Lattice Vibration As a Knob for Novel Quantum Criticality: Emergence Of

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, y 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 (jΨspini ⊗ jΨphononi) 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 jαihαj + P ! byb , where an β;fbq g 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 (! ), and phonon cre- q The relaxation and decay rates are estimated as τ −1 / g2 ation/annihilation operators (by=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- jGspini ' jGspini sion is invalid due to gapless critical excitations, and X hβ; fbqgjHs−ljα; fbpgi + g jβ; fbqgi 0 + ··· ; instead, lattice vibration opens up the possibility of a E − Eβ;fb g β;fbg 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 jβ; fbqgi 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 M!2 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 !0, 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 y 1 y 0.3 0.6 solvable and becomes H0 = k(f fk − ) + !k(b bk + 0.3 k k 2 k 0.2 0.4 1 y 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, !k = 2!0j sin( 2 )j, 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 y 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 = !0a) are asso- ciated with magnetic exchange and Debye energy scales, In Fig.

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