$su(2)$ symmetry of XX spin chains

TL;DR

This paper demonstrates that by adjusting a uniform transverse magnetic field, inhomogeneous open XX spin chains exhibit a two-fold degeneracy and an exact $su(2)$ symmetry with nonlocal generators, linked to discrete orthogonal polynomials.

Contribution

It reveals the presence of an exact $su(2)$ symmetry in inhomogeneous XX spin chains through magnetic field adjustment, with explicit generator computation.

Findings

Two-fold degeneracy in the spectrum

Existence of an exact $su(2)$ symmetry

Generators depend on discrete orthogonal polynomial coefficients

Abstract

We show that, after suitably adjusting a uniform transverse magnetic field, the generic inhomogeneous open XX spin chain has a two-fold degeneracy, and an exact $su(2)$ symmetry whose "inhomogeneous" nonlocal generators depend on coefficients that can be explicitly computed for models associated with discrete orthogonal polynomials.

Equations

$$\mathcal{H}=-\frac{1}{2}\sum _{n=0}^{N-1}{J}_n({\sigma }_n^x{\sigma }_{n+1}^x+{\sigma }_n^y{\sigma }_{n+1}^y)-\frac{1}{2}\sum _{n=0}^N{B}_n{\sigma }_n^z,$$

$$\left[\mathcal{H},S^z\right]=0,S^z=\frac{1}{2}\sum _{n=0}^N{\sigma }_n^z.$$

$$\left[\mathcal{H},{\mathbb{S}}^x\right]=\left[\mathcal{H},{\mathbb{S}}^y\right]=\left[\mathcal{H},{\mathbb{S}}^z\right]=0,$$

$${\mathbb{S}}^x=\frac{1}{2}\sum _{n=0}^N{a}_n{\sigma }_0^z\dots {\sigma }_{n-1}^z{\sigma }_n^x,{\mathbb{S}}^y=\frac{1}{2}\sum _{n=0}^N{a}_n{\sigma }_0^z\dots {\sigma }_{n-1}^z{\sigma }_n^y,$$

$${\mathbb{S}}^z:=-i\left[{\mathbb{S}}^x,{\mathbb{S}}^y\right].$$

$${\vec{\mathbb{S}}}^2=({\mathbb{S}}^x{)}^2+({\mathbb{S}}^y{)}^2+({\mathbb{S}}^z{)}^2=\frac{3}{4}\mathbb{I},$$

$$c_n^{†}={\sigma }_0^z\dots {\sigma }_{n-1}^z{\sigma }_n^{+},c_n={\sigma }_0^z\dots {\sigma }_{n-1}^z{\sigma }_n^{-},{\sigma }^{\pm }=\frac{1}{2}\left({\sigma }^x\pm i{\sigma }^y\right),$$

$$\mathcal{H}=\sum _{n=0}^{N-1}{J}_n(c_n^{†}{c}_{n+1}+c_{n+1}^{†}{c}_n)-\sum _{n=0}^N{B}_n{c}_n^{†}{c}_n,$$

$$\mathcal{H}=\sum _{m,n=0}^N{c}_m^{†}{H}_{mn}{c}_n,$$

$$H=\left(\begin{array}{cccccc}-B_0& J_0& & & & \\ J_0& -B_1& J_1& & & \\ & J_1& -B_2& J_2& & \\ & & \ddots & \ddots & \ddots & \\ & & & J_{N-2}& -B_{N-1}& J_{N-1}\\ & & & & J_{N-1}& -B_N\end{array}\right).$$

$$H|{\omega }_k⟩={\omega }_k|{\omega }_k⟩,k=0,1,\dots ,N.$$

$${\omega }_K=0.$$

$$|{\omega }_k⟩=\sum _{n=0}^N{\varphi }_n({\omega }_k)|n⟩,$$

$$\sum _{n=0}^N{\varphi }_n({\omega }_k){\varphi }_n({\omega }_p)={\delta }_{kp}.$$

$$\mathcal{H}=\sum _{k=0}^N{\omega }_k{\tilde{c}}_k^{†}{\tilde{c}}_k$$

$${\tilde{c}}_k=\sum _{n=0}^N{\varphi }_n({\omega }_k)c_n,c_n=\sum _{k=0}^N{\varphi }_n({\omega }_k){\tilde{c}}_k,$$

$$|\Psi ⟩⟩={\tilde{c}}_{k_1}^{†}\dots {\tilde{c}}_{k_r}^{†}|0⟩⟩,$$

$${\tilde{c}}_k|0⟩⟩=0,k=0,\dots ,N.$$

$$\left[\mathcal{H},{\tilde{c}}_K^{†}\right]=\left[\mathcal{H},{\tilde{c}}_K\right]=0.$$

$${\mathbb{S}}^x=\frac{1}{2}\left({\tilde{c}}_K^{†}+{\tilde{c}}_K\right),{\mathbb{S}}^y=-\frac{i}{2}\left({\tilde{c}}_K^{†}-{\tilde{c}}_K\right);$$

$$a_n={\varphi }_n({\omega }_K)={\varphi }_n(0).$$

$$\left[{\mathbb{S}}^x,{\mathbb{S}}^z\right]=-i{\mathbb{S}}^y,\left[{\mathbb{S}}^y,{\mathbb{S}}^z\right]=i{\mathbb{S}}^x.$$

$$\left[{\mathbb{S}}^x,S^z\right]$$

$$\left[{\mathbb{S}}^y,S^z\right]$$

$$\left[{\mathbb{S}}^z,S^z\right]$$

$${\left({\mathbb{S}}^x\right)}^2={\left({\mathbb{S}}^y\right)}^2={\left({\mathbb{S}}^z\right)}^2=\frac{1}{4}\mathbb{I},$$

$${\mathbb{S}}^x|\Psi ⟩⟩=\alpha |\Psi ⟩⟩$$

$$J_0=\dots =J_{N-1}=-\frac{1}{2},B_0=\dots =B_{N-1}=-\cos \left(\frac{\pi (K+1)}{N+2}\right),$$

$${\varphi }_n({\omega }_k)=\sqrt{\frac{2}{N+2}}\sin \left[\frac{\pi (k+1)(n+1)}{N+2}\right],$$

$${\omega }_k=\cos \left(\frac{\pi (K+1)}{N+2}\right)-\cos \left(\frac{\pi (k+1)}{N+2}\right),k=0,1,\dots ,N.$$

$$a_n=\sqrt{\frac{2}{N+2}}\sin \left[\frac{\pi (K+1)(n+1)}{N+2}\right].$$

Full text

$su(2)$ symmetry of XX spin chains

Nicolas Crampéa, Rafael I. Nepomechieb, Luc Vinetc,d,e 111email:luc.vinet@umontreal.ca (Corresponding author)

and Nabi Zare Haroftehb

aCNRS – Université de Montréal CRM - CNRS, France

bDepartment of Physics, PO Box 248046

University of Miami, Coral Gables, FL 33124 USA

cCentre de Recherches Mathématiques, Université de Montréal

P.O. Box 6128, Centre-ville Station

Montréal (Québec), H3C 3J7, Canada

dDépartement de Physique, Université de Montréal

Montréal (Québec), H3C 3J7, Canada

eIVADO, 6666 Rue Saint-Urbain

Montréal (Québec), H2S 3H1, Canada

We show that, after suitably adjusting a uniform transverse magnetic field, the generic inhomogeneous open XX spin chain has a two-fold degeneracy, and an exact $su(2)$ symmetry whose “inhomogeneous” nonlocal generators depend on coefficients that can be explicitly computed for models associated with discrete orthogonal polynomials.

1 Introduction

The XX spin chain has been widely studied, see e.g. [1, 2, 3, 4, 5] and references therein. We consider here the inhomogeneous open XX spin-chain Hamiltonian defined on a lattice with $N+1$ sites111Our notation closely follows that of [3].

[TABLE]

where $J_{n}\neq 0$ and $B_{n}$ are real parameters. This Hamiltonian has the evident $U(1)$ symmetry

[TABLE]

In this note, we observe that, for arbitrary values of the parameters $J_{n}$ and $B_{n}$ (up to a possible shift of the latter parameter by an $n$-independent constant, which corresponds to turning on a uniform magnetic field in the $z$ direction), this Hamiltonian possesses the additional exact symmetry

[TABLE]

where $\mathbb{S}^{x}$ and $\mathbb{S}^{y}$ are corresponding “inhomogeneous” nonlocal generators of the form

[TABLE]

whose coefficients $a_{n}$ (which depend on the model parameters $J_{n}$ and $B_{n}$) follow from a general expression given below, see (2.15). Explicit analytic expressions for these coefficients can be obtained for models associated with discrete orthogonal polynomials. Moreover,

[TABLE]

The algebra generated by $\mathbb{S}^{x}$, $\mathbb{S}^{y}$ and $\mathbb{S}^{z}$ is $su(2)$, with its Casimir element given by

[TABLE]

which follows from $\sum_{n=0}^{N}(a_{n})^{2}=1$, proven in the following. This result shows that this representation of the $su(2)$ algebra is a direct sum of two-dimensional irreducible representations. This symmetry implies that the spectrum of the Hamiltonian is two-fold degenerate.

We shall see that the $su(2)$ symmetry and two-fold degeneracy of the spectrum can be understood from the fact that the corresponding fermionic system has a zero-energy mode.

Generators of $su(2)$ that depend on model parameters and are spatially dependent appeared recently in [6]. However, the system considered there is homogeneous, and the symmetry is only up to boundary terms. The generators (1.4) resemble inhomogeneous versions of so-called semilocal charges considered in e.g. [7, 8].

We present the general construction of the symmetry generators and their algebra in Section 2. Two examples are worked out in Section 3. We conclude with a brief discussion in Section 4.

2 Fermionic zero modes and symmetry generators

In order to prove the symmetries (1.3) and determine the coefficients $a_{n}$ in the “inhomogeneous” generators (1.4), we use the Jordan–Wigner transformation [9]

[TABLE]

where $\{c_{m}^{\dagger}\,,c_{n}\}=\delta_{m,n}$ and $\{c_{m}\,,c_{n}\}=0$, to bring the Hamiltonian (1.1) to that of an inhomogeneous free-fermion model with open boundary conditions

[TABLE]

up to an additive constant. The Hamiltonian (2.2) can be rewritten in quadratic form

[TABLE]

where $H$ is the $(N+1)\times(N+1)$ tridiagonal matrix

[TABLE]

We denote by $|\omega_{k}\rangle$ the orthonormal eigenvectors of $H$ with eigenvalues $\omega_{k}$

[TABLE]

A key point is that one of the eigenvalues of $H$, say $\omega_{K}$, can always be set to 0

[TABLE]

Indeed, this can evidently be arranged by performing the constant shift $H\mapsto H-\omega_{K}\mathbb{I}$, where $\mathbb{I}$ is the identity matrix, which results in a shift of all the eigenvalues $\omega_{k}\mapsto\omega_{k}-\omega_{K}$. In view of (2.4), this means shifting $B_{n}\mapsto B_{n}+\omega_{K}$, which corresponds in (1.1) to turning on the uniform magnetic field $-\omega_{K}\hat{z}$. The eigenvectors can be expressed as

[TABLE]

where $\{|0\rangle,|1\rangle,\dots,|N\rangle\}$ is the canonical orthonormal basis of $\mathbb{C}^{N+1}$. The eigenfunctions $\phi_{n}(\omega_{k})$ are real, since the matrix $H$ is real and its eigenvalues are non-degenerate (see e.g. Lemma 3.1 in [10]). These eigenfunctions therefore satisfy the orthonormality conditions

[TABLE]

The free-fermion Hamiltonian $\mathcal{H}$ (2.2) becomes diagonal

[TABLE]

when expressed in terms of the annihilation operators $\tilde{c}_{k}$ defined by

[TABLE]

and the creation operators $\tilde{c}^{\dagger}_{k}$ given by Hermitian conjugation of (2.10), hence $\{\tilde{c}^{\dagger}_{k}\,,\tilde{c}_{p}\}=\delta_{k,p}$ and $\{\tilde{c}_{k}\,,\tilde{c}_{p}\}=0$. In view of (2.9), the eigenvectors of $\mathcal{H}$ are given by

[TABLE]

with corresponding energy eigenvalues $E=\sum_{i=1}^{r}\omega_{k_{i}}$, where $k_{1},\ldots,k_{r}\in\{0,\ldots,N\}$ are pairwise distinct, and the vacuum state $|0\rangle\!\rangle$ is annihilated by all the annihilation operators

[TABLE]

We now observe that, due to the zero mode (2.6), the Hamiltonian (2.9) commutes with the corresponding zero-mode operators

[TABLE]

It follows that the XX spin-chain Hamiltonian (1.1) indeed has the claimed symmetry (1.3), where the symmetry generators $\mathbb{S}^{x}$ and $\mathbb{S}^{y}$ are given by linear combinations of the fermionic zero-mode operators

[TABLE]

the $a_{n}$ coefficients in (1.4) are given, in view of (2.1) and (2.10), by

[TABLE]

The generators $\mathbb{S}^{x}$, $\mathbb{S}^{y}$ and $\mathbb{S}^{z}$ (the latter defined in (1.5)) close to an $su(2)$ algebra

[TABLE]

If we extend this algebra to include $S^{z}$ (1.2), then we have

[TABLE]

We further note the identities

[TABLE]

which follow from (2.14).

The symmetry (1.3) implies that the spectrum of the Hamiltonian has a two-fold degeneracy. Indeed, if $|\Psi\rangle\!\rangle$ is an eigenvector of $\mathcal{H}$, then $\mathbb{S}^{x}\,|\Psi\rangle\!\rangle$ is also an eigenvector of $\mathcal{H}$ with the same eigenvalue. Moreover, $|\Psi\rangle\!\rangle$ and $\mathbb{S}^{x}\,|\Psi\rangle\!\rangle$ are linearly independent222Recalling (1.2), let $|\Psi\rangle\!\rangle$ be a simultaneous eigenvector of $\mathcal{H}$ and $S^{z}$; and let us assume that $\mathbb{S}^{x}\,|\Psi\rangle\!\rangle$ and $|\Psi\rangle\!\rangle$ are not linearly independent, so that

$\mathbb{S}^{x}\,|\Psi\rangle\!\rangle=\alpha\,|\Psi\rangle\!\rangle$

for some constant $\alpha$. Applying (2.17) to $|\Psi\rangle\!\rangle=|\Psi\rangle\!\rangle$, we obtain $\mathbb{S}^{y}\,|\Psi\rangle\!\rangle=0$; and it follows from (2.20) that $|\Psi\rangle\!\rangle=0$, which is a contradiction., and $(\mathbb{S}^{x})^{2}$ is proportional to the identity operator.

3 Examples

We present here two examples with the symmetry (1.3)-(1.5), where the coefficients $a_{n}$ (2.15) can be computed explicitly. Both examples correspond to models associated with discrete orthogonal polynomials [3].

3.1 Homogeneous chain

For the homogeneous chain, the Hamiltonian is given by (1.1) with

[TABLE]

for which the eigenfunctions in (2.7) are given by [3]

[TABLE]

where

[TABLE]

Note that $B_{n}$ in (3.1) have been adjusted to ensure the zero mode $\omega_{K}=0$. The coefficients $a_{n}$ (2.15) are therefore given by

[TABLE]

One can check, by brute-force diagonalization for $N=1,2\ldots$ and $K=0,1,\ldots,N$, that all the eigenvalues of the Hamiltonian (1.1) with parameters (3.1) are (at least) two-fold degenerate.

3.2 Krawtchouk chain

For the Krawtchouk chain, the Hamiltonian is given by (1.1) with [3]

[TABLE]

The eigenfunctions in (2.7) are given by

[TABLE]

where $R_{n}(-k)$ are Krawtchouk polynomials that can be expressed in terms of the hypergeometric function

[TABLE]

and

[TABLE]

Note that here $\omega_{0}=0$, corresponding to the choice $K=0$. The coefficients $a_{n}$ (2.15) are therefore given by

[TABLE]

One can explicitly check that all the eigenvalues of the Hamiltonian (1.1) with parameters (3.5) and $N=1,2\ldots$ are (at least) two-fold degenerate.

4 Discussion

We have seen that, after suitably adjusting a uniform transverse magnetic field, the generic inhomogeneous open XX spin chain has a two-fold degeneracy, and an enhanced $su(2)$ symmetry (1.3). The “inhomogeneous” symmetry generators (1.4) depend on coefficients $a_{n}$ (2.15) that can be computed explicitly for models associated with discrete orthogonal polynomials. The expressions for the generators (1.4), with their non-local “tails” of $\sigma^{z}$ operators, are reminiscent of coproducts of quantum-group generators [11]. The exact two-fold degeneracy of all the levels of the open XX chain due to the $su(2)$ symmetry presented here is reminiscent of the approximate pairing of all the levels in the ordered phase of the one-dimensional open Ising chain due to the presence of a strong zero mode [12, 13, 14]. The edge spins of the latter model are known to have long coherence times [14]. For the former model, perhaps some (presumably nonlocal) observable could be found, which has suitable overlap with the $su(2)$ charges and is therefore conserved for long times. The symmetry uncovered here may provide further insight into quench dynamics [7, 8] and protected phases [15] in inhomogeneous spin chains. It would be very interesting if this type of symmetry could be extended to XXZ-type chains. It would also be interesting to find higher-rank algebras with such “inhomogeneous” generators, and corresponding physical models with such symmetries.

Acknowledgements

We thank Eric Vernier for helpful correspondence. NC is supported in part by the international research project AAPT of the CNRS. RN is supported in part by the National Science Foundation under grant PHY 2310594, and by a Cooper fellowship. LV is funded in part by a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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