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Quantum spin liquids SFB/TR 49 student seminar Hamburg, October 2013 Simon Trebst University of Cologne [email protected] © Simon Trebst Motivation – a paradigm H= interacting many-body system X z z i j hiji Spontaneous symmetry breaking © Simon Trebst • ground state has less symmetry than Hamiltonian • local order parameter • phase transition / Landau-Ginzburg-Wilson theory Every rule has an exception ◆ N ✓ ⇣ ⌘ X 2 1 e H= pj A(xj ) + eA0 (xj ) 2m c j=1 X + V (|xi xj |) electron gas B I i<j interacting many-body system ✓ 2 tr A ^ dA + A ^ A ^ A 3 ◆ Sometimes, the exact opposite happens © Simon Trebst • ground state has more symmetry than Hamiltonian • non-local order parameter • emergence of degenerate ground states, exotic statistics, ... V AlGaAs GaAs AlGaAs Topological quantum matter • • © Simon Trebst Spontaneous symmetry breaking • ground state has less symmetry than Hamiltonian • Landau-Ginzburg-Wilson theory • local order parameter Topological order • ground state has more symmetry than Hamiltonian • degenerate ground states • non-local order parameter concepts © Simon Trebst Quantum spin liquids Quantum spin liquids are exotic ground states of frustrated quantum magnets, in which local moments are highly correlated but still fluctuate strongly down to zero temperature. Classification a la Xiao-Gang Wen three spatial dimensions spatial dimensions energy spectrum two spatial dimensions gapped gapless Xiao-Gang Wen MIT/Perimeter topological spin liquids gapless spin liquids singular points e.g. Dirac cones © Simon Trebst spinon surface, Bose surface Topological quantum matter Energy Xiao-Gang Wen: A ground state of a – be fully + + system that + cannot many-body E characterized by a local order parameter. E Ep Often characterized by a variety of non-local “topological properties”. + – – © Simon Trebst + 0 Co ba B 0 0 • – n – • c + • b 0 – a ba Momentum A topological phase can Figure 2 | Topological insulators andbe bandpositively structures. a, The conduction and valence bands o A typical 3D solid (middle section). The shaded regions are the bands in the bulkBof the solid, a identified by its entanglement properties. thick black lines are the bands at the surface. (Similar behaviour is observed in a 2D system 1D boundary.) In general, the conduction band is symmetric (red), the valence band is antisy (blue), and spin-up and spin-down electrons (black arrows) have the same energy. En, E0 and Knots & edge states • Bringing a topological and a conventional state into spatial proximity will result in a gapless edge state – literally a knot in the wavefunction. • We know this: “Counterintuitive states” UK © Simon Trebst Ireland Australia Japan China Hong Kong Knots & edge states China Hong Kong Flipper bridge © Simon Trebst topological quantum spin liquids – gapped quantum spin liquids – © Simon Trebst the toric code the drosophila for lattice models of topologically ordered phases © Simon Trebst The toric code A. Kitaev, Ann. Phys. 303, 2 (2003). HTC = A ⇥ v j vertex(v) z j B ⇥ x j p j plaquette(p) similar to ring exchange introduces frustration i Hamiltonian has only local terms. All terms commute → exact solution! The vertex term A. Kitaev, Ann. Phys. 303, 2 (2003). HTC = A ⇥ v j vertex(v) z j B ⇥ x j p j plaquette(p) • is minimized by an even number of down-spins around a vertex. • Replacing down-spins by loop segments maps ground state to closed loops. • Open ends are (charge) excitations costing energy 2A. z = A ⇥ The plaquette term A. Kitaev, Ann. Phys. 303, 2 (2003). HTC = A ⇥ v j vertex(v) z j B ⇥ x j p j plaquette(p) • flips all spins on a plaquette. • favors equal amplitude superposition of all loop configurations. • Sign changes upon flip (vortices) cost energy 2B. fugacity = 1 free loop creation isotopy free local distortions surgery reconnections Toric code – quantum loop gases A. Kitaev, Ann. Phys. 303, 2 (2003). Ground-state manifold is a quantum loop gas. Ground-state wavefunction is equal superposition of loop configurations. Toric code – quantum loop gases A. Kitaev, Ann. Phys. 303, 2 (2003). Ground-state manifold is a quantum loop gas. Py = 1 Py = Topological sectors defined by winding number parity Py/x = z i i cx/y 1 Toric code – quantum loop gases A. Kitaev, Ann. Phys. 303, 2 (2003). Ground-state manifold is a quantum loop gas. Py = 1 Py = Topological sectors defined by winding number parity Py/x = z i i cx/y 1 Anyons in the toric code A. Kitaev, Ann. Phys. 303, 2 (2003). Two types of excitations • electric charges: open loop ends violate vertex constraint • magnetic vortices: plaquettes giving -1 when flipped We get a minus sign taking one around the other electric charges and magnetic vortices are mutual anyons = (-‐‑‒1) x quantum double models looking beyond the drosophila © Simon Trebst quantum double models Quantum double models form a larger family of lattice models harboring non-trivial topological order, e.g. non-Abelian string nets. loop gas configuration (toric code) • Quantum double models are generally constructed from an underlying anyon theory. • Key ingredient are so-called fusion rules of anyons. M. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 (2005); C. Gils et al., Nature Physics 5, 834 (2009). © Simon Trebst string net configuration A Ā } anyon theories 0, 1, 2, . . . k . quantum double models One can think of every degree of freedom as a representation of the quantum group su(2)k or in analogy to the spin representation of SU(2), which corresponds to the limit k ! 1, as generalized spins with even/odd integer labels corresponding to integer/half-integer spins. We can combine two representations of su(2)kdouble into one models form a larger family of lattice models Quantum joint representation – similar to combining two spin quantum harboring numbers into one joint spin quantum number fornon-trivial conventional topological order, e.g. non-Abelian string nets. SU(2) spins. This process, which for the anyon theories is often called fusion, has to obey very similar rules as those for combining two conventional SU(2) spins. In particular, they have to obey the so-called fusion rules which also incorporate the cut-off k in a consistent way min[i+j,2k i j] X i⇥j = (14) l, l=|i j| where l increases in steps of two. Eq. (14) can be written more compactly by introducing fusion coefficients Nijl , which are defined via i⇥j ⌘ k X l=0 Nijl l . (15) loop gas configuration Note that for the su(2)k anyon theories at hand these fusion coefficients are always either 0 or 1. The simplest example is probably the anyon theory su(2)1 with anyonic degrees of freedom 0 and 1, for which the above Z2 anyon theory fusion rules become and finally the Fibonacci theory, which is the even-integer subset of this su(2)3 anyonic theory anyon theory Fibonacci 0⇥0=0 0⇥1=1 1 ⇥ 1 = 0. 0⇥0=0 0⇥2=2 2 ⇥ 2 = 0 + 2. toric code 0 1 (16) 0 0 0 A slightly less trivial example is the anyon theory su(2)2 with anyonic degrees of freedom 0, 1 and 2, for which the fusion rules read © Simon Trebst string net configuration 1 0 2 (19) 0 0 0 non-Abelian One striking distinction between the fusion rules for su(2)1 in (16) and for the remaining ones in (17), (18), and (19) is the occurrence of representations, which fused with itself, gener- 2 possible labelings of ed of allowed vertices for ory are given (up to rota respectively. 2 Forming an entire gr tices and corresponding tude of lattice configura 2 2 configurations it becom translate into constraint For the case of the su(2) gapless quantum spin liquids – an example – © Simon Trebst The Kitaev model Jz gapped spin liquid Z2 topological order x y z Jx + Jy + Jz = const. z z i j x x i j HKitaev = y y i j X J Jx i j links Rare combination of a model of fundamental conceptual importance and an exact analytical solution. Jy gapless spin liquid w/ Majorana fermion excitations 2 Dirac cones A. Kitaev, Ann. Phys. 321, 2 (2006) © Simon Trebst Solving the Kitaev model Step 1: Majorana fermionization x y z bz bx † † fermions a" a" a# a# © Simon Trebst bz = a# + a†" ⇣ c = i a# y a†# ⌘ a†" ⌘ x z = i by c D= i x y z } Majorana fermions bx = a" + a†# ⇣ y b = i a" by c = i bx c = i bz c = bx b y bz c gauge operator [D, ]=0 physical subspace D = 1 Solving the Kitaev model Step 2: Diagonalization of Hamiltonian In the language of Majorana fermions, we now have j k = i bj cj (i bk ck ) = i ujk cj ck ujk = i bj bk which immediately allows us to write the Hamiltonian as iX H= Ajk cj ck 4 hjki Ajk = 2J ujk bz bx by c © Simon Trebst Hamiltonian is skew-symmetric, because ujk = ukj Ajk = Akj Solving the Kitaev model Step 2: Diagonalization of Hamiltonian In the language of Majorana fermions, we now have j k = i bj cj (i bk ck ) = i ujk cj ck ujk = i bj bk which immediately allows us to write the Hamiltonian as iX H= Ajk cj ck 4 hjki Ajk = 2J ujk bz bx by c Hamiltonian is skew-symmetric, because ujk = Ajk = Finally, there is a gauge choice to be made ujk © Simon Trebst ukj has eigenvalues ±1 Akj eight copies of each p the same translationa phase are algebraical We now consider t tum q is defined mod momentum space by the symmetric case (J Solving the Kitaev model Step 2: Diagonalization of Hamiltonian The Hamiltonian has turned into a free (Majorana) fermion problem iX H= Ajk cj ck 4 hjki which can readily be diagonalized by Fourier transformation nx i H(q) = A(q) = 2 ny ✓ ◆ 0If |Jx| if (q) and |Jy| decrea ⇤ i f(within (q) 0the paralle |Jiq·n x| + |Jy| = |Jz|. The p y f (q) = Jx e + Jyallelogram. e + Jz (Note tha At the points ±q* thus yielding a gapless energy spectrum of the form iq·nx bz b x by c nx = ny = p ! 1 3 , 2 2 p ! 1 3 , 2 2 ✏(q) = ±|f (q)| Dirac cones q⇤1 q⇤2 © Simon Trebst = = ✓ ✓ 2⇡ 2⇡ ,p 3 3 ◆ 2⇡ 2⇡ ,p 3 3 ◆ The Kitaev model Jz gapped spin liquid Z2 topological order x y z Jx + Jy + Jz = const. z z i j x x i j HKitaev = y y i j X J Jx i j links Rare combination of a model of fundamental conceptual importance and an exact analytical solution. Jy gapless spin liquid w/ Majorana fermion excitations 2 Dirac cones A. Kitaev, Ann. Phys. 321, 2 (2006) © Simon Trebst entanglement © Simon Trebst Identification of topological order Which observables can we measure numerically to identify topological order? • • © Simon Trebst bulk properties • entanglement entropy & spectrum • ground-state degeneracy edge properties • entanglement entropy • energy spectra bulk properties entanglement entropy © Simon Trebst Entanglement If two quantum mechanical objects are interwoven in such a way that their collective state cannot be described as a product state we say they are entangled. • " A B i + sin ↵| " | i = cos ↵| " i| A i| " i B Calculate the reduced density matrix for one of the two parts ⇢A = | iA h |A (traced over subsystem B) • A " A A ih " ⇢A = cos2 ↵| " ih " | + sin2 ↵| A | One quantitative measure of entanglement is the entanglement entropy S(⇢A ) = Tr( ⇢A log ⇢A ) von Neumann entropy = first Renyi entropy Sn (⇢A ) = 1 1 Renyi entropy © Simon Trebst n ln [Tr( ⇢nA )] S2 (⇢A ) = ln ⇥ second Renyi entropy Tr( ⇢2A ) ⇤ Entanglement If two quantum mechanical objects are interwoven in such a way that their collective state cannot be described as a product state we say they are entangled. " " Tr( ⇢A log ⇢A ) p i| " i) / 2 0.6 Renyi entropy 1 n ln [Tr( ⇢nA )] Renyi entropy Sn (⇢A ) = 1 i | p i| " i) / 2 S1 S2 S3 S4 S5 S6 S7 S8 0.4 0.2 | " second Renyi entropy (| " i| i| " i | " i| " ⇥ ⇤ ln Tr( ⇢2A ) B S = ln 2 von Neumann entropy = first Renyi entropy S2 (⇢A ) = A " S(⇢A ) = i+| B i| " i " (| " i| " A i + sin ↵| " | i = cos ↵| " i| i 0 0 © Simon Trebst π/4 π/2 α 3π/4 π Entanglement • Entanglement in quantum many-body systems • Consider bipartition of system into two parts A and B, and calculate the reduced density matrix for the two parts ⇢A = | iA h |A • (traced over subsystem B) One quantitative measure of entanglement is the entanglement entropy S(⇢A ) = Tr( ⇢A log ⇢A ) von Neumann entropy = first Renyi entropy Sn (⇢A ) = 1 1 n ln [Tr( ⇢nA )] Renyi entropy S2 (⇢A ) = ln ⇥ second Renyi entropy © Simon Trebst Tr( ⇢2A ) ⇤ B A Boundary law also called area law B A Entanglement scales with the length L of the boundary of the bipartition S = ln 2 © Simon Trebst S / aL Corrections to the boundary law Corrections to the boundary law can arise from • geometric aspects of the bipartition B • S = aL + b · (# of corners) A (for 2D gapped state) topological aspects of the bipartition A S = aL B © Simon Trebst · (# of disconnected parts) (for 2D gapped state) Corrections to the boundary law Corrections to the boundary law also originate from the specific character of the underlying quantum many-body state! • topological spin liquids this talk quantum double models Levin-Wen / Kitaev models S = aL • gapless spin liquids • gapless modes at singular point in momentum space S = aL + c (Lx , Ly ) • gapless modes on surface in momentum space “Bose metals” S = cL ln(L) • (Motrunich, Sheng & Fisher) critical points, conformal critical points, Goldstone modes, ... SQCP = aL + c (Lx , Ly ) © Simon Trebst Kitaev model (honeycomb) ScQCP = µL + cQCP SG = aL + b ln(L) + (Lx , Ly ) Topological entanglement entropy Distilling the topological correction by using a clever sequence of partitions A1 A3 A2 A4 2 2 S = aL + b · (# of corners) Stopo = · (# of disconnected parts) S A 1 + S A2 + S A 3 SA4 = A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006). © Simon Trebst 2 Topological entanglement entropy The topological correction is universal. v u n uX = ln t d2 i quantum dimension of excitation i=1 Examples: • toric code (loop gas) 1 e m di • 1 1 1 em 1 = ln p 1 + 1 + 1 + 1 = ln 2 ⇡ 0.693 Fibonacci theory (string net) 1 di 1 ⌧ p 1+ 5 = 2 p = ln 1 + A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006). © Simon Trebst 2 ⇡ 0.643 Let’s try this on the toric code ARTICLES a toric code (loop gas) 1 e m di 1 1 1 5 5 hx = hz = h h = 0.2 h = 0.3 h = 0.4 4 em 1 3 4 3 2 S(Ly) • S(Ly) Examples: N 1 1 0 = ln p 2 0 ¬ln2 1 + 1 + 1 + 1 = ln 2 ⇡ 0.693 ¬1 b 0 4 2 4 6 8 Ly 0 10 12 14 16 hx = 0.3, hz = 0.0 S(Ly) 2 ARTICLES 1 0 PUBLISHED ONLINE: 11 NOVEMBER 2012 | DOI: 10.1038/NPHYS2465 ¬ln2 ¬1 Hong-Chen Jiang1 , Zhenghan Wang2 and Leon Balents1 * Topological phases are unique states of matter that incorporate long-range quantum entanglement and host exotic excitations with fractional quantum statistics. Here we report a practical method to identify topological phases in arbitrary realistic models by accurately calculating the topological entanglement entropy using the density matrix renormalization group (DMRG). We argue that the DMRG algorithm systematically selects a minimally entangled state from the quasi-degenerate ground states in a topological phase. This tendency explains both the success of our method and the absence of ground-state degeneracy in previous DMRG studies of topological phases. We demonstrate the effectiveness of our procedure by obtaining the topological entanglement entropy for several microscopic models, with an accuracy of the order of 10 3 , when the circumference of the cylinder is around ten times the correlation length. As an example, we definitively show that the ground state of the quantum S = 1/2 antiferromagnet on the kagome lattice is a topological spin liquid, and strongly constrain the conditions for identification of this phase of matter. © Simon Trebst Nature Physics 8, 902 (2012). 2 second-neighbou 0 Identifying topological order by entanglement entropy 0 Figure 3 | The entan equation (2), with L = 0.698(8) at J2 = lattice with Lx = 12 a G = +¬1 G=0 3 ¬1 0 2 4 6 8 10 12 14 Ly Figure 2 | The von Neumann entropy S(Ly ) for the toric-code model in magnetic fields. a, S(Ly ) with Ly = 4–16 at Lx = 1 for symmetric magnetic fields at hx = hz = h = 0.2, 0.3 and 0.4. By fitting S(Ly ) = aLy , we get = 0.693(1), 0.691(4) and 0.001(5), respectively. b, The pure electric case, hx = 0.3,hz = 0, and comparison of S(Ly ) in the MES obtained in the large Lx limit (black squares) with that of the absolute ground state from systems of dimensions Lx ⇥ Ly = 20 ⇥ 4,24 ⇥ 6,24 ⇥ 8,24 ⇥ 10 (red circles). Extrapolation shows that the MES has the universal TEE, whereas the absolute ground state has zero TEE. around the entire cylinder. To take this into account, the DMRG must fully converge the entanglement of the opposite ends of the where Si is the s the nearest neighb simulation, we se DMRG studies17 substantial spin-li We take the ka along a bond dire the unit of lengt results for the en shown in Fig. 3 w spacing for both s We see that a linea gives = 0.698( both within one that this phase is quantum dimens was proposed for neighbour interac of the spectrum of ! properly, but also point to a finite value of ", and hence a topological ground state. The quantum dimension pffiffiffi is D ¼ 2, excluding chiral spin liquids (" ¼ 1=2 or D ¼ 2 [77]). Rigorously, DMRG only provides a lower bound on D [80], but the bound is essentially exact as DMRG is a method with low entanglement bias [81]. Conclusion.—Through a combination of a large number of DMRG states, large samples with small finite size effect, and the use of the SU(2) symmetry of the kagome model, we have been able to corroborate earlier evidence for a QSL as opposed to a VBC, due to energetic considerations and complete absence of breaking of space group invariance, although DMRG should be biased towards VBC due to its low-entanglement nature and the use of OBC. On the basis of the numerical evidence (spin gap, structure factor, spin, dimer and chiral correlations, topological entanglement entropy) numerous QSL proposals can be ruled out for the kagome system. On the system sizes reached, the spin gap is ARTICLES [85,86] albeit for a variational energy far from the groundstate energy. The concentration of weight of the structure factor at the hexagonal Brillouin zone edge with shallow maxima at the corners would also point to the Z2 QSL as proposed by Sachdev [27], and a Z2 QSL is also consistent with all other numerical findings. All this provides strong evidence for the Z2 QSL, whereas to our knowledge, no plausible scenario for the emergence of a double-semion phase in the KAFM has been discovered somodel far, making it Heisenberg implausible, but of course not impossible. An analysis of the degenerate ground-state manifold as proposed in Ref. [80], not possible with our data, would settle the issue. Even if the j answer provided final evidence for a Z2 QSL,i many questions regarding the detailed microscopic structure of the hi,ji ground-state wave functions and the precise nature of the Z2 QSL would remain for future research. kagomé lattice S. D. and U. S. thank F. on Essler, A. Läuchli, C. Lhuillier, D. Poilblanc, S. Sachdev, R. Thomale, and S. R. White for NATURE PHYSICS DOI: 10.1038/NPHYS2465 discussions. S. D. and U. S. acknowledge support by DFG. U. S. thanks the GGI, Florence, for its hospitality. I. P. M. 5 acknowledges support from the Australian Research Council Centre of Excellence for Engineered Quantum 4 Systems and the Discovery Projects funding scheme (Project No. DP1092513). 3 Note added.—Recently, we became aware of Ref. [81], which calculates topological entanglement entropy from 2 von Neumann entropy for a next-nearest neighbor modification of the KAFM, 1perfectly consistent with our results J2 = 0.10 of D ¼ 2 for the KAFM itself. J2 = 0.15 Kagomé antiferromagnet a S2 S3 5 S4 h = 0.2 h = 0.3 h = 0.4 3 fit to S2 fit to S3 3 hx = hz = h 4 S5 4 5 2 fit to S4 2 1 fit to S5 1 0 0 ¬ln2 0 -1 ¬1 0 2 4 0 ¬ln2 0 2 6 8 circumference c b 10 4 12 6 14 8 Ly 10 4 hz = 0.0 hx = 0.3,S FIG. 6 (color online). Renyi entropies # of infinitely long G = +¬1 cylinders for various # versus circumference c, extrapolated to G=0 3 c ¼ 0. The negative intercept is the topological entanglement entropy ". S(Ly) 2 S. Depenbrock, I.P. McCulloch, and U. Schollwöck, 1 Phys. Rev. Lett. 109, 067201 (2012). 0 © Simon Trebst ~ ·S ~ S S(Ly) 6 S(Ly) Renyi entropy Sα 7 H= X 12 14 ¬1 0 2 4 6 Ly 8 10 12 *[email protected] Figure 3 | The entanglement entropy S(Ly ) of the kagome J1 –J2 model in [1] L. Balents, Nature (London) (2010). equation (2), with Ly = 464, 4–12 at199 Lx = 1. By fitting S(Ly ) = aLy , we get [2] V. Elser, Phys. Rev. Lett. 62, 2405 (1989). = 0.698(8) at J2 = 0.10 and = 0.694(6) at J2 = 0.15. Inset: kagome Lx =Rev. 12 andBLy 83, = 8. 214406 (2011). [3] K. Matan etlattice al., with Phys. 067201-4 0 16 second-neighbour interactions, whose Hamiltonian is XL. Balents, H.-C. Jiang, X Z. Wang, and H = JPhysics Si ·Sj8, +902 J2 (2012). Si ·Sj 1 Nature hiji hhijii (2) where Si is the spin operator on site i, and hiji (hhijii) denotes the nearest neighbours (next-nearest neighbours). In the numerical bulk properties entanglement spectrum © Simon Trebst Entanglement spectrum • The entanglement entropy is one quantitative measure of entanglement ⇢A = | iA h |A (traced over subsystem B) B S(⇢A ) = Tr( ⇢A log ⇢A ) • However, a lot of information possibly contained in the density matrix is discarded in this simple measure. • The entanglement spectrum aims at unraveling some of this information ⇢A = | i A h | A = S(⇢A ) = X 2 ↵ ln 2 ↵ X ↵ ↵ 2 ↵ | ↵ iA h ↵ |A =e ⇠↵ /2 (Schmidt decomposition) ⇢A = e ↵ (HE = H. Li and F.D.M. Haldane, Phys. Rev. Lett. 101, 010504 (2008). © Simon Trebst A HE ln ⇢A ) “entanglement Hamiltonian” Entanglement spectrum H. Li and F.D.M. Haldane, Phys. Rev. Lett. 101, 010504 (2008). The entanglement spectrum aims at unraveling some of the information contained in the density matrix • ⇢A = | i A h | A = B X ↵ ↵ ⇠↵ /2 =e ⇢A = e (a) (b) HE “entanglement Hamiltonian” week ending (HE = 7lnMAY ⇢A )2010 PHYSICAL REVIEW LETTERS 180502 (2010) A 2 ↵ | ↵ iA h ↵ |A 8 20 20 15 15 6 4 2 10 45 0 1 0.04 0.08 0.12 N 10 1 1 23 5 53 2 11 40 5 counting related to number of modes of the gapless edge theory (CFT) 35 5 30 25 20 0 10 6 8 10 15 12 14 20 25 30 35 0 5 10 15 20 25 30 35 R. Thomale, A. Sterdyniak, N. Regnault, and B.A. Bernevig, Phys. Rev. Lett. 104, 180502 (2010). or online). Entanglement spectrum for the N ¼ 11 bosons, N# ¼ 20, $ ¼ 1=2 Coulomb state on the sphere. The cut is ¼ 10 orbitals and NA ¼ 5 bosons. (a) Standard normalization on the quantum Hall sphere. The inset show the remainder spectrum where the entanglement levels exceed % ¼ 24. (b) CL normalization. We observe that the CL separates a set of © Simon Trebst Entanglement summary • The entanglement entropy and entanglement spectrum are powerful bulk “observables” that allow • • to unambiguously distinguish topological order from conventional order to characterize the type of topological order to a good extent* * in some measures, e.g. the topological entanglement entropy, some unambiguities might remain and require additional work • The reduced density matrix is readily available in some numerical methods such as • • • exact diagonalization density matrix renormalization group ⇢A = | iA h |A Other numerical techniques have caught up, in particular • quantum Monte Carlo ➝ replica trick ➝ Renyi entropies / entanglement entropy You will hear about this in Peter Bröcker’s talk tomorrow morning. • © Simon Trebst numerical linked cluster expansion ➝ mutual information bulk properties ground-state degeneracy © Simon Trebst Ground-state degeneracy ST et al., Phys. Rev. Lett. 98, 070602 (2007). topological phase paramagnet 4 4 ⇥E / exp( energy splitting !E(L) 2 L) 2 0 0 L = 48 L = 40 L = 32 L = 24 L = 16 L = 12 L=8 -2 -4 -2 -4 -6 -6 E -8 0.54 © Simon Trebst 0.55 0.56 0.57 0.58 0.59 magnetic field hx L -8 0.6 0.61 0.62 edge properties entanglement entropy © Simon Trebst Edge states “vacuum” “vacuum” topologically ordered state edge state edge state • Edge states correspond to the gapless modes of a critical one-dimensional system, which are typically described by a conformal field theory (CFT). • The conformal field theory can again be identified via entanglement properties } ` central charge ✓ ✓ ◆◆ c ⇡` S = log L sin 3 L C. Holzhey et al., Nucl. Phys. B 424, 44 (1994); P. Calabrese and J. Cardy, J. Stat. Mech. P06002 (2004). © Simon Trebst ` = L/2 S= c log L 3 Xi Xi±1 Xi = Xi , |xi−1 xi xi+1 "} = −1, 0, 0}, where the straints arising from xi+1 = τ , the F an are the following • ϕ−2 ϕ−3/2 ϕ−3/2 ϕ−1 # $ . & ni + 1 + ϕ−2 , chain. In this exprese occupation on link tonian H acts on the We simulated this finite-size excitation f up to L = 32 sites. vanishes linearly in ynamical critical exal field theory. The ated from the finiteFor two subsystems odic (PBC) and open ement entropy scales c log L . 6 (3) up method (DMRG) lculating these quanccording to Eqs. (3) stimates of cPBC = 1 respectively. Since y of these estimates 5 we can unambiguent only with central © Simon Trebst week ending j j correspond " = e[i] " Edgee[i]|j states to|jthej jgapless ofE VaI Ecritical system, P H Ymodes SICAL R W L E T Tone-dimensional ERS 20 APRIL 2007 PRL 98, 160409 (2007) ' which %are typically described (CFT). " conformal field theory ! Y by a 6 6 ( j~ j & j i−1 i i+1 "% j! (2) onian, it can be writ• L) ∝ Edge states i ji+1 ji−1 &j ! i ji i−1 ! i i+1 0 0 0 0 i%1 i e( !i"j~ S:# (6) ji Sj ! i !j0m ;jm $e!i"j 'j : ) i&1 i (5) and = δji−1 ,ji+1 Sj0i−1 Sj0m!i ji i+1 The Hamiltonian of the so-defined lattice model is •obare known [12] to form atained representation the Temperley-Lieb from itsoftransfer matrix by taking, as usual [16], algebra* (4) for any value k, where anisotropic |ji − ji+1 | = 1/2 uand theofextremely limit, ( 1, j! (2j+1)(2j ! +1) 2 Sj := (k+2) sin[π ][13]. expf&a$H % c1 ' % O$a2 'g; k+2 T # Our model of interacting Fibonacci anyonsu’ can be cast into 1 a # xi = 1 → ji = ( this form at k = 3 by first mapping sin!"=$k % 2'" 0, and xi = τ → ji = 1, and then applying theP SU (2)3 fusion rule 1 yielding H # & sites. is ananyunimportant constant). 3/2 × j = 3/2 − j to the even-numbered i (c1maps i ’ eThis admissible labeling |&x"Since := |x1the , x2 ,operators . . ." uniquely j" := can |&be identified with ei , this Xi into demonstrates that the Hamiltonian of the Fibonacci chain is exactly that of the corresponding 1.5 1.5 k # 3 RSOS model which is a lattice description of the tricritical Ising model at its critical point. The latter is a well-known (supersymmet7 ric) CFT with central charge c # 10 [17,18]. Analogously one obtains [19] for general k the $k & 1'st unitary minimal 1 1 CFT [10] of central charge c # 1 & 6=$k % 1'$k % 2'. A periodic boundary conditions ferromagnetically coupled Fibonacci chain (energetically open boundary conditions favoring the fusion along the # channel) is described [20] by the critical 3-state Potts model with c # 45 and, for 0.5 0.5 general k, by the critical Zk -parafermion CFT [15,19] with central charge c # 2$k & 1'=$k % 2'. Excitation spectra.—We calculated the excitation spectra of chains up to size L # 37 0with open and periodic 0 conditions 20 30 40 boundary 80 100 120 using 200 240diagonalization, as shown 50 60 160 exact in system Fig. 3.size TheLnumerical results not only confirm the CFT predictions but also reveal some important details about the FIG. 2: (Color online) Entropy scaling for between interactingcontinuous Fibonacci correspondence fields and microanyons arranged along an open (open squares) or scopic observables. In periodic general,chain low-energy states on a (closed circles) versus the system size L. Logarithmic fits (solid ring are associated with local conformal fields [21], whose lines) give central charge estimates of cPBC = 0.701 ± 0.001 and holomorphic andtheantiholomorphic cOBC = 0.70 ± 0.01 respectively, where for open boundary con-parts belong to representations of the Virasoro algebra, described by conformal ditions only the et values the 5Rev. largest systems been taken into A. Feiguin al., for Phys. Lett. 98, have 160409 (2007). weights hL and hR . The energy levels are given by account due to large finite-size effects. # $ 2"v c E#E L% & %h %h ; (7) j e[i]ji+1 i−1 j! i The CFT can be identified via entanglement properties c S = log L 3 5 5 Even4 more information about the CFT 4 reveals itself in the energy spectrum L = 36 3 3 rescaled energy E(K) . (4) where the ‘d-isotopy’ parameter equals the golden ratio, d = ϕ. This representation can be seen to be identical to the standard Temperley-Lieb algebra representation associated with SU (2)k at level k = 3. For arbitrary k, the latter contains k + 1 anyon species labelled by j = 0, 1/2, 1, ..., k/2, satisfying the fusion rules of SU (2)k [11]. The operators ei defined by entropy S(L) 1 i+1 (Fxxi−1 τ τ )x! i [Xi , Xj ] = 0 for |i − j| ≥ 2 , 1/5 + 3 3 6/5 + 2 1/5 + 2 3/40 + 3 3/40 + 3 0+3 primary fields descendants 1/5 + 2 0+2 7/8 + 2 3/40 + 2 ⌘2 2⇡v ⇣ c E1 = E1 L + + xs 1 Neveu-Schwarz Ramond L 12 sector sector 0 0 2 6/5 + 1 6/5 3/40 + 2 7/8 + 1 1/5 + 1 7/8 3/40 + 1 1/5 0 3/40 0 rescaled energy E(K) |xi−1 x! i xi+1 " (1) i 2 4 6 8 10 12 momentum K [2π/L] 14 18 16 6 6 5 5 4 4 7/8 + 2 2 1 0+2 7/10 + 2 L = 37 primary fields descendants 3 3/40 + 2 3/2 + 1 7/8 + 1 7/8 3/40 0 0 7/10 + 1 3 2 3/2 3/40 + 1 Ramond sector 2 Neveu-Schwarz sector 4 6 8 10 12 momentum K [2π/L] 14 16 1 7/10 0 18 FIG. 3 (color online). Energy spectra for periodic chains of size L. Energies are rescaled and shifted such that the two lowest eigenvalues match the CFT assignments. Open boxes indicate 7 CFT. Open circles give positions of primary fields of the c # 10 positions of descendant fields as indicated. As a guide to the eye the solid line is a cosine-fit of the dispersion. A. Feiguin et al., Phys. Rev. Lett. 98, 160409 (2007). 00 factor exp!2"i$hL)" & hL & h"L '" matches the momenta 00 We are done! So what did we learn? © Simon Trebst classification of quantum spin liquids © Simon Trebst Quantum spin liquids Quantum spin liquids are exotic ground states of frustrated quantum magnets, in which local moments are highly correlated but still fluctuate strongly down to zero temperature. Classification a la Xiao-Gang Wen three spatial dimensions spatial dimensions energy spectrum two spatial dimensions gapped topological spin liquids gapless gapless spin liquids S = aL entanglement © Simon Trebst singular points spinon surface, e.g. Dirac cones Bose surface S = aL + c (Lx , Ly ) S = cL ln(L) Summary • The formation of quantum spin liquids in an interacting quantum many-body system is one of the most fascinating phenomena in condensed matter physics • The identification of topological order or gapless spin liquids builds on concepts from • • statistical physics van Neumann & Renyi entropies • quantum information theory entanglement • mathematical physics boundary laws, anyon theories The exploration of topological order is a rich and quickly evolving research field – just at its beginning. All slides of this presentation will become available on our group webpage at www.thp.uni-koeln.de/trebst © Simon Trebst