Entanglement entropy in low-energy field theories at finite chemical potential

TL;DR

This paper explores how entanglement entropy behaves in non-relativistic systems with O(2) symmetry and Lorentz invariance breaking, linking the Higgs gap to entanglement properties and validating predictions with numerical results.

Contribution

It establishes a theoretical connection between the Higgs gap and entanglement entropy in non-relativistic, symmetry-broken phases, supported by numerical comparisons.

Findings

Entanglement entropy follows an area-law in the studied systems.

The Higgs gap influences the entanglement entropy and correlation length.

Numerical results agree with theoretical predictions in specific quantum systems.

Abstract

We investigate the leading area-law contribution to entanglement entropy in a system described by a general Lagrangian with O(2) symmetry containing first- and second-order time derivatives, namely breaking the Lorentz-invariance. We establish a connection between the Higgs gap present in a symmetry-broken phase and the area-law term for the entanglement entropy in the general, non-relativistic case. Our predictions for the entanglement entropy and correlation length are successfully compared to numerical results in two paradigmatic systems: the Mott insulator to superfluid transition for ultracold lattice bosons, and the ground state of ferrimagnetic systems.

Equations

$$\mathcal{L}/K=|({\partial }_t-i{\mu }_r)\psi {|}^2-c^2|\nabla \psi {|}^2-m^2|\psi {|}^2-c_4|\psi {|}^4,$$

$$\begin{array}{rl}\mathcal{H}(\mathbf{k})=& \frac{1}{2K}({\pi }_1^2+{\pi }_2^2)+\frac{K}{2}(m^2+c^2{k}^2)({\varphi }_1^2+{\varphi }_2^2)\\ & +{\mu }_r({\varphi }_1{\pi }_2-{\varphi }_2{\pi }_1)+\frac{K}{4}{c}_4({\varphi }_1^2+{\varphi }_2^2{)}^2,\end{array}$$

$${\Psi }_{{\Omega }_{\mathbf{k}}}[\varphi ]\propto \exp \left\{-\frac{1}{2}\int d^D\mathbf{k}K{\Omega }_{\mathbf{k}}{\varphi }^2\right\}.$$

$$S/\mathcal{A}=a_0-2a_1(\xi ),$$

$$\begin{array}{r}{\mathcal{H}}_{ord.}^{(2)}(\mathbf{k})=\frac{1}{2K}({\pi }_1^2+{\pi }_2^2)+{\mu }_r({\varphi }_1{\pi }_2-{\varphi }_2{\pi }_1)\\ +\frac{K}{2}\left[{\varphi }_1^2(c^2{k}^2+3{\mu }_r^2-2m^2)+{\varphi }_2^2(c^2{k}^2+{\mu }_r^2)\right].\end{array}$$

$$H=-J\sum _{⟨i,j⟩}\left(b_i^{†}{b}_j+\text{h.c.}\right)+\frac{U}{2}\sum _i{b}_i^{†}{b}_i^{†}{b}_i{b}_i-\mu \sum _i{b}_i^{†}{b}_i,$$

$$\xi =2\sqrt{\frac{\rho }{\Delta }}.$$

$$\mathcal{L}/K=|({\partial }_t-i{\mu }_r)\psi {|}^2-c^2|\nabla \psi {|}^2-m^2|\psi {|}^2-c_4|\psi {|}^4.$$

$$\mathcal{H}(\mathbf{k})=\frac{1}{2K}({\pi }_1^2+{\pi }_2^2)+\frac{K}{2}(m^2+c^2{k}^2)({\varphi }_1^2+{\varphi }_2^2)+{\mu }_r({\varphi }_1{\pi }_2-{\varphi }_2{\pi }_1),$$

$$\begin{array}{rl}{\pi }_{\pm }& =\frac{1}{\sqrt{2}}\left[\sqrt{\frac{{\Gamma }_{\pm }}{K\Omega }}{\pi }_1\pm \sqrt{K\Omega {\Gamma }_{\pm }}{\varphi }_2\right],\\ {\varphi }_{\pm }& =\frac{1}{\sqrt{2}}\left[\sqrt{\frac{K\Omega }{{\Gamma }_{\pm }}}{\varphi }_1\mp \frac{1}{\sqrt{K\Omega {\Gamma }_{\pm }}}{\pi }_2\right].\end{array}$$

$$\mathcal{H}(\mathbf{k})=\frac{1}{2}\left({\pi }_{+}^2+{\pi }_{-}^2\right)+\frac{1}{2}({\Gamma }_{+}^2{\varphi }_{+}^2+{\Gamma }_{-}^2{\varphi }_{-}^2).$$

$$a_{\pm }=\sqrt{\frac{{\Gamma }_{\pm }}{2}}{\varphi }_{\pm }+\frac{i}{\sqrt{2{\Gamma }_{\pm }}}{\pi }_{\pm },$$

$$\mathcal{H}(\mathbf{k})={\Gamma }_{+}\left(a_{+}^{†}{a}_{+}+\frac{1}{2}\right)+{\Gamma }_{-}\left(a_{-}^{†}{a}_{-}+\frac{1}{2}\right).$$

$${\Psi }_0\left[{\varphi }_{+},{\varphi }_{-}\right]=\mathcal{N}{e}^{-\frac{1}{2}\int d^D\mathbf{k}\left[{\varphi }_{+}{\Gamma }_{+}{\varphi }_{+}+{\varphi }_{-}{\Gamma }_{-}{\varphi }_{-}\right]},$$

$${\Psi }_0\left[{\varphi }_1,{\pi }_2\right]=\mathcal{N}{e}^{-\frac{1}{2}\int d^D\mathbf{k}\left[{\varphi }_{1}K\Omega {\varphi }_1+{\pi }_2\frac{1}{K\Omega }{\pi }_2\right]},$$

$${\Psi }_0\left[{\varphi }_1,{\varphi }_2\right]=\mathcal{N}{e}^{-\frac{1}{2}\int d^D\mathbf{k}\left[{\varphi }_{1}K\Omega {\varphi }_1+{\varphi }_{2}K\Omega {\varphi }_2\right]},$$

$$S_E=-A\frac{N}{12}\int \frac{d^{D-1}{k}_{\perp }}{(2\pi {)}^{D-1}}\log \frac{k_{\perp }^2+{\xi }^{-2}}{k_{\perp }^2+a^{-2}},$$

$$\frac{\partial S_E}{\partial {\xi }^{-2}}=-A\frac{N}{12}\int \frac{d^{D-1}{k}_{\perp }}{(2\pi {)}^{D-1}}\frac{1}{k_{\perp }^2+{\xi }^{-2}},$$

$$S_E/A=-\frac{1}{6\xi }+a_0,$$

$$\mathcal{L}/K=|({\partial }_t-i{\mu }_r)\psi {|}^2-c^2|\nabla \psi {|}^2-m^2|\psi {|}^2-c_4|\psi {|}^4.$$

$$\mathcal{H}$$

$$=$$

$${\mathcal{H}}^{(2)}=\frac{1}{2K}({\pi }_1^2+{\pi }_2^2)+{\mu }_r({\pi }_2{\varphi }_1-{\pi }_1{\varphi }_2)+\frac{K}{2}{\mu }_r^2({\varphi }_1^2+{\varphi }_2^2)+K({\mu }_r^2-m^2){\varphi }_1^2+\frac{K{c}^2}{2}[(\nabla {\varphi }_1{)}^2+(\nabla {\varphi }_2{)}^2].$$

$${\cal H}^{(2)}(k)=\frac{1}{2}\left(\begin{array}[]{cccc}\phi_{1}&\phi_{2}&\pi_{1}&\pi_{2}\end{array}\right)H(k)\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\\ \pi_{1}\\ \pi_{2}\end{array}\right)~{},$$

$$H(k)=\left(\begin{array}[]{cccc}A_{1}&0&0&C\\ 0&A_{2}&-C^{\prime}&0\\ 0&-C^{\prime}&B_{1}&0\\ C&0&0&B_{2}\end{array}\right)~{},$$

$$W^{T}HW=\left(\begin{array}[]{cccc}\omega_{1}^{2}&0&0&0\\ 0&\omega_{2}^{2}&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{array}\right)~{}.$$

$${\mathcal{H}}^{(2)}(k)=\sum _{i=1}^2\frac{1}{2}\left[p_i(k)p_i(-k)+{\omega }_i(k{)}^2{q}_i(k)q_i(-k)\right],$$

$${\omega }_{\pm }^2=c^2{k}^2+3{\mu }_r^2-m^2\pm \sqrt{4{\mu }_r^2{c}^2{k}^2+(3{\mu }_r^2-m^2{)}^2}.$$

$${\omega }_{\mathrm{G}}^2={\omega }_{-}^2=c^2{k}^2\frac{{\mu }_r^2-m^2}{3{\mu }_r^2-m^2}+O(k^4),$$

$${\omega }_{\mathrm{H}}^2={\omega }_{+}^2=6{\mu }_r^2-2m^2+c^2{k}^2\frac{5{\mu }_r^2-m^2}{3{\mu }_r^2-m^2}+O(k^4).$$

$$\left\langle\left(\begin{array}[]{c}\phi_{1}\\ \pi_{2}\end{array}\right)\left(\begin{array}[]{cc}\phi_{1}&\pi_{2}\end{array}\right)\right\rangle=X\left(\begin{array}[]{cc}\frac{1}{2\omega_{1}}&0\\ 0&\frac{1}{2\omega_{2}}\end{array}\right)X^{T}$$

Full text

In condensed-matter physics, relativistic quantum field theories often arise as low-energy effective approximations. However, in multiple situations the local Lorentz invariance is lost. As a prominent example, near a quantum phase transition Sachdev (2011), a dynamical critical exponent different from one indicates the different scaling of space and time. A second example are non-Lorentz-invariant systems where the ground state spontaneously breaks a symmetry of the Hamiltonian. Goldstone’s theorem ensures the presence of Nambu-Goldstone (NG) bosons at low energy Goldstone (1961); Goldstone et al. (1962), but the lack of Lorentz invariance may dramatically change their dispersion relation Lange (1965, 1966); Leutwyler (1994); Watanabe and Murayama (2014). Non-relativistic NG bosons have been extensively studied recently Watanabe and Murayama (2014) and naturally appear in the many-body context Brauner (2010), e.g. in the presence of long-range interactions Frérot et al. (2017).

The nature of the low-energy excitations of a many-body system is deeply related to the quantum fluctuations in the ground state, and has a profund impact on the structure of quantum entanglement across the system. For instance, the scaling of ground-state entanglement entropy with the subsystem size displays a logarithmic violation of the so-called area law Srednicki (1993); Bombelli et al. (1986); Callan and Wilczek (1994); Calabrese and Cardy (2004); Casini and Huerta (2009); Eisert et al. (2010) in one-dimensional gapless systems with short-range interactions Calabrese and Cardy (2004), or in the presence of a Fermi surface in any dimension Gioev and Klich (2006); Wolf (2006); and in bosonic systems with spontaneous symmetry breaking, a subdominant universal additive logarithmic correction carries the nature of the Goldstone modes Metlitski and Grover (2011); Frérot et al. (2017).

In the present paper we study the entanglement present in a general non-relativistic field-theoretical description with O(2) symmetry. The latter appears both in many condensed-matter-physics phenomena Sachdev (2011), and in particle physics at finite chemical potential, e.g. kaon condensation Schäfer et al. (2001) with an enlarged U(2) symmetry. We show that the dominant area-law prefactor of entanglement entropy acquires a universal contribution throughout the phase diagram, associated to the finite correlation length in the gapped disordered phase, and to an elusive “Higgs correlation length” in the gapless ordered phase, associated to amplitude fluctuations of the order parameter. We discuss the relevance of our analytical field-theory predictions for two prominent examples of many-body phenomena: the superfluid to Mott insulator quantum phase transition for ultracold lattice bosons; and ferrimagnets, which are gapless yet short-range-correlated systems.

Non-relativistic low-energy theory. We consider the general O(2)-invariant Lagrangian density ${\mathcal{L}}$ describing the dynamics of a complex field $\psi(\vec{r},t)=\left(\phi_{1}+i\phi_{2}\right)/\sqrt{2}$ in $D+1$ dimensions:

[TABLE]

where the global factor $K$ plays the role of a kinetic mass for the field degrees of freedom [see Eq. (2)]. The (relativistic) chemical potential $\mu_{r}$ breaks the Lorentz invariance of the theory, and is relevant to many condensed-matter problems where Lorentz invariance is absent Hertz (1976); Sachdev (2011). For certain systems, such as superconductors, the equations of motion have to be symmetric under complex conjugation as a consequence of particle-hole symmetry Pekker and Varma (2015); Varma (2002). This imposes the coefficient of the first-order time derivative to vanish, and therefore $\mu_{r}=0$. On the contrary, in pure non-relativistic systems, the dynamics is driven by a Schrödinger equation and only contains first-order time derivatives, e.g. superfluid Helium. The Lagrangian Eq. (1) was studied in the context of relativistic Bose-Einstein condensates Kapusta (1981); Bernstein and Dodelson (1991), and more recently, in the study of non-relativistic NG bosons Leutwyler (1994); Watanabe and Murayama (2014); Alvarez-Gaume et al. (2017), which naturally appear in systems at finite chemical potential Hama et al. (2011); Nicolis and Piazza (2013); Watanabe et al. (2013), where the interplay between first- and second-order time derivatives can lead to the appearance of massive NG bosons Gongyo and Karasawa (2014).

Our purpose is to study the ground-state bipartite entanglement entropy for systems effectively described, at low energy, by the Lagrangian Eq. (1). In particular, we shall consider the influence of the Lorentz-invariance-breaking chemical potential, especially at the quantum phase transition between the disordered phase and the long-range-ordered phase. We consider a subsystem $A$ immersed in an infinite ground state, and compute the von Neumann entropy $S$ of its reduced state $\rho_{A}={\rm Tr}_{B}|\Psi_{0}\rangle\langle\Psi_{0}|$, where ${\rm Tr}_{B}$ denotes the trace over $B$ degrees of freedom (the complement of $A$), and $|\Psi_{0}\rangle$ is the ground state. In $D=2,3$, $S$ obeys a so-called area law over the whole phase diagram, namely it scales as $S=a{\cal A}$ up to subdominant corrections, where ${\cal A}$ is the area of the boundary between the $A$ and $B$ regions Eisert et al. (2010). In gapless $D=1$ systems, $S$ may display a logarithmic violation of the area law, namely $S\propto\log L$ where $L$ is the length of $A$ subsystem Calabrese and Cardy (2004). Furthermore, throughout the paper, we shall not consider the subdominant corrections Metlitski and Grover (2011), and focus instead on the area-law coefficient $a$. Our main purpose is to identify universal contributions to $a$.

For our purpose, it is more convenient to work with the Hamiltonian formulation of the theory. We introduce the canonical moments $\pi_{1/2}=\frac{\delta{\cal L}}{\delta(\partial_{t}\phi_{1/2})}=K(\partial_{t}\phi_{1/2}\pm\mu_{r}\phi_{2/1})$. The Hamiltonian operator reads, in Fourier space:

[TABLE]

where $[\phi_{a}({\bf k}),\pi_{b}({\bf k^{\prime}})]=i\delta_{a,b}\delta({\bf k}-{\bf k^{\prime}})$, and $[\phi_{a}({\bf k}),\phi_{b}({\bf k^{\prime}})]=[\pi_{a}({\bf k}),\pi_{b}({\bf k^{\prime}})]=0$. Throughout the paper, we shall consider a gaussian approximation to the ground state, accounting for harmonic quantum fluctuations around a saddle-point (mean-field) solution. Our focus is the entanglement content of such quantum fluctuations. At the mean-field level, two phases are found: a disordered phase for $m^{2}\geq\mu_{r}^{2}$ (such that $\phi_{1}^{(0)}=\phi_{2}^{(0)}=\pi_{1}^{(0)}=\pi_{2}^{(0)}=0$), and an ordered phase for $m^{2}<\mu_{r}^{2}$, where the O(2) symmetry is spontaneously broken (e.g. $\phi_{2}^{(0)}=\pi_{1}^{(0)}=0$, $\phi_{1}^{(0)}=\sqrt{(\mu_{r}^{2}-m^{2})/c_{4}}$, $\pi_{2}^{(0)}=-K\mu_{r}\phi_{1}^{(0)}$). The quadratic Hamiltonian governing the harmonic fluctuations around this mean-field solution is found upon replacing $\phi_{1/2}\rightarrow\phi_{1/2}^{(0)}+\phi_{1/2}$ and $\pi_{1/2}\rightarrow\pi_{1/2}^{(0)}+\pi_{1/2}$, and neglecting terms of order $O(\phi_{1/2}^{3},\pi_{1/2}^{3})$ and higher. We also subtract the mean-field ground-state energy contribution.

Disordered phase. In the disordered phase ($m^{2}\geq\mu_{r}^{2}$), the quadratic Hamiltonian is simply obtained from Eq. (2) by setting $c_{4}=0$. The excitation spectrum displays two gapped modes of engery $\omega_{\bf k}^{\pm}=\Omega_{\bf k}\pm\mu_{r}$, with $\Omega_{\bf k}=\sqrt{m^{2}+c^{2}k^{2}}$ Sup . Even though the Hamiltonian Eq. (2) depends on $\mu_{r}$, its ground state is in fact independent of $\mu_{r}$. This is a manifestation of the Silver Blaze problem, i.e. at zero temperature thermodynamical observables are independent of the chemical potential up to some critical value Cohen (2003), namely $\mu_{r}^{2}\leq m^{2}$. Specifically, as shown in Sup , we find a factorized ground-state wave-functional, $\Psi_{0}[\phi_{1},\phi_{2}]=\Psi_{\Omega_{\bf k}}[\phi_{1}]\Psi_{\Omega_{\bf k}}[\phi_{2}]$, with:

[TABLE]

This expression explicitly shows that quantum fluctuations in the ground-state are only sensitive to the combined excitation energy $\omega_{\bf k}^{+}+\omega_{\bf k}^{-}=2\Omega_{\bf k}$, as a consequence of the conservation of the charge associated to the O(2) invariance of the full theory. Furthermore, it shows that entanglement entropy $S$ is the sum of two contributions, stemming from the uncoupled gaussian fluctuations of the $\phi_{1}$ and $\phi_{2}$ fields. Entanglement entropy therefore obeys an area law containing a non-universal, UV-cutoff-dependent part $a_{0}$, and a universal contribution $a_{1}$ governed by the correlation length $\xi=c/m$ Metlitski et al. (2009); Calabrese and Cardy (2004); Sup :

[TABLE]

where $a_{1}(\xi)=1/(12\xi)$ in $D=2$ and $a_{1}(\xi)=(24\pi\xi^{2})^{-1}\log\xi$ in $D=3$.

Ordered phase. In this part ($\mu_{r}^{2}>m^{2}$), we focus on the ordered phase for systems in $D=2,3$ spatial dimensions, as in $D=1$ long-range order is not stable, and the physics is not captured by the gaussian approximation we consider. Here, $\phi_{1}$ and $\phi_{2}$ respectively capture amplitude and phase fluctuations of the order parameter $\psi^{(0)}=\sqrt{(\mu_{r}^{2}-m^{2})/c_{4}}$, and we find:

[TABLE]

We first discuss the relativistic case ($\mu_{r}=0$, $m^{2}<0$). In this situation, amplitude and phase fluctuations are decoupled. They are the normal modes of the theory, namely the gapped Higgs mode with frequency $\omega^{\rm H}_{\bf k}=\sqrt{c^{2}k^{2}-2m^{2}}$ (the Higgs gap is $\Delta_{\rm H}=|m|\sqrt{2}$) and the gapless Goldstone mode with $\omega_{\bf k}^{\rm G}=ck$. As a consequence, the ground-state wave-functional factorizes: $\Psi_{0}[\phi_{1},\phi_{2}]=\Psi_{\omega_{\bf k}^{\rm H}}[\phi_{1}]\Psi_{\omega_{\bf k}^{\rm G}}[\phi_{2}]$, so that entanglement entropy is again the sum of two contributions: $S=S_{\rm H}+S_{\rm G}$. Both contributions satisfy an area law in $D\geq 2$ spatial dimensions, and the area-law prefactor of $S_{\rm G}$ (stemming from the gapless Goldstone mode) is non-universal Hastings et al. (2010); Humeniuk and Roscilde (2012); Metlitski and Grover (2011); Frérot and Roscilde (2015); Swingle and McGreevy (2016). $S_{\rm H}$, on the other hand, contains a universal contribution $-a_{1}(\xi_{\rm H})$ to the area law prefactor Calabrese and Cardy (2004), governed by the “Higgs correlation length” $\xi_{\rm H}=c/\Delta_{\rm H}$. The expression of $a_{1}(\xi)$ is given after Eq. (4). This prediction is one of our main results. It shows that upon crossing the O(2) quantum-critical point, entanglement entropy displays a universal singularity, directly stemming from the Higgs mode going gapless at the critical point. Such a dependence on the correlation length was known in the (gapped) disordered phase Metlitski et al. (2009), but not in the (gapless) ordered phase, where it is a consequence of the Higgs gap. This singularity has been observed in previous numerical studies Singh et al. (2012); Helmes and Wessel (2014); Frérot and Roscilde (2016), and our paper provides its analytical explanation at a field-theory level.

In the general, non-relativistic case ($\mu_{r}\neq 0$), the normal modes (a gapped $\phi_{\rm H}$ and a gapless $\phi_{\rm G}$ mode) are linear combinations of $\phi_{1}$ and $\phi_{2}$ Sup . The Goldstone mode maintains a linear dispersion at low energy, $\omega_{\bf k}^{\rm G}\approx c_{\rm G}k$, albeit with a modified velocity $c_{\rm G}=c\sqrt{(\mu_{r}^{2}-m^{2})/(3\mu_{r}^{2}-m^{2})}$. The Higgs gap is $\Delta_{\rm H}=\sqrt{6\mu_{r}^{2}-2m^{2}}$, and most importantly, it remains finite at the phase transition ($\Delta_{H}=2|\mu_{r}|$). In this non-relativistic case, entanglement entropy cannot be separated into two additive contributions stemming from amplitude and phase fluctuations. We can, however, evaluate it numerically Sup , following well-established methods to compute entanglement entropy for gaussian states Peschel and Eisler (2009); Frérot and Roscilde (2015). On Fig. 1, we show entanglement entropy in the $(-m^{2},\mu_{r})$ phase diagram for $D=2$. Interestingly, the non-relativistic quantum “critical” points are actually not critical, as all correlation functions decay exponentially – with a correlation length governed by the gap, $\xi=c/(2|\mu_{r}|)$. This feature is clear from the discussion in the disordered phase: the ground state is independent of $\mu_{r}$ as long as $\mu_{r}\leq m$, including at the “critical” point $\mu_{r}=m$. Therefore, it is identical to the ground state with $\mu_{r}=0$ and $m>0$, namely, strictly inside the disordered phase. This observation is key to understand the low-energy properties of ferrimagnets, as discussed at the end of the paper.

The results presented above are very general and affect a large variety of models whose low energy dynamics is captured by the Lagrangian Eq. (1). In the following we provide two prominent examples where our results can be directly applied.

The Bose-Hubbard model. We first consider a paradigmatic instance of the O(2) quantum phase transition: the Mott-insulator (MI) to superfluid (SF) transition in the Bose-Hubbard model in $D=2$ dimensions Greiner et al. (2002). The Hamiltonian describes a square-lattice Bose gas with contact interactions at zero temperature Fisher et al. (1989):

[TABLE]

where $b_{i}$ ($b_{i}^{\dagger}$) are bosonic annihilation (creation) operators on site $i=1,...,N$, $\mu$ is the chemical potential, $J$ is the hopping amplitude and $U$ is proportional to the two-bosons interaction strength. Near the critical point an effective low-energy description of the system applies Fisher et al. (1989); Sengupta and Dupuis (2005); Sachdev (2011). This effective description coincides with the Lagrangian Eq. (1), where the order parameter is proportional to the vacuum expectation value of the bosonic annihilation operator $\psi(\vec{r},t)\propto\langle b_{i}(t)\rangle$. The coefficients of the Lagrangian Eq. (1) can be expressed in terms of the Bose-Hubbard parameters Sachdev (2011); Faccioli and Salasnich (2019), see Ref. Sup .

The transition between the MI and the SF may occur via two different mechanisms: either the interaction strength is varied at fixed, integer filling fraction, or the particle-number is changed by adjusting the chemical potential. In the former case, the low-energy description becomes relativistic Pekker and Varma (2015) so that $\mu_{r}=0$ Sup . The phase transition occurs across the O(2) quantum-critical point, where the Higgs correlation length $\xi_{\rm H}$ diverges. In the latter case, relativistic invariance is not present ($\mu_{r}\neq 0$), and the Higgs gap remains finite at the transition. The comparison between our results for the entanglement entropy along these two paths in the phase diagram, and the analytical formula Eq. (4) using $\xi=c/m$ in the disordered phase and $\xi_{\rm H}=c/\Delta_{\rm H}$ in the ordered one, is shown on Fig. 2 (panel 1). The agreement is in all cases extremely good. Finally, on the SF side, touching the O(2) point by varying $\mu$ at fixed $U=U_{c}$, we predict that entanglement entropy behaves linearly with $\mu$, in agreement with the numerical results of Ref. Frérot and Roscilde (2016) [Fig. 2 (panel 2)].

Non-relativistic Nambu-Goldstone bosons. The Lagrangian Eq. (1) with $c_{4}=0$ and $\mu_{r}^{2}=m^{2}$ has been proposed as the low-energy description of NG bosons without Lorentz invariance Leutwyler (1994); Watanabe and Murayama (2014), in a physical situation where the system has a rotational O(3) symmetry which is broken down to O(2), i.e. the ground state chooses a particular orientation. Following the general classification given in Nielsen and Chadha (1976); Watanabe and Murayama (2012); Hidaka (2013), the original complex scalar field can be identified with two NG fields $\psi(\vec{r},t)=\pi_{1}(\vec{r},t)+i\pi_{2}(\vec{r},t)$. One of them corresponds to a type-B NG boson and has a quadratic dispersion relation Watanabe and Murayama (2014); the other one is a so-called gapped partner Hama et al. (2011); Nicolis and Piazza (2013); Watanabe et al. (2013); Gongyo and Karasawa (2014).

From our previous discussion, one sees that these systems are at the critical point between the disordered and the ordered phase, at finite chemical potential $\mu_{r}=m$. They exhibit a finite correlation length $\xi=c/m$ although they present quadratic gapless excitations. Therefore, entanglement entropy should obey an area law [see Eq. (4)]. This should be compared with the relativistic case where (type-A) NG bosons acquire a linear dispersion relation, and a diverging correlation length.

Here, we focus on the $D=1$ case where entanglement entropy is expected to scale according to $S_{E}\sim\log\xi$ Callan and Wilczek (1994); Calabrese and Cardy (2004); Hertzberg and Wilczek (2011). We validate our approach by considering a particular family of systems, namely ferrimagnets, which are spin systems living on two sublattices $A_{1}\cup A_{2}$. On the $A_{1}$ sublattice, we define $\vec{S}_{i}$ as spin-$s_{1}$ operators, and on the $A_{2}$ sublattice, we define $\vec{\tau}_{j}$ as spin-$s_{2}$ operators. Spins interact via Heisenberg-type exchange interactions with Hamiltonian $H_{\rm ferrimag.}=J\sum_{<i,j>}\vec{S}_{i}\cdot\vec{\tau}_{j}$. Typically, the ground state of the system exhibits ferrimagnetic order, i.e. an anti-alignment of the spins living on different sublattices. In this case, the different magnitude of the spins induces a total magnetization density in the system $m=\langle S^{z}\rangle/N\sim|s_{1}-s_{2}|$. This is an indicator of the non-zero expectation value of the commutator of two broken charges, $\langle\left[S^{x},S^{y}\right]\rangle\neq 0$, which breaks Lorentz invariance Watanabe and Murayama (2014).

The low-energy effective description of ferrimagnets is the Lagrangian Eq. (1) with $c_{4}=0$ and $\mu_{r}^{2}=m^{2}$  Gongyo et al. (2016), whose low-energy excitations are $\omega_{\bf k}^{\pm}=\sqrt{m^{2}+c^{2}k^{2}}\pm m$. Thus, we expect quadratic type-B NG bosons $\omega_{\bf k}^{-}=\rho k^{2}$ and a gapped partner $\omega_{\bf k}^{+}=\Delta$, where $\rho$ and $\Delta$ are the spin stiffness and energy gap, respectively. In terms of the coefficients of the Lagrangian Eq. (1), we identify $\rho=c^{2}/(2m)$ and $\Delta=2m$, which allows us to write the expression for the correlation length (in units of the lattice spacing):

[TABLE]

We compare the prediction Eq. (7) for the correlation length with various numerical computations on $H_{\rm ferrimag.}$, where a very short correlation length was found together with gapless excitations Pati et al. (1997); Kolezhuk et al. (1997); Yamamoto et al. (1998), two key features which are clearly present at the field-theory level. Indeed, Eq. (7) allows us to predict the value of the correlation length for several ferrimagnets using previously obtained numerical results for $\rho$ and $\Delta$. The results are summarized on Fig. 3. Our prediction for $\xi$ is systematically smaller than those predicted by spin-wave theory, $\xi^{-1}=\log(s_{1}/s_{2})$ Pati et al. (1997), and closer to the (more accurate) value obtained using matrix product states Pati et al. (1997); Kolezhuk et al. (1997) for $(s_{1}=1,s_{2}=1/2)$. In addition to existing results in the literature, we also carried out iDMRG simulations Hauschild and Pollmann (2018) and computed the correlation lengths for different spin values $s_{1}$ and $s_{2}$. We found a good agreement with the prediction Eq. (7), using existing Monte-Carlo data for the values of $\rho$ and $\Delta$. Finally, we observe that these values of $\xi$ are also consistent with entanglement entropy scaling as $S\sim\log\xi$.

Outlook We have investigated the area-law prefactor of entanglement entropy in non-relativistic low-energy field theories with O(2) symmetry. Our predictions have been successfully confronted with two prominent examples from condensed-matter physics: the Mott insulator to superfluid transition and non-relativistic Nambu-Goldstone bosons in ferrimagnets. Our findings, which could be tested in quantum simulators Islam et al. (2015); Brydges et al. (2019), also apply to particle-physics models with a non-zero chemical potential.

SUPPLEMENTAL MATERIAL for Entanglement entropy in low-energy field theories at finite chemical potential

In this Supplemental Material we provide further details about:

1. The representation of the ground state wavefunctional in real in the disordered phase and its connection with entanglement entropy.

2. The gaussian approximation in the ordered phase and the computation of entanglement entropy.

3. The explicit relation with the parameters of the Bose-Hubbard model.

.1 1. Disordered phase

We consider the free non-relativistic theory describing the dynamics of a complex scalar field $\psi=\left(\phi_{1}+i\phi_{2}\right)/\sqrt{2}$, whose Lagrangian density in $D+1$-dimensional spacetime is given by,

[TABLE]

The theory is non-relativistic because of the presence of the term $\mu_{r}\neq 0$. We work with the Hamiltonian formulation of the theory. We introduce the canonical moments $\pi_{1/2}=\frac{\delta{\cal L}}{\delta(\partial_{t}\phi_{1/2})}=K(\partial_{t}\phi_{1/2}\pm\mu_{r}\phi_{2/1})$. The Hamiltonian operator reads, in Fourier space:

[TABLE]

where $[\phi_{a}({\bf k}),\pi_{b}({\bf k^{\prime}})]=i\delta_{a,b}\delta({\bf k}-{\bf k^{\prime}})$, and $[\phi_{a}({\bf k}),\phi_{b}({\bf k^{\prime}})]=[\pi_{a}({\bf k}),\pi_{b}({\bf k^{\prime}})]=0$.

We are now neglecting the interactions, setting $c_{4}=0$. Therefore, the results derived in this section can be seen as performing a Gaussian approximation in the disordered phase described in the main text. We look for a canonical transformation of the form:

[TABLE]

One can verify that choosing $\Gamma_{\pm}=\sqrt{m^{2}+c^{2}k^{2}}\mp\mu_{r}$ and $\Omega=\sqrt{m^{2}+c^{2}k^{2}}$, the Hamiltonian reads:

[TABLE]

Finally the Hamiltonian Eq. (11) can be diagonalized by introducing a set of normal modes ($\pm$) defined by the annihilation operators:

[TABLE]

which allows one to write the Hamiltonian in an harmonic-oscillator form,

[TABLE]

Because of this harmonic-oscillator form we know that the ground state $|\Psi_{0}\rangle$ is uniquely determined by the condition $a_{+}({\bf{k}})|\Psi_{0}\rangle=a_{-}({\bf{k}})|\Psi_{0}\rangle=0$, $\forall({\bf{k}})$. In the Schrödinger representation, the groundstate wavefunctional is the gaussian:

[TABLE]

where $\mathcal{N}$ is a normalization factor. Turning back to the original $\phi_{1}$ and $\phi_{2}$ fields using the canonical transformation Eq. (10), one obtains:

[TABLE]

or equivalently:

[TABLE]

which coincides with the groundstate wavefunctional of two relativistic scalar fields of kinetic mass $K$ and dispersion relation $\Omega({\bf k})=\sqrt{m^{2}+c^{2}k^{2}}$. This result explicitly shows that the ground state is independent of $\mu_{r}$, as long as we stay in the disordered phase (namely, as long as $\mu_{r}^{2}\leq m^{2}$). From this groundstate wavefunctional one can readily see that the two-point function $\langle 0|\psi(\vec{x},t)\psi^{\dagger}(\vec{y},t)|0\rangle$ has an exponential decay $\exp\{-r/\xi\}$ for large distance $r=|\vec{x}-\vec{y}|$ given by the correlation length $\xi=c/m$. Since we have been able to express the groundstate wavefunctional as the one of a relativistic complex scalar field with a mass $m$, the computation of entanglement entropy will be based on the known results obtained for the relativistic case Srednicki (1993); Bombelli et al. (1986); Callan and Wilczek (1994); Calabrese and Cardy (2004); Casini and Huerta (2009).

Consider that we perform a spatial partition of the system into two parts $\mathcal{A}$ and $\mathcal{B}$. The entanglement entropy is defined as $S_{E}=-\text{Tr}_{\mathcal{A}}\{\rho_{\mathcal{A}}\ln\rho_{\mathcal{A}}\}$, where $\rho_{\mathcal{A}}=\text{Tr}_{\mathcal{B}}\rho$ is the reduced density matrix obtained by tracing out the degrees of freedom living on $\mathcal{B}$ and $\rho=|0\rangle\langle 0|$ is the density matrix which can be represented using the groundstate wavefunctional Eq. (14). If the system presents a finite correlation length $\xi$ the leading area law term of entanglement entropy is given by Calabrese and Cardy (2004); Hertzberg and Wilczek (2011),

[TABLE]

where $\epsilon=1/a$ is the UV-cutoff, $A$ is the area of the boundary separating the two parts of the system $\mathcal{A}$ and $\mathcal{B}$, and $N=2$ is the number of free scalar fields. One can see that this integral contains a term which is UV-cutoff dependent and that makes it non-universal. On the other hand, there is a finite universal term that goes like $\xi^{1-D}$ which can be explicitly manifested by taking the derivative $\partial S_{E}/\partial\xi^{-2}$,

[TABLE]

which leads to the well known result for $D=2$ Calabrese and Cardy (2004); Hertzberg and Wilczek (2011),

[TABLE]

where $a_{0}$ stands for some non-universal divergent constant cutoff dependent. In this way one can compute $\Delta S_{E}=S_{E}(\xi\rightarrow\infty)-S_{E}(\xi)$ which should not depend on the UV cutoff and therefore it will be universal in this sense.

.2 2. Normal modes in the ordered phase

We consider the Lagrangian density:

[TABLE]

Introducing $\psi=(\phi_{1}+i\phi_{2})/\sqrt{2}$, and $\pi_{1/2}=\frac{\partial{\cal L}}{\partial(\partial_{t}\phi_{1/2})}=K(\partial_{t}\phi_{1/2}\pm\mu_{r}\phi_{2/1})$, we have the equivalent Hamiltonian density:

[TABLE]

We focus on the ordered phase, namely $m^{2}<\mu_{r}^{2}$. Imposing that $\delta{\cal H}=0$, we obtain the mean-field solution $\phi_{2}^{(0)}=0$, $\pi_{1}^{(0)}=0$, $\pi_{2}^{(0)}=-K\mu_{r}\phi_{1}^{(0)}$, and $\phi_{1}^{(0)}=\pm\sqrt{(\mu_{r}^{2}-m^{2})/c_{4}}$. Expanding around this mean-field solution (namely, $x\rightarrow x^{(0)}+x$ for $x=\pi_{1/2},\phi_{1/2}$) and neglecting terms of order $O(x^{3})$, we obtain:

[TABLE]

Clearly, when $\mu_{r}=0$, $\phi_{1}$ and $\phi_{2}$ are decoupled and represent the normal modes (namely, the Higgs and Goldstone modes of the ordered phase). They contribute independently to entanglement entropy of a subsystem, which therefore reads $S=S_{\rm Goldstone}+S_{\rm Higgs}$. However, for $\mu_{r}\neq 0$ they are coupled.

.2.1 Diagonalization of the Hamiltonian

Going to Fourier space, the Hamiltonian density reads:

[TABLE]

where $H$ is of the form:

[TABLE]

with $A_{1}=K(3\mu_{r}^{2}-2m^{2}+c^{2}k^{2})$, $A_{2}=K(\mu_{r}^{2}+c^{2}k^{2})$, $B_{1}=B_{2}=1/K$, $C=C^{\prime}=\mu_{r}$. In order to diagonalize the Hamiltonian, we look for a canonical transformation \left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\\ \pi_{1}\\ \pi_{2}\end{array}\right)=W\left(\begin{array}[]{c}q_{1}\\ q_{2}\\ p_{1}\\ p_{2}\end{array}\right), where $W$ is chosen of the form: W=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&0&0\\ 0&0&\beta_{1}&\beta_{2}\\ 0&0&\gamma_{1}&\gamma_{2}\\ \delta_{1}&\delta_{2}&0&0\\ \end{array}\right). $W$ represents a canonical transformation iff $WJW^{T}=J$ with J=\left(\begin{array}[]{cccc}0&0&1&0\\ 0&0&0&1\\ -1&0&0&0\\ 0&-1&0&0\end{array}\right). Introducing the matrices X=\left(\begin{array}[]{cc}\alpha_{1}&\alpha_{2}\\ \delta_{1}&\delta_{2}\end{array}\right) and Y=\left(\begin{array}[]{cc}\gamma_{1}&\gamma_{2}\\ -\beta_{1}&-\beta_{2}\end{array}\right), $W$ is a canonical transformation iff $XY^{T}={\rm Id}$. The aim of the canonical transformation $W$ is to bring the matrix $H$ into the form:

[TABLE]

This is equivalent to having X^{T}M_{1}X=\left(\begin{array}[]{cc}\omega_{1}^{2}&0\\ 0&\omega_{2}^{2}\end{array}\right) with M_{1}=\left(\begin{array}[]{cc}A_{1}&C\\ C&B_{2}\end{array}\right); and $Y^{T}M_{2}Y={\rm Id}$ with M_{2}=\left(\begin{array}[]{cc}B_{1}&C^{\prime}\\ C^{\prime}&A_{2}\end{array}\right). The diagonalization may be achieved by noticing that \left(\begin{array}[]{cc}\omega_{1}^{2}&0\\ 0&\omega_{2}^{2}\end{array}\right)=X^{T}M_{1}XY^{T}M_{2}Y=X^{T}M_{1}M_{2}Y, where we use the fact that $XY^{T}={\rm Id}$. Since $X^{T}=Y^{-1}$, we can now diagonalize $M_{1}M_{2}$ to obtain $X$, $Y$, $\omega_{1}$ and $\omega_{2}$. This is done in two steps. First, $M_{1}M_{2}$ is diagonalized, yielding $\tilde{X}^{T}M_{1}M_{2}\tilde{Y}={\rm diag}(\omega_{1}^{2},\omega_{2}^{2})$. $Y$ is then obtained by properly normalizing the columns of $\tilde{Y}$. We initially have $\tilde{Y}^{T}M_{2}\tilde{Y}={\rm diag}(\lambda_{1}^{2},\lambda_{2}^{2})$. Finally, we obtain the normal modes by choosing $Y=\tilde{Y}{\rm diag}(1/\lambda_{1},1/\lambda_{2})$, and $X^{T}=Y^{-1}$. Then the Hamiltonian reads:

[TABLE]

where \left(\begin{array}[]{c}\phi_{1}\\ \pi_{2}\end{array}\right)=X\left(\begin{array}[]{c}q_{1}\\ q_{2}\end{array}\right) and \left(\begin{array}[]{c}\pi_{1}\\ -\phi_{2}\end{array}\right)=Y\left(\begin{array}[]{c}p_{1}\\ p_{2}\end{array}\right).

.2.2 Eigenmodes

The eigenfrequencies are:

[TABLE]

For the gapless Goldstone mode, we find, expanding at low $k$:

[TABLE]

Hence a modified sound velocity $c_{\rm G}=c\sqrt{\frac{\mu_{r}^{2}-m^{2}}{3\mu_{r}^{2}-m^{2}}}$, such that $\omega_{\rm G}=c_{\rm G}k+O(k^{2})$. For the gapped Higgs mode, we find instead:

[TABLE]

This defines the Higgs gap: $\Delta_{\rm H}=\sqrt{6\mu_{r}^{2}-2m^{2}}$ and the Higgs velocity: $c_{\rm H}=c\sqrt{\frac{5\mu_{r}^{2}-m^{2}}{3\mu_{r}^{2}-m^{2}}}$, such that $\omega_{\rm H}=\sqrt{\Delta_{\rm H}^{2}+c_{\rm H}^{2}k^{2}}+O(k^{4})$.

.2.3 Correlation functions

Correlation functions in the ground state are obtained according to:

[TABLE]

[TABLE]

and $\langle\phi_{1}\phi_{2}\rangle=0=\langle\pi_{1}\pi_{2}\rangle$, $\langle\phi_{1}\pi_{1}\rangle=i/2=\langle\phi_{2}\pi_{2}\rangle$.

.2.4 Entanglement entropy

In order to compute the entanglement entropy of a subsystem $A$, one first forms the correlation matrix for the field degrees of freedom belonging to that subsystem: $\langle\phi_{1}(x)\phi_{2}(x^{\prime})\rangle$ for $x,x^{\prime}\in A$, etc. We assume that $A$ contains $N$ sites. The correlation matrix is:

[TABLE]

where $\langle\phi_{1}\phi_{1}\rangle$ represents the $N\times N$ matrix $\langle\phi_{1}(x)\phi_{2}(x^{\prime})\rangle$ for $x,x^{\prime}\in A$, and similarly for $\langle\phi_{1}\pi_{2}\rangle$, etc. Similarly to the diagonalization of the Hamiltonian, the canonical transformation to the normal modes may be chosen of the form: W=\left(\begin{array}[]{cccc}\alpha_{1}&\alpha_{2}&0&0\\ 0&0&\beta_{1}&\beta_{2}\\ 0&0&\gamma_{1}&\gamma_{2}\\ \delta_{1}&\delta_{2}&0&0\\ \end{array}\right), where now $\alpha_{i},\beta_{i},\gamma_{i},\delta_{i}$ are $N\times N$ matrices. The diagonalization of ${\cal C}$ goes along the same line as that of the Hamiltonian, except for the fact that each symbol now represents a $N\times N$ matrix. Indeed, introducing X=\left(\begin{array}[]{cc}\alpha_{1}&\alpha_{2}\\ \delta_{1}&\delta_{2}\end{array}\right) and Y=\left(\begin{array}[]{cc}\gamma_{1}&\gamma_{2}\\ -\beta_{1}&-\beta_{2}\end{array}\right), and denoting {\cal C}=\left(\begin{array}[]{cccc}A_{1}&0&(i/2){\rm Id}&C\\ 0&A_{2}&-C^{\prime}&(i/2){\rm Id}\\ -(i/2){\rm Id}&-C^{\prime T}&B_{1}&0\\ C^{T}&-(i/2){\rm Id}&0&B_{2}\end{array}\right), we have that ${\cal C}={\rm diag}(X^{T}M_{1}X,~{}Y^{T}M_{2}Y)$ with M_{1}=\left(\begin{array}[]{cc}A_{1}&C\\ C^{T}&B_{2}\end{array}\right) and M_{2}=\left(\begin{array}[]{cc}B_{1}&C^{\prime T}\\ C^{\prime}&A_{2}\end{array}\right). Diagonalizing the matrix $M_{1}M_{2}$, one obtains eigenvalues of the form $\lambda_{i}^{2}=\left[\frac{1}{2}+\frac{1}{e^{\tilde{\omega}_{i}}-1}\right]^{2}$, where $\tilde{\omega}_{i}$ forms the (one-body) entanglement spectrum. The entanglement entropy of $A$ is finally obtained as:

[TABLE]

As a further simplification, we notice that if $A$ is periodic along a certain direction (say $x$), the correlation matrix ${\cal C}$ is block-diagonal with respect to the momentum $k_{x}$ along that direction. The diagonalization may thus be achieved separately in each momentum sector $k_{x}$, and the entanglement entropy is the sum of the contributions from the different sectors $k_{x}$.

.3 3. Relation with Bose-Hubbard parameters

In this section we write down explicitly the relations between the Bose-Hubbard (BH) parameters and the effective parameters of our original model. We introduce the notations $K_{1}\equiv 2\mu_{r}K$, $c_{2}\equiv K\left(m^{2}-\mu_{r}^{2}\right)$ and $K_{3}\equiv Kc^{2}$. For the first lobe $n=1$ and setting $U=1$ the following relations can be applied Sachdev (2011); Faccioli and Salasnich (2019),

[TABLE]

In these relations the chemical potential of the BH model $\mu$ can be different from the chemical potential of the original model $\mu_{r}$ (we add the subscript $r$ to differentiate them). In order to simplify these relations we expand around the tip of the lobe $zJ_{c}=3-2\sqrt{2}$ and $\mu_{c}=\sqrt{2}-1$, and we obtain the following relations,

[TABLE]

where $K(\mu_{c})=1/\sqrt{2}$. If we compare these relations with the ones for the original model $c_{2}/K=m^{2}-\mu_{r}^{2}$ and $\mu_{r}=K_{1}/2K$, we see that at leading order,

[TABLE]

From Eq. (19) we can see that inside the Mott-insulator phase $\Delta S_{E}$ will be independent of the chemical potential,

[TABLE]

Inside the superfluid phase in the relativistic regime ($\mu=\mu_{c}$) we obtain,

[TABLE]