Lattice vibration as a knob on exotic quantum criticality

Control of quantum coherence in a many-body system is one of the key issues in modern condensed matter, and conventional wisdom is that lattice vibration is an innate source of decoherence. Much research has been conducted to eliminate lattice effects. Challenging this wisdom, 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 methods and theoretically discover a stable $\mathcal{N}=1$ 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 the perspective that lattice vibration may be a knob for exotic quantum many-body states.

Figure 1

Illustration of the transverse Ising model under lattice vibrations. The red arrow stands for the Ising spin, and the springs represent the vibrating lattices. The green arrow at the top shows the transverse field, and $\gamma$ is the coupling constant between the spin degree of freedom and lattice vibrations.

Figure 2

(a) Phase diagram of the transverse-field Ising model under lattice vibrations. $r<1$ corresponds to the ordered phase, and the spin degrees of freedom are aligned along the $z$ axis, while $r>1$ indicates that the system is in the quantum disordered phase and the spin degrees of freedom are along the $x$ axis. $r=1$ and $T=0$ are the quantum critical point which is described by the $\mathcal{N}=1$ supersymmetric conformal field theory (CFT) with central charge $c=3/2$. (b)–(d) DMRG calculation of the entanglement entropy of the system for three different values of $r$ as indicated in the phase diagram in (a). The values are represented by circles. Here, $l$ is the length of the subsystem. CFT predicts the scaling of entanglement entropy, and the results for central charge $c=1/2$ (dash-dotted line), $c=1$ (dotted line), and $c=3/2$ (dashed line) are plotted as a comparison. At the critical point [in (c)], the scaling suggests a central charge of $3/2$, while away from the critical point [in (b) and (d)], the central charge is 1.

Figure 3

Feynman diagrams for the RG analysis in the transverse Ising model under lattice vibration up to one-loop order. (a) and (b) are for the fermion and boson self-energies, and (c) is for the vertex correction. The solid line with the arrow and the wavy line stand for the fermion and boson, respectively.

Figure 4

(a) RG flow diagram with the two dimensionless parameters, the velocity ratio of phonons and spinons $\rho$ ($\equiv v_s/v_M$) and the spin-lattice coupling constant ${\alpha }_g$ $[\equiv g^2/\left(2\pi v_s^2{v}_M\right)]$. While the RG flow (red) is directed to $(\rho ,{\alpha }_g)\to (0,\infty )$ for $\rho <1$, the flow (blue) is directed to the fixed point $({\rho }^{*},{\alpha }_g^{*})=(1,0)$ for $\rho >1$. The fixed point is described by $\mathcal{N}=1$ superconformal field theory with central charge $c=3/2$. (b) and (c) DMRG calculations of the entanglement entropy with spin-lattice coupling $\gamma$. The two plots represent each side ($\rho <1$ and $\rho >1$) of the flow diagram as $\rho =0.2$ for (b) and $\rho =1.5$ for (c). One can observe the strong deviation from the original CFT for $\rho <1$. The inset shows how the linear entropy and the average phonon occupancy change with $\gamma$. The significant increase of the linear entropy for $\rho <1$ shows the state is flowing away from the fixed point, and a similar trend of phonon density indicates that the phonons are responsible for this.

Figure 5

The finite-size effect of the DMRG calculation. (a) and (b) Entanglement entropy scaling for $\gamma =0.5$ as shown in Figs. 3 (b) and (c). The deviation from the $c=3/2$ CFT (gray dashed line) is more prominent for larger systems. (c) Linear entropy as a function of $\gamma$ as in the inset in Fig. 3. The decrease of $S_{\text{lin}}$ in system size implies the decoupling of spin and lattice in the thermodynamic limit for this parameter regime ($\rho >1$).