Lattice vibration as a knob for novel quantum criticality : Emergence of supersymmetry from spin-lattice coupling

SangEun Han,1, ∗ Junhyun Lee,2, ∗ and Eun-Gook Moon1, † 1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea 2Department of Physics, Condensed Matter Theory Center and the Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA (Dated: November 6, 2019) Control of quantum coherence in many-body system is one of the key issues in modern condensed matter. Conventional wisdom is that lattice vibration is an innate source of decoherence, and amounts of research have been conducted to eliminate lattice effects. Challenging this wisdom, here we show that lattice vibration may not be a decoherence source but an impetus of a novel coherent quantum many-body state. We demonstrate the possibility by studying the transverse-field Ising model on a chain with renormalization group and density-matrix renormalization group method, and theoretically discover a stable N = 1 supersymmetric quantum criticality with central charge c = 3/2. Thus, we propose an Ising spin chain with strong spin-lattice coupling as a candidate to observe supersymmetry. Generic precursor conditions of novel quantum criticality are obtained by generalizing the Larkin-Pikin criterion of thermal transitions. Our work provides a new perspective that lattice vibration may be a knob for exotic quantum many-body states.

Quantum states on a lattice inevitably couple to lattice vibration, and the coupling is known to be one of the hxˆ main sources of decoherence of quantum states. To be ~si+1 z specific, let us consider a spin system on a lattice whose z s i+1 i Hamiltonian may be schematically written by, ~si -Js

0 0 H = Hspin + Hphonon + gHs−l, i+1 2 u 0 γ 0 where Hspin/phonon is for a pure spin/phonon system, and Mω Hs−l is for the spin-lattice coupling. We introduce a ui dimensionless coupling constant (g) to characterize the strength of the coupling. FIG. 1. Illustration of the transverse Ising model Phonons usually play the role of an environment, and under lattice vibrations. The red arrow stands for the a spin quantum state becomes decoherent due to the Ising spin and the springs represent the vibrating lattices. spin-lattice coupling. In other words, a disentangled The green arrow at the top shows the transverse field and γ quantum state (|Ψspini ⊗ |Ψphononi) is generically not is the coupling constant between the spin degree of freedom an eigenstate of the total Hamiltonian. For example, and lattice vibrations. magnon excitations may decay into acoustic phonons [1], and spin qubits may develop spin-relaxation time [2]. In terms of the spin-lattice model, the Hamiltonian With the energy gap (∆ > 0), the denominator of the sec- without the spin-lattice coupling may be expressed by ond term may be safely approximated as E − E0 H0 + H0 ' P E |αihα| + P ω b†b , where an β;{bq } G & spin lattice α α q q q q ∆, and it is apparent that the spin-lattice coupling acts eigenenergy (E ) with a quantum number (α) of spins, α as a decoherence source of a pure quantum state of spins. acoustic phonon energy spectrum (ω ), and phonon cre- q The relaxation and decay rates are estimated as τ −1 ∝ g2 ation/annihilation operators (b†/b ) with momentum q q q for small g. It is widely believed that elimination of the are introduced. The ground state with characteristic

arXiv:1911.01435v1 [cond-mat.str-el] 4 Nov 2019 lattice coupling is crucial to control coherence of quan- length/time scales such as an excitation energy gap of tum many-body states [4]. quasi-particle excitations [3] loses coherence by coupling with acoustic phonons, as manifested in the perturbative Here, we challenge the common belief by demonstrat- calculation, ing that lattice vibration may be used to realize a novel coherent quantum many-body state. Especially, for a 0 quantum critical state, the above perturbative discus- |Gspini ' |Gspini sion is invalid due to gapless critical excitations, and X hβ; {bq}|Hs−l|α; {bp}i + g |β; {bq}i 0 + ··· , instead, lattice vibration opens up the possibility of a E − Eβ;{b } β,{b} G q novel quantum critical state by entangling lattice vibra- (1) tion and quantum critical modes as we show below. In where disentangled excited states |β; {bq}i are used. other words, spin-lattice coupling may strongly drive a 2 system to be in a pure state as a whole. (a) T For a proof of principle, we consider the transverse-field Ising chain model with acoustic phonons. The Hamilto- nian without spin-lattice coupling is Supersymmetric Quantum X h P 2 Mω2 i Critical H = − Jszsz − hsx + i + 0 (u − u )2 , 0 i i+1 i 2M 2 i+1 i i Quantum (2) Ordered Disordered with a magnetic exchange interaction J, a transverse r magnetic field h, Debye frequencey ω0, and ion mass M (b) (c) (d) (Fig.1). The deviation of spin positions are captured by r = 0.5 r = 1 r = 2 ui and the quantum spins are represented by the Pauli 0.5 1.0 0.6

x,y,z 0.5 matrices (sj ) at site j. The Hamiltonian is exactly 0.4 0.8 0.4 P † 1 † 0.3 0.6 solvable and becomes H0 = k(f fk − ) + ωk(b bk + 0.3 k k 2 k 0.2 0.4 1 † 0.2 0.2 2 ). The bosonic operators (bk, bk) describe acoustic 0.1 0.1 ka 0.0 0.0 0.0 phonons with the energy spectrum, ωk = 2ω0| sin( 2 )|, 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 † and the fermionic ones (fk, fk ) are from the Jordan- (b) r = 0.5 (c) r = 1 (d) r = 2 Wigner transformation of spins and have energy spec- p − 2 trum of k = 2J 1 2r cos(ka) + r . Lattice spacing FIG. 2. Phase diagram and entanglement entropy plot a and the ratio r = h/J are introduced. Note that the for the transverse Ising model under lattice vibra- pure spin term may also be represented by two Majorana tions. (a) Phase diagram of the transverse Ising model under (1,2) lattice vibrations. r < 1 corresponds to the ordered phase and fermions at each site (ηj ). For example, the spin ex- z z (2) (1) the spin degrees of freedom are aligned along the z axis, while change term becomes sj sj+1 = −iηj ηj+1 in this repre- r > 1 indicates the system is in the quantum disordered phase sentation. At r = 1, gapless Majorana fermion excitation and the spin degrees of freedom are along to the x axis. r = 1 arises in the pure spin model, indicating the Ising univer- and T = 0 is the quantum critical point which is described sality class of central charge c = 1/2 [3]. On the other by the N = 1 supersymmetric conformal field theory (CFT) hand, the phonon spectrum is gapless because phonons with central charge c = 3/2. (b, c, d) DMRG calculation of are Goldstone bosons of translational symmetry. the entanglement entropy of the system for three different val- ues of r as indicated in the phase diagram (a). The x axis is The spin-lattice coupling appears with spatial modula- the location of the boundary of the two subsystems (L is the tion of the magnetic exchange interaction, J → Ji,i+1 = length of the system while l is that of the subsystem). CFT 2 J + γ(ui+1 − ui) + O((ui+1 − ui) )[5], and the leading predicts the scaling of entanglement entropy and the result for interaction term is central charge c = 1/2 (dash-dotted line), c = 1 (dotted line), and c = 3/2 (dashed line) are plotted as a comparison. At X z z the critical point [(c)], the scaling suggests central charge of H1 = γ (ui+1 − ui)s s . (3) i i+1 3/2 while away from the critical point [(b,d)], central charge i is 1. The fitted CFT scaling is shown as a solid line in all Away from the critical point (r =6 1), perturbative cal- three figures. culation indicates that decay rate of a quantum state is indeed proportional to τ −1 ∝ γ2, and the spin-lattice coupling becomes a source of decoherence. tified as Z Now, let us consider a quantum critical state. The g T scale invariance allows us to use the critical theory of S1 = (∂xu)η σyη. 2 τ,x spin and lattice degrees of freedom, whose form is, The total critical theory, S0 + S1, are analyzed by in- Z 1 1 v2 | 2 s 2 troducing the two dimensionless coupling constants, ρ ≡ S0 = η (∂τ + ivM σx∂x) η + (∂τ u) + (∂xu) , 2 2 2 2 2 τ,x vs/vM and αg ≡ g /(2πvs vM ). We perform the one loop renormalization group (RG) analysis and obtain the flow where the Pauli matrices (σx,y,z) are defined in the two equations, component Majorana spinor η| = (η(1) η(2)) space. The Majorana fields are rescaled to have the factor 1/2, and  − 2  −  R R dρ αg 1 ρ dαg 2 1 ρ the short-handed notation ≡ dτdx is used here- = − ρ, = αg . (4) τ,x d` 2 1 + ρ d` 1 + ρ after. The two velocities (vM = 2Ja, vs = ω0a) are asso- ciated with magnetic exchange and Debye energy scales, In Fig. 3(a), the flow diagram is illustrated, and the fixed respectively. Then, the spin-lattice coupling may be iden- point is at (ρ, αg) = (1, 0). The RG flow around the fixed 3

1.0 which is fermionic, proven by the Jordan-Wigner trans- 2 formation, and satisfies Hsc = JQˆ . Therefore, this 0.8 fermionic operator becomes a supercharge, [Q,ˆ Hsc] = 0, 0.6 and the N = 1 supersymmetry with central charge c = 3/2 is obtained. 0.4 Our analysis is also checked by the density-matrix renormalization group (DMRG) method. For a chain of 0.2 length L = 40, we calculate the ground state of the sys- 0.0 tem and its bipartite entanglement entropy for different 0.5 1.0 1.5 2.0 ratios of energy scales (J, h, γ). The central charge can be obtained from the entanglement entropy scaling with the subsystem size. Without spin-lattice coupling, we (a) find c = 3/2 at the critical point (r = 1) while c = 1 oth- 1.0 1.0 erwise as the spin sector becomes gapped and only the 0.8 0.8 acoustic phonon contributes (Fig.2(b)(c)(d)). When the 0.6 0.6 3 0.6 0.5 0.6 0.5 interaction is turned on and the system is at the critical 0.4 0.4 0.4 0.3 0.4 0.3 2

hnndensity phonon 0.2 density phonon 0.2 point, two distinct behaviors arise for regions ω0 > 2J 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.2 0.2 1 and ω0 < 2J. The system is stable for ω0 > 2J, mani-

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.0 1.5 fested2.0 by the central charge unchanged at c = 3/2, while big deviations from c = 3/2 occurs for ω0 < 2J, which is (b) (c) consistent with the results of the RG flow (Fig. 3(b) and 3(c)). We have also measured the phonon density in each FIG. 3. RG flow diagram with the two dimensionless calculation and observed that the phonon occupancy sig- parameters, the velocity ratio of phonons and spinons nificantly increases for nonzero coupling and ω0 < 2J, ρ (≡ vs/vM ) and the spin-lattice coupling constant αg supporting the conventional decoherence from phonons (≡ g2/(2πv2v )), and bipartite entanglement entropy s M in that parameter regime. plots in terms of ρ and γ. (a) While the RG flow (red) Few comments are as follows. First, the supersym- is directed to (ρ, αg) = (0, ∞) for ρ < 1, the flow (blue) is directed to (ρ, αg) = (1, 0), for ρ > 1. The fixed point is metric quantum state emerges from the spin-lattice cou- described by N = 1 superconformal field theory with central pling. Without the coupling, the state loses supersym- charge c = 3/2. (b, c) DMRG calculations of the entan- metry unless interactions are fine-tuned. Note that the glement entropy with spin-lattice coupling γ. The two plots interactions from spin-lattice are unique in a sense that represents each side (ρ < 1 and ρ > 1) of the flow diagram bosons have a shift symmetry, u → u + a, in con- as ρ = 0.2 for (b) and ρ = 1.5 for (c). One can observe the i i trast to ladder systems [6–8]. Second, the supersymmet- strong deviation from the original CFT for ρ < 1. The inset shows how the average phonon occupancy changes with γ. ric quantum criticality cannot be obtained by the stan- The significant increase of phonon density in ρ < 1 indicates dard quantum-classical mapping. It is because lattice that the phonons are responsible for the state flowing away vibrations are intrinsically tied to spatial dimensions. To from the decoupled theory. illustrate this, we consider the classical Ising model in P z z 2d, HIsing = −J hi,ji σi σj , which may be mapped to the transverse-field Ising chain model [3]. The 2d static point is intriguing. If the phonon is slower than the Ma- phonon Hamiltonian in cubic solids may be described by jorana fermion (ρ < 1), the flow goes to (ρ, αg) = (0, ∞) R 2 1  2 2 2  Hph = d x 2 C11(e11 + e22) + 2C12e11e22 + C44e12 , signaling a first order phase transition. In the oppo- 1 where eab ≡ (∂aub + ∂bua) is the strain field, ua is the site case where the phonon is faster than the Majorana 2 acoustic phonon field with a = 1, 2, and Cij’s are the elas- fermion (ρ > 1), the RG flow is directed to the stable tic stiffness constants [9]. There are two phonon modes fixed point, (ρ, αg) = (1, 0). along the two spatial directions in sharp contrast to the The Hamiltonian at the stable fixed point may be writ- one mode in the quantum model, and moreover, the bulk ten as modulus, K = (C11 + C12)/2, keeps decreasing under X h i the scale transformation, which indicates the instability − z z − x 2 − 2 Hsc = J si si+1 si + pi + (xi+1 xi) , (5) of the thermal Ising transition. Thus, the N = 1 su- i persymmetry at the critical point cannot appear in the thermal transition. Third, the origin of the supersymme- with√ rescaled momentum√ and position operators, pj ≡ Pj/ Mω0 and xj ≡ Mω0uj. We introduce an opera- try in our work is different from the previously suggested tor, ones in the literature [6,8, 10–15] where bosons are made of fermions, and special types of interactions or surface ˆ X Y x
 z y degrees of freedom are necessary. In contrast, bosons are Q = − sl (xj−1 − xj)sj + pjsj , (6) j l 0 tions at the critical point. 3 (4)-state Potts [17] 2 5/6 (2/3) > 0 Going beyond the spin chain system, let us consider a q-state clock (q > 4) [18] 2 ∞ < 0 generic Landau-Ginzburg Hamiltonian with a local order Ising [19] 3 0.63 > 0 parameter φ, q-state clock (q ≥ 4)[20] 3 0.67 < 0 O(N)(N ≥ 2) [19, 20] 3 ≥ 0.67 < 0 X X H = − t φiφj + λ φiφjφkφl + Hph. (7) N = 2 WZ SUSY [14] 3 0.917 < 0 hiji ijkl N = 2 XYZ SUSY [15] 3 1/2 + /4 < 0 O(N)[3] 4 1/2 0 Here the indices are for the positions of the order pa- rameters. For simplicity, we consider the case where TABLE I. Scaling dimensions of the coupling ([g] = 1/ν − the symmetry group of the order parameter is decou- deff/2) in models with continuous phase transitions. The ef- pled from that of the lattice in this work, leaving other fective dimension deff ≡ (d+1) is introduced for spatial dimen- cases for future works. The lattice Hamiltonian Hph gen- sion d. For [g] > 0, the original criticalities become unstable erally consists of harmonic and anharmonic terms with signaling the first-order phase transition under lattice vibra- an additional polarization index. As in the Ising model, tion, and for [g] < 0, the original criticalties are stable under we promote the coupling at the lowest order of φ to have lattice vibrations. For [g] = 0, a novel quantum criticality may appear as in the spin-chain model in the main text. a spatial modulation to introduce minimal interaction between the order parameter and the lattice vibrations: 2 t → tij = t + γ(uj − ui) + O((uj − ui) ). tice vibration. Second, as in the above spin-chain model, A quantum phase transition may be described by tun- a novel quantum criticality may appear. As an example, ing the parameter λ/t. To study the behavior near the the scaling dimensions of the lattice-order parameter in phase transition, we again consider the critical field the- several models which have z = 1 are presented in Ta- ory. The total action S = S + S + S describes the c ph ph-c ble.I. Note that our condition becomes the Larkin-Pikin interplay physics between quantum criticality and acous- criterion [16] in the limit of classical phase transitions. tic phonons, where S is the critical action for the original c Namely, setting z = 0, [g] < 0 becomes the negative theory for the order parameter, S is the action for the ph heat capacity critical exponent α = 2 − dν < 0, and the acoustic phonons, and S represents the interaction of ph-c corresponding classical criticality is stable. the two. The coupling term in the action, Sph-c, is solely determined by the symmetry of the theory. Since we are The generalized Larkin-Pikin criterion may be also considering the case where the order parameter and the applied to unconventional quantum criticalities. First, lattice represents different symmetries, the most relevant topological phase transitions in weakly correlated sys- R Pd tems are generically described by the Dirac/Weyl interaction term is Sph-c = g τ,x OE i=1 ∂iui. ui is the phonon field and d is the number of spatial dimen- fermions, whose Hamiltonian is written as HD/W = R d † − a ··· sions. The form of the energy operator OE depends on d xψ ( i∂aΓ )ψ with a = 1, , d, the Clifford alge- 2 the system, for example, OE = φ /2 in the conventional bra matrices Γa, and the spinor ψ [21]. The sign of the 4 R 1 2 2 2
r 2 λ 4 mass determines whether the system is in the topologi- φ theory, Sc = τ,x 2 (∂τ φ) + v (∇φ) + 2 φ + 4! φ . cal phase, and the correlation length critical exponent is The standard scaling analysis may be performed at the ν = 1. Setting z = 1, the coupling constant is marginal fixed point without the lattice-order parameter coupling. in d = 1 and irrelevant for d > 1. For criticalities with The strain tensor and the energy operator have [e ] = ij z > 1, the coupling becomes less irrelevant, but is still d+z and [O ] = z + d − 1 . The scaling dimension of g is 2 E ν irrelevant at higher dimensions such as d = 3. Thus, 1 d + z 2 − (d + z)ν new universality class or instability may appear at d = 1 [g] = − = , (8) while topological phase transitions in higher dimensions ν 2 2ν may be decoupled from the lattice vibration. Second, whose sign becomes the main criterion for the stability. quantum criticalities with an enlarged symmetry, such as For [g] < 0, the quantum criticality of Sc is stable, so the criticalities in a deconfined phase, may have different uni- ground state may be described by a disentangled state versality class from that of the Landau-Ginzburg-Wilson of order parameters and phonons. Perturbative calcula- paradigm [22, 23]. For example, a Z2 symmetry break- tions give rise to decoherence of quantum states of order ing transition with Z2 local gauge in d = 2 has the same parameters. But, for [g] ≥ 0, the disentangled state be- universality class as the one of U(1) symmetry transi- comes unstable indicating two possibilities. 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