At the core of every frustrated system, one can identify the existence of frustrated rings that are usually interpreted in terms of single-particle physics. We check this point of view through a careful analysis of the entanglement entropy of both models that admit an exact single-particle decomposition of their Hilbert space due to integrability and those for which the latter is supposed to hold only as a low energy approximation. In particular, we study generic spin chains made by an odd number of sites with short-range antiferromagnetic interactions and periodic boundary conditions, thus characterized by a weak, i.e. nonextensive, frustration. While for distances of the order of the correlation length the phenomenology of these chains is similar to that of the non-frustrated cases, we find that correlation functions involving a number of sites scaling like the system size follow different rules. We quantify the long-range correlations through the von Neumann entanglement entropy, finding that indeed it violates the area law, while not diverging with the system size. This behavior is well fitted by a universal law that we derive from the conjectured single-particle picture.

We show that a quantum spin circulator, a nonreciprocal device that routes spin currents without any charge transport, can be achieved in Y junctions of identical spin-1/2 Heisenberg chains coupled by a chiral three-spin interaction. Using bosonization, boundary conformal field theory, and density-matrix renormalization group simulations, we find that a chiral fixed point with maximally asymmetric spin conductance arises at a critical point separating a regime of disconnected chains from a spin-only version of the three-channel Kondo effect. We argue that networks of spin-chain Y junctions provide a controllable approach to construct long-sought chiral spin liquid phases.Introduction.-The spin-1/2 Heisenberg chain represents an analytically accessible model of basic importance in condensed matter theory [1]. By now, many experimental and theoretical works have contributed to a rather complete understanding of this model, including the effects of boundaries and junctions of two chains [2]. However, little attention has been devoted to quantum junctions formed by more than two Heisenberg chains. In fact, recent theoretical developments provide hints that interesting physics should be expected in that direction: First, multichannel Kondo fixed points have been predicted for junctions of anisotropic spin chains [3][4][5][6]. Second, electronic charge transport through junctions of three quantum wires is governed by a variety of nontrivial fixed points which cannot be realized in two-terminal setups [7][8][9][10][11][12][13][14][15][16]. As spin currents in antiferromagnets can be induced by spin pumping [17] or by the longitudinal spin-Seebeck effect [18], it is both an experimentally relevant and fundamental question to determine nontrivial fixed points governing spin transport in junctions of multiple spin chains. In particular, we are interested in the possibility of realizing a circulator for spin currents. While circulators have been discussed for photons [19][20][21] and for quantum Hall edge states [22,23], we are not aware of existing proposals for spin circulators. Once realized, a spin circulator has immediate applications in the field of spintronics [24], which has recently turned to the study of charge-insulating antiferromagnetic materials [25][26][27].In this paper, we study Y junctions of spin-1/2 Heisenberg chains coupled at their ends by spin-rotation [SU(2)] invariant interactions. We assume identical chains such that the junction is Z 3 -symmetric under a cyclic exchange. These conditions are respected by a chiral three-spin coupling J χ [see Eq. (1) below], which breaks time reversal (T ) symmetry and can be tuned from weak to strong coupling, e.g., by changing an Aharonov-Bohm flux [28][29][30]. Apart from condensed matter systems, such Y junctions can also be studied in ultracold atom platforms [31], where Heisenberg chains [32-34] and multi-spin exchange processes [35] have recently been realized. We use three complementary theoretical ap-

We investigate the N -leg spin-S Heisenberg ladders by using the density matrix renormalization group method. We present estimates of the spin gap s and of the ground-state energy per site e N ∞ in the thermodynamic limit for ladders with widths up to six legs and spin S 5 2 . We also estimate the ground-state energy per site e 2D ∞ for the infinite two-dimensional spin-S Heisenberg model. Our results support that for ladders with semi-integer spins the spin excitation is gapless for N odd and gapped for N even, whereas for integer spin ladders the spin gap is nonzero, independent of the number of legs. Those results agree with the well-known conjectures of Haldane and Sénéchal-Sierra for chains and ladders, respectively. We also observe edge states for ladders with N odd, similar to what happens in spin chains.

We study the influence of reflective boundaries on time-dependent responses of one-dimensional quantum fluids at zero temperature beyond the low-energy approximation. Our analysis is based on an extension of effective mobile impurity models for nonlinear Luttinger liquids to the case of open boundary conditions. For integrable models, we show that boundary autocorrelations oscillate as a function of time with the same frequency as the corresponding bulk autocorrelations. This frequency can be identified as the band edge of elementary excitations. The amplitude of the oscillations decays as a power law with distinct exponents at the boundary and in the bulk, but boundary and bulk exponents are determined by the same coupling constant in the mobile impurity model. For nonintegrable models, we argue that the power-law decay of the oscillations is generic for autocorrelations in the bulk, but turns into an exponential decay at the boundary. Moreover, there is in general a nonuniversal shift of the boundary frequency in comparison with the band edge of bulk excitations. The predictions of our effective field theory are compared with numerical results obtained by time-dependent density matrix renormalization group (tDMRG) for both integrable and nonintegrable critical spin-S chains with S = 1/2, 1 and 3/2.

We show that a honeycomb lattice of Heisenberg spin-1/2 chains with three-spin junction interactions allows for controlled analytical studies of chiral spin liquids (CSLs). Tuning these interactions to a chiral fixed point, we find a Kalmeyer-Laughlin CSL phase which here is connected to the critical point of a boundary conformal field theory. Our construction directly yields a quantized spin Hall conductance and localized spinons with semionic statistics as elementary excitations. We also outline the phase diagram away from the chiral point where spinons may condense. Generalizations of our approach can provide microscopic realizations for many other CSLs. Introduction.-Chiral spin liquids occupy a prominent position among the most exotic quantum phases of matter [1]. As examples of quantum spin liquids [2, 3], they occur in magnetic insulators with long-rangeentangled ground states that break time-reversal and reflection symmetries. The historically first proposal is the Kalmeyer-Laughlin CSL [4, 5], a topological phase of interacting spins equivalent to a bosonic fractional quantum Hall state. The non-Abelian phase of Kitaev's honeycomb model in a magnetic field provides another CSL example, with Ising anyons as elementary excitations [6]. Recent experiments have reported a quantized thermal Hall conductance for the Kitaev material α-RuCl 3 [7], compatible with the chiral Majorana edge mode expected for this CSL phase. Various other CSL phases have been theoretically investigated [8-22] and are actively searched for in experiments, including gapless CSLs with spinon Fermi surfaces [23-26].A major obstacle to the theory of CSLs comes from the shortage of analytical methods able to predict their occurrence and their physical properties in microscopic models. Apart from exactly solvable models [6,9,10], standard approaches employ parton mean-field theories that fractionalize the spin operator into fermionic or bosonic quasiparticles [5,27], or use variational wave functions obtained by a Gutzwiller projection scheme [2]. Such approaches are often able to capture the basic phenomenology when the CSL phase is indeed realized. However, since they rely on uncontrolled approximations, their predictions are often questionable, e.g., due to the neglect of interactions mediated by emergent gauge fields. In this Letter we establish a connection between chiral fixed points of boundary conformal field theory (BCFT) [28,29] and CSL phases, and use it to formulate a controlled analytical construction scheme for CSLs where chiral junctions of multiple spin chains serve as the elementary building blocks in two-dimensional (2D) networks of spin chains. Our approach markedly differs from standard coupled-wire constructions [13][14][15][16][30][31][32], where spin liquid phases emerge in models of coupled parallel chains. Our network construction instead uses

We study a Y junction of spin-1/2 Heisenberg chains with an interaction that breaks both time-reversal and chain exchange symmetries, but not their product nor SU(2) symmetry. The boundary phase diagram features a stable disconnected fixed point at weak coupling and a stable three-channel Kondo fixed point at strong coupling, separated by an unstable chiral fixed point at intermediate coupling. Using non-abelian bosonization and boundary conformal field theory, together with density matrix renormalization group and quantum Monte Carlo simulations, we characterize the signatures of these low-energy fixed points. In particular, we address the boundary entropy, the spin conductance, and the temperature dependence of the scalar spin chirality and the magnetic susceptibility at the boundary. h B n d Q h w Z Q E P A M r / D m P D o v z r v z M W 8 t O P n M I f y B 8 / k D h C + P n A = = < / l a t e x i t > J < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 I Z Z 6 U M H d 4 D 0 X 9 s v v f d t 9 + 0 / G f o = " > A A A B 7 X i c d V D L S s N A F J 3 U V 6 2 v q k s 3 g 0 V w F Z I Y 2 r o r u h F X F e w D 2 l A m 0 0 k 7 d j I T Z i Z C C f 0 H N y 4 U c e v / u P N v n L Q V V P T A h c M 5 9 3 L v P W H C q N K O 8 2 E V V l b X 1 j e K m 6 W t 7 Z 3 d v f L + Q V u J V G L S w o I J 2 Q 2 R I o x y 0 t J U M 9 J N J E F x y E g n n F z m f u e e S E U F v 9 X T h A Q x G n E a U Y y 0 k d r X g z 4 e 0 0 G 5 4 t j n 9 a r n V 6 F j O 0 7 N 9 d y c e D X / z I e u U X J U w B L N Q f m 9 P x Q 4 j Q n X m C G l e q 6 T 6 C B D U l P M y K z U T x V J E J 6 g E e k Z y l F M V J D N r 5 3 B E 6 M M Y S S k K a 7 h X P 0 + k a F Y q W k c m s 4 Y 6 b H 6 7 e X i X 1 4 v 1 V E 9 y C h P U k 0 4 X i y K U g a 1 g P n r c E g l w Z p N D U F Y U n M r x G M k E d Y m o J I J 4 e t T + D 9 p e 7 b r 2 O 6 N X 2 l c L O M o g i N w D E 6 B C 2 q g A a 5 A E 7 Q A B n f g A T y B Z 0 t Y j 9 a L 9 b p o L V j L m U P w A 9 b b J 6 z A j z E = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 I Z Z 6 U M H d 4 D 0 X 9 s v v f d t 9 + 0 / G f o = " > A A A B 7 X i c d V D L S s N A F J 3 U V 6 2 v q k s 3 g 0 V w F Z I Y 2 r o r u h F X F e w D 2 l A m 0 0 k 7 d j I T Z i Z C C f 0 H N y 4 U c e v / u P N v n L Q V V P T A h c M 5 9 3 L v P W H C q N K O 8 2 E V V l b X 1 j e K m 6 W t 7 Z 3 d v f L + Q V u J V G L S w o I J 2 Q 2 R I o x y 0 t J U M 9 J N J E F x y E g n n F z m f u e e S E U F v 9 X T h A Q x G n E a U Y y 0 k d r X g z 4 e 0 0 G 5 4 t j n 9 a r n V 6 F j O 0 7 N 9 d y c e D X / z I e u U X J U w B L N Q f m 9 P x Q 4 j Q n X m C G l e q 6 T 6 C B D U l P M y K z U T x V J E J 6 g E e k Z y l F M V J D N r 5 3 B E 6 M M Y S S k K a 7 h X P 0 + k a F Y q W k c m s 4 Y 6 b H 6 7 e X i X 1 4 v 1 V E 9 y C h P U k 0 4 X i y K U g a 1 g P n r c E g l w Z p N D U F Y U n M r x G M k E d Y m o J I J 4 e t T + D 9 p e 7 b r 2 O 6 N X 2 l c L O M o g i N w D E 6 B C 2 q g A a 5 A E 7 Q A B n f g A T y B Z 0 t Y j 9 a L 9 b p o L V j L m U P w A 9 b b J 6 z A j z E = < / l a t e x i t > < l a t e x i t s ...

We investigate the finite-size corrections of the entanglement entropy of critical ladders and propose a conjecture for its scaling behavior. The conjecture is verified for free fermions, Heisenberg and quantum Ising ladders. Our results support that the prefactor of the logarithmic correction of the entanglement entropy of critical ladder models is universal and it is associated with the central charge of the one-dimensional version of the models and with the number of branches associated with gapless excitations. Our results suggest that it is possible to infer whether there is a violation of the entropic area law in two-dimensional critical systems by analyzing the scaling behavior of the entanglement entropy of ladder systems, which are easier to deal.PACS numbers: 03.67.Mn, 64.60.an Introduction. Entanglement is a very peculiar property of composite systems which has intrigued the physicists since the beginning of quantum mechanics. The entanglement is a fundamental ingredient to teleport quantum states and it is also an important key in quantum computation and quantum information [1]. Among the various quantifiers of entanglement, the entanglement entropy (EE) is one of the most used since it is sensitive to the long-distance quantum correlations of critical systems.

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