Ground-state fidelity and bipartite entanglement in the Bose-Hubbard model

Information geometry and synchronization phase transition in the Kuramoto model

We discuss how the synchronization in the Kuramoto model can be treated in terms of information geometry. We argue that the Fisher information is sensitive to synchronization transition; specifically, components of the Fisher metric diverge at the critical point. Our approach is based on the recently proposed relation between the Kuramoto model and geodesics in hyperbolic space.

Quantum analog of the maximum power transfer theorem

We discover the quantum analog of the well-known classical maximum power transfer theorem. Our theoretical framework considers the continuous steady-state problem of coherent energy transfer through an N-node bosonic network coupled to an external dissipative load. We present an exact solution for optimal power transfer in the form of the maximum power transfer theorem known in the design of electrical circuits. Furthermore, we introduce the concept of quantum impedance matching with Thevenin equivalent networks, which are shown to be exact analogs to their classical counterparts. Our results are applicable to both ordered and disordered quantum networks with graph-like structures ranging from nearest-neighbor to all-to-all connectivities. This work points towards universal design principles adapting ideas from the classical regime to the quantum domain for various quantum optical applications in energy-harvesting, wireless power transfer, and energy transduction.

Information theoretical and thermodynamic view of the excited-state density functional theory of Coulomb systems

Excited states of Coulomb systems are studied within density functional theory with information theoretical quantities. The Ghosh-Berkowitz-Parr thermodynamic transcription is extended to excited states, and the concept of the local temperature is introduced. It is shown that extremization of information entropy or Fisher information results in a constant temperature. For Coulomb systems, there is a simple relation between the total energy and phase-space Fisher information. The phase-space fidelity between excited states is proportional to the position-space fidelity, with a factor of proportionality depending on total energies. The phase-space relative entropy is equal to the position-space relative entropy plus a term depending only on the total energies. The relationship between the phase-space fidelity susceptibility and Fisher information is also presented.

Universal Scaling Theory of the Boundary Geometric Tensor in Disordered Metals

We investigate the finite-size scaling of the boundary quantum geometric tensor (QGT) numerically close to the Anderson localization transition in the presence of small external magnetic fields. The QGT exhibits universal scaling and reveals the crossover between the orthogonal and unitary critical states in weak random magnetic fields. The flow of the QGT near the critical points determines the critical exponents. Critical distributions of the QGT are universal and exhibit a remarkable isotropy even in a homogeneous magnetic field. We predict universal and isotropic Hall conductance fluctuations at the metal-insulator transition in an external magnetic field.

Many-Body Multifractality throughout Bosonic Superfluid and Mott Insulator Phases

We demonstrate many-body multifractality of the Bose-Hubbard Hamiltonian's ground state in Fock space, for arbitrary values of the interparticle interaction. Generalized fractal dimensions unambiguously signal, even for small system sizes, the emergence of a Mott insulator that cannot, however, be naively identified with a localized phase in Fock space. We show that the scaling of the derivative of any generalized fractal dimension with respect to the interaction strength encodes the critical point of the superfluid to the Mott insulator transition, and provides an efficient way to accurately estimate its position. We further establish that the transition can be quantitatively characterized by one single wave function amplitude from the exponentially large Fock space.

Fidelity susceptibility of the anisotropic XY model: The exact solution

We derive several closed-form expressions for the fidelity susceptibility (FS) of the anisotropic XY model in the transverse field. The basic idea lies in a partial fraction expansion of the expression so that all the terms are related to a simple fraction or its derivative. The critical points of the model are reiterated by the FS, demonstrating its validity for characterizing the phase transitions. Moreover, the critical exponents ν associated with the correlation length in both critical regions are successfully extracted by the standard finite-size scaling analysis.

Excited-state fidelity as a signal for the many-body localization transition in a disordered Ising chain

In this work, we use exact matrix diagonalization to explore the many-body localization (MBL) transitions in quantum Ising chains with disordered nearest-neighbour couplings, disordered next-nearest-neighbour couplings and disordered external fields. It is demonstrated that the fidelity can be used to characterize the interaction-driven MBL transition in this closed spin system in a manner that is consistent with previous analytical and numerical results. We compute the fidelity for high-energy many-body eigenstates, namely, the excited-state fidelity. It is demonstrated that disordered nearest-neighbour couplings, disordered next-nearest-neighbour couplings and disordered external fields each have different effects on the MBL transition. Furthermore, we investigate the MBL transition of a quantum Ising chain with both disordered nearest-neighbour couplings and disordered next-nearest-neighbour couplings to see how these two types of disordered couplings drive the occurrence of the MBL transition.

Fidelity of the diagonal ensemble signals the many-body localization transition

In this work, we use exact matrix diagonalization to explore the many-body localization (MBL) transition in a random-field Heisenberg chain. We demonstrate that the fidelity and fidelity susceptibility can be utilized to characterize the interaction-driven many-body localization transition in this closed spin system which is in agreement with previous analytical and numerical results [S. Garnerone, N. T. Jacobson, S. Haas, and P. Zanardi, Phys. Rev. Lett. 102, 057205 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.057205; P. Zanardi and N. Paunkovic, Phys. Rev. E 74, 031123 (2006)PLEEE81539-375510.1103/PhysRevE.74.031123]. In particular, instead of ground-state fidelity, we test the fidelity between two diagonal ensembles related by a small parameter perturbation δh, it is special that here the parameter perturbation δh_{i} for each site are random variables like h_{i}. It shows that fidelity of the diagonal ensemble develop a pronounced drop at the transition. We utilize fidelity to estimate the critical disorder strength h_{c} for different system size, we get h_{c}∈ [2.5,3.9] and get a power-law decay with an exponent of roughly -1.49(2) for system size N, and can extrapolate h_{c}^{inf} of the infinite system is about 2.07 which all agree with a recent work by Huse and Pal, in which the MBL transition in the same model was predicted to be hc [2,4]. We also estimate the scaling of maximum of averaged fidelity susceptibility as a function of system size N, it shows a power law increase with an exponent of about 5.05(1).

Applications of fidelity measures to complex quantum systems

We revisit fidelity as a measure for the stability and the complexity of the quantum motion of single-and many-body systems. Within the context of cold atoms, we present an overview of applications of two fidelities, which we call static and dynamical fidelity, respectively. The static fidelity applies to quantum problems which can be diagonalized since it is defined via the eigenfunctions. In particular, we show that the static fidelity is a highly effective practical detector of avoided crossings characterizing the complexity of the systems and their evolutions. The dynamical fidelity is defined via the time-dependent wave functions. Focusing on the quantum kicked rotor system, we highlight a few practical applications of fidelity measurements in order to better understand the large variety of dynamical regimes of this paradigm of a low-dimensional system with mixed regular-chaotic phase space.

Entanglement Entropy across the Superfluid-Insulator Transition: A Signature of Bosonic Criticality

We study the entanglement entropy and entanglement spectrum of the paradigmatic Bose-Hubbard model, describing strongly correlated bosons on a lattice. The use of a controlled approximation-the slave-boson approach-allows us to study entanglement in all regimes of the model (and, most importantly, across its superfluid-Mott-insulator transition) at a minimal cost. We find that the area-law scaling of entanglement-verified in all the phases-exhibits a sharp singularity at the transition. The singularity is greatly enhanced when the transition is crossed at fixed, integer filling, due to a richer entanglement spectrum containing an additional gapless mode, which descends from the amplitude (Higgs) mode of the global excitation spectrum-while this mode remains gapped at the generic (commensurate-incommensurate) transition with variable filling. Hence, the entanglement properties contain a unique signature of the two different forms of bosonic criticality exhibited by the Bose-Hubbard model.

Nonergodic metallic and insulating phases of Josephson junction chains

Strictly speaking, the laws of the conventional statistical physics, based on the equipartition postulate [Gibbs J W (1902) Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics] and ergodicity hypothesis [Boltzmann L (1964) Lectures on Gas Theory], apply only in the presence of a heat bath. Until recently this restriction was believed to be not important for real physical systems because a weak coupling to the bath was assumed to be sufficient. However, this belief was not examined seriously until recently when the progress in both quantum gases and solid-state coherent quantum devices allowed one to study the systems with dramatically reduced coupling to the bath. To describe such systems properly one should revisit the very foundations of statistical mechanics. We examine this general problem for the case of the Josephson junction chain that can be implemented in the laboratory and show that it displays a novel high-temperature nonergodic phase with finite resistance. With further increase of the temperature the system undergoes a transition to the fully localized state characterized by infinite resistance and exponentially long relaxation.

Entanglement entropy and fidelity susceptibility in the one-dimensional spin-1 XXZ chains with alternating single-site anisotropy

We study the fidelity susceptibility in an antiferromagnetic spin-1 XXZ chain numerically. By using the density-matrix renormalization group method, the effects of the alternating single-site anisotropy D on fidelity susceptibility are investigated. Its relation with the quantum phase transition is analyzed. It is found that the quantum phase transition from the Haldane spin liquid to periodic Néel spin solid can be well characterized by the fidelity. Finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of second order. The results are confirmed by the second derivative of the ground-state energy. We also study the relationship between the entanglement entropy, the Schmidt gap and quantum phase transitions. Conclusions drawn from these quantum information observables agree well with each other.

A quantum fidelity study of the anisotropic next-nearest-neighbour triangular lattice Heisenberg model

Ground- and excited-state quantum fidelities in combination with generalized quantum fidelity susceptibilites, obtained from exact diagonalizations, are used to explore the phase diagram of the anisotropic next-nearest-neighbour triangular Heisenberg model. Specifically, the J'-J2 plane of this model, which connects the J1-J2 chain and the anisotropic triangular lattice Heisenberg model, is explored using these quantities. Through the use of a quantum fidelity associated with the first excited-state, in addition to the conventional ground-state fidelity, the BKT-type transition and Majumdar-Ghosh point of the J1-J2 chain (J'=0) are found to extend into the J'-J2 plane and connect with points on the J2=0 axis thereby forming bounded regions in the phase diagram. These bounded regions are then explored through the generalized quantum fidelity susceptibilities χρ, χ₁₂₀°, χD and χCAF which are associated with the spin stiffness, 120° spiral order parameter, dimer order parameter and collinear antiferromagnetic order parameter respectively. These quantities are believed to be extremely sensitive to the underlying phase and are thus well suited for finite-size studies. Analysis of the fidelity susceptibilities suggests that the J', J2≪J phase of the anisotropic triangular model is either a collinear antiferromagnet or possibly a gapless disordered phase that is directly connected to the Luttinger phase of the J1-J2 chain. Furthermore, the outer region is dominated by incommensurate spiral physics as well as dimer order.

Work and quantum phase transitions: quantum latency

We study the physics of quantum phase transitions from the perspective of nonequilibrium thermodynamics. For first-order quantum phase transitions, we find that the average work done per quench in crossing the critical point is discontinuous. This leads us to introduce the quantum latent work in analogy with the classical latent heat of first order classical phase transitions. For second order quantum phase transitions the irreversible work is closely related to the fidelity susceptibility for weak sudden quenches of the system Hamiltonian. We demonstrate our ideas with numerical simulations of first, second, and infinite order phase transitions in various spin chain models.

Complexity and instability of quantum motion near a quantum phase transition

We show that the number of harmonics of the Wigner function, recently proposed as a measure of quantum complexity, can also be used to characterize quantum phase transitions. The nonanalytic behavior of this quantity in the neighborhood of a quantum phase transition is illustrated by means of the Dicke model and is compared to two well-known measures of the (in)stability of quantum motion: the quantum Loschmidt echo and fidelity.

Three-site interacting spin chain in a staggered field: fidelity versus Loschmidt echo

We study the the ground state fidelity and the ground state Loschmidt echo of a three-site interacting XX chain in presence of a staggered field which exhibits special types of quantum phase transitions due to change in the topology of the Fermi surface, apart from quantum phase transitions from gapped to gapless phases. We find that, on one hand, the fidelity is able to detect only the boundaries separating the gapped from the gapless phase; it is completely insensitive to the phase transition from the two Fermi points region to the four Fermi points region lying within this gapless phase. On the other hand, the Loschmidt echo shows a dip only at a special point in the entire phase diagram and hence fails to detect any quantum phase transition associated with the present model. We provide appropriate arguments in support of this anomalous behavior.

Quantum phase transition in fiber-coupled quantum networks

We explore the quantum criticality in a one-dimensional quantum network, where adjacent nodes are connected by a fiber and each node consists of a cavity doped with N two-level atoms. We investigate the phase diagrams for cases of different atom numbers on resonance by making use of the mean-field approximation and find that both the fidelity and entanglement entropy for the ground state have dramatic changes near the critical points. Finally, the photon population imbalance of adjacent cavities is analytically explored via the two-node quantum network as the forerunner of phase transition.

Quantum phases of dimerized and frustrated Heisenberg spin chains with s = 1/2, 1 and 3/2: an entanglement entropy and fidelity study

We study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with frustration (next-nearest-neighbor interaction J2) and dimerization (δ). In particular, we analyze the behaviors of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2-δ plane. All the calculations in this work are based on numerical exact diagonalizations of finite systems.

Scaling behavior for a class of quantum phase transitions

We show that for quantum phase transitions with a single bosonic zero mode at the critical point, like the Dicke model and the Lipkin-Meshkov-Glick model, metric quantities such as fidelity, that is, the overlap between two ground states corresponding to two values λ(1) and λ(2) of the controlling parameter λ, depend only on the ratio η = (λ(1) -λ(c))/(λ(2) -λ(c)), where λ = λ(c) at the critical point. This scaling property is valid also for time-dependent quantities such as the Loschmidt echo, provided time is measured in units of the inverse frequency of the critical mode.

Quantum phases with differing computational power

The observation that concepts from quantum information has generated many alternative indicators of quantum phase transitions hints that quantum phase transitions possess operational significance with respect to the processing of quantum information. Yet, studies on whether such transitions lead to quantum phases that differ in their capacity to process information remain limited. Here we show that there exist quantum phase transitions that cause a distinct qualitative change in our ability to simulate certain quantum systems under perturbation of an external field by local operations and classical communication. In particular, we show that in certain quantum phases of the XY model, adiabatic perturbations of the external magnetic field can be simulated by local spin operations, whereas the resulting effect within other phases results in coherent non-local interactions. We discuss the potential implications to adiabatic quantum computation, where a computational advantage exists only when adiabatic perturbation results in coherent multi-body interactions.

Dipolar molecules in optical lattices

We study the extended Bose-Hubbard model describing an ultracold gas of dipolar molecules in an optical lattice, taking into account all on-site and nearest-neighbor interactions, including occupation-dependent tunneling and pair tunneling terms. Using exact diagonalization and the multiscale entanglement renormalization ansatz, we show that these terms can destroy insulating phases and lead to novel quantum phases. These considerable changes of the phase diagram have to be taken into account in upcoming experiments with dipolar molecules.

Quantum mechanics in metric space: wave functions and their densities

Hilbert space combines the properties of two different types of mathematical spaces: vector space and metric space. While the vector-space aspects are widely used, the metric-space aspects are much less exploited. Here we show that a suitable metric stratifies Fock space into concentric spheres on which maximum and minimum distances between states can be defined and geometrically interpreted. Unlike the usual Hilbert-space analysis, our results apply also to the reduced space of only ground states and to that of particle densities, which are metric, but not Hilbert, spaces. The Hohenberg-Kohn mapping between densities and ground states, which is highly complex and nonlocal in coordinate description, is found, for three different model systems, to be simple in metric space, where it becomes a monotonic and nearly linear mapping of vicinities onto vicinities.

Finite-temperature fidelity susceptibility for one-dimensional quantum systems

We calculate the fidelity susceptibility χf for the Luttinger model and show that there is a universal contribution linear in temperature T (or inverse length 1/L). Furthermore, we develop an algorithm--based on a lattice path integral approach--to calculate the fidelity F(T) in the thermodynamic limit for one-dimensional quantum systems. We check the Luttinger model predictions by calculating χf(T) analytically for free spinless fermions and numerically for the XXZ chain. Finally, we study χf at the two phase transitions in this model.

Entanglement and the Mott insulator--superfluid phase transition in bosonic atom chains

We analyze the developing of bipartite and multipartite entanglement through the Mott insulator-superfluid quantum phase transition. Starting from a Mott insulator state, where a filling factor ν = N/M = 1 per lattice site is considered, we derive an expression for a completely connected graph configuration of bosons and show how bipartite and multipartite entanglement evolve through the phase transition predicted in previous works. We show how, in the transition, bipartite entanglement is distributed through the system respecting monogamous relations.

Ground-state properties of the two-site Hubbard-Holstein model: an exact solution

We study the two-site Hubbard-Holstein model by using an extended phonon coherent state. For the nontrivial singlet bipolarons, the double occupancy probability, the fidelity and the entanglement entropy are calculated to characterize the ground-state properties in both two-site and single-site bipolaron-dominated regimes. We use the localized minimum of the fidelity to define a crossover and plot the bipolaron phase diagram, which separates the large and small entanglement region. Furthermore, the relation between the bipolaron entanglement and the correlation functions demonstrates that the large entanglement corresponds to the large magnitude of lattice deformations induced by electrons.

Pseudoclassical theory for fidelity of nearly resonant quantum rotors

Using a semiclassical ansatz we analytically predict for the fidelity of delta-kicked rotors the occurrence of revivals and the disappearance of intermediate revival peaks arising from the breaking of a symmetry in the initial conditions. A numerical verification of the predicted effects is given and experimental ramifications are discussed.

Many-body reduced fidelity susceptibility in Lipkin-Meshkov-Glick model

We study the reduced fidelity susceptibility chi(r) for an M -body subsystem of an N -body Lipkin-Meshkov-Glick model with tau=M/N fixed. The reduced fidelity susceptibility can be viewed as the response of subsystem to a certain parameter. In noncritical region, the inner correlation of the system is weak, and chi(r) behaves similar with the global fidelity susceptibility chi(g), the ratio eta=chi(r)/chi(g) depends on tau but not on N. However, at the critical point, the inner correlation tends to be divergent, and we find chi(r) approaches chi(g) with increasing the N. It is interesting to note that, eta=1 in the thermodynamic limit, which means the susceptibilities of the local and global system are the same. Finally, we make numerical computations, and they are in perfect agreement with the analytical predictions.

Quantum phase transitions in a two-dimensional quantum XYX model: ground-state fidelity and entanglement

A systematic analysis is performed for quantum phase transitions in a two-dimensional anisotropic spin-1/2 antiferromagnetic XYX model in an external magnetic field. With the help of an innovative tensor network algorithm, we compute the fidelity per lattice site to demonstrate that the field-induced quantum phase transition is unambiguously characterized by a pinch point on the fidelity surface, marking a continuous phase transition. We also compute an entanglement estimator, defined as a ratio between the one-tangle and the sum of squared concurrences, to identify both the factorizing field and the critical point, resulting in a quantitative agreement with quantum Monte Carlo simulation. In addition, the local order parameter is "derived" from the tensor network representation of the system's ground-state wave functions.

Quantum fidelity and thermal phase transitions

We study the quantum fidelity approach to characterize thermal phase transitions. Specifically, we focus on the mixed-state fidelity induced by a perturbation in temperature. We consider the behavior of fidelity in two types of second-order thermal phase transitions (based on the type of nonanaliticity of free energy), and we find that usual fidelity criteria for identifying critical points is more applicable to the case of lambda transitions (divergent second derivatives of free energy). Our study also reveals that for fixed perturbations, the sensitivity of fidelity at high temperatures (where thermal fluctuations wash out information about the transition) is reduced. From the connection to thermodynamical quantities we propose slight variations to the usual fidelity approach that allow us to overcome these limitations. In all cases we find that fidelity remains a good precriterion for testing thermal phase transitions, and we use it to analyze the nonzero temperature phase diagram of the Lipkin-Meshkov-Glick model.

Reduced fidelity susceptibility and its finite-size scaling behaviors in the Lipkin-Meshkov-Glick model

We derive a general formula for the reduced fidelity susceptibility when the reduced density matrix is 2x2 block diagonal. By using this result and a continuous unitary transformation, we study finite-size scaling of the reduced fidelity susceptibility in the Lipkin-Meshkov-Glick model. It is found that it can well characterize quantum phase transitions, that is, we can extract information about quantum phase transitions from only the fidelity of a subsystem, which is of practical use in experiments.

Quantum criticality of the Lipkin-Meshkov-Glick model in terms of fidelity susceptibility

We study the critical properties of the Lipkin-Meshkov-Glick model in terms of the fidelity susceptibility. By using the Holstein-Primakoff transformation, we obtain explicitly the critical exponent of the fidelity susceptibility around the second-order quantum phase transition point. Our results provide a rare analytic case for the fidelity susceptibility in describing the universality class in quantum critical behavior. The different critical exponents in two phases are nontrivial results, indicating that the fidelity susceptibility is not always extensive.

Concurrence and fidelity of a Bose-Fermi mixture in a one-dimensional optical lattice

We study the ground-state fidelity and entanglement of a Bose-Fermi mixture loaded in a one-dimensional optical lattice. It is found that the fidelity is able to signal quantum phase transitions between the Luttinger liquid phase, the density-wave phase, and the phase separation state of the system, and the concurrence, as a measure of the entanglement, can be used to signal the transition between the density-wave phase and the Ising phase.

Ground state fidelity from tensor network representations

For any D-dimensional quantum lattice system, the fidelity between two ground state many-body wave functions is mapped onto the partition function of a D-dimensional classical statistical vertex lattice model with the same lattice geometry. The fidelity per lattice site, analogous to the free energy per site, is well defined in the thermodynamic limit and can be used to characterize the phase diagram of the model. We explain how to compute the fidelity per site in the context of tensor network algorithms, and demonstrate the approach by analyzing the two-dimensional quantum Ising model with transverse and parallel magnetic fields.

Macroscopic distinguishability between quantum states defining different phases of matter: fidelity and the Uhlmann geometric phase

We study the fidelity approach to quantum phase transitions (QPTs) and apply it to general thermal phase transitions (PTs). We analyze two particular cases: The Stoner-Hubbard itinerant electron model of magnetism and the BCS theory of superconductivity. In both cases we show that the sudden drop of the mixed state fidelity marks the line of the phase transition. We conduct a detailed analysis of the general case of systems given by mutually commuting Hamiltonians, where the nonanalyticity of the fidelity is directly related to the nonanalyticity of the relevant response functions (susceptibility and heat capacity), for the case of symmetry-breaking transitions. Further, on the case of BCS theory of superconductivity, given by mutually noncommuting Hamiltonians, we analyze the structure of the system's eigenvectors in the vicinity of the line of the phase transition showing that their sudden change is quantified by the emergence of a generically nontrivial Uhlmann mixed state geometric phase.

Fidelity and quantum phase transition for the Heisenberg chain with next-nearest-neighbor interaction

In this paper, we investigate the fidelity for the Heisenberg chain with the next-nearest-neighbor interaction (or the J1-J2 model) and analyze its connections with quantum phase transition. We compute the fidelity between the ground states and find that the phase transition point of the J1-J2 model cannot be well characterized by the ground-state fidelity for finite-size systems. Instead, we introduce and calculate the fidelity between the first excited states. Our results show that the quantum transition can be well characterized by the fidelity of the first excited state even for a small-size system.

Quantum critical scaling of the geometric tensors

Berry phases and the quantum-information theoretic notion of fidelity have been recently used to analyze quantum phase transitions from a geometrical perspective. In this Letter we unify these two approaches showing that the underlying mechanism is the critical singular behavior of a complex tensor over the Hamiltonian parameter space. This is achieved by performing a scaling analysis of this quantum geometric tensor in the vicinity of the critical points. In this way most of the previous results are understood on general grounds and new ones are found. We show that criticality is not a sufficient condition to ensure superextensive divergence of the geometric tensor, and state the conditions under which this is possible. The validity of this analysis is further checked by exact diagonalization of the spin-1/2 XXZ Heisenberg chain.

Fidelity, dynamic structure factor, and susceptibility in critical phenomena

Motivated by the growing importance of fidelity in quantum critical phenomena, we establish a general relation between the fidelity and structure factor of the driving term in a Hamiltonian through the concept of fidelity susceptibility. Our discovery, as shown by some examples, facilitates the evaluation of fidelity in terms of susceptibility using well-developed techniques, such as density matrix renormalization group for the ground state, or Monte Carlo simulations for the states in thermal equilibrium.