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arXiv:2110.09723 · Physical Review A

Discrete scale-invariant boson-fermion duality in one dimension

Satoshi Ohya

Published Mar 21, 2022• 10 citations
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Abstract

We introduce models of one-dimensional $n(\ensuremath{\ge}3)$-body problems that undergo phase transition from a continuous scale-invariant phase to a discrete scale-invariant phase. In this paper, we focus on identical spinless particles that interact only through two-body contacts. Without assuming any particular cluster-decomposition property, we first classify all possible scale-invariant two-body contact interactions that respect unitarity, permutation invariance, and translation invariance in one dimension. We then present a criterion for the breakdown of continuous scale invariance to discrete scale invariance. Under the assumption that the criterion is met, we solve the many-body Schr\"odinger equation exactly; we obtain the exact $n$-body bound-state spectrum as well as the exact $n$-body $S$-matrix elements for arbitrary $n\ensuremath{\ge}3$, all of which enjoy discrete scale invariance or log periodicity. Thanks to the boson-fermion duality, these results can be applied equally well to both bosons and fermions. Finally, we demonstrate how the criterion is met in the case of $n=3$; we determine the exact phase diagram for the scale-invariance breaking in the three-body problem of identical bosons and fermions. The zero-temperature transition from the unbroken phase to the broken phase is the Berezinskii-Kosterlitz-Thouless-like transition discussed in the literature.

DOI
10.1103/physreva.105.033312

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We introduce models of one-dimensional n(\ensuremath≥3)n(\ensuremath{\ge}3)n(\ensuremath≥3)-body problems that undergo phase transition from a continuous scale-invariant phase to a discrete scale-invariant phase. In this paper, we focus on identical spinless particles that interact only through two-body contacts. Without assuming any particular cluster-decomposition property, we first classify all possible scale-invariant two-body contact interactions that respect unitarity, permutation invariance, and translation invariance in one dimension. We then present a criterion for the breakdown of continuous scale invariance to discrete scale invariance. Under the assumption that the criterion is met, we solve the many-body Schr"odinger equation exactly; we obtain the exact nnn-body bound-state spectrum as well as the exact nnn-body SSS-matrix elements for arbitrary n\ensuremath≥3n\ensuremath{\ge}3n\ensuremath≥3, all of which enjoy discrete scale invariance or log periodicity. Thanks to the boson-fermion duality, these results can be applied equally well to both bosons and fermions. Finally, we demonstrate how the criterion is met in the case of n=3n=3n=3; we determine the exact phase diagram for the scale-invariance breaking in the three-body problem of identical bosons and fermions. The zero-temperature transition from the unbroken phase to the broken phase is the Berezinskii-Kosterlitz-Thouless-like transition discussed in the literature.
Scale invariance in quantum mechanics can be broken in several ways. A well-known example is the breakdown of continuous scale invariance to discrete scale invariance, whose typical realization is the Efimov effect of three-body problems. Here we discuss yet another discrete symmetry to which continuous scale invariance can be broken: discrete phase invariance. We first revisit the one-body problem on the half line in the presence of an inverse-square potential -- the simplest example of nontrivial scale-invariant quantum mechanics -- and show that continuous scale invariance can be broken to discrete phase invariance in a small window of coupling constant. We also show that discrete phase invariance manifests itself as circularly distributed simple poles on Riemann sheets of the S-matrix. We then present three examples of few-body problems that exhibit discrete phase invariance. These examples are the one-body Aharonov-Bohm problem, a two-body problem of nonidentical particles in two dimensions, and a three-body problem of nonidentical particles in one dimension, all of which contain a codimension-two ``magnetic'' flux in configuration spaces.
The disorder-induced quantum phase transition between superfluid and nonsuperfluid states of bosonic particles in one dimension is generally expected to be of the Berezinskii-Kosterlitz-Thouless (BKT) type. Here, we show that hard-core lattice bosons with power-law hopping decaying with distance as $1/{r}^{\ensuremath{\alpha}}$ with a finite integral over space---corresponding in spin language to an XYXYXY model with power-law couplings---undergo a non-BKT continuous phase transition instead. We use exact quantum Monte Carlo methods to determine the phase diagram for different values of the exponent $\ensuremath{\alpha}$, focusing on the regime $\ensuremath{\alpha}>2$. We find that the scaling of the superfluid stiffness with the system size is scale invariant at the transition point for any $\ensuremath{\alpha}\ensuremath{\le}3$---a behavior incompatible with the BKT scenario and typical of continuous phase transitions in higher dimension. By a scaling analysis near the transition point, we find that our data are consistent with a correlation length exponent satisfying the Harris bound $\ensuremath{\nu}\ensuremath{\ge}2$ and demonstrate a universal behavior of disordered bosons in one dimension. For $\ensuremath{\alpha}>3$ our data are consistent with a BKT scenario where the liquid is pinned by infinitesimal disorder.
We study boson–fermion dualities in one-dimensional many-body problems of identical particles interacting only through two-body contacts. By using the path-integral formalism as well as the configuration-space approach to indistinguishable particles, we find a generalization of the boson–fermion duality between the Lieb–Liniger model and the Cheon–Shigehara model. We present an explicit construction of n-boson and n-fermion models which are dual to each other and characterized by n−1 distinct (coordinate-dependent) coupling constants. These models enjoy the spectral equivalence, the boson–fermion mapping, and the strong–weak duality. We also discuss a scale-invariant generalization of the boson–fermion duality.
Abstract Because of the absence of indistinguishability constraint, interparticle interactions between nonidentical particles have in general much more variety than those between identical particles. In particular, it is known that there exists a U (2) family of two-body contact interactions between nonidentical particles in one spatial dimension. This paper studies breakdown of continuous scale invariance to discrete scale invariance under this U (2) family of two-body contact interactions in two-body problems of nonidentical particles on the half line. We show that, in contrast to the corresponding identical-particle problem, there exist two distinct channels that admit geometric sequences of two-body bound states.
The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of many-body bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in three-body problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of many-body problems in various dimensions. Here, we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semi-infinite line and interact with each other through a pairwise δ-function potential with a particular position-dependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together, and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of two-body bound states. We also study the two-body scattering off the boundary and derive the exact reflection amplitude that exhibits discrete scale invariance.
We study three particles on the square lattice with contact interactions of infinite strength, for all total quasimomenta, for three statistics: three identical bosons, two identical fermions plus a third particle, and two identical bosons plus a third particle. Previous work treated only the trimer and zero quasimomentum. Here the atom-dimer sector is analyzed on the whole Brillouin zone. A Feshbach-Schur reduction onto the infinitely degenerate eigenvalue 1 of the pair-contact operator gives explicit effective Hamiltonians with proved first-order error. The threshold problem reduces to a 4x4 Hermitian matrix criterion built from the two-point resistances of the square lattice, and hence to a quartic polynomial. This yields the complete atom-dimer zone and the critical mass ratio on the Brillouin zone, with bound between 1 and pi/(pi-2), and closed forms at all symmetry points. We prove: the zones in closed form; the critical mass ratio with exact value pi/(pi-2) at zero quasimomentum, reproducing a numerical value; at most two atom-dimer states for every quasimomentum and mass ratio; the onset laws, an essential singularity on the zone boundary and a linear-over-logarithm onset in the pure p-wave; two exact dualities showing that a change of statistics is equivalent to a shift of quasimomentum by pi; and the global range of the second fermionic threshold. The onset laws admit a renormalization-group reading: the inverse lattice Green function is a marginal s-wave coupling, binding energies arise by dimensional transmutation with lattice scale 16, and the zone boundary is of marginal (BCS/Kondo) type rather than of Berezinskii-Kosterlitz-Thouless type. All constants are verified independently by lattice quadrature, computer algebra, exact diagonalisation, and infinite-lattice solvers.