Localization Properties of the Disordered XY Spin Chain

A general XY spin chain in transversal field is described by the Hamiltonian

where $\sigma_{j}^{X}$, $\sigma_{j}^{Y}$ and $\sigma_{j}^{Z}$ are the standard Pauli matrices acting on the $j$-th spin. The parameter sequences $\mu_{j}$, $\gamma_{j}$ and $\nu_{j}$, describing the interactions strength, anisotropy and field strength, are viewed as random parameters reflecting the disorder in the system. We will generally assume that all three sequences are independent and identically distributed, with bounded distribution, although this could be weakened in several ways (as long as the required localization properties of the effective Hamiltonians, see below, can be shown).

We will state all results for the finite XY chain, i.e. restricted to the interval $\Lambda=[1,n]$, although with constants bounded uniformly in $n$. This allows to draw conclusions for the infinite chain, although we will not disucss this here.

The Jordan-Wigner transform

with the spin lowering operators $a_{j}=(\sigma_{j}^{X}-i\sigma_{j}^{Y})/2$, satisfies the canonical anti-commutation relations (CAR) $\{c_{j},c_{k}^{*}\}=\delta_{jk}$, $\{c_{j},c_{k}\}=\{c_{j}^{*},c_{k}^{*}\}=0$, and maps the XY chain onto a system of quasi-free Fermions governed by

Here $A$ and $B$ are the tridiagonal $n\times n$-matrices

If $\gamma_{j}=0$, $j\in\Lambda$, i.e. for the isotropic XY chain, we have $B=0$ and the matrix $A$ has the role of an effective Hamiltonian. In the general anisotropic case the effective Hamiltonian is the $2n\times 2n$-matrix $\begin{pmatrix}A&B\\ -B&-A\end{pmatrix}$. However, to correctly reflect locality properties, we re-index this matrix and write the spin Hamiltonian as the quadratic form

in terms of the column vector ${\mathcal{C}}=(c_{1},c_{1}^{*},\ldots,c_{n},c_{n}^{*})^{t}$ and effective Hamiltonian given as the $2\times 2$-block Jacobi matrix

Here $\sigma^{Z}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ is the third Pauli matrix and $S(\gamma)=\begin{pmatrix}1&\gamma\\ -\gamma&-1\end{pmatrix}$.

We note that, instead of the fermionic modes $c_{j}$ and $c_{j}^{*}$, commonly used are also the Majorana fermions $a_{j}^{+}=c_{j}^{*}+c_{j}$, $a_{j}^{-}=i(c_{j}^{*}-c_{j})$, which lead to a slightly different (but unitarily equivalent) form of the effective Hamiltonian.

The quadratic Hamiltonian (5) can be diagonalized via a Boguliubov transformation implemented by a $2n\times 2n$ Bogoliubov matrix $W$ (meaning that $W$ is orthogonal with $WJW^{t}=J$ for $J=(\sigma^{X})^{\oplus n}$). $W$ diagonalizes $M$ as

with $0\leq\lambda_{1}\leq\ldots\leq\lambda_{n}$ (which can be identified as the singular values of $A+B$, e.g.[29]). This gives a new set of fermionic modes $b_{j}$, $b_{j}^{*}$, represented in vector form as ${\mathcal{B}}=W{\mathcal{C}}$, which expresses $H$ as a free fermion system,

with $E_{0}=\sum_{j}\lambda_{j}$. The Fock basis

with respect to the vacuum vector $\psi_{0}$ of this system gives a full set of eigenvectors of $H$ with corresponding eigenvalues $\sum_{j:\alpha_{j}=1}\lambda_{j}-E_{0}$.

The phenomena of single-particle localization and many-body localization should be clearly distinguished. The latter describes properties of group waves as opposed to waves associated with individual particles in a many-body system. For the XY chain, however, MBL properties of $H$ are consequences of single-particle localization properties of the effective Hamiltonian $M$ (but note that $M$ does not act on a spin, but on a particle with an infinite degree of freedom as $n\to\infty$, e.g. a one-dimensional ‘electron’).

While this is the main mechanism behind the results presented here, we stress that it makes the XY chain a rather unrealistic example of a many-body quantum system. One other example which shares the property of being fully reducible to an effective one-particle Hamiltonian are interacting harmonic quantum oscillators, see [44, 45]. For other classes of disordered quantum many-body systems the relation between one-particle and many-body localization, and the variety of possible MBL regimes and phenomena, will likely be much more involved.

We will state results under the assumption of exponential eigencorrelator localization for $M$, i.e. the existence of a finite constant $C$ and a positive constant $\eta$, not depending on $n$, such that

for all $1\leq j,k\leq n$. Here ${\mathord{\mathbb{E}}}(\cdot)$ denotes disorder averaging and $g(M)$ is defined via the spectral theorem for symmetric matrices, i.e. $g(M)=\sum_{j}g(\lambda_{j})|\phi_{j}\rangle\langle\phi_{j}|$, where $\phi_{j}$ and $\lambda_{j}$ are a full system of eigenvectors and corresponding eigenvalues for $M$. For later convenience, we think of the $2n\times 2n$-matrix $g(M)$ and an $n\times n$-matrix with $2\times 2$-matrix-valued entries $g(M)_{jk}$ and of $\|\cdot\|$ as a $2\times 2$-matrix norm.

In the $n\to\infty$ limit, eigencorrelator localization (10) implies that $M$ has pure point spectrum with exponentially decaying eigenfunctions (with $\eta$ giving the inverse localization length), but is a mathematically stronger property. In particular, with the choice $g(M)=e^{-itM}$, it includes uniform exponential decay of time evolution amplitudes

Also covered by (10) are the Fermi projectors $\chi_{(-\infty,E]}(M)$, i.e. eigenprojectors onto energies below $E$. Eigencorrelator localization in the above strong form allows to derive all the MBL properties to be discussed below.

For the isotropic case, $M$ can be replaced by $A$ in (10) and is known to hold for large classes of random parameters $\nu_{j}$ and $\mu_{j}$. In particular, if $\mu_{j}=1$ and the $\nu_{j}$ are i.i.d. with sufficiently smooth distribution (for example if they have bounded compactly supported density), then $A$ is the one-dimensional Anderson model and (10) is known since the work of Kunz and Souillard [37] and, for larger classes and under weaker assumptions, can also be proven using the fractional moments method of Aizenman and Molchanov [3]. See [56, 4] for general introductions to the mathematical theory of Anderson localization.

Proving (10) in the anisotropic case is mathematically more challenging due to the lack of monotonicity of (6) in the random parameters. A general result of [24] covers the case of a magnetic field at large disorder, i.e. $\nu_{j}$ is multiplied with a sufficiently parameter $\lambda$. For additional results see [16].

For this work we use the assumption of exponential decay of eigencorrelators mostly for ease of presentation and for its physical relevance. Many of the results discussed below, in suitably modified form, also hold under weaker assumptions on eigencorrelator decay, such as sufficiently fast power law decay.

Another assumption needed for some of the results below is that

This is true, for example, if the random variables $\nu_{j}$ have continuous distribution, see Appendix A in [2].

A local observable acting on the $j$-th spin via the $2\times 2$-matrix $A$ will be denoted by $A_{j}$. For $j\not=k$ local observables commute: $[A_{j},B_{k}]=0$. Lieb-Robinson bounds, initially introduced in [38], provide upper bounds on the propagation speed of group waves through the spin system by bounding the norm of the commutators under the time-evolution $\tau_{t}(A_{j})=e^{itH}A_{j}e^{-itH}$ of one of the observables. In general, a Lieb-Robinson bound of the form

with constants $C$ and $\eta>0$ which do not depend on $n$, $j$, $k$ and $t$ and hold for all local observables $A$ and $B$, show that $v$ is an upper bound on the group velocity in the system, see [43] for a survey of recent results and applications.

It was shown in [29] that the introduction of disorder into the $XY$-chain leads to Lieb-Robinson velocity zero in the disorder average: Under the one-particle localization condition (10) (in fact (11) suffices here) it holds that

uniformly in $n$ and $1\leq j,k\leq n$.

For the case of the isotropic $XY$-chain in random transversal field a slightly weaker form of this bound, with right hand side growing linearly in $t$ and quadratically in $n$, was discussed earlier in [15]. The key fact behind the proof of (14) is the relation

where the time evolution of the vector $\mathcal{C}$ is understood componentwise. This relates the one-particle dynamics of $M$, given by the $2n\times 2n$-matrix $e^{-2itM}$, to the many body dynamics of $\mathcal{C}$ and thus, through Jordan-Wigner, of local observables in the spin chain. Here some care is needed to account for the non-locality of the Jordan-Wigner transform: Assume $k\geq j$, then (11), (15) and a geometric summation give

Writing the local spin lowering operators as $a_{j}=\sigma_{1}^{Z}\ldots\sigma_{j-1}^{Z}c_{j}$, an iterative argument based on the Leibnitz rule followed by another geometric sum gives

Using that $a_{j}$, $a_{j}^{*}$, $a_{j}^{*}a_{j}$ and $a_{j}a_{j}^{*}$ generate the local operators at site $j$, this leads to the claim (14) with $C^{\prime}=96C/(1-e^{-\eta})^{2}$. More details on this proof are in [29]; see also the review [54].

More recent results of this type for disordered systems can be found in [27], which establishes an MBL transition for the XY chain in decaying (and thus non-ergodic) random field in the $z$-direction. Related bounds for quasi-periodic models are contained in e.g. [18], [35], and references therein.

One of the benefits of Lieb-Robinson bounds (with finite velocity $v$) is that they can be used to prove exponential clustering of the ground state for gapped spin systems, i.e. exponential decay of the ground state correlations

in $|j-k|$ for local observables $A_{j}$ and $B_{k}$, see [30, 42].

In [29] it was shown that in the presence of a zero-velocity Lieb-Robinson bound the assumption of a uniform ground state gap can be relaxed, leading to a correction of the exponential clustering bound which is logarithmic in the inverse gap size. For the case of the isotropic $XY$-chain in random transverse field this leads to a bound of the form $C\|A\|\|B\|ne^{-\eta|j-k|}$ for the disorder average of ground state correlations (Theorem 4.2 in [29]). The dependence on the length $n$ of the chain is due to the use of a Wegner estimate for the Anderson model, giving a volume dependent ground state gap. For earlier work on this question, employing bounds on the Anderson model found by multiscale analysis, see [36].

We stress that rapid decay of ground state correlations, by itself, is not sufficient to indicate MBL, as it holds for many translation invariant systems, even without requiring a spectral gap [41, 25]. Rather, MBL should reflect a mobility gap, i.e. a range of energies near the bottom of the spectrum which is entirely many-body localized. For correlations in the disordered XY chain this was recently settled in full generality by Sims and Warzel [55]. They consider time dependent correlations

for the anisotropic disordered $XY$-chain (1), where $\langle\cdot\rangle=\operatorname{tr}(\rho\,\cdot)$ and $\rho=|\psi_{\alpha}\rangle\langle\psi_{\alpha}|$ for an arbitrary eigenstate $\psi_{\alpha}$ of $H$ or a thermal state $\rho=e^{-\beta H}/\operatorname{tr}e^{-\beta H}$. Assuming eigencorrelator localization (10) it is shown that

where the constants $C<\infty$ and $\eta>0$ can be chosen uniformly in the eigenstate label $\alpha$ and the inverse temperature $\beta>0$, respectively. In fact, having exponential clustering for all eigenstates means that the disordered XY chain in fully many-body localized at all energies, i.e. in the infinite temperature limit.

Let us now discuss a few of the central ideas that are used in proving (20). In order to do so, we first review some basics about quasi-free states on the CAR algebra and refer the interested reader to e.g. [14] for more details. Recall that for any Hilbert space $\mathcal{H}$, one can associate to each $f\in\mathcal{H}$ annihilation and creation operators, which we label by $c(f)$ and $c^{*}(f)$ respectively. These operators act on the fermionic (i.e. anti-symmetric) Fock space $\mathcal{F}(\mathcal{H})$ corresponding to $\mathcal{H}$ and satisfy canonical anti-commutation relations (CAR), i.e.

with $\{A,B\}=AB+BA$. The $C^{*}$-algebra $\mathcal{A}(\mathcal{H})$ generated by the identity ${\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}$ and the operators $c(f)$ and $c^{*}(g)$ for all $f,g\in\mathcal{H}$ is called the CAR algebra associated to $\mathcal{H}$.

A state $\omega$ on $\mathcal{A}(\mathcal{H})$ is said to be quasi-free if all its correlation functions can be computed using Wick’s rule. An important sub-class of quasi-free states $\omega_{\varrho}$ on $\mathcal{A}(\mathcal{H})$ are uniquely determined by a one-particle density operator $0\leq\varrho\leq{\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}$ acting on $\mathcal{H}$. In this case, multipoint correlation functions corresponding to these states have a determinantal structure, e.g.

for any $m\geq 1$ and $f_{1},\cdots,f_{m},g_{1},\cdots,g_{m}\in\mathcal{H}$. Both classes of states considered in (20), i.e. eigenstates and thermal states of the XY-model, are quasi-free states on $\mathcal{A}(\ell^{2}([1,n]))$ with this particular form. We remark that a full characterization of quasi-free states on $\mathcal{A}(\mathcal{H})$ can be found in [6], but this goes beyond the scope of our applications.

A key result in [55] estimates structured determinants; in particular, the result applies to multi-point correlation functions associated to these quasi-free states, see (22), in the case that $\mathcal{H}=\ell^{2}(\mathbb{Z})$. (The result also applies in the case of $\mathcal{H}=\ell^{2}([1,n])$ for any $n\geq 1$.) To state a version of this result, first introduce for $m\geq 1$, an ordered configuration as ${\rm x}=(x_{1},\cdots,x_{m})\in\mathbb{Z}^{m}$ with $x_{1}<\cdots<x_{m}$. For any two such configurations ${\rm x}$ and ${\rm y}$, denote a configuration distance by

The following is proven in [55].

In words, this result shows that decay in the entries of a matrix, i.e. (25), does lead to a form of decay in the corresponding matrix determinant, see (26). The decay in the determinant is expressed in terms of the configuration distance $D({\rm x},{\rm y})$, see (23).

With these preliminaries, we can now give some rough ideas for how (20) is proven. Consider a quasi-free state $\omega_{\varrho}$ on $\mathcal{A}(\ell^{2}([1,n]))$ of the form described in (22) above and let $\tau_{t}$ denote the dynamics corresponding to the isotropic XY-model. For any pair of ordered configurations ${\rm x}=(x_{1},\cdots,x_{m})$ and ${\rm y}=(y_{1},\cdots,y_{m})$, with components in $[1,n]$, one has

where $A$ is the effective one-particle Hamiltonian and $t\in\mathbb{R}$ is arbitrary. Given the assumptions of exponential eigencorrelator localization and non-degeneracy, i.e. (10) and (12), one concludes that the matrix entries on the right-hand side of (5) decay and as a result, these multi-point correlations decay as well, by an application of Theorem 5.1. A technical point here is that the assumption of exponential eigencorrelator localization only guarantees that the disorder-averaged matrix entries, found on the right-hand side of (5), decay. However, a further result in [55], see Theorem 1.2 therein, proves a generalization of Theorem 5.1 which covers this random situation. In words, the proof shows that if the averaged matrix entries decay similarly to (25), then the averaged determinant decays similarly to (26) In particular, both results obtain estimates which are uniform in $m,n,$ and $t\in\mathbb{R}$.

Now, using the Jordan-Wigner transform, the dynamic correlations considered in (19) can be re-written in terms of time-dependent multi-point correlations, similar to those on the left-hand-side of (5), in specific quasi-free states with the form discussed above. In full generality, the evaluation of these multi-point correlations, i.e. the application of Wick’s rule, results in a pfaffian; not a determinant. Since pfaffians and determinants share similar mathematical properties, it is not surprising that analogs of the theorems mentioned above hold in this case as well. Such technical matters are discussed in detail and proven in [55]. It is with these ideas that (20) is proven.

Another property which reflects many-body localization are area laws for the bipartite entanglement of eigenstates and for the time evolution of initially unentangled states, e.g. [9, 11, 12, 13, 32, 53]. In fact, such bounds are closely related to the rapid decay of correlations discussed in the previous section. Hastings showed in his seminal work [31] that the ground state of a uniformly gapped one-dimensional spin system satisfies an area law, i.e. its bipartite entanglement with respect to a subchain is uniformly bounded, independent of the length of the chain and the subchain. As mentioned above such systems also satisfy exponential clustering of the ground state. In fact, Brandao and Horodecki [12, 13] proved a general result saying that exponential decay of correlations of one-dimensional quantum states implies an area law. This does not only provide a new proof of Hastings’ result, but also applies in cases where exponential decay of correlations is known for excited states, without knowledge on spectral gaps, and thus is applicable to the situation one wants to see in MBL. In particular, it is likely that the results of [29] and [55] on correlation decay in the disordered XY chain combine with the result of [13] to provide area laws for this example. Apart from the fact that in [13] the authors work with periodic boundary conditions, a potential obstacle arises from the fact that the correlation bounds required in [13] are stronger than those provided in [55]. The issue is, once again, the prefactor. The theorem in [13] assumes not only a uniform correlation length but also that the prefactor is just the product of the norms of two arbitrary observables and neither depends on the size of the system nor on the size of the supports of the two observables.

This is one of the reasons why it is desirable to have a more direct argument for an area law of the eigenstate entanglement in the example of the disordered XY chain. Towards this, decompose the chain $\Lambda$ into left and right ends $A=[1,\ell]$ and $B=[\ell+1,n]$. For a pure state $\rho=|\psi\rangle\langle\psi|$, let $\rho_{A}$ and $\rho_{B}$ be the corresponding reduced states. The bipartite entanglement of $\rho$ with respect to this decomposition is the von Neumann entropy of the reduced state,

Generic states will satisfy the volume law ${\mathcal{E}}(\rho)\sim\min\{\ell,n-\ell\}$ (using ${\mathcal{S}}(\rho_{A})={\mathcal{S}}(\rho_{B})$). However, under the assumptions of one-particle localization (10) and non-degeneracy (12) on the XY chain, we have the uniform area law

where the supremum is taken over $\rho=|\psi\rangle\langle\psi|$ for all normalized eigenstates $\psi$ of $H$ and the constant $C$ can be chosen uniformly in $n$ and $\ell$. As before, this includes the disorder average ${\mathord{\mathbb{E}}}(\cdot)$.

We may also consider the dynamical growth of entanglement under a quantum quench. For this, let $\phi_{A}$ and $\phi_{B}$ be eigenstates of $H_{A}$ and $H_{B}$, the restrictions of the XY Hamiltonian to the left and right end chains, respectively. Starting with the product state $\rho=|\varphi_{A}\otimes\varphi_{B}\rangle\langle\varphi_{A}\otimes\varphi_{B}|$, we consider the time evolution $\tau_{t}(\rho)=e^{itH}\rho e^{-itH}$ under the full XY chain Hamiltonian. Under assumptions as above, the dynamical entanglement remains bounded,

uniform in $n$ and $\ell$, with supremum taken over all times $t\in{\mathord{\mathbb{R}}}$ as well as all eigenstates of $H_{A}$ and $H_{B}$. In fact, this result extends to any finite number of quenches in the initial state, with bounds not depending on the number of quenches.

The proofs of (29) and (30) in [2] and [1] rely heavily on the fact that the eigenstates $\rho$ of $H$ are quasi-free, as already used in the previous section, so that they are entirely determined by their two-point function with respect to the fermionic modes $c_{j}$, $c_{j}^{*}$, summarized in the correlation matrix

which we use as vector notation for the $2n\times 2n$-matrix with $2\times 2$-matrix-valued entries

As observed in [57], the entropy of quasi-free states can be calculated via the trace identity $\operatorname{Tr}\rho\log\rho=\operatorname{tr}\Gamma_{\rho}\log\Gamma_{\rho}$. Here we distinguish the traces $\operatorname{tr}$ and $\operatorname{Tr}$ in the one-particle space and many-particle space, respectively. Most importantly, the reduction $\rho_{A}$ of $\rho$ to the left end $[1,\ell]$ of the chain is again quasi-free, so that

Here the correlation matrix $\Gamma_{\rho}^{A}$ of the reduced state is simply the upper left $2\ell\times 2\ell$-block of $\Gamma_{\rho}$ (by the ‘left locality’ of the operators $c_{j}$ and $c_{j}^{*}$).

By a calculation due to [48] this leads to the bound

The eigenstates of $H$ are given by the fermionic basis vectors $\psi_{\alpha}$ in (9). This allows to calculate the correlation matrices of $\rho=\rho_{\alpha}=|\psi_{\alpha}\rangle\langle\psi_{\alpha}|$ via the Bogoliubov transform ${\mathcal{B}}=W{\mathcal{C}}$ and (7) as

Here the right hand side is a spectral projection for the effective Hamiltonian $M$ onto $\Delta_{\alpha}:=\{\lambda_{j}:\alpha_{j}=0\}\cup\{-\lambda_{j}:\alpha_{j}=1\}$. Thus (33) and exponential eigencorrelator localization (10) lead to

proving (29).

Similar arguments lead to the bound (30) on dynamical entanglement in [1]. In particular, this uses that the product states $\rho=|\varphi_{A}\otimes\varphi_{B}\rangle\langle\varphi_{A}\otimes\varphi_{B}|$ as well as their time evolution $\tau_{t}(\rho)$ under $H$ are still quasi-free, so that ${\mathcal{E}}(\tau_{t}(\rho))$ can be calculated using (33). In this case the correlation matrix is given by

where $M_{A}$ and $M_{B}$ are restrictions of the effective Hamiltonian to $A$ and $B$, respectively, and spectral projections analogous to (35) appear. Similar to above, now using eigencorrelator localization for $M$, $M_{A}$ and $M_{B}$, this leads to (30).

Several remarks are in order here:

(i) The bound (36) on the entanglement grows as $O(\xi^{2})$ as the one-particle localization length $\xi=1/\eta$ diverges. This is substantially stronger than what one gets by combining the general result of [13] with the bound on correlation decay from [55], giving the bound $O(e^{c\xi\log\xi})$ on the entanglement.

(ii) The argument leading to (34) was previously used in [48] to bound the entanglement in disordered $d$-dimensional free Fermion systems, where the general area law $O(\ell^{d-1})$ is found, both as upper and lower bound. Moreover, it was shown in [23] that the fermionic entanglement is self-averaging for $d>1$ but not for $d=1$. Note, however, that the fermionic entanglement has to be interpreted differently from entanglement in spin systems due to the non-locality of the fermionic modes, e.g. [8, 49], and that the connection to the XY chain via Jordan-Wigner only holds for $d=1$.

(iii) In principle, the above result also applies to thermal states $\rho_{\beta}=e^{-\beta H}/\operatorname{tr}e^{-\beta H}$. In this case, instead of (35) the correlation matrix is $\Gamma_{\rho_{\beta}}=({\mathchoice{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.0mul}}{\mathrm{1\mskip-4.5mul}}{\mathrm{1\mskip-5.0mul}}}+e^{-2\beta M})^{-1}$, so that the eigencorrelator bound (10) still applies and gives exponential decay of matrix elements and an area law for the bipartite entanglement ${\mathcal{S}}(\rho_{\beta})$. However, for the mixed state $\rho_{\beta}$ this is generally not considered a good entanglement measure (in particular, vanishing of ${\mathcal{S}}(\rho_{\beta})$ does not mean that $\rho_{\beta}$ is a product state). A better quantity to consider is the logarithmic negativity of $\rho_{\beta}$, e.g. [58, 50], but it remains an open problem if the logarithmic negativity of mixed states of the disordered XY chain satisfies an area law.

On the other hand, it follows easily from (29) that the entanglement of formation of thermal states satisfies an area law. The latter is the convex roof extension of the entanglement entropy [61], defined for a general mixed state $\rho$ as

with arbitrary $p_{k}$ and unit vectors $\phi_{k}$ such that $\rho=\sum_{k}p_{k}|\phi_{k}\rangle\langle\phi_{k}|$.

The thermal states are given by

where $\psi_{\alpha}$ and $q_{\alpha}$ are the eigenstates and eigenvalues of $H$ and $Z_{\beta}=\sum_{\alpha}e^{-\beta q_{\alpha}}$. Using the uniform area law (29) for the eigenstates $\psi_{\alpha}$ this immediately leads to

an area law for the averaged entanglement of formation of the thermal states $\rho_{\beta}$, uniform in the inverse temperature $\beta$.

One also gets a similar extension of (30) to the case where $\rho=\rho_{A}\otimes\rho_{B}$ and $\rho_{A}$ and $\rho_{B}$ are thermal states of the subsystems.

Here we consider the isotropic XY chain, i.e. we set $\gamma_{j}=0$ for all $j$ in (1). Thus the matrix $B$ in (4) vansihes and, using the CAR in (2), the Hamiltonian can be written as

where $\tilde{E}_{0}=\sum_{j}\nu_{j}$. This simpler form of the quadratic Fermionic Hamiltonian (as compared to (2)) allows diagonalization via $(\tilde{b}_{1},\ldots,\tilde{b}_{n})^{t}:=U(c_{1},\ldots,c_{n})^{t}$, i.e. a Bogoliubov transformation which does not mix creation and annihilation modes. Here $U$ is the orthogonal matrix which diagonalizes $A$:

with $\tilde{\lambda}_{j}$ the eigenvalues of $A$. One gets the free Fermion system

Let $\psi_{0}$ be the vacuum vector associated with this free Fermion system and

the corresponding Fermionic basis of eigenvectors of $H_{iso}$. Due to the non-mixing of modes, $\psi_{0}$ coincides with the all spins down vector $e_{0}$ and, for each $0\leq k\leq n$, the $k$-particle space span$\{\psi_{\beta}:\#\{j:\beta_{j}=1\}=k\}$ coincides with the space spanned by the spin basis vectors $e_{\alpha}$ with $k$ up-spins. Thus the number of up-spins is interpreted as the number of particles and ${\mathcal{N}}=\sum_{j=1}^{n}a_{j}^{*}a_{j}=\sum_{j=1}^{n}c_{j}^{*}c_{j}$ as the total particle number operator. Preservation of the particle number if reflected in the fact that $H_{iso}$ commutes with ${\mathcal{N}}$.

For subsets $S\subset[1,n]$ the local particle number operators are ${\mathcal{N}}_{S}=\sum_{j\in S}a_{j}^{*}a_{j}$. For any state $\rho$ the expected number of particles (up-spins) in $S$ is given by $\langle{\mathcal{N}}\rangle_{\rho}=\operatorname{Tr}\rho{\mathcal{N}}_{S}$.

In the many-body localized phase one expects absence of particle transport. In fact, let $S_{1}$ and $S_{2}$ be two disjoint subsets of $\Lambda$ such that $S_{2}\subset\Lambda\setminus[\min S_{1},\max S_{1}]$, i.e., $S_{1}$ is entirely surrounded by $S_{2}$. As initial state consider a particle profile

and assume $\eta_{j}=0$ for all $j\in\Lambda\setminus S_{2}$, meaning only down-spins outside $S_{2}$. Let $\rho_{t}=e^{-iHt}\rho e^{iHt}$ be the Schrödinger evolution. If exponential eigencorrelator localization (10) is assumed, then it can be shown that

Here $d(S_{1},S_{2})=\min\{|x-y|:x\in S_{1},y\in S_{2}\}$ is the distance of $S_{1}$ and $S_{2}$. This bound follows as a special case of Theorem 1.1 in [1]. The argument is similar to the proof of Theorem 7.1 below, provided in Appendix A.

Thus, if at time $t=0$ all particles are concentrated in $S_{2}$ (corresponding to $\eta_{j}=1$ for $j\in S_{2}$), then the number of particles at distance more than $d$ from $S_{2}$ remains exponentially small in $d$, uniformly for all times $t$. As in the previous section, the bound diverges as $O(\xi^{2})$ in the one-particle localization length $\xi=1/\eta$. The proof of (45) can be seen from the proofs of Theorems 7.1 and 7.2 below.

Additional results of this type, establishing the absence of particle transport in the disordered isotropic XY chain, can be found in [1]. Similar results for the disordered Tonks-Girardeau gas, which can be viewed as a continuum analogue of the XY chain, can be found in [52].

This result does not extend to the anisotropic XY chain, were the particle number is not preserved and thus particles can be created by local properties of the dynamics, even if the system if fully in the MBL phase.

In essentially the same way one can show the absence of energy transport in the MBL phase. For this let $S_{1}$, $S_{2}$ be as above, but assume that $S_{1}=[a,b]$ is an interval. By $H_{S_{1}}$ we denote the restriction of the isotropic XY chain to $S_{1}$, given as in (40) with $[1,n]$ replaced by $[a,b]$ (but still acting on the full spin chain). Thus $\langle H_{S_{1}}\rangle_{\rho}$ is the expected energy of a state $\rho$ in $S_{1}$.

The uniform matrix bound $D$ can be chosen to be for example $D=2\mu_{max}+\nu_{max}$, where $\mu_{max}$ and $\nu_{max}$ are the maximal values of the distributions of $|\mu_{j}|$ and $|\nu_{j}|$, respectively. To interpret (46), note that by construction of $\rho$ at time $t=0$ all its interaction energy is concentrated in $S_{2}$, so that the amount of energy which can be transported from $S_{2}$ to $S_{1}$ is exponentially small in the distance of the two subsystems, uniformly in time.

A weakened version of (7.1) can be shown to hold for the more general anisotropic XY chain (1).

The constant bound in (47) means that the energy fluctuations within the subsystem $S_{1}$ are uniformly bounded from above, while the full energy $\langle H\rangle_{\rho}$ will grow linearly in the size of $\Lambda$.

Detailed proofs of Theorems 7.1 and 7.2 are provided in Appendix A.

Many-body localization of eigenstates is generally expected to correspond to localization in Fock space, not in physical space, e.g. [5, 10, 11]. This refers to a picture where the Fock space basis of Slater determinants of the (non-interacting) single-particle eigenstates is used as a reference lattice, and eigenstates of the interacting system can be labelled by the Fock space basis and decay rapidly (exponentially) in the distance from some finite subset of the lattice. Here we demonstrate this form of MBL for the disordered XY chain.

As in Section 7 we consider the isotropic XY quantum spin chain in transversal field