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arXiv:0803.2114 · Physical Review A

Entanglement and fidelity signatures of quantum phase transitions in spin liquid models

Amit Tribedi, Indrani Bose

Published Mar 5, 2008• 14 citations
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Abstract

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 $u=0$ and $\ensuremath{\infty}$. We study the variation of entanglement and fidelity measures in the ground states as a function of $u$ and specially look for signatures of quantum phase transitions at $u=0$ and $\ensuremath{\infty}$. The two different entanglement measures used are $S(i)$ (the single-site von Neumann entropy) and $S(i,j)$ (the two-body entanglement). At the quantum critical point (QCP) $u=\ensuremath{\infty}$, the entanglement measure $E$ $[=S(i),S(i,j)]$ vanishes but remains nonzero at the other QCP $u=0$. The first and second derivatives of $E$ with respect to the parameter $u$ and the entanglement length associated with $S(i,j)$ are further calculated to identify special features, if any, near the QCPs. We further determine the GS fidelity $F$ and a quantity $\text{ln}|D|$ related to the second derivative of $F$ and show that these quantities calculated for finite-sized systems are good indicators of QPTs occurring in the infinite system.

DOI
10.1103/physreva.77.032307

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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∞\ensuremath{\infty}\ensuremath∞. 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∞\ensuremath{\infty}\ensuremath∞. 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∞u=\ensuremath{\infty}u=\ensuremath∞, 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.
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We consider a two-chain, spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg spin ladder in an external magnetic field HHH. The spin ladder is known to undergo second-order quantum phase transitions (QPTs) at two critical values Hc1{H}_{c1}Hc1​ and Hc2{H}_{c2}Hc2​ of the magnetic field. There are now known examples of strongly coupled (where the rung exchange interaction is much stronger than the nearest-neighbor intrachain exchange interaction) organic ladder compounds in which QPTs have been experimentally observed. In this paper, we investigate whether well-known bipartite entanglement meaures such as one-site von Neumann entropy, two-site von Neumann entropy, and concurrence develop special features close to the quantum critical points. As suggested by an earlier theorem, the first derivatives of the measures with respect to magnetic field are expected to diverge as $H\ensuremath{\rightarrow}{H}_{c1}$ and $H\ensuremath{\rightarrow}{H}_{c2}$. Based on numerical diagonalization data and a mapping of the strongly coupled ladder Hamiltonian onto the XXZXXZXXZ chain Hamiltonian, for which several analytical results are known, we find that the derivatives of the entanglement measures diverge as $H\ensuremath{\rightarrow}{H}_{c2}$ but remain finite as $H\ensuremath{\rightarrow}{H}_{c1}$. The reason for this discrepancy is analyzed. We further calculate two recently proposed quantum information theoretic measures, the reduced fidelity and reduced fidelity susceptibility, and show that these measures provide appropriate signatures of the QPTs occuring at the critical points H=Hc1H={H}_{c1}H=Hc1​ and H=Hc2H={H}_{c2}H=Hc2​.
We consider a one-parameter family of matrix-product states of spin-1 particles on a periodic chain and study in detail the entanglement properties of such a state. In particular, we calculate exactly the entanglement of one site with the rest of the chain, and the entanglement of two distant sites with each other, and show that the derivative of both these properties diverge when the parameter ggg of the states passes through a critical point. Such a point can be called a point of quantum phase transition, since at this point the character of the matrix-product state, which is the ground state of a Hamiltonian, changes discontinuously. We also study the finite size effects and show how the entanglement depends on the size of the chain. This later part is relevant to the field of quantum computation where the problem of initial state preparation in finite arrays of qubits or qutrits is important. It is also shown that the entanglement of two sites have scaling behavior near the critical point.
According to our scheme to construct quantum phase transitions (QPTs) in spin chain systems with matrix product ground states, we first successfully combine matrix product state (MPS) QPTs with spontaneous symmetry breaking. For a concrete model, we take into account a kind of MPS QPTs accompanied by spontaneous parity breaking, though for either side of the critical point the GS is typically unique, and show that the kind of MPS QPTs occur only in the thermodynamic limit and are accompanied by the appearance of singularities, diverging correlation length, vanishing energy gap and the entanglement entropy of a half-infinite chain not only staying finite but also whose first derivative discontinuous.
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.
Utilizing the matrix product formalism, we have studied the variation of entanglement and fidelity measures in the MP ground states of a generic anisotropic spin 2 chain with nearest neighbour interactions and shared symmetries. These MP states represent the exact Ground State solutions of the system and display distinctive characteristics determined by a maximum of three parameters exhibiting one antiferromagnetic Haldane phase (AFMH), together with a secondary weak antiferromagnetic (WAFM) and a weak ferromagnetic phase (WFM). Using Transfer Matrix Method (TM), we have obtained nontrivial features in the variation of Quantum Correlation measures and QIT Measures around the VBS point for an acritical choice of the parameters, and a Quantum Critical Point, for a critical choice of the parameters, of the AFMH spin liquid phase of the system.