Entanglement and Fidelity: Statics and Dynamics

Abstract

Herein, aspects of entanglement and fidelity and their use in condensed matter systems are briefly reviewed. Both static and time-dependent situations are considered. Different signatures of phases and phase transitions are discussed, including the dynamic aspects of the evolution across a critical point. Some emphasis is placed on the use of entanglement in phase transitions with no clear order parameters and no symmetry breaking.

1. Introduction

Symmetry plays an important role in physics. A specific example is in the framework of the Landau theory of phase transitions [1]. These are often associated with symmetry breaking, as one crosses from a high-temperature phase to a low-temperature and more ordered phase. Symmetry plays an important role in establishing the type of terms that are allowed in the Ginzburg–Landau functional that describes the vicinity of the phase transition point. The theory involves the identification of a quantity, the order parameter, that distinguishes the two phases, typically vanishing in the disordered phase and taking finite values in the ordered phase, either continuously or discontinuously. Due to fluctuations in low spatial dimensions, the Mermin–Wagner theorem [2] shows the absence of a stable ordered phase, in the sense that it is not possible to determine a stable local order parameter. Phase transitions still occur in some cases such as that of the Kosterlitz–Thouless transition [3,4], where no local-order parameter acquires a non-vanishing average value across the transition, but the phases are distinguished by the different asymptotic values of correlation functions. Topological phase transitions may also be seen as not fitting into Landau’s theory due to the absence of symmetry breaking at the transition. Moreover, topological systems [5] are generically understood as involving long-range correlations and different topological phases are usually characterized and distinguished by non-local quantities, the topological invariants, or by the existence of edge states. In topological phases, symmetry reappears, protecting the topological phases and defining universality classes such as in integer spin chains [6,7].

There is a complementarity between the concepts and methods of quantum information and the methods of condensed matter physics in the study of quantum phases and their phase transitions. This is particularly interesting to explore in the context of systems where an order parameter is not easy to determine or it may not be defined, such as in topological systems. In this context, the relevance of entanglement and its long-range nature may lead to new insights into condensed matter problems.

Entanglement has been used to characterize phases and entanglement differences have been used to detect transitions [8]. Examples include von Neumann entropy [9] and other entanglement measures. The central quantities are the density matrix and the reduced density matrix. Interesting information about the entanglement is obtained from the eigenvalues of the density matrix and from the entanglement between the different parts of a system, using the reduced density matrix. Summing up all the eigenvalues in appropriate ways, such as in the definition of the von Neumann entropy, or studying the set of eigenvalues (entanglement spectrum) in detail, has led to an interesting characterization of different phases [10]. Interesting information is also encoded in the eigenvectors themselves, which may be used to clarify the nature of a phase. It has recently been shown that combining the information on the eigenvectors and eigenvalues quantities that play, to some extent, a similar role to order parameters, which are called topological correlators [11], may be used to distinguish and characterize phases. Interacting systems are more complex and in general one expects complicated entanglement, which suggests that the use of quantum information techniques may play a significant role. In addition to the entanglement in a given state of the system, the state overlap, or the general fidelity [12], has emerged as a natural quantity to compare states. The distinguishability between states across a quantum phase transition using the fidelity also allows the detection of a transition.

The use of entanglement measures in condensed matter physics has been extensively considered [8], particularly in quantum phase transitions [13] including the use of fidelity [14]. Different information measures have been used [15,16,17,18,19,20,21] such as for a bipartite system, the concurrence [22], the mutual information [23,24,25], the negativity [26], or the Meyer–Wallach [27] and generalized global entanglement measures [28,29]. Furthermore, the cases of free electrons [30], bosons [31] and a BCS superconductor [32] have been considered.

The notion of fidelity between two pure ground states has been proposed as a method of detection of quantum phase transitions [12] and has been applied to several problems [14,33,34,35,36,37,38,39,40,41,42,43,44]. The more elusive Kosterlitz–Thouless transition was also successfully detected in the calculation of the fidelity susceptibility of the $XXZ$ spin chain [38,45] illustrating the power of the method in a case that does not fall in the Landau theory of phase transitions. The method was also applied to other systems such as the one-dimensional Hubbard model [46,47].

In the case of topological phases, such as in topological insulators and topological superconductors (TSCs), various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity and fidelity spectrum, have been used. Examples include quantum Hall phases, spin liquids, topological insulators and topological superconductors. Other systems were considered such as spin-$1/2$ particles on a torus [48], the toric code model, the quantum eight-vertex model [49], the spin honeycomb Kitaev model [50,51,52,53,54], as well as other spin systems [55,56].

The fidelity between pure states was generalized to the case of mixed states and was used as a signature of thermal phase transitions distinguishing states at finite temperatures [57,58,59]. The concept was also generalized to consider partial states, by showing that quantum phase transitions can be equally detected using reduced density matrices (of a subsystem of the full system) [41,60].

The utility of the quantum information measures is not limited to static or equilibrium situations. The dynamics of entangled systems has also been considered. In general, one finds entanglement growth in time-dependent phenomena. A suitable generalization of the fidelity concept to dynamic situations is the Loschmidt amplitude. Its time evolution has been shown to reflect some of the properties of the time evolution of the system. Indeed, it has been further shown that in a dynamical situation, it is possible to construct a dynamical order parameter that reflects the topological properties of the evolving system. These issues will be detailed ahead together with the study of the time evolution of the topological correlators, which are average values of the projectors to the dominant eigenvectors of suitable reduced density matrices of a given system. This provides further evidence of the significance and utility of quantum information measures, even in systems away from equilibrium.

In Section 2, some entanglement measures are discussed, such as the von Neumann entropy, mutual information, negativity and concurrence. Furthermore, the entanglement spectrum is considered. In Section 3, the usefulness of entanglement in topological phases is specified, particularly the notion of the topological entanglement entropy as a signature of the topological correlations in a system. In Section 4, the notion of topological correlators that arises from the analysis of the reduced density matrix eigenproblem is briefly reviewed, and the ability of the topological correlators to distinguish topological phases is shown. In Section 5, the fidelity between different density matrices, the fidelity susceptibility and its scaling near a critical point are considered. The partial state fidelity and the notion of the fidelity spectrum (eigenvalues of the fidelity operator, whose trace is the fidelity itself) are considered, and the study to translational invariant non-interacting systems, factorizable in momentum space is specified. In Section 6, we consider systems far from equilibrium and their dynamical evolution, and discuss entanglement growth, the Loschmidt rate, the Berry, and Pancharatnam phases and the dynamical topological order parameter, recalling the connection between geometric quantities and quantum information measures. We finish with the study of the time evolution of the topological correlators and how it compares with the evolution of the geometric quantities and correlation functions.

2. Entanglement Signatures

A system is generally represented by a density matrix [61]. Consider a system at a temperature of zero. If the system is in a pure state, the density matrix is given by the projector

$$\rho =|\mathrm{\Phi }\rangle \langle \mathrm{\Phi }|,$$

(1)

where $|\mathrm{\Phi }\rangle$ is the ground state. In the case of mixed states, the density matrix includes contributions from different states. An example is a system at a finite temperature where the density matrix takes the canonical form

$$\rho =\frac{e^{-\beta H}}{Z},$$

(2)

where H is the Hamiltonian and Z is the partition function.

In the case of a non-interacting translationally invariant system, the density matrix operator for a momentum k may be defined as usual as

$${\widehat{\rho }}_k=\frac{{\mathrm{e}}^{-\beta {\widehat{H}}_k}}{Z_k},$$

(3)

with $\widehat{H}={\sum }_k{\widehat{H}}_k$, the density matrix is obtained as $\rho ={\prod }_k\otimes {\rho }_k$.

The system may also be divided into two parts, namely A and B (for instance, two spatial regions or some other form of partition), and the reduced density matrix integrating over the degrees of freedom of subsystem B is defined as

$${\rho }_A=T{r}_B\rho .$$

(4)

The parts A and B may have any relative sizes and, if the division is made in real space, the set of sites of either subsystem may be contiguous or not. If the sites are contiguous, the reduced density matrix has information on the two subsystems with special relevance on the border between the two subsystems. If, for instance, the subsystem A is composed of two spatially separate regions (for instance subsystem A is composed of two sites that are a distance r from each other) the reduced density matrix also has interesting information on the correlations between the two parts of subsystem A.

2.1. Von Neumann Entropy

The von Neumann entropy [9] is defined as the information contained on part A of the system

$$S_A=-Tr{\rho }_A\ln {\rho }_A$$

(5)

and measures the entanglement between subsystems A and B. Keeping a lattice system in mind, the simplest choice is to select A as a single site and B as the remaining $N-1$ sites. In a translationally invariant system, the entropy $S_A$ will be the same for every site, but in a non-homogeneous system, the entropy is site-dependent.

A connection between the quantum information quantity, von Neumann entropy and thermodynamic quantities, can be established since the matrix elements of the density matrix are defined in terms of correlation functions of the whole system. As an example, we may consider a generic spin-$1/2$ fermionic system with normal or anomalous (pairing) terms. In the case of the single-site entanglement, the reduced density matrix can be written as (see for example [62])

$${\rho }_A=\left(\begin{array}{cccc}\langle (1-n_{\uparrow })(1-n_{\downarrow })\rangle & \langle c_{\uparrow }^{†}{c}_{\downarrow }^{†}\rangle & 0& 0\\ \langle c_{\downarrow }{c}_{\uparrow }\rangle & \langle n_{\uparrow }{n}_{\downarrow }\rangle & 0& 0\\ 0& 0& \langle n_{\uparrow }(1-n_{\downarrow })\rangle & \langle c_{\downarrow }^{†}{c}_{\uparrow }\rangle \\ 0& 0& \langle c_{\uparrow }^{†}{c}_{\downarrow }\rangle & \langle (1-n_{\uparrow })n_{\downarrow }\rangle \end{array}\right).$$

using the local basis $|0\rangle ,|\uparrow ,\downarrow \rangle ,|\uparrow \rangle ,|\downarrow \rangle$, for unoccupied, double occupied, singly occupied with an electron with spin ↑ and singly occupied with an electron with spin ↓, respectively. The matrix elements involve correlation functions that may be calculated using standard condensed matter methods. The correlation functions that do not conserve the number of fermions will generally vanish, unless pairing is to be considered. This matrix can be diagonalized and the von Neumann entropy is obtained establishing the connection to usual correlation functions. Using this representation, it is clear that any singularities that may occur in the system, such as a phase transition between different phases, is generally reflected in the correlation functions, and therefore, in the reduced density matrix and in the von Neumann entropy.

Evidently, the entries of the reduced density matrix may be independently obtained from the calculation of the correlation functions. For instance, starting from a pure system (such as a non-degenerate ground state) for the whole system, and tracing over the subsystem B, we directly obtain the reduced density matrix, if we know the ground state.

The connection between entanglement and information theory and quantum condensed matter physics was considered in the context of spin chains and in other systems. For instance, in Refs. [63,64], single-spin entropies and two-spin entanglement were considered, showing that these quantities exhibit a peak at a critical point. In Ref. [65], the entanglement of finite extent blocks of size L with the rest of the system was considered. The interplay between the existence of entanglement and critical correlations is the main insight. Considering the behavior of the entanglement entropy in parameter ranges away from a critical point, it was found that the entropy saturates to a finite value (which is a function of the block size). On the other hand, if the parameters are such that the system is at a critical point, the entropy has a logarithmic dependence on the block size

$$S_L=-Tr\left({\rho }_L\ln {\rho }_L\right)\sim \frac{c}{3}\ln \left(L\right),$$

(6)

such that the constant (central charge of the conformal field theory) depends on the universality class of the critical behavior. Therefore, the entropy identifies the vicinity of the critical point. This type of result has also been shown to hold in a relativistic critical theory [66]. In case the system is away from a critical regime, has a gap and a corresponding finite, but a large correlation length, $\xi$, the correlation length replaces the length of the system in the expression of the entanglement entropy. This leads to a result of the type [66]

$$S_L\sim n_A\frac{c}{6}\ln \xi ,$$

(7)

where $n_A$ is the number of boundary points of the subsystem A. General results for the entanglement entropy for random pure states for both ergodic [67] and non-ergodic states [68] have also been obtained and have also been applied to random quadratic Hamiltonians [69].

2.2. Mutual Information

While von Neumann entropy measures the entanglement between a subsystem and the rest of the system, the entanglement between two sites of the system may be obtained integrating, as before, the degrees of freedom of the remaining ($N-2$ sites). The entangled states of the two sites i and j are represented by a density matrix ${\rho }_{ij}$ which includes the correlations of the two sites with the remaining $N-2$ sites and the entanglement between the two sites themselves. In order to access the correlation between the two sites, we may define the mutual information as [9,23,24,25]

$$M_{ij}=S_i^{\left(1\right)}+S_j^{\left(1\right)}-S_{ij}^{\left(2\right)},$$

(8)

where $S_i^{\left(1\right)}$ is a single-site von Neumann entropy and $S_{ij}^{\left(2\right)}$ is a two-site von Neumann entropy. This quantity is also an interesting signature of any phase transition since it probes the correlations between two sites, which may be arbitrarily far from each other, and is therefore sensitive to the short- or long-range character of the correlations in a given system.

2.3. Negativity

The negativity [26] distinguishes entangled from disentangled states (product states are a typical example of disentangled states). One considers two subsystems A and B and the joint density matrix of the two subsystems, ${\rho }_{AB}$. Take the transposed A density matrix while the basis states of part B remain the same. It turns out that the eigenvalues of the transformed density matrix may have negative values if the system is entangled [26] and the finiteness of the sum of the negative eigenvalues identifies an entangled state (the sum vanishes if the system is not entangled).

2.4. Concurrence

Starting from a pure state $|\psi \rangle$ for two qubits, it has been shown that the von Neumann entropy may be rewritten as a function of a quantity called the concurrence [22] defined by

$$C\left(\psi \right)=|\langle \psi |\tilde{\psi }\rangle |,$$

(9)

where $|\tilde{\psi }\rangle$ is a state that results from the original state by applying a time-reversal operation, which amounts to flipping spins (spin-$1/2$ qubits, or in general two-level systems). It may be shown that

$$S_A=h\left(\frac{1+\sqrt{1-C^2}}{2}\right),$$

(10)

with

$$h\left(x\right)=-x\ln x-(1-x)\ln (1-x).$$

(11)

The concurrence may also be used as a measure of entanglement, vanishing if a state is unentangled and equal to one, for instance, if one considers the singlet state between the two spins.

2.5. Entanglement Spectrum

The possible factorization of the entanglement entropy in terms of the eigenvalues of the reduced density matrix, to be inserted in Equation (5), suggests the use of the information contained in the density matrix spectrum itself, in addition to its use in the trace operation. The set of the eigenvalues of the density matrix (or its logarithm) is called the entanglement spectrum. The eigenvectors are also interesting and provide information on the correlations of the system, as will be discussed ahead.

2.5.1. Momentum Space Description

If the Hamiltonian is separable in momentum space, the density matrix operator is separable for each momentum k as in Equation (3). In a multiband system, the Hamiltonian matrix, $H_k$, may be diagonalized as

$$H_k{\mathbf{Q}}_{k,n}={\lambda }_{k,n}{\mathbf{Q}}_{k,n},$$

(12)

where n runs over the number of bands. The eigenvalues of the density matrix at a finite temperature may be written as

$${\rho }_k{\mathbf{Q}}_{k,n}={\mathrm{\Lambda }}_{k,n}{\mathbf{Q}}_{k,n},$$

(13)

where

$${\mathrm{\Lambda }}_{k,n}=\frac{{\mathrm{e}}^{-\beta {\lambda }_{k,n}}}{{\displaystyle \sum _{n^{\prime }}}{\mathrm{e}}^{-\beta {\lambda }_{k,n^{\prime }}}}.$$

(14)

The expression for the ground state may be obtained considering the zero temperature limit. The entropy for each momentum can be obtained using that

$$S_k=-\sum _n{\mathrm{\Lambda }}_{k,n}\ln \left({\mathrm{\Lambda }}_{k,n}\right)$$

(15)

and the total entropy is obtained summing over the Brillouin zone

$$S=\sum _k{S}_k.$$

(16)

2.5.2. Real Space Description

In general, if there is no translational symmetry, a real-space description is adopted. If the system is also non-interacting or described by a quadratic Hamiltonian (such as free electrons or a superconductor in the mean-field approximation), it is enough to know the eigenvalues of the single-particle correlation matrix defined in the subregion A [70,71,72,73,74].

The thermal reduced density matrix of any subsystem A of a quadratic many-body Hamiltonian is ${\rho }_A=e^{-1/2C_A^{†}{\mathrm{\Omega }}_A{C}_A}/Z_A$ (with $C_A$ being a vector of annihilation and creation operators restricted to subsystem A). The matrix ${\mathrm{\Omega }}_A$ can be obtained from the correlations matrix ${\chi }_{i,j}=\langle C_i{C}_j^{†}\rangle$ (restricted to the sites in the subsystem A) as [70]:

$$\begin{array}{ccc}\hfill {\mathrm{\Omega }}_A& =& -\ln \left[{\chi }_A^{-1}-1\right]\hfill \end{array}$$

(17)

and $Z_A=\sqrt{det\left({\chi }_A^{-1}\right)}$. Therefore, the entanglement entropy of a subsystem is given by

$$\begin{array}{ccc}\hfill S\left[{\rho }_A\right]& =& -\frac{1}{2}\sum _{\alpha }\left[\left(1-{\lambda }_{\alpha }\right)\ln \left(1-{\lambda }_{\alpha }\right)+{\lambda }_{\alpha }\ln {\lambda }_{\alpha }\right],\hfill \end{array}$$

(18)

where ${\lambda }_{\alpha }$ are the eigenvalues of ${\chi }_A$.

As argued in Ref. [10], the analysis of the entanglement spectrum provides more detailed information about a system than just an overall quantity, such as the entanglement entropy. In the context of the quantum Hall effect [10,75] and in the context of coupled spin chains [76], it was shown that the entanglement spectrum of a subsystem A contains information about excited energy states located on the frontier between subsystem A and complementary subsystem B. Other partitionings of the system have been proposed that lead to further information [77]. Considering a partitioning in momentum space, information about the energy excitations of a single Heisenberg chain is contained in the ground state wave function through the entanglement spectrum [78].

The study of the entanglement spectrum of two partitions in one-dimensional systems has also allowed to distinguish topological phases. The entanglement spectrum of topological insulators and superconductors provides information about partition boundary states [79] and that the gapless edge states are linked with the degeneracies of the entanglement spectrum. In the presence of inversion symmetry [80] the entanglement spectrum is a more reliable confirmation of topology than the edge states themselves. This robustness extends to the analysis of the spectral flow of the entanglement spectrum [81,82]. An experimental measurement of the entanglement spectrum has been proposed in the context of cold atoms [83].

The existence of fractionalized modes at the edges of the system, such as dangling $S=1/2$ states (using open boundary conditions) in the $S=1$ Haldane topological phase of an Heisenberg chain, translates into a double degeneracy of the entanglement spectrum [84]. Moreover, this spectrum degeneracy is protected by the same symmetries that protect the Haldane phase, such as inversion symmetry, time-reversal symmetry or dihedral symmetry, establishing the entanglement properties as an additional way to detect topological phases. The various types of phases in one-dimensional interacting fermionic systems [85] also have a correspondence with the different transformation laws of the entanglement spectrum under the symmetry operations of the system [86].

3. Topological Entanglement Entropy

The ground state entanglement entropy of a subsystem A of an infinite system with a gap in the spectrum follows the so-called area law [87]. It is therefore proportional to the boundary area to its complementary subsystem. The result is a consequence of the finite correlation length associated with the gap in the system [88]. If the system is critical (gapless), then the behavior is different, with logarithmic corrections. Considering the mutual information, but in the case of a subsystem A and its complementary subsystem B, it gives the entanglement between subsystems A and B and their correlations. If there are no correlations between the two subsystems, then the mutual information vanishes. If the correlations decay with distance, it is expected that the difference between the sum of the entropies of the two regions and the entropy of their union should only reflect the sites that are in the neighborhood of the frontier that separates A from B. Therefore, one expects it to scale with the surface of the interface instead of with the volumes of the two regions. This has been shown for different systems [88]. If we consider a temperature of zero, one expects the ground state of the system to be non-degenerate, which leads to an entropy of the whole system that vanishes. Therefore, the mutual information is just twice the entanglement entropy of either subsystem, which leads to the conclusion that the entanglement entropy of a given subsystem also follows an area law. This is non-trivial since states generally follow a volume law. It turns out that if we consider finite temperatures, an area law is also obeyed.

For a two-dimensional system with a perimeter P, the area law [87] leads to:

$$\begin{array}{c}\hfill S_A={\xi }_{A}P-{\gamma }_A+\cdots .\end{array}$$

(19)

Here, ${\xi }_A$ is a non-universal constant term, ${\gamma }_A$ contains the non-extensive contributions and ⋯ denotes other contributions that vanish as the perimeter becomes infinite. If a phase has a topological order, one expects an entropy reduction with respect to the area law due to the long-range entanglement.

The non-extensive corrections have different origins [89]. Some contributions to ${\gamma }_A$ are universal and can be assigned to a given phase of matter [90,91], and some non-universal terms may also contribute [92,93]. An alternative view, which provides a clear relation between the entanglement entropy and a non-trivial topological system, is the relation established with a non-trivial Berry phase in a topological phase [94]. The bulk contribution in a gapped phase gives information on the correlation length (associated with some non-local quantity such as a string or twist operator). On the other hand, the boundary contribution is related to the Berry phase (and edge states) which are characteristic of a non-trivial phase.

Kitaev and Preskil [90] and Levin and Wen [91]. isolated the topological contribution to the entropy from the non-extensive terms (such as terms that arise from the edges and corners of a finite system). This contribution (topological entanglement entropy (TEE)) is related to the quantum dimension D of the system: ${\gamma }_{\mathrm{TEE}}=\ln \left(D\right)$ [90,91]. This quantity signals the existence of a topological order in several systems such as in frustrated quantum dimer models and in the Kitaev honeycomb model [48,95], and has been proposed to detect a topological order in spin liquid states [89,96]. In the case of gapped systems that spontaneously break a discrete symmetry, one finds extra negative corrections to the area law [97], since there is a positive entropy associated with the logarithm of the number of degenerate ground states [97]. In the presence of gapless collective modes, one finds logarithmic corrections as the system size grows [98].

In non-interacting topological insulators or topological superconductors, the TEE vanishes [93], since these systems have only short-range entanglement [99,100] and no topological order with long-range entanglement [101,102]. Non-trivial projection operators may turn the system to one with topological order [103]. In some cases, it is not straightforward to identify the topological features, e.g., due to geometry effects, including effects of corners and edges [92], or in the case of a Kitaev model where a positive contribution to the entanglement entropy ($\ln \sqrt{2}$) was obtained [104].

The method described in Ref. [90] enables one to compute the topological term by minimizing finite size effects. The TEE is obtained considering three regions $A,B,C$ (defined in Figure 3 of Ref. [90]) and calculating

$$\begin{array}{ccc}\hfill S_{\mathrm{TEE}}& =& S_A+S_B+S_C-S_{AB}-S_{BC}-S_{AC}+S_{ABC}\hfill \\ & =& -2{\gamma }_{\mathrm{TEE}}.\hfill \end{array}$$

(20)

For example, the method was applied to a two-dimensional triplet superconductor [105,106] considering the three regions $A,B,C$ immersed in an infinite system. For a finite system, the transition lines between the various topological phases are clearly seen, particularly if one considers the derivative of the entropy with respect to some parameter that leads to a change of topological phase. Far from the transition lines, ${\gamma }_{\mathrm{TEE}}\to 0$, as the system size grows, as expected [101,107].

Considering, however, a finite system with cylindrical ($C_y$) geometry allows a better understanding [89,104]. In this geometry, there are no corners and contributions only arise from edges states. The contributions to the TEE [108] can be written as ${\gamma }_{\mathrm{Cy}}=-(1/2\ln 2)g_{\mathrm{edge}}$, where $g_{\mathrm{edge}}$ is the number of edge state pairs. Therefore, trivial phases can be distinguished from topological phases by identifying the presence of edge states. Each contribution is therefore $\ln \sqrt{2}$ and is characteristic of a half-fermion (Majorana). A similar result was found for the Majorana mode of the n-channel Kondo model with $n=2,S=1/2$ due to the impurity contribution [109,110,111,112]. In general, for $n>2S$, the entropy is fractional.

4. Reduced Density Matrix and Order Parameters of a Topological Insulator

Continuous quantum phase transitions are usually described using Landau’s symmetry breaking theory, which involves the identification of an order parameter. This includes quantum phase transitions (QPT) [13,113]. Usually, an appropriate choice of order parameter implies knowledge of the symmetry breaking of the system. However, for instance, topological quantum phase transitions do not involve a change of symmetry [114]. A notable difference of using methods of quantum information is that a prior knowledge of the system’s symmetry is not required.

Starting from the mutual information, knowing the ground state of a system and the spectra of the corresponding reduced density matrix, it has been proposed that the order parameter may be found [115]. Identifying the most significant eigenvalues of the appropriate reduced density matrix, operators are constructed based on the eigenvectors of the reduced density matrix, and using general conditions, an order parameter may be obtained, as confirmed for the spin–density wave, charge–density wave, bond-order wave and the phase separation phase in the extended Hubbard model [116] using considerations that involve the single site and two sites reduced density matrices in a quite unbiased way.

Other independent proposals to derive the order parameter have been presented. A variational method was proposed [117] by investigating a set of low-energy “quasi-degenerate” states using the argument that an order parameter should distinguish the degenerate states associated with the symmetry breaking. The optimization procedure for the order parameter leads to an expression in terms of the projectors to the eigenvectors of the difference between the reduced matrices for the degenerate states. The method was further elaborated later [118]. Another suggestion [119] was based on the study of the singular-value decomposition of the correlation density matrix to gain information on the correlation function and the order parameter. Considering two small disjoint clusters A and B, one can also define the cluster that is the union of the two small clusters. Defining the reduced density matrices of the union ${\rho }_{AB}$, and of each cluster, ${\rho }_A,{\rho }_B$, by tracing over all other sites, as usual, the correlation density matrix is defined as

$${\rho }_C={\rho }_{AB}-{\rho }_A\otimes {\rho }_B.$$

(21)

This vanishes and ${\rho }_C=0$ if there are no correlations between the clusters A and B. Otherwise, if correlations are present and the correlation density matrix is non-vanishing and using a singular-value decomposition, it is possible to identify the dominating correlations in the system and their spatial dependence, without any previous knowledge of the system’s behavior.

4.1. Reduced Density Matrix and Order Parameters

The method proposed by Gu et al. [115] is a non-variational approach that establishes a connection between the mutual information, the reduced density matrix, and the order parameters. In this method, the first step is to identify the minimal subsystem A in a given system, where B is the remaining subsystem that describes some sort of long-range correlations, and introduces the reduced density matrix ${\rho }_A=T{r}_B\rho$ that leads to a finite mutual information at long distances, and is supposedly associated with a long-range order (or QLRO or some sort of entanglement) in the system [24,88]. The essential information is contained in the eigenvalues and eigenvectors of the reduced density matrices of the minimal block size.

The method can also be applied to a problem which has topological properties [11]. A considered model was a one-dimensional spinless fermions SSH-like model with explicit dimerization. The model consists of a chain with alternating hoppings with values of $t_1=t(1+\eta )$ and $t_2=t(1-\eta )$, such that $t_1$ is the hopping between sites A and B on a given cell and $t_2$ is the hopping between site B at cell j and site A at cell $j+1$. The eigenspectrum of the reduced density matrix calculated with a block consisting of an atom A and an atom B at site j, is such that for $\eta >0$ (trivial regime), using the basis of $|n_{j,A},n_{j,B}\rangle =\{|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \}$, the eigenstate $|A\rangle =\frac{1}{\sqrt{2}}\left(|10\rangle +|01\rangle \right)$ has the highest eigenvalue, that saturates to one when $\eta \to 1$. Choosing a block with an atom B at site j and an atom A at site $j+1$, and using these points as the new basis, the same eigenstate (defined in terms of this new basis) has the highest eigenvalue for $\eta <0$, which also saturates to one when $\eta \to -1$.

When $\eta =1$ or $\eta =-1$, the Hamiltonian is easily analytically diagonalizable by rotating the fermionic operators to new fermionic operators, calling them $d\left(\eta \right)$ and $f\left(\eta \right)$ fermions. At the points $\eta =\pm 1$ the state $|n_{f,\eta }=0,n_{d,\eta }=0\rangle$ dominates. The Hamiltonian is diagonal and the eigenvector of the reduced density matrix is just the eigenvector of the state for which both $d\left(\eta \right)$ and $f\left(\eta \right)$ are empty. One may define operators that are the projectors to these dominant eigenstates. In the trivial phase the order parameter is fully local and involves the two bands at a given site. In the topological phase, the order parameter involves two neighboring lattice sites. It turns out that calculating the average of these operators as we continuously change $\eta$ (even though the operators are defined at $\eta =\pm 1$) enables a crossing of the two average values at $\eta =0$, which is the point of the quantum phase transition between the trivial and the topological region (at $\eta <0$). These operators may be considered as “order parameters” in the sense that their crossing identifies the transition and the largest average value corresponds to the trivial or topological regime, but not in the sense that they vanish in some phase and become finite in the other. They are also called topological correlators.

4.2. Topological Correlators in Equilibrium

Another model that also has a topological nature generalizes the SSH-model to include superconductivity, for which we may define topological correlators, which is given by the Hamiltonian [120] $H=H_{\mu }+H_t+H_{\mathrm{\Delta }}$ with $H_{\mu }=-\mu {\sum }_j\left(c_{j,A}^{†}{c}_{j,A}+c_{j,B}^{†}{c}_{j,B}\right)$ and

$$H_t=-\sum _j{t}_1\left[c_{j,B}^{†}{c}_{j,A}+c_{j,A}^{†}{c}_{j,B}\right]+t_2\left[c_{j+1,A}^{†}{c}_{j,B}+c_{j,B}^{†}{c}_{j+1,A}\right]$$

(22)

$$H_{\mathrm{\Delta }}=\sum _j{\mathrm{\Delta }}_1\left[c_{j,B}^{†}{c}_{j,A}^{†}+c_{j,A}{c}_{j,B}\right]+{\mathrm{\Delta }}_2\left[c_{j+1,A}^{†}{c}_{j,B}^{†}+c_{j,B}{c}_{j+1,A}\right],$$

(23)

where $t_1=t(1+\eta ),t_2=t(1-\eta )$ are the hoppings, ${\mathrm{\Delta }}_1=\mathrm{\Delta }(1+\eta ),{\mathrm{\Delta }}_2=\mathrm{\Delta }(1-\eta )$ are the pairing amplitudes of the dimerized lattice and $\mu$ is the chemical potential. The model with no superconductivity ($\mathrm{\Delta }=0$) is related to the SSH model [121]. The region of $\eta >0$ corresponds to $t_1>t_2$, and vice versa for $\eta <0$, and with no dimerization, $(\eta =0)$ reduces to the Kitaev model [122]. To simplify the discussion, we take $\mu =0$. Both models have topological states that in the SSH case are found for $\eta <0$ (as discussed above). The phases are characterized by their winding numbers or by the presence of edge states. For $\eta =0$, we have two topological phases that are of the Kitaev type. At the points $\eta =0,\mathrm{\Delta }=\pm t$ and $\mathrm{\Delta }=0,\eta =\pm 1$, the model is easily diagonalizable in terms of new fermionic operators that can be expressed in terms of the original fermionic operators on neighboring lattice sites. For instance, taking $\mathrm{\Delta }=0$ and $\eta =-1$, we may rewrite the Hamiltonian in terms of new operators as $H=t{\sum }_{j=1}^{N-1}\left(2d_j^{†}{d}_j-1+2f_j^{†}{f}_j-1\right)$. The ground state is then simply the state of these operators and the projector at a given site, j, not being occupied, which can be written as [11], $O=|00\rangle \langle 00|$. The projector at this point in the phase diagram for $\eta =-1$ can be expressed as

$$O_{-}^{SSH}=\frac{1}{2}\left(c_{j+1,A}^{†}{c}_{j,B}+c_{j,B}^{†}{c}_{j+1,A}\right)-n_{j,B}{n}_{j+1,A}+\frac{1}{2}\left(n_{j,B}+n_{j+1,A}\right)$$

(24)

and the projector to the ground state at $\eta =1$ can be written as

$$O_{+}^{SSH}=\frac{1}{2}\left(c_{j,A}^{†}{c}_{j,B}+c_{j,B}^{†}{c}_{,A}\right)-n_{j,A}{n}_{j,B}+\frac{1}{2}\left(n_{j,A}+n_{j,B}\right).$$

(25)

Similar expressions can be found when $\eta =0$ and $\mathrm{\Delta }=\pm t$. They read

$$\begin{array}{ccc}\hfill O_{+}^{SK}& =& \frac{1}{4}\left[c_{j,B}^{†}\left(c_{j,A}+c_{j+1,A}\right)+\left(c_{j,A}^{†}+c_{j+1,A}^{†}\right)c_{j,B}\right.\hfill \\ & +& \left.c_{j,B}\left(c_{j,A}-c_{j+1,A}\right)+\left(c_{j,A}^{†}-c_{j+1,A}^{†}\right)c_{j,B}^{†}\right]\hfill \\ & -& \frac{1}{4}\left(2n_{j,B}-1\right)\left(c_{j+1,A}^{†}{c}_{j,A}+c_{j,A}^{†}{c}_{j+1,A}+c_{j+1,A}{c}_{j,A}+c_{j,A}^{†}{c}_{j+1,A}^{†}\right)+\frac{1}{4},\hfill \end{array}$$

(26)

for $\mathrm{\Delta }=t$ and

$$\begin{array}{ccc}\hfill O_{-}^{SK}& =& \frac{1}{4}\left[c_{j,B}^{†}\left(c_{j,A}+c_{j+1,A}\right)+\left(c_{j,A}^{†}+c_{j+1,A}^{†}\right)c_{j,B}\right.\hfill \\ & +& \left.c_{j,B}\left(-c_{j,A}+c_{j+1,A}\right)+\left(-c_{j,A}^{†}+c_{j+1,A}^{†}\right)c_{j,B}^{†}\right]\hfill \\ & -& \frac{1}{4}\left(2n_{j,B}-1\right)\left(c_{j+1,A}^{†}{c}_{j,A}+c_{j,A}^{†}{c}_{j+1,A}-c_{j+1,A}{c}_{j,A}-c_{j,A}^{†}{c}_{j+1,A}^{†}\right)+\frac{1}{4},\hfill \end{array}$$

(27)

for $\mathrm{\Delta }=-t$. Note that these operators are quasi-local in real space. It was shown in [11,123] that these topological correlators (average values of the projector operators) identify the phase transitions and that the amplitude of the correlators is maximal in each corresponding phase. Even though the topological correlators were identified considering the non-interacting problem, it was shown that they also identify the various phases, including topological phases, when the effect of interactions is included in the SSH model (taking $\mathrm{\Delta }=0$), such as adding terms including

$$H_{int}=U\sum _j{n}_{j,A}{n}_{j,B}+V\sum _j{n}_{j,B}{n}_{j+1,A}.$$

(28)

The correlation between the phases and the topological correlators was also confirmed by calculating the Berry phase (using twisted boundary conditions in the many-body system) and showing that the topological region is identified by the topological correlator [11]. Interestingly, in the case of the Kitaev model, the projector may be identified with an energy bond. It has also been shown that the first derivative of the average value of the quasi-local operators associated with the bonds in the Hamiltonian, with respect to the driving parameter across a critical point, may detect the topological transitions in a way that is more sensitive in comparison to a derivative of the energy itself, since it involves a second-order derivative (or susceptibility) [124].

The usefulness of the topological correlators was also established in the context of a topological $QE{D}_2$ lattice gauge theory that can be seen as a correlated version of the SSH topological insulator, which has been shown [125] to display symmetry-protected topological phases. Accordingly, it was shown that the topological order parameter may be used to identify a non-trivial topological phase.

Another proposal to detect topological phases is the result of the introduction of the concept of strange correlators defined in terms of a correlation function, for some local operator in two different spatial points, defined as a matrix element between a state with short-range entanglement, as in a symmetry-protected topological system and a reference trivial state. It turns out that this correlator may saturate to a constant or have a power law decay instead of an exponential decay if there is no topological short-range entanglement [126,127,128,129,130]. A true long-range topological order may also be identified considering non-local order parameters [124,131].

5. Fidelity

5.1. Pure State Fidelity

In the case of pure states with density matrices ${\rho }_1=|G{S}_1\rangle \langle G{S}_1|$ and ${\rho }_2=|G{S}_2\rangle \langle G{S}_2|$, the fidelity is defined as the norm of the overlap

$$F({\rho }_1,{\rho }_2)=|\langle G{S}_1|G{S}_2\rangle |.$$

(29)

The magnitude of this overlap allows one to identify states which are indistinguishable from states that are distinguishable. If we change a given parameter upon which the Hamiltonian of a system depends, then we cross a critical point and one expects that two quantum states defining different macroscopic phases have enhanced distinguishability with respect to that taken between the states from the same phase. Therefore, taking Hamiltonians $\widehat{H}\left(q\right)$, where q is a parameter, and taking two parameter values that are close by, the overlap may indicate the points of quantum phase transitions associated with a decrease in the fidelity if one point is in one phase and the other in some other neighboring phase.

The drop in the (ground) state fidelity [22] has been applied, starting in [12], to the Dicke and the $XY$ models, and later to free Fermi systems [33], Bose–Hubbard models [35,36], spin chains [37,38] and in [39,40]. The method also produced interesting results in transitions that are not of the Landau–Ginzburg–Wilson type [46,48,132]. The connection between the Berry phase descriptions of phase transitions and the fidelity was established [42,43,133,134,135,136] (as can also be seen in [44]).

While the fidelity between ground states provides a generally sensitive way to detect a quantum phase transition, some transitions escape this criterion. An example is the $J_1-J_2$ one-dimensional Heisenberg model given by

$$H=J_1\sum _{\langle i,j\rangle }{\mathbf{S}}_i·{\mathbf{S}}_j+J_2\sum _{\langle \langle i,j\rangle \rangle }{\mathbf{S}}_i·{\mathbf{S}}_j,$$

(30)

where ${\mathbf{S}}_i$ is a spin-$1/2$ operator at site i, the first term couples nearest neighbors and the second term couples the next nearest neighbors, respectively. The system has a Berezinsky–Kosterlitz–Thouless transition when $J_2/J_1\sim 0.24$. It has been shown that this transition occurs between the first and second excited states, and therefore, does not directly involve the ground state [137], and is well detected even for a finite system. Even though the transition occurs within excited states, it has been shown that the fidelity also detects this transition if one considers the fidelity between the first excited states [37]. A similar result was obtained considering the behavior of the energy bond topological correlator [124], taking its average with respect to the excited state.

5.2. Fidelity Susceptibility and Scaling

As an example, let us consider a two-level system described by a Hamiltonian of the form

$$H=\mathbf{d}·\sigma ,$$

(31)

where $\sigma$ are Pauli matrices. Let us consider the case where two close points in parameter space are selected, specified by two close values of vector $\mathbf{d}$. The fidelity between the states may be written as [14]

$$F=\frac{1}{\sqrt{2}}\sqrt{1+{\mathit{n}}_1·{\mathit{n}}_2},$$

(32)

where the vectors are unitary ($\mathit{n}=\mathbf{d}/|\mathbf{d}|$). If we consider two points which are very close to each other, then the unit vectors are also very close to each other and we may write ${\mathit{n}}_2={\mathit{n}}_1+\delta \mathit{n}$. Therefore, we obtain

$$F=\frac{1}{\sqrt{2}}\sqrt{2+\mathit{n}·\delta \mathit{n}},$$

(33)

where we omit the vector index. The unit vector generally depends on the various parameters in the Hamiltonian. Supposing that the transition is obtained by varying a single parameter h, we write $\mathit{n}=\mathit{n}\left(h\right)$. Then, the fidelity is given by (where we closely follow [14])

$$F=\frac{1}{\sqrt{2}}\sqrt{1+\mathit{n}\left(h\right)·\mathit{n}(h+\delta h)}.$$

(34)

The Taylor expansion,

$$\mathit{n}(h+\delta h)=\mathit{n}\left(h\right)+\delta h\frac{\partial \mathit{n}}{\partial h}+\frac{1}{2}{\left(\delta h\right)}^2\frac{{\partial }^2\mathit{n}}{\partial h^2}+\cdots$$

(35)

thus allowing to see that the fidelity may be approximated by

$$F=1-\frac{{\left(\delta h\right)}^2}{2}{\chi }_F,$$

(36)

where

$${\chi }_F=-\frac{1}{4}\mathit{n}\left(h\right)·\frac{{\partial }^2\mathit{n}}{\partial h^2}.$$

(37)

The linear term vanishes because the normalization of $\mathit{n}$ implies that $\mathit{n}·(\partial \mathit{n})/(\partial h)=0$. The quantity ${\chi }_F$ is called fidelity susceptibility [14]. If, as we change the parameter, we cross a transition, then the fidelity susceptibility has a singularity on the transition line and may be used efficiently to detect quantum phase transitions.

To illustrate, let us consider the simple case of a zero temperature transition only involving pure states. The fidelity may be written as

$$F=|\langle {\psi }_0\left(h\right)|{\psi }_0(h+\delta h)\rangle |,$$

(38)

that is, the absolute value of the overlap between two ground states, for the two parameters h and $h+\delta h$. We may use perturbation theory to write the wave function ${\psi }_0(h+\delta h)$ from ${\psi }_0\left(h\right)$. Assuming that the variation of h can be seen as a perturbation, $H_I$, then

$$|{\psi }_0(h+\delta h)\rangle =|{\psi }_0\left(h\right)\rangle +\delta h\sum _{n\ne 0}\frac{\langle {\psi }_n\left(h\right)|H_I|{\psi }_0\left(h\right)\rangle }{E_0\left(h\right)-E_n\left(h\right)}|{\psi }_n\left(h\right)\rangle .$$

(39)

The state (39) is not yet normalized. Indeed,

$$\langle {\psi }_0(h+\delta h)|{\psi }_0(h+\delta h)\rangle =1+{\left(\delta h\right)}^2\sum _{n\ne 0}\frac{|\langle {\psi }_n\left(h\right)|H_I|{\psi }_0\left(h\right){\rangle |}^2}{{\left(E_n\left(h\right)-E_0\left(h\right)\right)}^2}.$$

(40)

Normalizing the wave function and inserting in the fidelity, we obtain

$$F^2=1-{\left(\delta h\right)}^2\sum _{n\ne 0}\frac{|\langle {\psi }_n\left(h\right)|H_I|{\psi }_0\left(h\right){\rangle |}^2}{{\left(E_n\left(h\right)-E_0\left(h\right)\right)}^2}.$$

(41)

The fidelity susceptibility

$$\begin{array}{ccc}\hfill {\chi }_F\left(h\right)& =& -\frac{{\partial }^{2}F}{\partial {\left(\delta h\right)}^2}\hfill \\ & =& \sum _{n\ne 0}\frac{|\langle {\psi }_n\left(h\right)|H_I|{\psi }_0\left(h\right){\rangle |}^2}{{\left(E_n\left(h\right)-E_0\left(h\right)\right)}^2}.\hfill \end{array}$$

(42)

This expression clearly shows that a singularity arises in ${\chi }_F\left(h\right)$ if the gap closes, i.e., when some eigenenergy coincides with the energy of the ground state. The calculation of the fidelity susceptibility for the Kitaev model, for instance, when $\mu \approx 2t$ and keeping $\mathrm{\Delta }=t$, leads to the result

$${\chi }_F(k=\pi )\sim \frac{1}{{(\frac{\mu }{2t}-1)}^2},$$

(43)

displaying a divergence at the critical point.

While the fidelity vanishes in the thermodynamic limit due to the Anderson orthogonality catastrophe, the fidelity susceptibility shows interesting scaling behavior with the system size [43]. Different physical systems have often different scaling properties. Considering a d-dimensional system with linear dimension L, described by an interacting Hamiltonian composed of short-range local terms, it can be shown that the fidelity susceptibility scales as

$$\frac{{\chi }_F}{L^d}\sim L^{d+2z-2{\mathrm{\Delta }}_V},$$

(44)

where z is the dynamic exponent and ${\mathrm{\Delta }}_V$ is the scaling dimension of the interacting term. If a system has a gap, the fidelity susceptibility is regular. Near a critical point, where the system is gapless, then, depending on $d+2z-2{\mathrm{\Delta }}_V$, the fidelity susceptibility ${\chi }_F/L^d$ may be divergent if $d+2z>2{\mathrm{\Delta }}_V$ or intensive otherwise [14,138]

In systems that are non-trivial from the topological point of view, such as a symmetry-protected topological phase, a quantity called the boundary fidelity susceptibility, ${\chi }_B$, defined from the fidelity susceptibility ${\chi }_F$ as

$${\chi }_F={\chi }_N+\frac{{\chi }_B}{N},$$

(45)

where the first term is the bulk contribution and N is the total number of sites in the system, has been shown to detect the existence of edge states. Indeed, it gives a different contribution to the total fidelity susceptibility with respect to the trivial phase, for which there are no edge states. This has been shown for the SSH dimerized model [139].

5.3. Mixed State Fidelity

In general, the state of a system is not pure and some mixture of states occurs. The system is then characterized by a density matrix $\widehat{\rho }$. Different measures of quantum state distinguishability have been introduced [9,140], such as the generalization of the fidelity [22] to include mixed states

$$F({\widehat{\rho }}_a,{\widehat{\rho }}_b)=Tr\sqrt{\sqrt{{\widehat{\rho }}_a}{\widehat{\rho }}_b\sqrt{{\widehat{\rho }}_a}}.$$

(46)

The fidelity vanishes in the case of fully distinguishable states and it takes the value $F=1$, when the two quantum states are identical. The classical fidelity $F_c$ for probability distributions $\left\{p_a\left(i\right)\right\}$ and $\left\{p_b\left(i\right)\right\}$ is defined as $F_c(p_a,p_b)={\sum }_i\sqrt{p_a\left(i\right)p_b\left(i\right)}$. Using the modulus of an operator $\widehat{R}$, $|\widehat{R}|={\left(\widehat{R}{\widehat{R}}^{†}\right)}^{1/2}$, the fidelity can be written as [9]:

$$F({\widehat{\rho }}_a,{\widehat{\rho }}_b)=Tr\left|\sqrt{{\widehat{\rho }}_a}\sqrt{{\widehat{\rho }}_b}\right|=Tr\sqrt{\sqrt{{\widehat{\rho }}_a}\sqrt{{\widehat{\rho }}_b}{\left(\sqrt{{\widehat{\rho }}_a}\sqrt{{\widehat{\rho }}_b}\right)}^{†}}=Tr\sqrt{\sqrt{{\widehat{\rho }}_a}{\widehat{\rho }}_b\sqrt{{\widehat{\rho }}_a}}.$$

(47)

One example of a mixed state density matrix is that corresponding to a system in thermal equilibrium, such as the canonical or grand canonical ensembles. In this case, in the fidelity expression Equation (46), the reduced density matrices are those of equilibrium ensembles, for instance, for two sets of parameters, such as Hamiltonian parameters or temperature, as studied in [57,58] and in [42,44,59]. It was shown that a thermal phase transition is clearly detected by the fidelity through a sudden drop along the transition line [59], considering as examples the Stoner–Hubbard transition to magnetism or the BCS theory of superconductivity. A distinction between systems with mutually commuting Hamiltonians (as in the first case) and mutually non-commuting Hamiltonians (as in the second case) was established and where the singular behavior of the fidelity was related to the singular behavior of thermodynamic response functions such as the susceptibility and heat capacity.

5.4. Uhlman’s Phase

In the case of pure states, there is a link between the Berry phase and the fidelity [42,43,133,134,135,136]. The generalization of the Berry geometric phase to mixed quantum states, the Uhlmann mixed state geometric phase [141], was introduced [59] to analyze the structural change of the system eigenvectors as a transition takes place. The geometric phase is determined by the unitary operator ${\widehat{V}}_{ab}$, given by the polar decomposition [9] $\sqrt{{\widehat{\rho }}_a}\sqrt{{\widehat{\rho }}_b}=|\sqrt{{\widehat{\rho }}_a}\sqrt{{\widehat{\rho }}_b}|{\widehat{V}}_{ab}$ with $|\widehat{R}|={\left(\widehat{R}{\widehat{R}}^{†}\right)}^{1/2}$. The fidelity can be written as $F({\widehat{\rho }}_a,{\widehat{\rho }}_b)=Tr|\sqrt{{\widehat{\rho }}_a}\sqrt{{\widehat{\rho }}_b}|$. A nonzero difference $H({\widehat{\rho }}_a,{\widehat{\rho }}_b)-F({\widehat{\rho }}_a,{\widehat{\rho }}_b)$, where $H({\widehat{\rho }}_a,{\widehat{\rho }}_b)=Tr\left[\sqrt{{\widehat{\rho }}_a}\sqrt{{\widehat{\rho }}_b}\right]$, implies the emergence of a non-trivial Uhlmann geometric phase. In the case of mutually commuting Hamiltonians, the system eigenvectors do not change with the driving parameter and the Uhlmann geometric phase becomes trivial.

The Uhlmann phase was later used to detect topological phase transitions [142,143,144]. Since it was developed to consider mixed states, it is suited to consider, for instance, the effects of temperature on different systems. In topological systems, one expects zero temperature quantizations of the topological invariants and possible physical quantities that they are related to, such as the Hall conductance in the integer quantum Hall effect. As the temperature rises, there are populations of excited states (across the gap that characterizes the topological regime) and one does not expect the quantization of the physical quantities. Furthermore, one does not expect that traditional topological invariants will remain quantized (one expects partial occupation of bands instead of fully occupied or fully empty bands). However, surprisingly, it was found in one-dimensional topological systems [142] and in two-dimensional phases [143,144] that using the Uhlmann phase (which reduces at a temperature of zero to a Berry phase or to a Chern number in two-dimensions) non-trivial regimes are found at finite temperatures above the zero temperature topological regimes. This raised the question of the possible topological nature at a finite temperature. The topological indices were associated in [143] to the winding of the phases of the eigenvalues of the Uhlmann connection between the initial and final points of a trajectory in parameter space, under the conditions of parallel transport. The physical nature of the result was questioned and other works did not find topological transitions at finite temperatures and found that the finite temperature just smears the topological properties [145,146]. At the very least, the bulk-correspondence does not hold at finite temperatures [142,145]. Recent studies have also considered the use of Uhlmann phases with respect to interferometric phases and their abilities to detect the possible finite temperature topological transitions [147,148].

5.5. Partial State Fidelity

Considering reduced density matrices such as ${\widehat{\rho }}_A\left(q\right)={\mathrm{Tr}}_B|\mathrm{\Phi }\left(q\right)\rangle \langle \mathrm{\Phi }\left(q\right)|$, we may define the mixed-state fidelity $F({\widehat{\rho }}_A\left(q\right),{\widehat{\rho }}_A\left(\tilde{q}\right))$. This has led to the notion of a partial state fidelity approach to quantum phase transitions [41,60].

One early example of the application of the partial state fidelity [60] was on a conventional superconductor with a magnetic impurity inserted at its center. In the vicinity of the point of the (first-order) quantum phase transition, induced by the change in the coupling between the impurity spin and the electron spin density, a sudden drop in the fidelity involving a single-site or a two-site partial state, if one of the sites is the impurity site, was observed. Other systems were considered [54,56,149,150,151,152] where it was confirmed that the partial state fidelity adequately detects quantum phase transitions.

5.6. Fidelity Spectrum

Quantum fidelity between two density matrices $F({\rho }_1,{\rho }_2)$ is usually defined as the trace of the fidelity operator $\mathcal{F}=\sqrt{\sqrt{{\rho }_1}{\rho }_2\sqrt{{\rho }_1}}$. The logarithmic spectrum of this operator has been denoted fidelity spectrum [153], reducing to the entanglement spectrum if the two density matrices are equal. In the case of two equal mixed states, the operator $\mathcal{F}$ has a set of eigenvalues, $f_n={\mathrm{\Lambda }}_n$, such that $-\ln {\mathrm{\Lambda }}_n$ reduces to the entanglement spectrum. The fidelity spectrum was studied in the cases of the $XX$ spin chain in a magnetic field, a magnetic impurity inserted in a conventional superconductor and a bulk superconductor at finite temperature [153,154]. The information provided by the fidelity spectrum on the dominant modes that contribute to the distinguishability of phases forms parallels with the extra information provided by the entanglement spectrum [10] on the modes of a single phase compared to the von Neumann entropy.

The fidelity operator $\mathcal{F}$ can be studied using different basis states such as position, momentum, energy, or charge and spin. Rewriting the fidelity operator in these different representations allows us to more completely characterize the phase transition. While the entanglement spectrum has some relation to the energy spectrum of the edge states or even bulk states, the fidelity spectrum contains information about which eigenvalues have a larger contribution to the distinguishability between quantum states [153].

Considering two density matrices that result from the momentum space partition for two points in parameter space, we can write that the eigenstates satisfy

$${{\rho }_1}_k{Q_1}_k={Q_1}_k{{\mathrm{\Lambda }}_1}_k;{{\rho }_2}_k{Q_2}_k={Q_2}_k{{\mathrm{\Lambda }}_2}_k.$$

(48)

As mentioned above, the fidelity between two states, characterized by two density matrices ${{\rho }_1}_k$ and ${{\rho }_2}_k$, may then be defined as the trace of the fidelity operator, ${\mathcal{F}}_k$,

$$F_k({{\rho }_1}_k,{{\rho }_2}_k)=\mathrm{Tr}{\mathcal{F}}_k=\mathrm{Tr}\sqrt{\sqrt{{{\rho }_1}_k}{{\rho }_2}_k\sqrt{{{\rho }_1}_k}}.$$

(49)

This quantity may be calculated using the diagonalization of each matrix using Equation (13). The total fidelity is obtained as

$$F=\prod _k{F}_k.$$

(50)

As one approaches a transition, the factorization of the fidelity in terms of momentum contributions allows a more detailed understanding of the distinguishability between two states. The transitions are signaled by the decrease in the highest eigenvalue of the k-fidelity spectrum at the transition point. This allows one to identify points in momentum space associated with the quantum phase transition [153].

Interesting information can be obtained by taking two density matrices that correspond to points in parameter space that are far apart. This provides interesting information about the momenta values responsible for the transitions [155] and the physical nature of the transition. This was studied in the case of a two-dimensional triplet superconductor. The deviations from unity of the k-space fidelity throughout the Brillouin zone are significant, even when the phases have the same Chern number as a consequence of the different band-fillings. Singular transition points translate to zeros in the k-fidelity operator spectrum. As shown before [155], the k-fidelity operator spectrum vanishes at some momenta $\mathit{k}$. It turns out that the k-fidelity operator spectrum signals the minimum number of transitions that must occur when going from one phase to another.

Taking the case of an Hamiltonian with a $2\times 2$ structure [59,145,146] as an example, previously written as

$$H\left(\mathit{k}\right)=\mathit{h}\left(\mathit{k}\right)·\sigma ,$$

(51)

the k fidelity between two states ${\rho }_1$ and ${\rho }_2$ at zero temperature can be written as

$$F_{12}\left(\mathit{k}\right)=\sqrt{\frac{1}{2}\left(1+\frac{{\mathit{h}}_{\mathit{k},1}}{|{\mathit{h}}_{\mathit{k},1}|}·\frac{{\mathit{h}}_{\mathit{k},2}}{|{\mathit{h}}_{\mathit{k},2}|}\right)}.$$

(52)

In this case, $E_{\mathit{k}}=|{\mathit{h}}_{\mathit{k}}|$. It is easy to see that, if ${\mathit{h}}_{\mathit{k},1}={\mathit{h}}_{\mathit{k},2}$ the fidelity is one. Furthermore, we see that when the vectors ${\mathit{h}}_{\mathit{k},1}$ and ${\mathit{h}}_{\mathit{k},2}$ are antiparallel for a given momentum value, the k-space fidelity vanishes. This typically occurs as we vary some parameter across a critical point (gapless point in energy space) which leads to a vanishing contribution to the fidelity (see Ref. [156] for a discussion of the necessary and sufficient conditions to associate a vanishing k-space fidelity with the crossing of one or several quantum critical points along a certain line in the phase diagram). The condition for gapless points is

$$E_{q_c}\left(\mathit{k}\right)=|{\mathit{h}}_{q_c}\left(\mathit{k}\right)|=0,$$

(53)

and the vanishing of the fidelity leads to

$${\mathit{h}}_{q_1}\left(\mathit{k}\right)·{\mathit{h}}_{q_2}\left(\mathit{k}\right)=-|{\mathit{h}}_{q_1}\left(\mathit{k}\right)||{\mathit{h}}_{q_2}\left(\mathit{k}\right)|,$$

(54)

which sets that the angle between ${\mathit{h}}_{q_1}\left(\mathit{k}\right)$ and ${\mathit{h}}_{q_2}\left(\mathit{k}\right)$ which is $\pi$.

Considering for instance the Sato and Fujimoto model [106] simplified to a topologically equivalent case with no s-wave pairing and no spin-orbit coupling, the transitions between the various phases occur at the momentum points $\mathit{k}=(0,0),(0,\pi ),(\pi ,\pi )$ (and equivalent points). At $\mathit{k}=(0,0)$, and similarly for other momentum values [156], the gapless condition implies that $4t+\mu +M_z=0$, where t is the hopping, $\mu$ is the chemical potential, and $M_z$ is the magnetization (or external magnetic field). When

$$-1=\mathrm{sgn}(4t_1+{\mu }_1+M_{z,1})\mathrm{sgn}(4t_2+{\mu }_2+M_{z,2}),$$

(55)

the fidelity vanishes.

6. Entanglement Dynamics

6.1. Entanglement Growth

As a result of a sudden quench (change) of some Hamiltonian parameter(s), the entanglement entropy of a subsystem of size L [157] grows linearly in time until a time of the order of $(L/2v)$, where v is the propagation velocity of the excitations of the system. Considering an initial time such that the state of the system is not an eigenstate of the new Hamiltonian, the state will evolve in time under the action of the new Hamiltonian. This propagation, conditioned by the finite limit on the excitations velocity [158], establishes the causal relation between the quench and its effect on the state propagation.

After a quantum quench, a system evolves in time under the action of the unitary time evolution operator and, therefore, no relaxation to a thermodynamic equilibrium ensemble is expected. However, local or quasi-local quantities may behave differently. Even if the system is isolated from an environment, one may see the rest of the system as a reservoir, which leads to some relaxation of a finite subsystem. This convergence to some equilibrium state generally leads to a Gibbs thermal ensemble, but integrable systems behave differently, as a result of the extensive number of conservation laws. Indeed, it has been proposed that the asymptotic state is a so-called generalized Gibbs ensemble, which is characterized, in addition to the conserved Hamiltonian, by all the conservation laws [159,160,161]. This ensemble is naturally reflected in the reduced density matrix of integrable systems.

The time evolution of a reduced density matrix, taking the transverse field Ising chain [162] as an example, has been shown to converge for long times to a generalized Gibbs ensemble. The entanglement dynamics after quantum quenches in integrable systems has been studied in detail for various integrable systems, as can be seen, e.g., in [163], such as free fermionic and bosonic systems, Heisenberg spin chains and the Lieb–Liniger bosonic model. The entanglement and its dynamics in the context of cold atom systems has been experimentally measured [164,165,166,167].

Again focusing on the transverse field Ising model as a function of the magnetic field, two phases are present. Performing quenches within the same phase, the gap between the two highest eigenvalues of the entanglement spectrum decreases nearly monotonically [168]. On the other hand, carrying out a quench from the paramagnetic to the ferromagnetic phase, the gap has oscillations as a function of time that closes the gap at certain times, suggesting dynamical phase transitions detected in the entanglement spectrum. The case of ramps, where some parameter linearly changes in time across a quantum critical point, has also been studied (see for instance, Ref. [169]).

It has also been established that the time evolution of the single-particle entanglement spectrum crossing in quenches [170] may be related to the quench dynamics of edge states [171,172,173,174,175] (in this context, the zero mode in the entanglement spectrum).

An alternative way to characterize the entanglement, both in equilibrium and in a non-equilibrium situation, is to use the Rényi entropies. Their definition originates in the moments of the distribution of the eigenvalues of the reduced density matrix, ${\lambda }_i$, defined as

$$M_n=\sum _{i=1}^D{\lambda }_i^n=T{r}_A\left({\rho }_A^n\right).$$

(56)

Here, D is the size of the reduced density matrix of subregion A. The moment of order $n=1$ is the von Neumann entropy

$$S_1=-\sum _{i=1}^D{\lambda }_i\ln {\lambda }_i$$

(57)

and the Rényi entropies

$$S_n=\frac{1}{1-n}\ln M_n.$$

(58)

Considering a quench of the system, the Rényi entropies have an interesting time dependence that is related to the behavior of physical quantities [176,177]. For instance, the existence of conservation laws affects the entanglement growth. For example, the conservation of charge or magnetization leading to diffusive dynamics implies a long-time behavior, $t\to \infty$, of $S_2$ that scales as $\sqrt{t}$. On the other hand, in general, for shorter times, the behavior linearly scales in time (a ballistic-like regime as in a system with no conservation laws) and only at larger times, when the entropy $S_2$ becomes extensive with the system size, the entropy crosses over to the diffusive (or sub-ballistic) behavior. A possible exception is a system of two-levels that is diffusive for all times [177]. There is, however, some controversy on the time behavior [178].

6.2. Loschmidt Echo and Loschmidt Rate

As we discussed above, in equilibrium, an efficient way to detect a quantum phase transition between different phases is to use the fidelity [12,14]. Furthermore, the transition line may be detected considering the overlap between states that correspond to points that are far from the transition line [155,156] or considering the 2D fidelity map on overlap between states at different driving parameters [179].

In the case of a dynamical transition across some transition line, we may consider a related quantity called the Loschmidt amplitude. While the fidelity compares the eigenstates of the initial and final points, the Loschmidt amplitude calculates the overlap amplitude of some initial state with its time evolution, which results from the Hamiltonian with a time-dependent perturbation, which is a quench (evolution with the quenched Hamiltonian) or a ramp (evolution with the time-independent initial Hamiltonian plus the perturbation)

$$\mathcal{G}\left(t\right)=\langle {\psi }_0|{\psi }_0\left(t\right)\rangle =\langle {\psi }_0|e^{-iHt}|{\psi }_0\rangle ,$$

(59)

or its square $L\left(t\right)={|\mathcal{G}\left(t\right)|}^2$. In the thermodynamic limit, such an overlap vanishes due to the orthogonality catastrophe, and it is more convenient to calculate the rate

$$\lambda \left(t\right)=-\underset{N\to \infty }{lim}\frac{1}{N}\ln L\left(t\right),$$

(60)

where N is the size of the system.

A way to understand the time dependence of the Loschmidt rate is to recall that traditional phase transitions in statistical mechanics are present when there are singularities in the free energy. These are associated with the zeros of the partition function (Lee–Yang zeros). There is a formal similarity between the free energy per particle and the Loschmidt rate if we express the Loschmidt rate in terms of its zeros, defined in the complex space [180,181]. In the case of a two-band non-interacting translationally invariant Hamiltonian defined by the matrix $H\left(k\right)=\mathbf{d}\left(k\right)·\sigma$, the Loschmidt amplitude, associated with a quench from some initial parameters defined by the vector ${\widehat{\mathbf{d}}}_k^0$ to a final state defined by ${\widehat{\mathbf{d}}}_k^1$, may be written as [182]

$$L\left(t\right)=\prod _k\left[\cos \left({ϵ}_k^{1}t\right)+i{\widehat{\mathbf{d}}}_k^0·{\widehat{\mathbf{d}}}_k^1\sin \left({ϵ}_k^{1}t\right)\right],$$

(61)

where ${ϵ}_k^1=\pm |{\widehat{\mathbf{d}}}_k^1|$. The zeros, in the complex space, are given by

$$t_n\left(k\right)=\frac{\pi }{{ϵ}_k^1}\left(n+\frac{1}{2}\right)+\frac{i}{{ϵ}_k^1}{\tanh }^{-1}\left({\widehat{\mathbf{d}}}_k^0·{\widehat{\mathbf{d}}}_k^1\right).$$

(62)

The zeros define a momentum $k^{*}$ which, if ${\widehat{\mathbf{d}}}_{k^{*}}^0·{\widehat{\mathbf{d}}}_{k^{*}}^1=0$, leads to a set of real times $t_n=\frac{\pi }{2{ϵ}_{k^{*}}^1}(2n-1)$, which are associated with the transition between the two different phases. It turns out that the imaginary part only vanishes if, in the quench, the transition line is crossed. At these points, the Lochmidt rate has singularities. Note that the condition for the transition is different from that found in equilibrium ${\widehat{\mathbf{d}}}_{k_c}^0·{\widehat{\mathbf{d}}}_{k_c}^1=-1$, which was obtained from the vanishing of the fidelity [156]. There is also a relation to the oscillations found in the entanglement spectrum [168]. Furthermore, writing the Loschmidt amplitude as

$$L\left(t\right)=det\left(M\right),$$

(63)

where M is called the Loschmidt matrix, an analysis of the Loschmidt rate was carried out in terms of the eigenvalues of the Loschmidt matrix [182] and it has been shown that its spectrum contains information about the edge states of a topological system. Moreover, it was shown that there is a link between the spectrum and the long-range entanglement between the edges of the system. As time evolves after a quench, there are oscillations between a state that is close to a trivial state (for which the edges are not correlated) and a state that is close to a non-trivial state (for which the edges are correlated).

Considering more general multiband models, the oscillations of the various quantities become richer, including various aperiodic oscillations. Oscillations also occur for quenches within the same phase, and this behavior is well observed both in the Loschmidt amplitude and in quantum information measures, such as the entanglement entropy [183].

6.3. Berry Phase, Pancharatnam Phase and Dynamical Topological Order Parameter

The Loschmidt amplitude may be factorized as

$$\mathcal{G}\left(t\right)=\prod _{k>0}{\mathcal{G}}_k\left(t\right),$$

(64)

with

$${\mathcal{G}}_k\left(t\right)=r_k\left(t\right)e^{i{\varphi }_k\left(t\right)}.$$

(65)

The phase may be decomposed, defining a Pancharatnam geometric phase [184]

$${\varphi }_k^G\left(t\right)={\varphi }_k\left(t\right)-{\varphi }_k^{dyn}\left(t\right),$$

(66)

with the dynamical phase

$${\varphi }_k^{dyn}\left(t\right)=-{\int }_0^{t}ds\langle {\psi }_k\left(s\right)|H\left(s\right)|{\psi }_k\left(s\right)\rangle .$$

(67)

A dynamical topological order parameter is defined as

$${\nu }_D\left(t\right)=\frac{1}{2\pi }{\int }_0^{\pi }dk\frac{\partial {\varphi }_k^G\left(t\right)}{\partial k}.$$

(68)

This quantity has jumps of $\pi$ at the times $t_n\left(k\right)$ associated with the dynamical quantum transitions and may be considered as a dynamical topological order parameter (DTOP).

In addition to the use of the Loschmidt rate and the DTOP to identify the dynamical quantum phase transitions, other quantities have been proposed, including local measures such as the magnetization in the transverse field Ising model as one quenches to the ferromagnetic phase from the paramagnetic phase [180]. Furthermore, the real (or momentum)-local effective free energy [185], has been shown to detect dynamical quantum phase transitions. In this context, the Loschmidt echo

$${|L\left(t\right)|}^2=|\langle \psi |e^{-iHt}{|\psi \rangle |}^2=\langle \psi \left(t\right)||\psi \rangle \langle \psi ||\psi \left(t\right)\rangle ,$$

(69)

is replaced by

$$|L_M{\left(t\right)|}^2=\langle \psi \left(t\right)|P_M|\psi \left(t\right)\rangle ,$$

(70)

where $P_M$ is a projector to a finite, small, subspace of sites. A similar projection is carried out in momentum-space when one considers a local in momentum measure.

Other possibilities include the string order parameter in the $S=1$ spin Haldane phase [186] or one-dimensional Ising models [187], where it was shown that a finite length string operator is enough to detect the dynamical transition, similarly to ref. [185], or Loschmidt cumulants [188].

The case of finite temperature has also been considered, generalizing the formalism from pure states to mixed states. As for equilibrium systems, the sensitivity of the fidelity-associated Uhlmann phase and the interferometric phase were compared in the detection of dynamical phase transitions [189,190]. It has been shown that, while a finite temperature rounds off the sharp peaks observed in the Loschmidt rate at zero temperatures, it may still be used to identify the existence of dynamical transitions [191,192].

6.4. Time Evolution of Topological Correlators

Recently, the quasi-local quantities called topological correlators that have been proposed to identify and characterize equilibrium topological phase transitions [11,123,124], as discussed above, have been applied to the case of dynamical topological transitions [193].

Taking the dynamical phase transition in the non-interacting SSH model as an example, as one quenches from the trivial regime $\eta =0.9$ to the topological regime $\eta =-0.9$, signatures of the singularities in the Loschmidt rate are found in oscillations of the topological correlators, $O_{+}^{SSH}$ and $O_{-}^{SSH}$, which are particularly noticeable in the $O_{+}^{SSH}$ correlator [193]. The results show agreement between the minima (maxima) of $O_{+}^{SSH}$ ($O_{-}^{SSH}$) at the locations of the singularities of the Loschmidt rate. For all symmetric quenches $\eta \to -\eta$, one finds a clear correlation between the behavior of the topological correlators and the Loschmidt rate. Considering asymmetric quenches ${\eta }_0\to -|{\eta }_1|$, both the Loschmidt rate and the agreement are less well defined, but the first maximum or minimum of the correlator agree well with the singularity in the Loschmidt rate, which shows that a good description of the short-term behavior is achieved. Similar agreement was also found in the interacting case [193], particularly when the transition occurs between two phases that are also present in the non-interacting case, but a reasonable agreement was also found in the case where the transition occurs to a phase that is a consequence of the interactions, such as the CDW and phase separation [193]. In some cases, there is no coincidence of the peaks of the Loschmidt rate and the maxima/minima of the topological correlator, but the deviation in time is small. The results show that the local or quasi-local topological correlators provide an efficient way to detect topological transitions in both equilibrium and non-equilibrium situations.

7. Summary

This short review focused on the fruitful connections between quantum information techniques and condensed matter physics systems. The entanglement of a system’s constituents allows a complementary way to understand the physical nature of different phases. Different entanglement measures have been briefly reviewed. A topic not reviewed here is the use of the real space renormalization group to study entanglement and its applications in tensor network representations of quantum many-body systems [194,195]. Furthermore, the real space renormalization group was used by establishing a generalized notion of holography [196]. The utility of concepts associated with fidelity, to distinguish different phases and identify critical points and regimes, were also briefly reviewed. Furthermore, the impact of quantum information techniques extends to dynamical situations and their association with the presence of a phase transition, as one crosses between different phases, which may be due to quantum quenches. In this context of the time evolution of entanglement, several points are less clear, such as the observation of Loschmidt rate peaks in transitions within the same phase or, in some cases, its absence when crossing a critical point. The roles of interactions and long-range couplings deserve further study and analyzing entanglement and its dynamics will likely provide a deeper understanding of non-trivial many-body systems.