Entanglement entropy for scale-invariant states: universal finite-size scaling

Symmetry offers a powerful means to understand fundamental phenomena in physics and beyond symmetry ; andersonbook ; SSBbook . A specific example concerns scale invariance, which was believed to imply conformal invariance, as first speculated by Polyakov polyakov . Historically, this speculation led to the creation of conformal field theory cft . Since then much effort has been made in an attempt to clarify a deep connection between scale and conformal invariance morecf . However, a systematic investigation of scale-invariant, but not conformally invariant, quantum states, relevant to a classification of quantum critical points, is still lacking, even for one-dimensional quantum many-body systems.

A recent conceptual development has revealed that highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes in one-dimensional quantum many-body systems are scale-invariant FMGM ; golden ; LLspin1 . Somewhat surprisingly, a paradigmatic example is the ${\rm SU}(2)$ Heisenberg ferromagnet, with the highly degenerate ground states being the familiar ferromagnetic states. As demonstrated, the highly degenerate ground states admit an exact singular value decomposition, thus exhibiting self-similarities in the ground state subspace. As a consequence, an abstract fractal is revealed, living in a Hilbert space, which may be characterized in terms of the fractal dimension, introduced earlier by Castro-Alvaredo and Doyon doyon for the ${\rm SU}(2)$ Heisenberg ferromagnetic states in the context of a field-theoretic approach. The fractal dimension is identified with the number of the type-B Goldstone modes FMGM . In other words, there is a fascinating connection between scale-invariant states and the counting rule of the Goldstone modes FMGM . The latter in turn is established with the introduction of the Watanabe-Brauner matrix brauner-watanabe ; watanabe , as a result of an insightful observation made by Nambu nambu , thus leading to the classification of type-A and type-B Goldstone modes.

However, a natural question arises as to whether or not it is possible to distinguish scale-invariant states from conformally invariant states, given that the entanglement entropy scales logarithmically with the block size in the thermodynamic limit, with the prefactor being half the number of the type-B Goldstone modes FMGM for scale-invariant states and $c/3$ for conformally invariant states, where $c$ is the central charge. One may anticipate that a finite system-size scaling behavior of the entanglement entropy is different for scale-invariant states and conformally invariant states, thus furnishing a sensible means for distinguishing them. Physically, such a distinct finite system-size scaling behavior is deeply rooted in the fact that highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes only concerns a scale transformation, instead of being a scale transformation plus the Lorentz boost for conformally invariant states.

This question is addressed here using an heuristic physical argument, which leads to three constraints imposed on the entanglement entropy: first, it must be identical for a block and for its environment; second, it should reproduce a logarithmic scaling function with the block size in the thermodynamic limit; third, as a function of both the system size and the block size, it must be homogeneous, with the order being one, to render consistency with a scale transformation. As it turns out, the universal scaling function for such a type of scale-invariant states is the most fundamental solution satisfying the three constraints, with further confirmation coming from an asymptotic analysis of the entanglement entropy for the ${\rm SU}(2)$ spin-$1/2$ ferromagnetic states. To illustrate our scheme, we choose the ${\rm SU}(2)$ spin-$s$ Heisenberg ferromagnetic model, the ${\rm SU}(N+1)$ ferromagnetic model, and the staggered ${\rm SU}(3)$ spin-1 ferromagnetic biquadratic model.

Consider a quantum many-body system on a lattice, described by Hamiltonian $\mathscr{H}$. Suppose the system is partitioned into a block $B$ and its environment $E$, with the block consisting of $n$ (contiguous) lattice sites, with the other $L-n$ lattice sites constituting the environment $E$. If the symmetry group $G$ is spontaneously broken into $H$, then the counting rule is established as $N_{A}+2N_{B}=N_{BG}$ watanabe , where $N_{A}$ and $N_{B}$ are the numbers of the type-A and type-B GMs, and $N_{BG}$ is equal to the dimension of the coset space $G/H$. In particular, for a quantum many-body system, no type-A GM survives in one spatial dimension, as follows from the Mermin-Wagner-Coleman theorem mwc . Hence the number of the type-A GMs $N_{A}$ is always zero, if one is restricted to one-dimensional quantum many-body systems.

Here we present an heuristic argument to establish a universal scaling behavior of the entanglement entropy $S(L,n)$ with $L$ and $n$ for a scale-invariant quantum state, which is one of the highly degenerate ground states for a quantum many-body system undergoing spontaneous symmetry breaking with type-B Goldstone modes. For completeness, an alternative derivation is also suggested for conformally invariant quantum states.

For both scale-invariant and conformally invariant states FMGM , the entanglement entropy $S(n)$ exhibits a logarithmic scaling behavior with the block size $n$ in the thermodynamic limit (up to an additive constant):

Here, $S(n)$ denotes $S(L,n)$, when $L\rightarrow\infty$. Note that the prefactor is $N_{B}/2$ for a scale-invariant quantum state and $c/3$ for a conformally invariant quantum state, where $N_{B}$ denotes the number of the type-B Goldstone modes. We remark that, physically, both central charge $c$ and the number of the type-B Goldstone modes $N_{B}$ count the number of gapless low-lying excitations for the quantum many-body system under investigation.

Keeping this fact in mind, one may assume that, when the system size $L$ and the block size $n$ are finite but large enough, the entanglement entropy $S(L,n)$ scales with $L$ and $n$ as follows,

Here, $g(L,n)$ is introduced as a universal scaling function of $L$ and $n$, yet to be determined. Three constraints imposed on $g(L,n)$ emerge from the following three physical requirements, to which the entanglement entropy $S(L,n)$ is subject.

First, the fact that for any pure quantum state, the entanglement entropy for a block $B$ is identical to that for its environment $E$ nielsen implies that $S(L,n)=S(L,L-n)$. Hence, we have

In addition, $S(L,n)$ must be monotonically increasing with $n$ until it reaches $L/2$, where $L$ is assumed to be even.

Second, as the thermodynamic limit is approached, i.e., $L\rightarrow\infty$, Eq. (2) reduces to Eq. (1). Therefore, one may expect that in this limit $g(L,n)$ becomes $n$:

Third, $g(L,n)$ must be a first-order homogeneous function of $L$ and $n$ as a result of a scale transformation: $n\rightarrow\lambda\,n$ and $L\rightarrow\lambda\,L$, given that it becomes $n$ in the thermodynamic limit [cf. Eq. (4)]. Thus

As it turns out, the three constraints imposed on $g(L,n)$ allow us to determine its specific forms. In fact, if one chooses $\lambda=1/L$, then we have

This implies that $g(L,n)$ is, up to a multiplicative factor $L$, a function of $n/L$:

Here, $k(n/L)$ is a function of $n/L$, yet to be determined. Our task is therefore reduced to the determination of the function $k(x)$, with $x=n/L$.

To proceed, let us first make an intriguing observation. Suppose that a function $k(x)$ yields a solution satisfying the three constraints, then any function $F(k(x))$ of $k(x)$ also yields a solution, subject to two conditions for $F(x)$: (i) its constant term vanishes, $F(0)=0$ and (ii) its first-order derivative is nonzero, $F^{\prime}(0)\neq 0$. This observation drastically simplifies our task, since we may restrict ourselves to searching for the most fundamental solution, corresponding to a plain scale transformation, in the sense that no other symmetry transformation is involved, if one excludes any internal symmetry.

For our purpose, it is plausible to assume that $k(x)=u(x)u(1-x)$, with $u(x)$ being a polynomial function of $x$, in order to ensure that the first constraint (3) is satisfied. Then, the second constraint (4) implies that $u(x)=x$ as the simplest choice. Hence, we have $k(x)=x(1-x)$. This yields the simplest universal scaling function, thus constituting the most fundamental solution to the three constraints for scale-invariant states. Here, we remark that there exists another solution $k_{c}(x)$, which may be expressed in terms of elementary functions, if one assumes that $k_{c}(x)=v(x)+v(1-x)$, with $v(x)$ being a non-polynomial function of $x$, in order to ensure that the first constraint (3) is satisfied. Then, the second constraint (4) implies that $v(x)=1/(2i\pi)\exp(i\pi x)$. Hence, we have $k_{c}(x)=1/\pi\sin(\pi x)$. Indeed, $k_{c}(x)$ may be re-expressed as a function of $k(x)$, namely $k_{c}(x)=1/\pi\cos[\pi/2\sqrt{1-4k(x)}]$, as one may have anticipated from the observation above. Actually, $k_{c}(x)$ constitutes a universal scaling function for conformally invariant states, as predicted by Calabrese and Cardy cardy (more details on the universal scaling functions $k(x)$ and $k_{c}(x)$ are given in Section A of the Supplemental Material (SM)).

In order to lend further support to our claim that $k(x)$ is the desired universal scaling function for scale-invariant states arising from spontaneous symmetry breaking with type-B Goldstone modes, we may resort to exactly solvable quantum many-body systems. One may choose the ${\rm SU}(2)$ spin-$1/2$ ferromagnetic Heisenberg model, with the number of the type-B Goldstone mode $N_{B}$ being 1. For this model, the entanglement entropy $S(L,n)$ has been derived by Popkov and Salerno popkov . As it turns out, our findings reproduce their results for the ${\rm SU}(2)$ spin-$1/2$ ferromagnetic states (cf. Section B of the SM). For a conformally invariant state, one may choose the transverse-field spin-$1/2$ Ising chain at its critical point, with central charge $c=1/2$. For this model, the entanglement entropy $S(L,n)$ has been derived by Jin and Korepin korepin , thus allowing us to reproduce the prediction from conformal field theory cardy .

More precisely, for a scale-invariant state, with a given filling $f$, the entanglement entropy $S_{\!\!f}(L,n)$ takes the form

On the other hand, for a conformally invariant state, the entanglement entropy $S(L,n)$ takes the form

Note that an additive nonuniversal constant $S_{\!\!f0}$ or $S_{\!0}$ has been introduced for a scale-invariant or conformally invariant state, respectively.

Three remarks are in order. First, it is necessary to introduce the filling $f$ to indicate distinct degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes, in contrast to conformally invariant states. Note that, generically, $f$ refers to a set of fillings $f_{1},f_{2},\ldots,f_{r}$, with $r$ being the rank of a symmetry group for a quantum many-body system. Second, the block does not necessarily consist of contiguous sites for a scale-invariant state, which appears as one of highly degenerate ground states in a quantum many-body system, as already discussed for the entanglement entropy in the thermodynamic limit FMGM . Meanwhile, the scaling relation (8) is left intact as the boundary conditions vary from periodic to open. In contrast, a conformally invariant state appears to be a non-degenerate ground state for a quantum many-body system at criticality, and the scaling relation (9) does depend on the boundary conditions adopted cardy ; korepin1 ; zhou . Third, other choices for the universal scaling function $g(L,n)$ are possible for distinct types of scale-invariant states, but remain to be clarified.

For the ${\rm SU}(2)$ spin-$s$ ferromagnetic Heisenberg model, the entanglement entropy $S(L,n,M)$ for the highly degenerate ground states $|L,M\rangle$ have been derived in Ref. FMGM , with $f$ being the filling $f=M/L$. For convenience, the explicit expression for $S(L,n,M)$ has been collected in Section C of the SM.

We plot the entanglement entropy $S(L,n,M)$ vs $n$ against the universal finite-size scaling $S_{\!\!f}(L,n)$ vs $n$ in Fig. 1 for $s=1/2$, $1$, $3/2$ and $2$. Our numerical data for $S(L,n,M)$ fall on the curve $S_{\!\!f}(L,n)$, when $n$ ranges from 10 to 90, with the relative errors being less than $1.5\%$. Here and hereafter, we have regarded $S_{\!\!f}(L,n)$ as a function of $n$ for fixed $L$ and $f$.

For the ${\rm SU}(N+1)$ ferromagnetic model, the entanglement entropy $S(L,n,M_{1},\ldots,M_{N})$ for the highly degenerate ground state $|L,M_{1},M_{2},\ldots,M_{N}\rangle$ has been derived in Ref. FMGM , with the filling factors $f_{1}=M_{1}/L$, $f_{2}=M_{2}/L$ and $f_{3}=M_{3}/L$. The explicit expression for $S(L,n,M_{1},\ldots,M_{N})$ has been collected in Section C of the SM.

We plot the entanglement entropy $S(L,n,M_{1},M_{2})$ and the entanglement entropy $S(L,n,M_{1},M_{2},M_{3})$ vs $n$, with the selected fillings (a) $f_{1}=1/4$ and $f_{2}=1/4$ and (b) $f_{1}=1/4$, $f_{2}=1/4$ and $f_{3}=1/4$, respectively, against the universal finite-size scaling $S_{\!\!f}(L,n)$ vs $n$, in Fig. 2, with $N_{B}=2$ and $N_{B}=3$. Our numerical data for $S(L,n,M_{1},M_{2})$ and $S(L,n,M_{1},M_{2},M_{3})$ fall on the curve $S_{\!\!f}(L,n)$, when $n$ ranges from 10 to 90, with the relative errors being less than $1\%$.

Finally for the staggered ${\rm SU}(3)$ spin-1 ferromagnetic biquadratic model, the entanglement entropy $S(L,n,M_{1},M_{2})$ and the entanglement entropy $S(L,n,M_{2},M_{3})$ for the two degenerate ground states $|L,M_{1},M_{2}\rangle_{2}$ and $|L,M_{2},M_{3}\rangle_{4}$ have been derived in Ref. golden , with the filling factors $f_{1}=M_{1}/L$, $f_{2}=M_{2}/L$ and $f_{3}=M_{3}/L$. Here the explicit expressions for $S(L,n,M_{1},M_{2})$ and $S(L,n,M_{2},M_{3})$ have been collected in Section C of the SM.

For this model we plot the entanglement entropy $S(L,n,M_{1},M_{2})$ and $S(L,n,M_{2},M_{3})$ vs $n$ against the universal finite-size scaling $S_{\!\!f}(L,n)$ vs $n$ in Fig. 3, with the selected filling factors (a) $f_{1}=1/4$ and $f_{2}=1/4$, and (b) $f_{2}=1/4$ and $f_{3}=1/8$. Our numerical data for $S(L,n,M_{1},M_{2})$ and $S(L,n,M_{2},M_{3})$ are seen to fall on the curve $S_{\!\!f}(L,n)$, when $n$ ranges from 10 to $L-10$, with the relative errors being less than $1\%$.

A universal finite-size scaling behavior of the entanglement entropy has been shown for scale-invariant, but not conformally invariant, degenerate ground states, which arise from spontaneous symmetry breaking with type-B Goldstone modes in quantum many-body systems. As it turns out, scale-invariant states exhibit a universal finite system-size scaling behavior, distinct from conformally invariant states, given both are reduced to a logarithmic scaling function in the thermodynamic limit. This fact enables us to distinguish scale-invariant states from conformally invariant states. It follows that scale invariance does not necessarily imply conformal invariance, in contrast to the speculation made by Polyakov polyakov , even for the familiar ${\rm SU}(2)$ spin-$1/2$ ferromagnetic states – a paradigmatic example for spontaneous symmetry breaking with type-B Goldstone modes. Our findings shed further light on a complete classification of quantum phase transitions and quantum states of matter entropy . In this sense, the possibility for formulating a fully-fledged theory of scale-invariant states remains largely unexplored.

Instead, a simple but heuristic physical argument has been proposed, which imposes three given constraints on the entanglement entropy. This in turn makes it possible to classify distinct types of scale-invariant states. Our results clearly suggest that highly degenerate ground states arising from spontaneous symmetry breaking with type-B Goldstone modes fall into the simplest category for all possible scale-invariant states, featuring the most fundamental universal scaling function. All other possible universal scaling functions might be expressed in terms of this fundamental universal scaling function, as illustrated for conformally invariant states.

We thank John Fjaerestad for enlightening discussions.