The density-matrix renormalization group: a short introduction

Schollwöck, Ulrich

doi:10.1098/rsta.2010.0382

Cited by 53 publications

(50 citation statements)

References 35 publications

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“…Our numerical calculations in this section are based on the density-matrix renormalization group (DMRG) algorithm [41][42][43][44][45][46]. We use finite-system DMRG [47,48] The values of z obtained in this fashion can be independently corroborated in order to check for any dependence (or lack thereof) on the particular system sizes over which FSS is applied. While our former approach relied on considering ∆ as a function of f , one can alternatively study the scaling of ∆ as a function of L instead, at f = f c .…”

Section: Numerical Resultsmentioning

confidence: 99%

Quantum field theory for the chiral clock transition in one spatial dimension

2018

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We describe the quantum phase transition in the N -state chiral clock model in spatial dimension d = 1. With couplings chosen to preserve time-reversal and spatial inversion symmetries, such a model is in the universality class of recent experimental studies of the ordering of pumped Rydberg states in a one-dimensional chain of trapped ultracold alkali atoms. For such couplings and N = 3, the clock model is expected to have a direct phase transition from a gapped phase with a broken global Z N symmetry, to a gapped phase with the Z N symmetry restored. The transition has dynamical critical exponent z = 1, and so cannot be described by a relativistic quantum field theory. We use a lattice duality transformation to map the transition onto that of a Bose gas in d = 1, involving the onset of a single boson condensate in the background of a higher-dimensional N -boson condensate.We present a renormalization group analysis of the strongly coupled field theory for the Bose gas transition in an expansion in 2 − d, with 4 − N chosen to be of order 2 − d. At two-loop order, we find a regime of parameters with a renormalization group fixed point which can describe a direct phase transition. We also present numerical density-matrix renormalization group studies of lattice chiral clock and Bose gas models for N = 3, finding good evidence for a direct phase transition, and obtain estimates for z and the correlation length exponent ν.arXiv:1808.07056v2 [cond-mat.str-el] 19 Oct 2018 42 4. I (M ) 4 44 C. Renormalization constants for the Z N dilute Bose gas 45 D. The self-dual phase boundary in the chiral clock model 46 E. Analysis as λ → 0 48

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Section: Numerical Resultsmentioning

confidence: 99%

Quantum field theory for the chiral clock transition in one spatial dimension

2018

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“…These phase boundaries are also reproduced using finite-system DMRG [44,45] with a bond dimension up to D = 60 for a chain of L = 51 atoms and open boundary conditions. The first three energy levels are individually targeted, which, in turn, gives us access to the energy gap.…”

Section: Numerical Computation Of the Phase Diagrammentioning

confidence: 99%

Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator

Keesling

Omran

Levine

et al. 2019

Nature

493

379

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Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations [1]. These fluctuations play a dominant role in the quantum critical region surrounding the transition point, where the dynamics are governed by the universal properties associated with the QPT. While time-dependent phenomena associated with classical, thermally driven phase transitions have been extensively studied in systems ranging from the early universe to Bose Einstein Condensates [2-5], understanding critical real-time dynamics in isolated, non-equilibrium quantum systems is an outstanding challenge [6]. Here, we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations while crossing the QPT, we experimentally verify the quantum Kibble-Zurek mechanism (QKZM) [7-9] for an Ising-type QPT, explore scaling universality, and observe corrections beyond QKZM predictions. This approach is subsequently used to measure the critical exponents associated with chiral clock models [10,11], providing new insights into exotic systems that have not been understood previously, and opening the door for precision studies of critical phenomena, simulations of lattice gauge theories [12,13] and applications to quantum optimization [14,15].The celebrated Kibble-Zurek mechanism [2, 3] describes nonequilibrium dynamics and the formation of topological defects in a second-order phase transition driven by thermal fluctuations, and has been experimentally verified in a wide variety of physical systems [4,5]. Recently, the concepts underlying the Kibble-Zurek description have been extended to the quantum regime [7][8][9]. Here, the typical size of the correlated regions, ξ, after a dynamical sweep across the QPT scales as a power-law Critical point gc Control parameter g Correlation length » Phase 1 Phase 2 » » jg ¡ gcj ¡º Time ¿ » jg ¡ gcj ¡zº Fast ramp Slow ramp 0 1 2 3 4 5 Detuning ¢= 1 2 3 4 Interaction range RB=a ²±²±²±²±²±²±² ²±±²±±²±±²±±² ²±±±²±±±²±±±² ℤ2 ℤ3 ℤ4 t ¢ -0.05 0 0.05 -0.1 0 0.1 Density-density correlator G(r) 0 4 8 12 16 20 Distance r (sites) -0.2 -0.1 0 0.1 a b c d Ω Ω FIG. 1: Quantum Kibble-Zurek mechanism (QKZM) and phase diagram. a, Illustration of the QKZM. As the control parameter approaches its critical value, the response time, τ , given by the inverse energy gap of the system, diverges. When the temporal distance to the critical point becomes equal to the response time, as marked by red crosses, the correlation length, b, stops growing due to nonadiabatic excitations. c, Numerically calculated ground-state phase diagram. Circles (diamonds) denote numerically obtained points along the phase boundaries calculated using (infinite-size) Density-Matrix Renormalization Group techniques (Methods). The shaded regions are a guide to the eye. Dashed lines show the experimental trajectories across the phase transitions determined by the pulse diagram shown...

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“…We employed the matrix product states (MPS) time evolution method with matrix product operator (MPO) form of time-evolution operator e −itH [61] not only to simulate the real dynamics of the spin chain but also to obtain the cumulant generating function. For general information about MPS and MPO in DMRG, see for instance the reviews [62,63]. The large deviation function Ψ(a) can be calculated from the cumulant generating function of the integrated spin current [64].…”

Section: Numerical Analysismentioning

confidence: 99%

Exact large deviation function of spin current for the one dimensional XX spin chain with domain wall initial condition

Hiroki¹,

Nagao²,

Sasamoto³

2019

J. Stat. Mech.

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We investigate the fluctuations of the spin current for the onedimensional XX spin chain starting from the domain wall initial condition. The generating function of the current is shown to be written as a determinant with the Bessel kernel. An exact analytical expression for the large deviation function is obtained by applying the Coulomb gas method. Our results are also compared with DMRG calculations. * hmoriya@stat.phys.titech.ac.jp

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The density-matrix renormalization group: a short introduction

Cited by 53 publications

References 35 publications

Quantum field theory for the chiral clock transition in one spatial dimension

Quantum field theory for the chiral clock transition in one spatial dimension

Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator

Exact large deviation function of spin current for the one dimensional XX spin chain with domain wall initial condition

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