High-Temperature Criticality in Strongly Constrained Quantum Systems

Transcription

1 High-Temperature Criticality in Strongly Constrained Quantum Systems Claudio Chamon Collaborators: Claudio Castelnovo - BU Christopher Mudry - PSI, Switzerland Pierre Pujol - ENS Lyon, France PRB 2006 Ann. of Phys. 2005, 2006 DMR

2 Correlated Quantum Systems For a system to be in the quantum regime: T Γ, J, U Ex.: transverse field Ising model Couplings in the quantum Hamiltonian T/J H = J ij σ z i σ z j Γ i σ x i F QC How far does the QC regime extend? Kopp & Chakravarty, Nature Phys Γ/J

3 Quantum criticality here, there, everywhere? Look up, not down T QC AF SC Scaling of optical conductivity? van der Marel et al, Nature 2004 Not with single parameter scaling Phillips & Chamon, PRL 2005 Electrons interacting with a broad spectrum of bosons Norman & Chubukov, PRB 2006 Anderson, cond-mat/ Gutzwiller projection

4 Strongly Constrained Quantum Systems If there is a dominant, very high energy scale in the problem that imposes a severe kinematic constraint in the system: T Γ, J U then, we find that A hierarchy of scales can open such that: A) the constrained thermodynamics becomes classical even at low temperatures B) the dynamics is quantum in origin C) the system can display critical behavior at very high temperatures, which is unrelated to quantum criticality

5 Examples of constrained models

6 Spin Ice Dy 2 Ti 2 O 7 (Ho 2 Ti 2 O 7 ) Snyder et al, Nature (2001) Source: Snyder et al, Nature (2001)

7 Josephson junction arrays of T-breaking superconductors Sr 2 RuO 4 p x ± ip y Constrained Ising model σ i = ±1 chirality p x ± ip y Φ P = 2π/3 i P σ i = 2π/3 σ P σ P = i P σ i = ±6, 0 H = J ij σ i σ j with σ P = ±6, 0 Moore & Lee, PRB (2004) Castelnovo, Pujol, and Chamon, PRB (2004)

8 Generic constrained systems Ĥ = JĤJ ΓĤΓ UĤU U J, Γ with [ĤU, ĤJ] = 0 and [ĤU, ĤΓ] 0 For T U the system is effectively constrained to a restricted Hilbert space spanned by the eigenvectors of. Ĥ U H U H U is extensive in the system size H U is not extensive The non-interacting Γ = J = 0 system exhibits power-law correlations (criticality) when restricted to H U (e.g., dimer models, ice models, constrained Ising models) The correlations of the purely projected system are trivial (e.g., Hubbard model)

9 blank line A spin-1/2 example: ˆσ i z ˆσ j Γ Ĥ = J ˆσ i z ˆσ j z Γ i,j i ij ˆσ i x + U [ ( cos 2π ) ] 2 N ˆσ ˆσ i z /3 1 i x U ( cos 2π ) ˆσ i z /3 hex i hex hex i=1 i hex with U J, Γ, and for temperatures T U the systems is mostly confined to a superposition of GS eigenvectors of ĤU The non-commuting ĤΓ term has a vanishing first-order contribution and needs to be expanded in (degenerate) perturbation!

10 blank line Introducing the constraint in a quantum system: projection by a large energy coupling U J, Γ onto its (highly-degenerate) ground states. Ĥ = J ij Ĥ = J i,j ˆσ z i ˆσ z j Γ ˆσ z i ˆσ z j Γ i ˆσ i x + U [ ( N ˆσ cos 2π i x U ( cos 2π hex hex i hex i=1 ˆσ z i /3 i hex ) ˆσ z i /3 1 ) ] 2 (at temperatures much smaller than U) Ĥ eff = J i,j ˆσ z i ˆσ z j n=1 Γ n Ĥ(n) U n 1 Γ/U on the restricted Hilbert space span of GS basis vectors of U term First non-vanishing contributions of the (degenerate) perturbation theory appear only at sixth order (third order if we used Heisenberg Hamiltonian instead of Ising Hamiltonian): n = 6 = Γ eff = Γ6 U 5 1 Γ

11 Different temperature regimes T U - all terms in the Hamiltonian become negligible, free-spin paramagectic phase T Γ eff - quantum regime T Γ eff - classical hard-constrained regime with infinitesimal perturbation J and quantum dynamics A classical ordered phase can appear depending on the effects of J T U Hierarchy opens a large window in which the thermodynamics is classical Γ Γ eff Γ (Γ/U) n 1

12 T J, Γ eff quantum regime where the system generically sits in a shortrange correlated phase Dynamics is still quantum: why? J, Γ eff T U classical hard-constrained regime with infinitesimal perturbation J and quantum dynamics (Γ): consider coupling the system to a thermal bath, e.g., a la Caldeira-Leggett: ) γ ˆσ i (â x λi + â λi, i,λ i with 0 < γ (bath coupling) Γ (kinetic energy)

13 blank line Relaxation time for thermally activated processes (over the defect-creation barrier exp(u/t )): τ T = γ 1 e (U/T ) Relaxation time for virtual processes (via quantum tunneling of at least 6-th order in perturbation): τ Q = min( Γ 1 eff, γ 1 eff ), where γ eff γ(γ/u) n 1 γ Γ eff Γ(Γ/U) n 1 τ Q τ T = phantom of quantum mechanics in the form of sporadic tunneling events between which coherence is lost provides the fastest mechanism for the system to reach classical thermodynamic equilibrium when Γ eff T U.

14 Effects of the constraint at the classical behavior Ĥ = J ˆσ i z ˆσ j z ij N [ h + ( 1) i ] N h s ˆσ z i Γ ˆσ i x U ( cos 2π i=1 hex i=1 i hex ˆσ z i /3 ) How does the constraint alters the Ising model? The model is critical in the absence of interactions (Baxter), and its long distance behavior is captured by an SU(3) k=1 Wess-Zumino-Novikov-Witten conformal field theory S = ( π d 2 x 2 h 2 + V ( ) h) 2 component height field What are now the effects of J, h, h s in the constrained model?

15 Qualitative phase diagram T/U T/U T/U= g/t=0 * paramagnetic fixed point T/U= g/t=0 * paramagnetic fixed point constrained entropic fixed point T/U=0 g/t=0 * disordered phase decreasing g/u (g/t) c classical ordered fixed point ordered phase * T/U=0 g/t= g/t constrained entropic fixed point T/U=0 g/t=0 * disordered phase decreasing g/u classical ordered fixed point ordered phase * T/U=0 g/t= g/t g = h or h s or J

16 2.2. The phase diagram in presence of local interactions 1) The (staggered/uniform) magnetic field case From Monte Carlo simulations, Cluster Mean Field Method results, and Transfer Matrix calculations of the free energy and magnetization of the system, as well as analytical results in the staggered field case (Baxter, Kondev): E = i h i σ i i hex σ i = ±6, 0 The uniform magnetic field case exhibits an exotic first order phase transition to the ferromagnetically ordered phase where the magnetization per spin jumps discontinuously from 0 to 1 The staggered magnetic field case exhibits an infinite order phase transition at infinite temperature to the Néel phase

17 blank line Characteristic features: the criticality is robust in presence of a uniform magnetic field up to the first order phase transition the SU(3) symmetry is preserved up to the first order transition (same central charge c = 2 and same exponents)!

18 blank line 2) The nearest-neighbor interaction case From Monte Carlo simulations, Cluster Mean Field Method results, and Transfer Matrix calculations of the free energy and magnetization of the system: E = J i,j σ i σ j i hex σ i = ±6, 0 In the case of ferromagnetic interactions, the system undergoes an exotic first order phase transition to the ferromagnetically ordered phase where the magnetization per spin jumps discontinuously from 0 to 1 In the case of antiferromagnetic interactions, the system undergoes a second order phase transition to the Néel phase Monte Carlo simulations are incapable of detecting the AF transition

19 blank line Characteristic features: criticality with c = 2 for a ferromagnetic coupling J until the first order phase transition is reached (exp( L/ξ) finite size correction close to the transition) criticality with a varying central charge on the antiferromagnetic side approximately from βj = (βj) c AF (c 1.5) to βj = 0 (c = 2)

20 HiTc (High-temperature criticality) in strongly constrained quantum systems 2.3. HiTC (High-temperature criticality) in strongly constrained quantum systems blank Static properties of the effective Hamiltonian Ĥ eff = J i,j ˆσ z i ˆσ z j Γ eff Ĥ eff Γ/U where Γ eff = Γ6 U 5 most of order 1 in Γ. U and Ĥeff Γ/U is at U T classical paramagnetic phase (U relevant) J, Γ eff T U constraint fully enforced; classical phase equivalent to classical constrained model with nearest-neighbor interaction J: universal critical behavior at large enough temperature! T J, Γ eff ordered phases that depend on the model-specific Hamiltonian details Large temperature universal scaling regime with no required underlying critical point!

21 More hard-constrained systems T/t Classical dimer model v/t = 1 v/t Ĥ RK = P v ( + ) t ( + ) Rokhsar & Kivelson, PRL 1988 Ψ = 1 N C C at v = t ˆρ = 1 N increase T C,C C C ˆρ classical = 1 N C C C

22 Z 2 gauge theory Constraint: i S σ z i = 1 Ĥ U = U S Â S Â S = with σi z i S Ĥ = Γ i σ x i U S Â S Ĥ eff = Γ eff P spin 1/2 ˆB P U S Ψ = 1 N C C ˆρ classical = 1 N Â S ˆBP = with σi x Kitaev, Ann. Phys i P Wen, PRL 2003 ˆρ = 1 N C C C C,C C C

23 Classical correlation entropy of constrained systems vs. von Neumann entropy A: B: the rest S A = tr A (ρ A ln ρ A ) S B = tr B (ρ B ln ρ B ) where ρ A = tr B ˆρ ρ B = tr A ˆρ The von Neumann entropy depends on the boundary for pure states, but on the bulk as well for mixed states To subtract the bulk term, we define S boundary = 1 2 (S A + S B S A B ) S mixed boundary = 1 2 Spure boundary Non-trivial correlation entropy even at large T!

24 Summary 2.3. HiTC (High-temperature criticality) in strongly constrained quantum systems systems blank Static properties of the effective Hamiltonian HiTc (High-temperature criticality) in strongly constrained quantum Ĥ eff = J i,j ˆσ z i ˆσ z j Γ eff Ĥ eff Γ/U where Γ eff = Γ6 U 5 most of order 1 in Γ. U and Ĥeff Γ/U is at U T classical paramagnetic phase (U relevant) J, Γ eff T U constraint fully enforced; classical phase equivalent to classical constrained model with nearest-neighbor interaction J: universal critical behavior at large enough temperature! T J, Γ eff ordered phases that depend on the model-specific Hamiltonian details Large temperature universal scaling regime with no required underlying critical point!