)σ2 in the comes the lowest lying mode in this case, there is no de- A cay channel of the Higgs mode even including the higher or- counterterm SC, as shown in AppendixC. Going beyond the one-loop order, we introduce the random phase approximation der processes. Therefore, the Higgs mode can be stable in (RPA) of bubble diagrams and the self energy becomes anisotropic quantum magnets. In Figs.4, we show the self energy and spectral function Π (q) of the Higgs mode for D = 3. At q = 0, the finite Re(Σσ) in Σ (q) = π , (38) σ 2 2 Fig.4(a) shifts the spectral peak downward slightly [see Fig. 1 + gΠπ(q)/m0r 2 2 4(f)]. For A < 2r m0, the Im(Σσ) shown in Fig.4(b) remains as shown in Fig.3. The spectral function of the Higgs mode finite at ωc, which broadens the spectral peak due to the decay is of Higgs mode into magnon modes, as shown in Figs.4(c)- 2 2 4(e). On the other hand, when A > 2r m0, Im(Σσ) vanishes 00 χσσ(q) ≡ Im[χσσ(q)] at ωc due to the absence of a decay channel. In this case, a g2Im[Σ (q)] pronounced spectral peak of Higgs mode is identified as dis- = σ , (39)  2 played in Figs.4(c) and4(f). Here we add a tiny imaginary 2 2 2 − 2 2 2 2 q + r m0 gRe[Σσ(q)] + g Im[Σσ(q)] part to the frequency such that the spectral peak for A > 2r m0 7

(a) (b) (c)

(d) (e) (f)

FIG. 5. (a) and (b) Real and imaginary parts of the self energy of Higgs mode at q = 0 and D = 3 + 1. The other parameters are same as those used in Fig.4. (c) The spectral function at q = 0. (d)-(f) The intensity plot of the spectral functions of the Higgs mode for A = 0, 0.02, and 0.3, respectively, with the red and white dash lines representing the bare dispersion of Higgs and magnon modes. The spectral peak of the Higgs mode is sharp in D = 3 + 1. has a finite width. The self energy and spectral function for thereby define the Binder parameter, D = 4, shown in Fig.5, are similar to that for D = 3. How- ever, due to the weaker quantum fluctuation in higher dimen- hM4i sion, the self energy in Figs.5(a) and5(b) is much smaller U s , = 2 2 (40) than that in Figs.4(a) and4(b). As a consequence, even when hMs i 2 2 A < 2r m0, the spectral peak of Higgs mode is still apparent, as shown in Figs.5(c)-5(f). The spectral peak width decreases 2 2 with as A increases to A = 2r m0 above which Im[Σσ(wc)] = 0 and the spectral peak has zero width. T 2 ∂2Z T 4 ∂4Z hM2i = , hM4i = , (41) s N2 Z ∂h2 s N4 Z ∂h4 site s hs=0 site s hs=0

IV. QUANTUM MONTE CARLO RESULTS 2 where Z is the partition function and Nsite = 2L is the to- tal number of sites (L denotes the system size). U is a di- In the field theoretical calculations, we have considered the mensionless scaling parameter and is expected to be asymp- large N  1 limit in order to make controlled approximation. totically size independent at the QCP. In Figs.6(a) and6(b), 2 To connect to physical spin with N = 3, below, we show the we show the Jz/J dependence of hMs i and U, respectively, results of our unbiased quantum Monte Carlo simulation to for Jz/Jxy = 3 and 4 ≤ L ≤ 16. To investigate quantum address the excitation spectrum near the QCP in the Ising-like critical behaviors, the inverse temperature β = 1/T is set to bilayer XXZ model. Especially, we focus on d = 2, where the βJz = 2.5 × L, anticipating the D = 2 + 1 Ising universality anisotropy effect to stabilize the Higgs mode is expected as class where the dynamical scaling exponent is z = 1; this tem- more drastic than d = 3. Our quantum Monte Carlo simula- perature is low enough to study ground state properties. We 2 tion is based on the directed-loop algorithm [27, 28] with the find that hMs i increases with increasing Jz/J. Furthermore, 2 continuous imaginary time world-line scheme. The analytical in the region where rapid increase of hMs i suggests a QCP, U continuation from the imaginary to real frequency is the core shows a clear tendency towards crossing for different L. By part of our numerical study, which we perform utilizing the using the finite-size scaling analysis, we obtain the estimate recently developed stochastic optimization method [29]. of the QCP as (Jz/J)c = 0.332(1) for Jz/Jxy = 3 based on We simulated the bilayer square-lattice Hamiltonian the data collapse shown in Figs.6(c) and6(d). The enlarged [Eq. (1)] by adopting periodic boundary conditions in the a disordered phase relative to the prediction of the mean-field and b directions. First, to determine the QCP induced by theory, (Jz/J)c,MF = 1/4, is a normal observation. changing Jz/J, we consider a source term of a longitudi- To study the excitation spectrum near the QCP, we mea- P l iG0·ri z nal staggered field −hs l,i(−1) e S l,i with G0 = (π, π), sure the imaginary-time dynamical correlation function in the 8

0.25 (a) 1.00 L = 4 0.20 L = 6 0.80 L = 8

0.15 L = 4 2 s 0.60 L = 12 2 s M

η M

L = 6 1+ 0.10 L L = 16 0.40 L = 8

0.05 L = 12 0.20

L = 16 (c) 0.00 0.00 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 1/ν Jz / J L [Jz / J - (Jz / J)c ] 3.00 3.00 L = 4 3.00 (b) 2.50 L = 6 2.50 2.50 2.00 L = 8 L = 4 L = 12 U U 2.00 1.50 L = 6 2.00 L = 16 0.32 0.33 0.34 0.35 L = 8 1.50 1.50 L = 12

L = 16 (d) 1.00 1.00 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 1/ν Jz / J L [Jz / J - (Jz / J)c ]

2 FIG. 6. (a) Quantum Monte Carlo results of hMs i [Eq. (41)] and (b) U [Eq. (40)] for Jz/Jxy = 3 as a function of Jz/J at βJz = 2.5 × L. 2 Finite-size scaling of (c) hMs i and (d) U where we assume the critical exponents of the D = 2 + 1 Ising universality class: ν = 0.63012(16), η = 0.03639(15), and z = 1 [30]. (Jz/J)c = 0.332(1) is obtained by the data collapse of the presented data.

quantum Monte Carlo simulation, for Jz/J = 1/3.25 = 0.927(3) × (Jz/J)c shows the gapped z-component of the triplet excitation in the dimerized phase. zz 1  −βH z z  Figure7(b) corresponds to the spectrum in the vicinity of the C (τ) = Tr Tτ e S (τ)S (0) , (42) l,i j Z l,i l, j QCP and shows the quantum critical soft mode at k = (π, π) 1  − H  for J /J = 1/3 = 1.004(3) × (J /J) . Finally, the result in Cxx (τ) = Tr T e β S x (τ)S x (0) , (43) z z c l,i j Z τ l,i l, j Fig.7(c) shows the gapped Higgs excitations in the magneti- cally ordered phase for J /J = 1/2.75 = 1.095(3) × (J /J) . ≤ ≤ z z c where 0 τ β, Tτ denotes the time ordering operator, and The observed Higgs excitations are relatively sharp (smeared) S α (τ) = eτH S α e−τH (α = z, x). We evaluate the τ dependence l,i l,i in the long-wavelength limit k ' (π, π) [away from k ' (π, π)], of Cαα(τ) at equally-spaced discrete sample points, τ = m∆τ, l,i j m consistent with our field theory predictions. 0 ≤ m < Nτ, where Nτ ≡ β/∆τ is taken as Nτ = 200L in αα our study. By taking the input of Cl,i j(τ), the stochastic opti- mization method [29] can numerically execute the analytical continuation and the Fourier transformation to yield the cor- xx We also show the results of Cl,k(ω) for the same set of pa- responding dynamical spin structure factor, rameters in Figs.7(d)–7(f). We find that the spectral weight 1 X Z ∞ D E corresponding to the xy-components of the triplet excitation Cαα(ω) = dt S α (t)S α (0) e−i[k·(ri−r j)−ωt], in the dimerized phase and the one corresponding to the l,k 2πL2 l,i l, j T i, j −∞ magnons in the ordered phase seem to evolve continuously (44) into each other by varying Jz/J. These excitations are gapped and with small bandwidths all the way through the QCP. Re- α iHt α −iHt zz xx with α = z, x, S l,i(t) = e S l,ie , and h...iT denotes ther- markably, by comparing Cl,k(ω) and Cl,k(ω) in the ordered mal average at temperature T. Our simulations for these dy- phase, we find that the stable Higgs excitations emerge below namical correlation functions are carried out at βJz = 0.833 × the gapped magnon band near k = (π, π), whereas the smeared L, which is still a low enough temperature to address the Higgs excitations away from k ' (π, π) are within the en- ground-state spectral function. ergy range of the less dispersive magnon band [Figs.7(c) and zz We show the results of Cl,k(ω) in the intensity plot for 7(f)]. This observation confirms the predicted mechanism of L = 16 in Figs.7(a)–7(c). The consistency with the result for the protection of the long-wavelength Higgs mode through the L = 12 has been checked (not shown). The result in Fig.7(a) violation of the kinematic condition near the QCP. 9

(a) 0 0.1 0.2 0.3 0 0.1 0.2 0.3 2

2 ω / / J J 1 ω 1 K

Γ M (d) 0 0 Γ M K Γ Γ M K Γ

(b) 0 0.1 0.2 0.3 0 0.1 0.2 0.3 2

2 ω / / J J

ω 1 1

(e) 0 0 Γ M K Γ Γ M K Γ 3 3

(c) 0 0.1 0.2 0.3 0 0.1 0.2 0.3 2

2 ω J / J /

ω 1 1

(f) 0 0 Γ M K Γ Γ M K Γ

zz FIG. 7. Results of Cl,k(ω) [Eq. (44)] obtained by the quantum Monte simulation and the analytical continuation for L = 16 and (a) Jz/J = 1/3.25 = 0.927(3) × (Jz/J)c, (b) Jz/J = 1/3 = 1.004(3) × (Jz/J)c, and (c) Jz/J = 1/2.75 = 1.095(3) × (Jz/J)c, corresponding to the dimerized xx phase, a vicinity of the QCP, and the magnetically ordered phase, respectively. The results of Cl,k(ω) in the same parameters are shown in (d) Jz/J = 1/3.25, (e) Jz/J = 1/3, and (f) Jz/J = 1/2.75. The results are shown along the line in the Brillouin zone shown in (a).

V. CONCLUSIONS the bilayer square-lattice XXZ model around k = (π, π) near the QCP. Hence, our theory and simulation establish a new In this work, we show the existence of a stable Higgs mode mechanism to stabilize the Higgs mode in anisotropic quan- in an anisotropic quantum spin system near the QCP. The tum magnets near a QCP, which can be an ideal platform to easy axis anisotropy gaps out magnons, while the Higgs mode study the Higgs physics. gap vanishes at the QCP between the magnetically ordered and dimerized phases. Therefore, close to the QCP, the en- ergy of the long-wavelength Higgs mode is lower than that of ACKNOWLEDGMENTS magnons. As a consequence, the decay of the Higgs mode to the magnon modes is forbidden due to the energy conserva- The authors would like to thank Anders W. Sandvik, tion. In the quantum field theory perspective, the system can Cristian D. Batista, Ziyang Meng, Marc Janoschek, and be described by a coarse-grained O(N) nonlinear σ model [12] Filip Ronning for helpful discussions. This work was car- with an easy axis anisotropy term. The anisotropy completely ried out under the auspices of the U.S. DOE NNSA un- suppresses the damping of Higgs mode above a critical value. der contract No. 89233218CNA000001 through the LDRD In this case, the Higgs mode is the lowest lying mode and be- Program and the U.S. DOE Office of Basic Energy Sci- comes stable even in d = 2. Our quantum Monte Carlo simu- ences Program E3B5 (S.-Z. L. and J.-X. Z.). A.M.-K. lation indeed demonstrates the stability of the Higgs mode in used computational resources of the HPCI system through 10 the HPCI System Research Project (Project IDs.: hp170213, Appendix B: Polarization bubble hp180098, and hp180129). Y.K. acknowledges the support from NSFC Research Fund for International Young Scien- To evaluate the loop integral of the polarization bubble, we tists No. 11950410507 as well as the support from Min- use the Feynman parametrization istry of Science and Technology (MOST) with the Grants No. 2016YFA0300500 and No. 2016YFA0300501. W.Z. is 1 Z 1 du , supported by the start-up funding from Westlake University. = 2 (B1) AB 0 [uA + (1 − u)B] such that the loop integral in Eq. (37) becomes

m4r2 Z 1 Z dDk 1 Π (q) = 0 du π D  2 2 0 Λ (2π) k2 + C(u, q)2 R π R 2π 4 2 Z 1 Z Λ D−1 QD−2 j Appendix A: Conditions for hσi = 0 m r ρ dρ j=1 sin θ jdθ j dφ = 0 du 0 0 D  2 2 0 0 (2π) ρ2 + C(u, q)2 √  R 1     m4r2  1 du 1 tan−1 Λ − Λ , D = 3 The stable ground state Φg = (r N, 0) condition requires 0  4π2 0 C C Λ2+C2 = R 1   2 2  2  2  1 du Λ +C − Λ , D 16π2 0 log C2 Λ2+C2 = 4 (B2) R −S DΦ(x)σ(x)e p hσi = = 0, (A1) where C(u, q) = q2u(1 − u) + A/2 and dDk = Z D−1 QD−2 j ρ dρ j=1 sin θ jdθ jdφ in the spherical coordinate. For the ultraviolet cutoff Λ  C, the integrals in Eq. (B2) R reduce to Eq. (37) that reproduces the results in Ref. 12 in where DΦ(x) denotes the functional integral and Z = the limit A → 0. R −S DΦ(x)e is the partition function. Under the perturbation Appendix C: Cancellation of tadpole diagrams expansion to the leading order and in the large N limit, the sum√ 2 of the two 1PI√ diagrams originated from the term rσπ / N The full Higgs mode susceptibility is in S and 2 Nrσ in S must vanish as A c R Z DΦ(x)σ(x)σ(0)e−S χ (q) = dD xe−iq·x . (C1) σσ Z 2 Z m0r D 0 (0) 0 h (0) 2 i √ d x Gσσ(x − x ) (N − 1)Gππ (0) + (r − 1)N = 0, Employing the perturbation expansion to the first-order, we 2g N obtain (A2) (0) 2 where Gαα(x) = hα(x)α(0)i0 is the bare propagator for α = σ Z  Z (1) D −iq·x m0 D 0 (0) 0 (0) (0) 0 π χσσ(q) = d xe d x Gσσ(x − x )Gππ (0)Gσσ(x ) or . Eq. (A2) yields 2g (r2 − 1)m2 Z  + 0 dD x0G(0) (x − x0)G(0) (x0) , 2g σσ σσ Z dDk 1 g r2 − g − , (C2) = 1 D 2 = 1 (A3) Λ (2π) k + A/2 gc where the first term is the tadpole diagram from SA and the second term comes from the counterterm SC. According to (0) 2 that vanishes at the QCP g = gc and gives Eq. (35). Here we Eq. (A2), we have Gππ (0) = 1−r in the large N limit such that 0 −D R D 0 have used the identity Gππ(0) = (2π) d kχππ(k), where the tadpole diagram in Eq. (C1) is cancelled by the countert- 0 (1) χππ(k) is the bare susceptibility of magnon mode in Eq. (34). erm. Then the first order susceptibility vanishes χσσ(q) = 0.

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