Fidelity Mechanics: Analogues of the Four Thermodynamic Laws and Landauer's Principle.

Fidelity mechanics is formalized as a framework for investigating critical phenomena in quantum many-body systems. Fidelity temperature is introduced for quantifying quantum fluctuations, which, together with fidelity entropy and fidelity internal energy, constitute three basic state functions in fidelity mechanics, thus enabling us to formulate analogues of the four thermodynamic laws and Landauer's principle at zero temperature. Fidelity flows, which are irreversible, are defined and may be interpreted as an alternative form of renormalization group flows. Thus, fidelity mechanics offers a means to characterize both stable and unstable fixed points: divergent fidelity temperature for unstable fixed points and zero-fidelity temperature and (locally) maximal fidelity entropy for stable fixed points. In addition, fidelity entropy behaves differently at an unstable fixed point for topological phase transitions and at a stable fixed point for topological quantum states of matter. A detailed analysis of fidelity mechanical-state functions is presented for six fundamental models-the quantum spin-1/2 XY model, the transverse-field quantum Ising model in a longitudinal field, the quantum spin-1/2 XYZ model, the quantum spin-1/2 XXZ model in a magnetic field, the quantum spin-1 XYZ model, and the spin-1/2 Kitaev model on a honeycomb lattice for illustrative purposes. We also present an argument to justify why the thermodynamic, psychological/computational, and cosmological arrows of time should align with each other, with the psychological/computational arrow of time being singled out as a master arrow of time.

Critical exponents of block-block mutual information in one-dimensional infinite lattice systems.

We study the mutual information between two lattice blocks in terms of von Neumann entropies for one-dimensional infinite lattice systems. Quantum q-state Potts model and transverse-field spin-1/2 XY model are considered numerically by using the infinite matrix product state approach. As a system parameter varies, block-block mutual information exhibit singular behaviors that enable us to identify the critical points for the quantum phase transitions. As happens with von Neumann entanglement entropy of single block, at critical points, block-block mutual information for two adjacent blocks show a logarithmic leading behavior with increasing the size of the blocks, which yields the central charge c of the underlying conformal field theory, as it should be. It seems that two disjoint blocks show a similar logarithmic growth of the mutual information as a characteristic property of critical systems but the proportional coefficients of the logarithmic term are very different from the central charges. As the separation between the two lattice blocks increases, the mutual information reveals a consistent power-law decaying behavior for various truncation dimensions and lattice-block sizes. The critical exponent of block-block mutual information in the thermodynamic limit is estimated by extrapolating the exponents of power-law decaying regions for finite truncation dimensions. For a given lattice-block size ℓ, the critical exponents for the same universality classes seem to have very close values each other. Whereas the critical exponents have different values to a degree of distinction for the different universality classes. As the lattice-block size becomes bigger, the critical exponent becomes smaller. We discuss a relation between the exponents of block-block mutual information and correlation with the Shatten one-norm of block-block correlation.

Universal order parameters and quantum phase transitions: a finite-size approach.

We propose a method to construct universal order parameters for quantum phase transitions in many-body lattice systems. The method exploits the H-orthogonality of a few near-degenerate lowest states of the Hamiltonian describing a given finite-size system, which makes it possible to perform finite-size scaling and take full advantage of currently available numerical algorithms. An explicit connection is established between the fidelity per site between two H-orthogonal states and the energy gap between the ground state and low-lying excited states in the finite-size system. The physical information encoded in this gap arising from finite-size fluctuations clarifies the origin of the universal order parameter. We demonstrate the procedure for the one-dimensional quantum formulation of the q-state Potts model, for q = 2, 3, 4 and 5, as prototypical examples, using finite-size data obtained from the density matrix renormalization group algorithm.

Degenerate ground states and multiple bifurcations in a two-dimensional q-state quantum Potts model.

We numerically investigate the two-dimensional q-state quantum Potts model on the infinite square lattice by using the infinite projected entangled-pair state (iPEPS) algorithm. We show that the quantum fidelity, defined as an overlap measurement between an arbitrary reference state and the iPEPS ground state of the system, can detect q-fold degenerate ground states for the Z_{q} broken-symmetry phase. Accordingly, a multiple bifurcation of the quantum ground-state fidelity is shown to occur as the transverse magnetic field varies from the symmetry phase to the broken-symmetry phase, which means that a multiple-bifurcation point corresponds to a critical point. A (dis)continuous behavior of quantum fidelity at phase transition points characterizes a (dis)continuous phase transition. Similar to the characteristic behavior of the quantum fidelity, the magnetizations, as order parameters, obtained from the degenerate ground states exhibit multiple bifurcation at critical points. Each order parameter is also explicitly demonstrated to transform under the Z_{q} subgroup of the symmetry group of the Hamiltonian. We find that the q-state quantum Potts model on the square lattice undergoes a discontinuous (first-order) phase transition for q=3 and q=4 and a continuous phase transition for q=2 (the two-dimensional quantum transverse Ising model).

Universal construction of order parameters for translation-invariant quantum lattice systems with symmetry-breaking order.

For any translation-invariant quantum lattice system with a symmetry group G, we propose a practical and universal construction of order parameters which identify quantum phase transitions with symmetry-breaking order. They are defined in terms of the fidelity between a ground state and its symmetry-transformed counterpart, and are computed through tensor network representations of the ground-state wave function. To illustrate our scheme, we consider three quantum systems on an infinite lattice in one spatial dimension, namely, the quantum Ising model in a transverse magnetic field, the quantum spin-1/2XYX model in an external magnetic field, and the quantum spin-1 XXZ model with single-ion anisotropy. All these models have symmetry group Z(2) and exhibit broken-symmetry phases. We also discuss the role of the order parameters in identifying factorized states.

Spontaneous symmetry breaking and bifurcations in ground-state fidelity for quantum lattice systems.

Spontaneous symmetry breaking occurs in a system when its Hamiltonian possesses a certain symmetry, whereas the ground-state wave functions do not preserve it. This provides such a scenario that a bifurcation, which breaks the symmetry, occurs when some control parameter crosses its critical value. It is unveiled that the ground-state fidelity per lattice site exhibits such a bifurcation for quantum lattice systems undergoing quantum phase transitions. The significance of this result lies in the fact that the ground-state fidelity per lattice site is universal, in the sense that it is model independent, in contrast to (model-dependent) order parameters. This fundamental quantity may be computed by exploiting the developed tensor network algorithms on infinite-size lattices. We illustrate the scheme in terms of the quantum Ising model in a transverse magnetic field and the spin-1/2 XYX model in an external magnetic field on an infinite-size lattice in one spatial dimension.

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.

Is the nature of itinerant ferromagnetism playing a role in the competition between spin polarization and singlet pair correlations?

We consider the competition between spin singlet pairing and itinerant ferromagnetism whose magnetization is yielded by a relative shift of the bands with opposite spin polarization or by asymmetric spin-dependent bandwidths. Within the framework of the exact solution of an extended version of the reduced BCS model, the structure of the coexisting state is shown to have general features that are not related to the character of the ferromagnetism. The role of different types of ferromagnet is then investigated for the proximity effect in a system made of a bilayer junction with a spin singlet superconductor interfaced with a ferromagnet in the clean limit. We show that the qualitative behaviour of the proximity effect does not depend on the nature of the ferromagnetism. Differences emerge at the borderline with the half-metallic regime. For the spin-dependent bandwidth type of ferromagnetism the pairing amplitude exhibits an oscillating behaviour until the density of the minority spin carrier becomes almost zero. The crossover from an oscillating to an exponentially damped profile occurs away from the half-metallic limit when a spin exchange type ferromagnet is considered.

Exact solution for a trapped Fermi gas with population imbalance and BCS pairing.

The problem of a two-component Fermi gas in a harmonic trap, with an imbalanced population and a pairing interaction of zero total momentum, is mapped onto the exactly solvable reduced BCS model. For a one-dimensional trap, the complete ground state diagram is determined with various topological features in ground state energy spectra. In addition to the conventional two-shell density profile of a paired core and polarized outer wings, a three-shell structure as well as a double-peak superfluid distribution are unveiled.

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.

Ground-state entanglement of the BCS model.

The concept of local concurrence is used to quantify the entanglement between a single qubit and the remainder of a multiqubit system. For the ground state of the BCS model in the thermodynamic limit the set of local concurrences completely describes the entanglement. As a measure for the entanglement of the full system we investigate the average local concurrence (ALC). We find that the ALC satisfies a simple relation with the order parameter. We then show that for finite systems with a fixed particle number, a relation between the ALC and the condensation energy exposes a threshold coupling. Below the threshold, entanglement measures besides the ALC are significant.

Gauge fields, geometric phases, and quantum adiabatic pumps.

Quantum adiabatic pumping of charge and spin between two reservoirs (leads) has recently been demonstrated in nanoscale electronic devices. Pumping occurs when system parameters are varied in a cyclic manner and sufficiently slowly that the quantum system always remains in its ground state. We show that quantum pumping has a natural geometric representation in terms of gauge fields (both Abelian and non-Abelian) defined on the space of system parameters. Tunneling from a scanning tunneling microscope tip through a magnetic atom could be used to demonstrate the non-Abelian character of the gauge field.