Interaction-induced particle-hole symmetry breaking and fractional exclusion statistics

Abstract

Quantum statistics plays a fundamental role in the laws of nature. Haldane fractional exclusion statistics (FES) generalizes the Pauli exclusion statistics, and can emerge in the properties of elementary particles and hole excitations of a quantum system consisting of conventional bosons or fermions. FES has a long history of intensive studies, but its simple realization in interacting physical systems is rare. Here we report a simple non-mutual FES that depicts the particle-hole symmetry breaking in interacting Bose gases at a quantum critical point. We show that the FES distribution directly comes from particle-hole symmetry breaking. Based on exact solutions, quantum Monte Carlo simulations and experiments, we find that, over a wide range of interaction strengths, the macroscopic physical properties of these gases are determined by non-interacting quasi-particles that obey non-mutual FES of the same form in one and two dimensions. Whereas strongly interacting Bose gases reach full fermionization in one dimension, they exhibit incomplete fermionization in two dimensions. Our results provide a generic connection between interaction-induced particle-hole symmetry breaking (depicted by FES) and macroscopic properties of many-body systems in arbitrary dimensions. Our work lays the groundwork for using FES to explore quantum criticality and other novel many-body phenomena in strongly correlated quantum systems.

Near a quantum critical point, the macroscopic properties of interacting many-body Bose gases in one and two dimensions are determined by ideal particles that obey non-mutual fractional exclusion statistics.

INTRODUCTION

Bose-Einstein and Fermi-Dirac statistics constitute two cornerstones of quantum statistical mechanics. However, they are not the only possible forms of quantum statistics [1]. In two dimensions, anyonic excitations can carry fractional charges and obey fractional statistics [2–7]. To generalize fractional statistics, Haldane formulated a theory of fractional exclusion statistics (FES) that continuously interpolates between Bose and Fermi statistics in arbitrary spatial dimensions [8]. This theory depicts how much the Hilbert space dimensionality for available single-particle states, namely the ‘number of holes’ (N_{h, α}) of species α decreases as particles of species β are added to a system [8–10]:

(1)

Here the FES parameter g_αβ is independent of the particle number N_{P, β}. Bose and Fermi statistics correspond to the non-mutual FES where g_αβ = gδ_αβ with g = 0 and 1, respectively. FES has found exact realizations in a few one-dimensional (1D) systems, including the Calogero-Sutherland model of particles interacting through a 1/r² potential [11–14], Lieb-Liniger Bose gases [14,15] and anyonic gases with delta-function interaction [16,17].

FES reveals the statistical nature of a system with respect to its energy spectrum regardless of whether the constituent particles interact or not. On the other hand, particle-hole symmetry breaking (PHSB) [18] emerges as a key mechanism for understanding strongly correlated quantum materials including high-T_c superconductors [19,20] and fractional quantum Hall systems [21]. This symmetry breaking significantly influences physical properties such as equations of state [22], optical properties [23], dynamical evolutions [24], transport properties [25] and non-Fermi-liquid behaviors [26]. However, it remains challenging to identify emergent FES for depicting the particle-hole symmetry breaking in generic interacting many-body systems.

In this article, we show that FES naturally emerges as a result of particle-hole symmetry breaking in quantum many-body systems (see Fig. 1(a) and Equation (2)). In particular, we demonstrate interaction-induced non-mutual FES at a quantum critical point. We consider a repulsively interacting Bose gas that undergoes a quantum phase transition under zero temperature (T = 0) from a vacuum to a quantum liquid when the chemical potential μ exceeds a critical value μ_c (Fig. 1(b)). Here ‘quantum liquid’ denotes a Tomonaga-Luttinger liquid (TLL) [27] in one dimension or a superfluid in higher dimensions [28]. Based on exact solutions in one dimension and high-precision quantum Monte Carlo (QMC) simulations in two dimensions, we report evidence for emergent particle-hole symmetry breaking (Equation (2)) in these many-body systems. Our results are further supported by existing experimental data given in [27–32]. We establish a one-to-one correspondence between interaction and particle-hole symmetry breaking over a wide range of interaction strengths, and further observe that, remarkably, such symmetry breaking determines the macroscopic properties of interacting gases in a unified manner, as summarized by the logic flow shown in Fig. 1(c).

Figure 1.

Particle-hole symmetry breaking and Haldane FES in interacting many-body systems. (a) Particle-hole symmetry breaking in a quasi-momentum cell at (see Equation (2)) in an interacting system. (b) A many-body system near a quantum critical point. (c) Schematic of the logic flow: interaction determines the particle-hole symmetry breaking (depicted by FES) that in turn governs the macroscopic physical properties of a many-body system at and near a quantum critical point.

Particle-hole symmetry breaking and FES

In a quantum many-body system, interaction dresses the constituent particles to form quasi-particles that are statistically distributed over the quasi-momentum space. In each quasi-momentum cell that defines the species in Equation (1), the number of unoccupied states depends on the numbers of occupied states in the same cell and in other cells, forming a net of correlated cells in general. Such correlated cells can be depicted by quasi-momentum-dependent particle-hole symmetry breaking; see Equation S43 within the online supplementary material. At a quantum critical point where the system is strongly correlated with large characteristic lengths in real space, these quasi-momentum cells become decoupled into nearly independent cells (see Section 3 within the online supplementary material), which leads to a simplified form of particle-hole symmetry breaking equation that we expect to hold in arbitrary dimensions (Fig. 1(a)):

(2)

Here, the species label α is given by quasi-momentum , g_αβ by , N_P and N_h are scaled into distribution functions, and , of occupied states and of holes, and d_sp = 1/(2π)^D is a bare dimensionality of states in a phase-space unit cell for a D-dimensional system.

Accordingly, a non-mutual FES distribution [9,14] of quasi-particles naturally emerges at a quantum critical point. We prove (for details, see Section 4 within the online supplementary material) that Equation (2) directly gives rise to a non-mutual FES distribution with the following occupation number f in a state with energy ε:

(3)

This generic connection between Equations (2) and (3) enables understanding interacting systems from the perspective of emergent FES phenomena. Based on the interaction-induced particle-hole symmetry breaking (Equation (2)), the macroscopic physical properties of many-body systems can be obtained through non-interacting quasi-particles that obey the non-mutual FES distribution in Equation (3). Specifically, the number density and energy density are given by n = ∫G(ε)f(ε)dε and e = ∫G(ε)f(ε)εdε, where the density of states per volume is given by in one dimension and 1/(4π) in two dimensions for non-relativistic particles. We set 2m = k_B = ℏ = 1, where m is the particle mass, k_B the Boltzmann constant and ℏ the reduced Planck constant.

To connect Equation (2) to physical interacting systems, we introduce an interaction-FES correspondence hypothesis:

(4)

Here is a properly scaled interaction strength, is the corresponding FES parameter in Equations (2) and (3), and is a coefficient. This hypothesis is inspired by an analytical result for the FES in 1D strongly interacting Bose gases [33] as well as a Ginzburg-Landau theory for 2D superfluids [31,34], and is found to apply over a large interaction range (see Fig. 2). It provides a simple proportionality relation between the interaction strength and the resulting Hilbert space dimensionality ratio () of g − g₀ (for the Hilbert space occupied by one single particle because of interaction) to g_∞ − g (for the ‘remaining’ Hilbert space that is occupiable but yet unoccupied because has not reached infinity). For interacting Bose gases, g₀ = 0 and we denote g_∞ as g_max. Equations (2) and (4) together enable quantitative predictions of macroscopic physical properties. In the following, we provide evidence for the emergence of such interaction-induced particle-hole symmetry breaking and non-mutual FES at and near a quantum critical point.

Figure 2.

Demonstration of the proportionality relation in the interaction-FES correspondence hypothesis (Equation (4)). Based on exact solutions for 1D Bose gases with delta-function interaction, we compute that depicts the particle-hole symmetry breaking in low-energy excitations. The numerical data (circles) show the proportionality relation with fitted . The solid line shows Equation (4) with c₁ = 0.772 (see Equation (10) for details). Under strong couplings (), the numerical data can be described by an analytical form (dashed line; see Equation (7)).

FES in one dimension

The 1D Bose gas with delta-function interaction is an integrable model with great relevance in both theoretical and experimental contexts (see reviews [35,36]). Such gases are described by the Hamiltonian [15,37]

(5)

where c is the repulsive elastic interaction strength and N the particle number. In its dilution limit, the discrete 1D Bose-Hubbard model used in QMC simulations corresponds to Equation (5) with c = U/(2t^1/2) (see Section 9 within the online supplementary material), where U and t are the onsite interaction and tunneling parameters, respectively. We exactly solve such 1D gases at the vacuum-to-TLL transition (μ_c = 0) [27] based on the thermodynamic Bethe ansatz (TBA) equation [37,38],

(6)

where a(x) = 2c/(c² + x²), and the pressure is given by p(μ, T) = (T/2π)∫ln (1 + e^−ϵ(k)/T)dk. For convenience, we present thermodynamic observables and parameters in dimensionless forms (see Section 1 within the online supplementary material). We compute the critical entropy per particle S_c/N ≡ (S/N)(μ = μ_c), scaled critical density and scaled critical pressure by numerically solving Equation (6).

The S_c/N increases with a scaled interaction strength (Fig. 3(a)). It reaches A_{∞, 1D} ≈ 1.89738 at (see Section 1 within the online supplementary material), exactly matching the S_c/N of non-interacting fermions [37] (g_{max, 1D} = 1), as predicted and observed for Tonks-Girardeau gases [15,39–42]. These solutions agree with data extracted from experiments performed by the Kaiserslautern group [29] and the USTC group [27], and agree with our 1D QMC simulations (see Sections 10 and 11 within the online supplementary material).

Figure 3.

Evidence for interaction-induced FES in 1D Bose gases at a quantum critical point. (a) Critical entropy per particle S_c/N as a function of . Exact solutions (circles) agree excellently with QMC computations (diamonds) and agree with experiments (squares and triangles, from [27,29]). (b) Power-law scaling of S_c/N with respect to . Dotted line denotes the fermionization limit A_{∞, 1D}. (c)–(e) Under Equation (8) with g_{max, 1D} = 1, thermodynamic observables of interacting gases agree well with those of non-interacting quasi-particles that obey non-mutual FES: (c) S_c/N; (d) scaled critical density ; (e) scaled critical pressure . Error bars represent 1σ statistical uncertainties.

We now verify the particle-hole symmetry breaking equation (Equation (2)) and the interaction-FES correspondence hypothesis (Equation (4)) based on ab initio computations (see Section 5 within the online supplementary material). We compute and and define a -dependent FES parameter based on . We observe that is almost homogeneous within a finite range of (see Section 5 within the online supplementary material), corresponding to low-energy elementary excitations that obey simple particle-hole symmetry breaking (Equation (2)) depicted by a non-mutual FES distribution (Equation (3)). Equation (3) associated with captures the essential behaviors of and (see Section 5 within the online supplementary material). Furthermore, the Hilbert space dimensionality ratio shows a proportionality relation to the interaction strength over a large range (Fig. 2), with a fitted coefficient . Thus, Equation (4) provides a powerful approximation that depicts the particle-hole symmetry breaking for low-energy excitations in interacting gases. We note that, based on Equation (5), particle-hole symmetry breaking can in general be depicted by -dependent mutual FES [14]; such mutual FES reduces to non-mutual FES under strong coupling [33], which we here derive to be for finite temperatures (see Section 5 within the online supplementary material)

(7)

As shown in Fig. 2, Equation (4) not only agrees with this strong-coupling analytical form, but also depicts well the numerically computed over a significantly larger range covering strong, intermediate and weak interactions.

The particle-hole symmetry breaking (depicted by Equations (2) and (4)) dictates the distribution functions (Equation (3)) of elementary excitations and thereby determines the macroscopic properties of interacting gases. For Bose gases, Equation (4) predicts a one-to-one mapping to non-interacting quasi-particles with FES parameter

(8)

where is a transformed interaction parameter,

(9)

and g_{max, 1D} = 1. Based on the computed critical entropy per particle, we observe two scaling functions (Fig. 3(b) and (c)) that are similar to each other, which is characteristic of the interaction-FES correspondence (Equation (8)). For non-interacting FES quasi-particles, the S_{c, FES}/N at μ_c = 0 exhibits a power-law scaling with respect to g (Fig. 3(c), blue curve): , with β_{FES, 1D} = 0.298(2) fitted for 0.05 < g ≤ 1. This scaling suggests, and we indeed observe, that the S_c/N for interacting Bose gases accordingly obeys a power-law scaling with respect to (Fig. 3(b)):

(10)

Here β_1D = 0.298(1) and are fitted parameters. Under Equation (8), these two scaling functions match each other, and the numerical data agree within 4% (Fig. 3(c)). We note that Equation (10) agrees excellently with exact solutions within over , and the precisely determined conforms well to the determined earlier. Thus, identifying the power-law scaling for S_c/N provides a smoking-gun signature and a precise determination of the interaction-FES correspondence.

We find agreement within and for and , respectively (Fig. 3(d) and (e)). The overall good agreement for S_c/N, and shows that macroscopic thermodynamic observables are determined by interaction-induced particle-hole symmetry breaking. Therefore, these observables can serve as a practical gauge for conveniently measuring the corresponding non-mutual FES, especially when ab initio computations are difficult or unavailable.

FES in two dimensions

In higher dimensions, while the Bethe ansatz in general does not apply, particle-hole symmetry breaking remains a key characteristic [43] that governs measurable macroscopic properties of interacting gases. Here we benchmark the applicability of Equations (2) and (4) in 2D gases. Using QMC simulations [44,45], we study a 2D Bose-Hubbard lattice gas that has a vacuum-to-superfluid quantum phase transition at μ_c = −4t [28]. The Bose-Hubbard Hamiltonian is given by

(11)

where and are the creation and annihilation operators at site i, and 〈i, j〉 runs over all nearest neighboring sites. We define a scaled interaction strength (see Section 9 within the online supplementary material) that is the lattice-gas equivalence [28,31] of the interaction parameter for 2D Bose gases without lattices, where a is the scattering length and l_z is an oscillator length [32].

To obtain physical properties that are insensitive to the lattice structure, we perform QMC simulations for each at a series of temperatures down to T = 0.1t. We extract scaled quantities S_c/N, and for each T, and perform extrapolation towards T = 0 for each quantity (see Section 10 within the online supplementary material). We test this extrapolation protocol on a 1D Bose-Hubbard system (see Section 10 within the online supplementary material) and find excellent agreement with exact solutions (Fig. 3).

In two dimensions, we identify the same interaction-FES correspondence (Equation (8)) as in one dimension. The S_c/N increases with and reaches A_{∞, 2D} = 1.988(14) at (Fig. 4(a)), matching the S_{c, FES}/N of non-interacting FES quasi-particles with g_{max, 2D} = 0.432(14). Our QMC data agree well with a non-perturbative renormalization group (NPRG) computation [46], and with experiments by the Chicago [28] and ENS [30] groups. Based on Equation (9) and , S_c/N shows an excellent power-law scaling with respect to (Fig. 4(b)):

(12)

with A_{∞, 2D}, , and β_2D = 0.20(1) fitted for (). We then test the predictions of Equations (2) and (4) (with g_{max, 2D} and determined above) using the QMC data.

Figure 4.

Evidence for interaction-induced FES in 2D Bose gases at a quantum critical point. (a) Critical entropy per particle S_c/N as a function of . QMC results (circles) agree with NPRG computations [46] and experiments [28,30]. (b) Power-law scaling of S_c/N with respect to . (c)–(e) Given Equation (8) with g_{max, 2D} = 0.432(14), thermodynamic observables of interacting gases agree well with those of non-interacting quasi-particles that obey non-mutual FES: (c) S_c/N; (d) scaled critical density ; (e) scaled critical pressure , with the dotted line denoting the non-interacting boson limit  [46]. Our results agree with existing experiments [28,30,31]. Horizontal and vertical gray bands mark A_{∞, 2D} and g_{max, 2D}, respectively. Error bars represent 1σ statistical uncertainties.

We find evidence for emergent non-mutual FES based on good agreement for S_c/N, and . The FES quasi-particles also exhibit a power-law scaling, , with A_{FES, 2D} ≡ (S_{c, FES}/N)(g = 1) ≈ 2.373. The exponent β_{FES, 2D} = 0.2122(1) is fitted for 0.02 ≤ g ≤ 1 and agrees with β_2D. Given Equation (8), the two power-law scaling functions for S_c/N versus and for S_{c, FES}/N versus g agree well within (Fig. 4(c)). Accordingly, and show agreement within and , respectively (Fig. 4(d) and (e)). Our simulations agree with existing experiments [28,30,31] at and near the quantum critical point (see also Fig. S5 within the online supplementary material). These numerical and experimental data together are fully consistent with, and thereby strongly support, the emergence of interaction-induced particle-hole symmetry breaking and non-mutual FES in two dimensions (Equations (2) and (8)). The less-than-unity g_{max, 2D} = 0.432(14) shows incomplete fermionization of strongly interacting 2D Bose gases.

In summary, we established a generic connection between particle-hole symmetry breaking and FES and then found strong evidence for interaction-induced non-mutual FES at and near a quantum critical point. Our non-perturbative approach holds promise for studying the dynamical evolutions [24], transport properties [25] and hydrodynamics [47] of quantum systems in arbitrary dimensions, and for studying other integrable models such as multi-component systems. In particular, such simple non-mutual FES is expected to emerge in the charge degree of freedom at the quantum critical region of the multi-component ultracold atoms in one and higher dimensions, whereas the spin degrees of freedom are frozen out at quantum criticality. Moreover, our approach provides a route to understanding strongly interacting quantum materials where experiments can be both enriched and complicated by inelastic collisional losses and finite temperature effects [31,48,49].

METHODS

Quantum Monte Carlo simulations

In our work, we apply the worm algorithm in the path-integral representation to simulate the Bose-Hubbard model in both one and two dimensions using the QMC method. Here, we mainly focus on three observables: particle density n, pressure p and entropy per particle S/N. Our QMC data are obtained by spending about 2 × 10⁵ CPU hours. Further details of the QMC simulation can be found in Sections 8 to 11 within the online supplementary material.

Statistical methods for data analysis

The error bars in figures and texts represent 1σ statistical uncertainties of the QMC simulations or experimental measurements. Based on a set of numerical or experimental data and a specific model, a least-square fit can be performed to determine the best fitting parameters as well as the standard errors of these parameters.

DATA AVAILABILITY

The data supporting the findings of this study are available within the paper and accompanying online supplementary material.

ACKNOWLEDGEMENTS

We are grateful to Cheng Chin for insightful discussions. We acknowledge Chun-Jiong Huang, Zhen-Sheng Yuan, Chen-Lung Hung, Li-Chung Ha, Biao Wu, Hui Zhai, Shina Tan for discussions and technical support.

FUNDING

This work was supported by the National Key Research and Development Program of China (2018YFA0305601, 2016YFA0300901, 2017YFA0304500 and 2016YFA0301604), the National Natural Science Foundation of China (11874073, 12134015, 11874393, 11625522 and 12104372), the Chinese Academy of Sciences Strategic Priority Research Program (XDB35020100), the Chinese Academy of Sciences Innovation Team (12121004), and the Hefei National Laboratory.

AUTHOR CONTRIBUTIONS

X.Z., Y.D. and X.G. conceived the project and wrote the paper. Y.-Y.C. and L.L. performed the numerical and analytical studies on the 1D and 2D models, respectively. X.Z. also performed analytical and numerical computations for this paper.

Conflict of interest statement. None declared.