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arXiv:2108.09966 · Physical review. B./Physical review. B

Fidelity and entanglement entropy for infinite-order phase transitions

Jin Zhang

Published Nov 11, 2021• 13 citations
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Abstract

We study the fidelity and the entanglement entropy for the ground states of quantum systems that have infinite-order quantum phase transitions. In particular, we consider the quantum O(2) model with a spin-$S$ truncation, where there is an infinite-order Gaussian (IOG) transition for $S=1$ and there are Berezinskii-Kosterlitz-Thouless (BKT) transitions for $S\ensuremath{\ge}2$. We show that the height of the peak in the fidelity susceptibility (${\ensuremath{\chi}}_{F}$) converges to a finite thermodynamic value as a power law of $1/L$ for the IOG transition and as $1/ln(L)$ for BKT transitions. The peak position of ${\ensuremath{\chi}}_{F}$ resides inside the gapped phase for both the IOG transition and BKT transitions. On the other hand, the derivative of the block entanglement entropy with respect to the coupling constant (${S}_{vN}^{\ensuremath{'}}$) has a peak height that diverges as ${ln}^{2}(L)$ for $S=1$ and ${ln}^{3}(L)$ for $S\ensuremath{\ge}2$ and can be used to locate both kinds of transitions accurately. We include higher-order corrections for finite-size scalings and obtain the value of the central charge consistent with $c=1$ predicted by conformal field theory. The crossing point of ${\ensuremath{\chi}}_{F}$ between different system sizes is at the IOG point for $S=1$ but is inside the gapped phase for $S\ensuremath{\ge}2$, while those of ${S}_{vN}^{\ensuremath{'}}$ are at the phase-transition points for all $S$ truncations. Our work elaborates on how to use the finite-size scaling of ${\ensuremath{\chi}}_{F}$ or ${S}_{vN}^{\ensuremath{'}}$ to detect infinite-order quantum phase transitions and discusses the efficiency and accuracy of the two methods.

DOI
10.1103/physrevb.104.205112

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We study the fidelity and the entanglement entropy for the ground states of quantum systems that have infinite-order quantum phase transitions. In particular, we consider the quantum O(2) model with a spin-SSS truncation, where there is an infinite-order Gaussian (IOG) transition for S=1S=1S=1 and there are Berezinskii-Kosterlitz-Thouless (BKT) transitions for S\ensuremath≥2S\ensuremath{\ge}2S\ensuremath≥2. We show that the height of the peak in the fidelity susceptibility (\ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​) converges to a finite thermodynamic value as a power law of 1/L1/L1/L for the IOG transition and as 1/ln(L)1/ln(L)1/ln(L) for BKT transitions. The peak position of \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ resides inside the gapped phase for both the IOG transition and BKT transitions. On the other hand, the derivative of the block entanglement entropy with respect to the coupling constant (SvN\ensuremath′{S}_{vN}^{\ensuremath{'}}SvN\ensuremath′​) has a peak height that diverges as ln2(L){ln}^{2}(L)ln2(L) for S=1S=1S=1 and ln3(L){ln}^{3}(L)ln3(L) for S\ensuremath≥2S\ensuremath{\ge}2S\ensuremath≥2 and can be used to locate both kinds of transitions accurately. We include higher-order corrections for finite-size scalings and obtain the value of the central charge consistent with c=1c=1c=1 predicted by conformal field theory. The crossing point of \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ between different system sizes is at the IOG point for S=1S=1S=1 but is inside the gapped phase for S\ensuremath≥2S\ensuremath{\ge}2S\ensuremath≥2, while those of SvN\ensuremath′{S}_{vN}^{\ensuremath{'}}SvN\ensuremath′​ are at the phase-transition points for all SSS truncations. Our work elaborates on how to use the finite-size scaling of \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ or SvN\ensuremath′{S}_{vN}^{\ensuremath{'}}SvN\ensuremath′​ to detect infinite-order quantum phase transitions and discusses the efficiency and accuracy of the two methods.
We clarify the long-standing controversy concerning the behavior of the ground-state fidelity in the vicinity of a quantum phase transition of the Berezinskii-Kosterlitz-Thouless type in one-dimensional systems. Contrary to the prediction based on the Gaussian approximation of the Luttinger-liquid approach, it is shown that the fidelity susceptibility does not diverge at the transition but has a cusplike peak ${\ensuremath{\chi}}_{c}\ensuremath{-}\ensuremath{\chi}(\ensuremath{\lambda})\ensuremath{\sim}\sqrt{|{\ensuremath{\lambda}}_{c}\ensuremath{-}\ensuremath{\lambda}|}$, where $\ensuremath{\lambda}$ is a parameter driving the transition and ${\ensuremath{\chi}}_{c}$ is the peak value at the transition point $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$. Numerical claims of the logarithmic divergence of fidelity susceptibility with the system size (or temperature) are explained by logarithmic corrections due to marginal operators, which is supported by numerical calculations for large systems.
The potential of fidelity in accurately detecting unconventional quantum phase transitions (QPTs) remains an open question. Specific attention is directed towards high-order phase transitions, including the infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition and the Gaussian transition. In this study, we extend the concept of fidelity susceptibility to fixed-point fidelity susceptibility in the thermodynamic limit and assess this measure using tensor-network algorithms. Our results demonstrate that the fixed-point fidelity susceptibility effectively identifies the BKT and Gaussian phase transitions, a capability lacking in the traditional finite-fidelity approach. Furthermore, we establish a correlation between the fixed-point fidelity method and the entanglement method, demonstrating their equivalence. This research not only confirms the efficacy of fidelity in detecting QPTs but also enhances the understanding of fidelity-related methodologies.
In the present work, we study the fidelity susceptibility and the entanglement entropy in an antiferromagnetic spin-1 chain with additional next-nearest neighbor interactions and three-site interactions, which are given by H=(J1SiSi+1+ J2SiSi+2)+[J3(SiSi+1)(Si+1Si+2)+ h.c.]. By using the density matrix renormalization group method, the ground-state properties of the system are calculated with very high accuracy. We investigate the effect of the three-site interaction J3 on the fidelity susceptibility numerically, and then analyze its relation with the quantum phase transition (QPT). The fidelity measures the similarity between two states, and the fidelity susceptibility describes the associated changing rate. The QPT is intuitively accompanied by an abrupt change in the structure of the ground-state wave function, so generally a peak of the fidelity susceptibility indicates a QPT and the location of the peak denotes the critical point. For the case of J2=0, a peak of the fidelity susceptibility is found by varying J3, and the height of the peak grows as the system size increases. The location of the peak shifts to a slightly lower J3 up to a particular value as the system size increases. Through a finite size scaling, the critical point J3c=0.111 of the QPT from the Haldane spin liquid to the dimerized phase is identified. We also study the effect of the three-site interaction on the entanglement entropy between the right half part and the rest. It is noted that the peak of the entanglement entropy does not coincide with the critical point. Instead, the critical point is determined by the position at which the first-order derivative of the entanglement entropy takes its minimum, since a second-order QPT is signaled by the first derivative of density matrix element. Moreover, the entanglement entropy disappears when J3=1/6, which corresponds to the size-independent Majumdar-Ghosh point. The positions of quantum critical points extracted from these two quantum information observables agree well with those obtained by the string order parameters, which characterizes the topological order in the Haldane phase. Secondly, we also study the case of J20, and obtain the critical points by both the fidelity susceptibility and the entanglement entropy. Finally we provide a ground-state phase diagram of the system. To sum up, the quantum information observables are effective tools for detecting diverse QPTs in spin-1 models.
We study slightly generalized quantum fidelity susceptibilities where the differential change in the fidelity is measured with respect to a different term than the one used for driving the system toward a quantum phase transition. As a model system we use the spin-1/21/21/2 J1{J}_{1}J1​-J2{J}_{2}J2​ antiferromagnetic Heisenberg chain. For this model, we study three fidelity susceptibilities, ${\ensuremath{\chi}}_{\ensuremath{\rho}}$, ${\ensuremath{\chi}}_{D}$, and ${\ensuremath{\chi}}_{\mathrm{AF}}$, which are related to the spin stiffness, the dimer order, and antiferromagnetic order, respectively. All these ground-state fidelity susceptibilities are sensitive to the phase diagram of the J1{J}_{1}J1​-J2{J}_{2}J2​ model. We show that they all can accurately identify a quantum critical point in this model occurring at ${J}_{2}^{c}\ensuremath{\sim}0.241{J}_{1}$ between a gapless Heisenberg phase for J2<J2c{J}_{2}<{J}_{2}^{c}J2​<J2c​ and a dimerized phase for J2>J2c{J}_{2}>{J}_{2}^{c}J2​>J2c​. This phase transition, in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by a marginal operator and is therefore particularly difficult to observe.
We consider a spin ladder model which is known to have matrix product states as exact ground states with spin liquid characteristics. The model has two critical-point transitions at the parameter values u=0u=0u=0 and $\ensuremath{\infty}$. We study the variation of entanglement and fidelity measures in the ground states as a function of uuu and specially look for signatures of quantum phase transitions at u=0u=0u=0 and $\ensuremath{\infty}$. The two different entanglement measures used are S(i)S(i)S(i) (the single-site von Neumann entropy) and S(i,j)S(i,j)S(i,j) (the two-body entanglement). At the quantum critical point (QCP) $u=\ensuremath{\infty}$, the entanglement measure EEE [=S(i),S(i,j)][=S(i),S(i,j)][=S(i),S(i,j)] vanishes but remains nonzero at the other QCP u=0u=0u=0. The first and second derivatives of EEE with respect to the parameter uuu and the entanglement length associated with S(i,j)S(i,j)S(i,j) are further calculated to identify special features, if any, near the QCPs. We further determine the GS fidelity FFF and a quantity ln∣D∣\text{ln}|D|ln∣D∣ related to the second derivative of FFF and show that these quantities calculated for finite-sized systems are good indicators of QPTs occurring in the infinite system.
We investigate Berezinskii-Kosterlitz-Thouless (BKT) transitions in one-dimensional Rydberg chains, where commensurate critical regimes associated with the melting of crystalline orders with period larger than five and incommensurate floating phases are both described by Luttinger liquid theory. The central quantity is the Luttinger liquid parameter KKK, which characterizes the universal low-energy theory and controls the relevance of perturbations driving BKT transitions. We extract KKK using Friedel oscillations and the recently developed crosscap method introduced in Phys. Rev. Lett. 134, 076501 (2025). As benchmarks, we first apply the crosscap method to a Z3\mathbb{Z}_3Z3​ dual hard-core boson chain and a spin-1 XY chain with single-ion anisotropy, obtaining BKT transition points consistent with previous results after finite-size extrapolation. We then compute KKK in the Rydberg chain along lines with fixed correlation-oscillation period near BKT transitions. Along the commensurate period-five line, the critical values of KKK predicted by sine-Gordon theory reveal two BKT transitions separating the disordered phase, the critical phase, and the Z5\mathbb{Z}_5Z5​ crystalline phase. The results from Friedel oscillations and the crosscap method agree with each other and are further supported by energy-gap scaling and Binder-cumulant analysis. To obtain reliable values of KKK from Friedel oscillations, we use a multi-harmonic fitting scheme throughout the analysis. Along incommensurate lines, the BKT points obtained from Friedel oscillations agree with those extracted from entanglement entropy. Finally, we show that the values of KKK obtained from the two methods are mutually consistent inside the incommensurate floating phase.